Content-Type: multipart/mixed; boundary="-------------0511090440945" This is a multi-part message in MIME format. ---------------0511090440945 Content-Type: text/plain; name="05-381.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-381.keywords" bose fields, local states ---------------0511090440945 Content-Type: application/postscript; name="nordfjord2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nordfjord2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: nordfjord2.dvi %%Pages: 46 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR10 CMR9 CMMI10 CMSY10 CMR7 CMTI10 MSBM10 %%+ CMMI7 CMBX10 CMSY7 CMEX10 CMBX9 CMMI9 CMR6 MSBM7 CMR5 CMBXTI10 %%+ CMBSY10 CMMI5 CMSY5 CMTI9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o nordfjord2.ps nordfjord2.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2005.11.04:1148 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put dup 121 /dagger put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 15 /epsilon1 put dup 67 /C put dup 78 /N put dup 99 /c put dup 107 /k put dup 109 /m put dup 110 /n put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 F992DEA5FC3897E2CC8B7A90238E61E41622DE80F438DD994C73275CC52249D9 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBSY10 %!PS-AdobeFont-1.1: CMBSY10 1.00 %%CreationDate: 1992 Jul 23 21:21:18 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMBSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /H put readonly def /FontBBox{-27 -940 1332 825}readonly def /UniqueID 5000762 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBXTI10 %!PS-AdobeFont-1.1: CMBXTI10 1.0 %%CreationDate: 1991 Aug 18 17:46:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBXTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMBXTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 101 /e put dup 108 /l put dup 114 /r put readonly def /FontBBox{-29 -250 1274 754}readonly def /UniqueID 5000771 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put dup 99 /c put dup 105 /i put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 119 /w put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 33 /omega put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 70 /F put dup 103 /g put dup 105 /i put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec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%!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 63 /perpendicular put dup 80 /P put dup 106 /bar put dup 107 /bardbl put dup 121 /dagger put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 39 /quoteright put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 78 /N put dup 79 /O put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 10 /Omega put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 72 /H put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 88 /X put dup 89 /Y put dup 93 /sharp put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 52 /four put dup 55 /seven put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 75 /K put dup 77 /M put dup 79 /O put dup 80 /P put dup 83 /S put dup 86 /V put dup 87 /W put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 5 /Pi put dup 8 /Phi put dup 10 /Omega put dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 93 /sharp put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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%%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 15 /ffl put dup 18 /grave put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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b(of)g(quan)n(tum)h (\014eld)f(theory)g(can)g(b)r(e)h(explained,)f(understo)r(o)r(d)g(and)g (tested.)h(Second,)523 382 y(they)g(pro)n(vide)e(the)i(building)f(blo)r (c)n(ks)g(for)g(the)h(study)f(of)h(more)e(complicated)h(systems)g(in) 523 482 y(quan)n(tum)29 b(\014eld)g(theory)e(and)i(\(non-equilibrium\)) g(statistical)f(mec)n(hanics,)g(where)g(one)523 581 y(or)e(more)f(suc)n (h)i(\014elds)f(are)g(\(nonlinearly\))g(coupled)g(to)h(eac)n(h)e(other) h(or)g(to)g(other,)g(p)r(ossi-)523 681 y(bly)i(\014nite)g(dimensional)g (systems.)f(Bose)g(\014elds)h(are)f(for)g(example)h(a)f(p)r(opular)h (to)r(ol)f(for)523 780 y(mo)r(delling)22 b(heath)f(baths.)h(The)g(m)n (uc)n(h)f(studied)i(spin-b)r(ose)e(mo)r(del)h(and)f(more)g(generally) 523 880 y(the)28 b(P)n(auli-Fierz)e(mo)r(dels)h(are)g(all)g(of)h(this)g (t)n(yp)r(e.)648 980 y(In)33 b(Section)g(2,)g(I)g(shall)g(\014rst)g (brie\015y)f(describ)r(e)h(the)h(classical)e(mec)n(hanics)g(of)h(suc)n (h)523 1079 y(systems)d(in)g(a)g(uni\014ed)h(w)n(a)n(y)-7 b(.)29 b(This)i(will)f(then)h(allo)n(w)e(us)h(in)h(Section)f(3)g(to)g (write)g(do)n(wn)523 1179 y(the)39 b(corresp)r(onding)d(quan)n(tum)i (mec)n(hanical)g(systems)g({)f(the)i(free)f(b)r(ose)g(\014elds)g({)g (in)523 1279 y(a)f(straigh)n(tforw)n(ard)d(manner,)j(for)f Fp(b)l(oth)j(in\014nite)g(and)g(\014nite)f(dimensional)h Fv(systems.)523 1378 y(In)e(particular,)f(if)i(y)n(ou)e(are)g(familiar) g(with)i(the)f(quan)n(tum)g(mec)n(hanical)f(description)523 1478 y(of)i(\014nite)g(dimensional)f(systems,)g(y)n(ou)g(should)g (conclude)g(after)g(reading)g(these)g(t)n(w)n(o)523 1577 y(sections)28 b(that)h(the)g(description)f(of)h(the)g(in\014nite)g (dimensional)f(systems)h(can)f(b)r(e)h(done)523 1677 y(quite)f(analogously)-7 b(.)648 1777 y(A)n(t)25 b(that)g(p)r(oin)n(t,) h(w)n(e)f(will)g(b)r(e)g(ready)f(to)h(start)g(studying)g(the)g(systems) g(constructed,)523 1876 y(and)g(to)f(analyze)g(their)h(ph)n(ysical)e (prop)r(erties.)h(The)h(only)f(issue)h(I)g(will)g(address)e(here,)h(in) 523 1976 y(Section)f(4,)g(is)f(not)h(one)g(that)g(features)f(prominen)n (tly)h(in)g(quan)n(tum)g(\014eld)g(theory)f(b)r(o)r(oks,)523 2076 y(but)32 b(it)f(has)g(generated)e(a)i(fair)f(amoun)n(t)h(of)g (debate)g(and)g(ev)n(en)f(con)n(tro)n(v)n(ersy)-7 b(.)28 b(It)j(is)g(the)523 2175 y(one)23 b(of)f(lo)r(cal)h(observ)-5 b(ables,)21 b(and)i(of)g(lo)r(cal)f(states,)h(essen)n(tial)f(for)g(the) i(ph)n(ysical)e(in)n(terpre-)523 2275 y(tation)27 b(of)h(the)f(theory) -7 b(.)27 b(Other)g(topics)g(will)h(b)r(e)f(discussed)g(in)h([DB2].)f (I)h(will)f(adopt)g(the)523 2374 y(de\014nition)c(of)f(Knigh)n(t)g(of)h (\\strictly)f(lo)r(cal)g(excitation)g(of)g(the)h(v)-5 b(acuum")22 b(\(De\014nition)i(4\),)523 2474 y(that)31 b(I)h(will)f(refer)f(to)h(as)g(a)g(strictly)g(lo)r(cal)f(or)g(a)h (strictly)g(lo)r(calized)g(state)g(for)f(brevit)n(y)-7 b(.)523 2574 y(I)30 b(will)g(then)g(state)f(and)h(pro)n(v)n(e)e(a)h (generalization)f(of)i(Knigh)n(t's)f(Theorem)f([Kn])i(\(Sect.)523 2673 y(4.5\))g(whic)n(h)h(asserts)e(that)i(\014nite)h(particle)e (states)g(cannot)h(b)r(e)g(p)r(erfectly)g(lo)r(calized.)f(It)523 2773 y(will)37 b(furthermore)f(b)r(e)h(explained)f(ho)n(w)g(Knigh)n (t's)g(a)g(priori)g(coun)n(terin)n(tuitiv)n(e)g(result)523 2873 y(can)c(b)r(e)g(readily)g(understo)r(o)r(d)f(if)i(one)f(remem)n(b) r(ers)f(the)i(analogy)d(b)r(et)n(w)n(een)i(\014nite)h(and)523 2972 y(in\014nite)24 b(dimensional)f(harmonic)f(systems)h(alluded)h(to) f(ab)r(o)n(v)n(e.)f(I)h(will)h(also)e(discuss)h(the)523 3072 y(link)g(b)r(et)n(w)n(een)g(the)g(ab)r(o)n(v)n(e)e(result)h(and)h (the)g(so-called)e(Newton-Wigner)h(p)r(osition)h(op)r(er-)523 3171 y(ator)k(thereb)n(y)g(illuminating,)i(I)f(b)r(eliev)n(e,)g(the)g (di\016culties)g(asso)r(ciated)f(with)i(the)f(latter)523 3271 y(\(Sect.)k(4.7\).)f(I)h(will)f(in)h(particular)e(argue)g(that)i (those)f(di\016culties)h(do)f(not)h(\014nd)g(their)523 3371 y(origin)d(in)h(sp)r(ecial)f(relativit)n(y)g(or)g(in)h(an)n(y)f (form)g(of)h(causalit)n(y)f(violation,)f(as)h(is)h(usually)523 3470 y(claimed.)24 b(It)h(will)g(indeed)f(b)r(e)h(seen)f(that)h(the)g (Newton-Wigner)e(p)r(osition)i(op)r(erator)d(has)523 3570 y(an)30 b(immediate)h(analog)d(for)i(a)g(\014nite)h(or)f (in\014nite)h(system)f(of)g(oscillators,)e(and)j(that)f(it)523 3670 y(mak)n(es)g(absolutely)h(no)g(sense)f(there)i(since)f(it)g(is)g (at)h(o)r(dds)f(with)h(basic)e(ph)n(ysical)h(in)n(tu-)523 3769 y(ition)22 b(and)g(since)g(it)g(is)g(not)g(compatible)g(with)g (the)g(ph)n(ysically)f(reasonable)f(de\014nition)j(of)523 3869 y(Knigh)n(t.)29 b(The)g(conclusion)f(I)h(will)h(dra)n(w)e(is)h (that)g Fp(the)j(Newton-Wigner)f(op)l(er)l(ator)h(do)l(es)523 3968 y(not)c(pr)l(ovide)i(an)e(appr)l(opriate)j(to)l(ol)e(to)f(describ) l(e)i(the)e(strict)g(lo)l(c)l(alization)i(pr)l(op)l(erties)g(of)523 4068 y(the)e(states)f(of)h(extende)l(d)f(systems)g(of)h(the)g(typ)l(e)g (discusse)l(d)g(her)l(e)p Fv(.)d(It)h(sho)n(ws)e(up)h(only)g(b)r(e-)523 4168 y(cause)e(of)g(an)g(understandable)g(but)h(ill-fated)f(desire)g (to)g(force)g(to)r(o)g(stringen)n(t)f(a)h Fp(p)l(article)523 4267 y(interpr)l(etation)32 b Fv(with)f(all)g(its)g(usual)f(attributes) h(on)g(the)g(states)f(of)h(a)g Fp(\014eld)p Fv(.)g(The)g(righ)n(t)523 4367 y(notion)e(of)g(a)f(\(strictly\))i(lo)r(calized)e(state)h(is)g (the)g(one)g(giv)n(en)f(b)n(y)h(Knigh)n(t.)g(These)f(issues)523 4467 y(ha)n(v)n(e)21 b(generated)g(some)h(debate)g(in)h(the)g(con)n (text)f(of)g(relativistic)g(quan)n(tum)g(\014eld)g(theory)523 4566 y(o)n(v)n(er)k(the)i(y)n(ears,)e(up)r(on)i(whic)n(h)f(I)h(shall)f (commen)n(t)g(in)h(Sect.)g(4.7.)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free)h (b)r(ose)f(\014elds)228 b(3)648 282 y Fv(The)31 b(text)h(is)f(written)h (at)f(the)h(graduate)e(lev)n(el)h(and)g(is)g(aimed)g(at)h(an)f (audience)g(of)523 382 y(mathematicians)37 b(with)i(a)e(taste)g(for)h (ph)n(ysics)f(and)g(of)h(ph)n(ysicists)f(with)h(a)g(taste)f(for)523 482 y(mathematics.)28 b(A)g(bac)n(kground)e(in)i(the)h(classical)d(and) i(quan)n(tum)g(theory)f(of)h Fp(\014nite)g Fv(di-)523 581 y(mensional)33 b(systems)g(is)g(assumed,)g(although)f(the)i(text)g (is)f(basically)g(self-con)n(tained.)523 681 y(The)28 b(approac)n(h)d(to)j(the)g(sub)5 b(ject)28 b(c)n(hosen)e(here)i (di\013ers)f(b)r(oth)h(from)f(the)h(usual)g(\\second)523 780 y(quan)n(tization")d(and)i(\\canonical)e(quan)n(tization")h (treatmen)n(ts)g(of)h(quan)n(tum)f(\014eld)i(the-)523 880 y(ory)c(prev)-5 b(alen)n(t)25 b(in)g(the)h(ph)n(ysics)e(literature) g(\(although)h(it)h(is)f(v)n(ery)f(close)g(to)h(the)h(latter\).)523 980 y(It)e(is)g(not)g(axiomatic)e(either.)i(I)g(feel)g(it)g(is)g (fruitful)h(b)r(ecause)e(it)h(allo)n(ws)f(one)g(to)h(apply)f(the)523 1079 y(in)n(tuition)33 b(gained)f(from)g(the)h(study)g(of)f(\014nite)h (dimensional)f(systems)g(in)h(the)g(in\014nite)523 1179 y(dimensional)26 b(case.)g(This)h(helps)f(in)h(dev)n(eloping)f(a)g(go)r (o)r(d)g(understanding)g(of)g(the)h(basic)523 1279 y(ph)n(ysics)32 b(of)h(quan)n(tum)g(\014eld)h(theory)-7 b(,)32 b(and)h(in)g(particular) f(to)h(do)g(a)n(w)n(a)n(y)e(with)i(some)g(of)523 1378 y(the)h(confusion)f(surrounding)f(ev)n(en)h(some)f(of)i(the)g(simplest) f(asp)r(ects)g(of)h(this)f(theory)-7 b(,)523 1478 y(as)34 b(I)h(hop)r(e)g(to)g(illustrate)g(with)g(the)h(discussion)e(of)h(\\lo)r (calization")e(in)i(this)g(con)n(text.)523 1577 y(Although)f(m)n(y)g (approac)n(h)f(here)g(is)h(resolutely)f(non-relativistic,)g(I)i(hop)r (e)f(to)g(sho)n(w)f(it)523 1677 y(still)25 b(sheds)e(an)h(in)n (teresting)g(and)g(illuminating)g(ligh)n(t)g(on)g(relativistic)g (theories)f(as)h(w)n(ell.)523 1777 y(Indeed,)32 b(the)g(main)f(feature) g(of)g(the)h(systems)f(under)g(consideration)f(is)i(their)f(in\014nite) 523 1876 y(spatial)c(extension,)h(and)g(it)g(is)g(this)g(feature)f (that)i(distinguishes)e(them)i(from)e(systems)523 1976 y(with)34 b(a)e(\014nite)i(n)n(um)n(b)r(er)e(of)h(particles)f(suc)n(h)h (as)f(atomic)h(or)f(molecular)g(systems,)g(that)523 2076 y(ha)n(v)n(e)26 b(a)i(\014nite)g(spatial)f(extension.)648 2175 y(Related)49 b(topics)g(will)g(b)r(e)g(discussed)g(in)h(a)e(m)n (uc)n(h)h(extended)h(v)n(ersion)d(of)i(this)523 2275 y(man)n(uscript,)27 b(whic)n(h)h(is)f(in)h(preparation)e([DB2].)523 2586 y Ft(2)42 b(Classical)c(free)g(harmonic)h(systems)523 2781 y Fm(2.1)i(The)32 b(Hamiltonian)e(structure)523 2959 y Fv(Let)c(us)g(no)n(w)f(turn)h(to)g(the)g(systems)g(describ)r(ed) g(brie\015y)f(in)h(the)h(Preface.)d(My)i(\014rst)g(goal)523 3059 y(is)i(to)f(describ)r(e)g(in)h(detail)g(the)g(Hamiltonian)f (structure)g(underlying)1693 3236 y(\177)-48 b Fs(q)21 b Fv(+)d Fs(\012)1896 3202 y Fq(2)1934 3236 y Fs(q)26 b Fv(=)c(0)p Fs(:)1057 b Fv(\(1\))523 3413 y(F)-7 b(or)29 b(\014nite)h(dimensional)f(systems,)g(it)g(is)h(w)n(ell)f(kno)n(wn)f (ho)n(w)h(to)g(view)h(\(1\))f(as)g(a)g(Hamil-)523 3513 y(tonian)g(system,)g(and)h(w)n(e)f(will)h(no)n(w)e(sho)n(w)h(ho)n(w)g (to)g(do)g(this)h(for)f(in\014nite)h(dimensional)523 3612 y(systems)25 b(using)h(as)f(only)g(ingredien)n(t)g(the)h(p)r (ositiv)n(e)f(op)r(erator)f Fs(\012)2560 3582 y Fq(2)2623 3612 y Fv(on)i Fr(D)r Fv(\()p Fs(\012)2903 3582 y Fq(2)2941 3612 y Fv(\))d Fr(\032)g(K)q Fv(.)j(W)-7 b(e)523 3712 y(need)29 b(to)f(iden)n(tify)h(a)f(phase)h(space)e(on)i(whic)n(h)f(the) h(solutions)f(to)h(this)f(equation)g(de\014ne)523 3812 y(a)j(Hamiltonian)g(\015o)n(w)g(for)f(a)h(suitable)g(Hamiltonian.)g(F) -7 b(or)31 b(that)g(purp)r(ose,)g(note)g(that,)523 3911 y(formally)c(at)g(least,)g(\(1\))h(is)g(equiv)-5 b(alen)n(t)27 b(to)1612 4088 y(_)-39 b Fs(q)27 b Fv(=)22 b Fs(p;)58 b Fv(_)-39 b Fs(p)22 b Fv(=)h Fr(\000)p Fs(\012)2139 4054 y Fq(2)2176 4088 y Fs(q)s(;)523 4266 y Fv(whic)n(h)28 b(are)e(Hamilton's)i(equations)e(of)i(motion)f(for)g(the)h(Hamiltonian) g(\()p Fs(X)h Fv(=)23 b(\()p Fs(q)s(;)14 b(p)p Fv(\)\))1436 4482 y Fs(H)7 b Fv(\()p Fs(X)g Fv(\))23 b(=)1773 4426 y(1)p 1773 4463 42 4 v 1773 4539 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y(considered.)648 2536 y(Something)e(in)n(teresting)g(happ)r(ens,) g(ho)n(w)n(ev)n(er,)e(if)j(w)n(e)f(consider)g(the)h(case)e Fs(\027)j Fv(=)24 b(1)p Fs(=)p Fv(2.)523 2636 y(Remem)n(b)r(er)34 b(that)g(this)g(corresp)r(onds)e(to)h(setting)h Fs(!)2200 2648 y Fq(w)2286 2636 y Fv(=)e(0,)i(whic)n(h)g(w)n(as)e(not)i(allo)n(w) n(ed)523 2735 y(in)27 b(the)h(\014nite)f(ring)f(b)r(ecause)h(of)g(the)g (existence)g(of)f(the)i(zero)e(mo)r(de.)h(Some)f(remnan)n(t)h(of)523 2835 y(this)h(problem)f(sho)n(ws)f(up)i(here.)g(Indeed,)f(consider)g Fr(K)2249 2847 y Fn(\025)2293 2835 y Fv(,)h(for)f Fs(\025)c(<)g Fv(0.)k(Since)1422 3011 y Fs(!)1477 2977 y Fq(2)1514 3011 y Fv(\()p Fs(k)s Fv(\))c(=)g(\(2)p Fs(\031)s(!)1911 3023 y Fq(0)1948 3011 y Fv(\))1980 2977 y Fq(2)2017 3011 y Fs(k)2063 2977 y Fq(2)2119 3011 y Fv(+)18 b(o\()p Fs(k)2322 2977 y Fq(2)2359 3011 y Fv(\))p Fs(;)523 3188 y(q)26 b Fr(2)e(K)728 3200 y Fn(\025)792 3188 y Fv(if)e(and)e(only)h(if)g Fr(j)p Fs(k)s Fr(j)1353 3158 y Fn(\025)1402 3188 y 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Fv(is)g(no)n(w)f(di\013eren)n(t,)h(as)f(a)h(set,)523 3587 y(from)k(the)g(phase)f(space)g(when)i Fs(\027)h Fr(6)p Fv(=)c(1)p Fs(=)p Fv(2)g(and)i(in)g(addition,)g(one)f(phase)h (space)f(is)h(not)523 3686 y(included)f(in)f(the)h(other.)f(T)-7 b(o)29 b(see)g(this)h(has)e(noticeable)h(ph)n(ysical)g(consequences,)f (note)523 3786 y(the)34 b(follo)n(wing.)g(It)g(seems)g(lik)n(e)f(a)h (reasonable)e(thing)i(to)g(wish)g(to)g(study)g(the)h(motion)523 3885 y(of)41 b(the)g(c)n(hain)g(when)g(initially)g(only)f(one)h(of)g (the)g(degrees)e(of)i(freedom)g(is)g(excited.)523 3985 y(Supp)r(ose)f(therefore)f(y)n(ou)g(wish)g(to)h(pic)n(k)g(the)g (initial)g(condition)f Fs(q)s Fv(\()p Fs(i)p Fv(\))44 b(=)f Fs(p)p Fv(\()p Fs(i)p Fv(\))g(=)g(0,)523 4085 y(for)35 b(all)h Fs(i)h Fr(6)p Fv(=)f(0,)g Fs(q)s Fv(\(0\))h(=)f(0)h Fr(6)p Fv(=)f Fs(p)p Fv(\(0\).)g(In)g(other)g(w)n(ords,)e(the)j (oscillator)d(at)h(the)i(origin)523 4184 y(starts)f(from)g(its)g (equilibrium)h(p)r(osition)f(with)h(a)f(non-zero)f(initial)i(sp)r(eed,) f(while)h(all)523 4284 y(other)23 b(oscillators)f(are)h(at)g(rest)g(at) h(their)g(equilibrium)f(p)r(ositions.)h(The)f(trouble)h(is)f(that,)523 4384 y(when)32 b Fs(\027)k Fv(=)30 b(1)p Fs(=)p Fv(2,)g(this)i(initial) g(condition)g(do)r(es)g(not)g(b)r(elong)f(to)h(the)g(phase)g(space!)f (So)523 4483 y(it)h(should)f(b)r(e)h(remem)n(b)r(ered)f(that)g(the)h(c) n(hoice)f(of)g(phase)g(space)g(I)g(made)h(here,)f(whic)n(h)523 4583 y(is)d(reasonable)f(from)h(man)n(y)f(a)h(p)r(oin)n(t)h(of)f(view,) g(seems)g(to)g(nev)n(ertheless)g(b)r(e)g(somewhat)523 4682 y(to)r(o)c(restrictiv)n(e)e(in)j(this)f(particular)f(case,)g (since)h(it)g(excludes)g(certain)f(v)n(ery)g(reasonable)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 523 100 a Fu(8)229 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(initial)36 b(conditions)f (from)g(the)h(state)f(space)g(of)h(the)g(system.)f(This)h(is)f(one)g (asp)r(ect)h(of)523 382 y(the)f(so-called)d(infrared)i(problem)f(and)h (it)h(will)f(b)r(e)h(relev)-5 b(an)n(t)33 b(when)i(discussing)e(lo)r (cal)523 482 y(observ)-5 b(ables)26 b(in)i(Section)f(4.)648 581 y(The)c(generalization)f(of)i(the)g(preceding)f(considerations)f (to)i Fs(d)p Fv(-dimensional)f(trans-)523 681 y(lationally)g(in)n(v)-5 b(arian)n(t)23 b(lattices)g(of)h(oscillators)e(is)i(immediate.)g(One)g (has)f Fr(K)i Fv(=)d Fs(`)3020 651 y Fq(2)3057 681 y Fv(\()p Fo(Z)3144 651 y Fn(d)3183 681 y Fs(;)14 b Fo(R)p Fv(\))523 780 y(and,)28 b(for)f(all)g Fs(j)h Fr(2)23 b Fo(Z)1145 750 y Fn(d)1184 780 y Fv(,)596 975 y(\()p Fs(\012)696 941 y Fq(2)734 975 y Fs(q)s Fv(\)\()p Fs(j)5 b Fv(\))24 b(=)f Fs(!)1076 941 y Fq(2)1073 996 y(w)1125 975 y Fs(q)s Fv(\()p Fs(j)5 b Fv(\))19 b Fr(\000)f Fs(!)1425 941 y Fq(2)1422 996 y(n)1528 896 y Fk(X)1476 1078 y Fn(i)p Fl(2)p Fq(nn\()p Fn(j)s Fq(\))1700 975 y Fv(\()p Fs(q)s Fv(\()p Fs(i)p Fv(\))h Fr(\000)f Fs(q)s Fv(\()p Fs(j)5 b Fv(\)\))24 b(=)f Fs(!)2309 941 y Fq(2)2306 996 y(0)2346 975 y Fs(q)s Fv(\()p Fs(j)5 b Fv(\))19 b Fr(\000)f Fs(!)2646 941 y Fq(2)2643 996 y(n)2749 896 y Fk(X)2697 1078 y Fn(i)p Fl(2)p Fq(nn\()p Fn(j)s Fq(\))2935 975 y Fs(q)s Fv(\()p Fs(i)p Fv(\))p Fs(;)74 b Fv(\(10\))523 1255 y(where)25 b(nn\()p Fs(j)5 b Fv(\))26 b(designates)e(the)h(set)h(of)f(nearest)f (neigh)n(b)r(ours)g(of)h Fs(j)30 b Fv(and)25 b(where)g(this)g(time)1062 1490 y Fs(!)1117 1456 y Fq(2)1114 1511 y(0)1177 1490 y Fv(=)e Fs(!)1320 1456 y Fq(2)1317 1511 y(w)1387 1490 y Fv(+)18 b(2)p Fs(d!)1610 1456 y Fq(2)1607 1511 y(n)1814 1490 y Fv(and)165 b(0)23 b Fr(\024)g Fs(\027)28 b Fv(=)2433 1434 y Fs(!)2488 1404 y Fq(2)2485 1455 y(n)p 2433 1471 93 4 v 2433 1547 a Fs(!)2488 1519 y Fq(2)2485 1569 y(0)2558 1490 y Fr(\024)2677 1434 y Fv(1)p 2656 1471 85 4 v 2656 1547 a(2)p Fs(d)2750 1490 y(:)523 1728 y Fv(Using)20 b(the)h(F)-7 b(ourier)19 b(transform)g(to)h(diagonalize)e Fs(\012)2122 1698 y Fq(2)2180 1728 y Fv(one)i(\014nds)h(the)f(disp)r (ersion)g(relation)1243 1988 y Fs(!)s Fv(\()p Fs(k)s Fv(\))1408 1954 y Fq(2)1468 1988 y Fv(=)j Fs(!)1611 1954 y Fq(2)1608 2009 y(0)1662 1846 y Fk(")1710 1988 y Fv(1)18 b Fr(\000)g Fv(2)p Fs(\027)1955 1846 y Fk( )2063 1884 y Fn(d)2020 1909 y Fk(X)2027 2086 y Fn(i)p Fq(=1)2154 1988 y Fv(cos)13 b(2)p Fs(\031)s(k)2414 2000 y Fn(i)2442 1846 y Fk(!)o(#)2570 1988 y Fs(:)523 2248 y Fv(This)34 b(time)h(the)g(critical)f(v)-5 b(alue)34 b(of)g Fs(\027)40 b Fv(is)34 b(1)p Fs(=)p Fv(2)p Fs(d)g Fv(but)h(it)g(leads)e(to)i(less)f (sev)n(ere)e(infrared)523 2348 y(b)r(eha)n(viour.)26 b(Indeed,)i(if)g Fs(\027)h Fv(=)22 b(1)p Fs(=)p Fv(2)p Fs(d)p Fv(,)27 b(then)865 2612 y Fs(!)s Fv(\()p Fs(k)s Fv(\))1030 2578 y Fq(2)1091 2612 y Fv(=)22 b Fs(!)1233 2578 y Fq(2)1230 2633 y(0)1281 2556 y Fv(1)p 1280 2593 44 4 v 1280 2669 a Fs(d)1390 2508 y Fn(d)1347 2533 y Fk(X)1353 2710 y Fn(i)p Fq(=1)1467 2612 y Fv(\(2)p Fs(\031)s(k)1634 2624 y Fn(i)1662 2612 y Fv(\))1694 2578 y Fq(2)1750 2612 y Fv(+)c(o\()p Fr(j)p Fs(k)s Fr(j)1999 2578 y Fq(2)2036 2612 y Fv(\))23 b(=)2189 2556 y(\(2)p Fs(\031)s(!)2365 2568 y Fq(0)2402 2556 y Fv(\))2434 2526 y Fq(2)p 2189 2593 283 4 v 2309 2669 a Fs(d)2481 2612 y Fr(j)p Fs(k)s Fr(j)2573 2578 y Fq(2)2629 2612 y Fv(+)18 b(o\()p Fr(j)p Fs(k)s Fr(j)2878 2578 y Fq(2)2915 2612 y Fv(\))p Fs(:)523 2872 y Fv(But)37 b(no)n(w)g(all)g(compactly)f(supp)r(orted)h Fs(q)j Fv(b)r(elong)d(to)g Fr(K)2339 2884 y Fn(\025)2382 2872 y Fv(,)h(for)e(all)h Fr(\000)p Fs(d=)p Fv(2)g Fs(<)i(\025)p Fv(,)e(as)g(is)523 2972 y(easily)h(c)n(hec)n(k)n(ed.)f(As)i(a)f (result,)g(this)h(time)f(the)h(phase)f(space)g Fr(H)h Fv(con)n(tains)f(all)g(suc)n(h)523 3071 y(initial)e(conditions)f(as)g (so)r(on)g(as)g Fs(d)h Fr(\025)g Fv(2.)g(W)-7 b(e)36 b(shall)f(refer)g(to)g(them)h(as)f Fp(strictly)j(lo)l(c)l(al)523 3171 y(p)l(erturb)l(ations)28 b(fr)l(om)h(e)l(quilibrium)f Fv(and)d(study)i(their)f(quan)n(tum)g(analogues)e(in)j(Section)523 3271 y(4.)37 b(T)-7 b(o)37 b(b)r(e)h(more)f(precise,)g(if)h Fs(d)i Fr(\025)f Fv(2,)e(and)h(if)g(w)n(e)f(denote)g(b)n(y)h Fs(C)2631 3283 y Fq(c)2665 3271 y Fv(\()p Fo(Z)2752 3240 y Fn(d)2791 3271 y Fv(\))g(the)g(space)f(of)523 3370 y(compactly)31 b(supp)r(orted)h(sequences,)e(then)j Fs(C)1977 3382 y Fq(c)2011 3370 y Fv(\()p Fo(Z)2098 3340 y Fn(d)2137 3370 y Fv(\))21 b Fr(\002)g Fs(C)2335 3382 y Fq(c)2369 3370 y Fv(\()p Fo(Z)2456 3340 y Fn(d)2495 3370 y Fv(\))30 b Fr(\032)g(H)q Fv(,)i(for)f(all)g(p)r(ossible)523 3470 y(v)-5 b(alues)25 b(of)g Fs(\027)5 b Fv(.)25 b(If)h Fs(X)j Fv(=)23 b(\()p Fs(q)s(;)14 b(p)p Fv(\))23 b Fr(2)g Fs(C)1564 3482 y Fq(c)1598 3470 y Fv(\()p Fo(Z)1685 3440 y Fn(d)1725 3470 y Fv(\))13 b Fr(\002)g Fs(C)1907 3482 y Fq(c)1941 3470 y Fv(\()p Fo(Z)2028 3440 y Fn(d)2068 3470 y Fv(\),)25 b(then)h Fs(X)31 b Fv(describ)r(es)25 b(an)g(initial)g(state)523 3570 y(in)f(whic)n(h)g(only)f(a)h(\014nite)g(n)n(um)n(b)r(er)g(of)f (oscillators)f(is)i(displaced)f(from)h(their)g(equilibrium)523 3669 y(p)r(osition)38 b(and)p Fs(=)p Fv(or)f(mo)n(ving.)g(So)h(for)g (this)h(rather)e(large)g(and)h(v)n(ery)g(natural)f(class)h(of)523 3769 y(initial)27 b(conditions,)g(the)g(dynamics)g(can)f(b)r(e)i(in)n (v)n(estigated)e(as)g(a)h(function)g(of)g Fs(\027)5 b Fv(,)27 b(for)g(all)523 3868 y(p)r(ossible)g(v)-5 b(alues)28 b(of)f Fs(\027)5 b Fv(.)648 3968 y(Lattices)19 b(of)h(oscillators)f (are)g(used)h(to)g(describ)r(e)f(the)i(thermal)f(and)g(acoustic)f(prop) r(er-)523 4068 y(ties)k(of)f(v)-5 b(arious)21 b(solids,)h(suc)n(h)g(as) g(metals,)h(crystals)e(of)h(all)g(sorts,)g(amorphous)f(materials)523 4167 y Fp(etc.)p Fv(.)30 b(Putting)f Fs(!)1064 4179 y Fq(n)1130 4167 y Fv(=)d(0)j(in)g(the)h(expressions)d(ab)r(o)n(v)n(e,)h (one)h(obtains)g(the)g(so-called)f(Ein-)523 4267 y(stein)e(mo)r(del,)h (in)f(whic)n(h)g(the)h(oscillators)d(represen)n(ting)h(the)h(ions)g(of) g(the)h(solid)e(are)h(not)523 4367 y(coupled.)33 b(The)f(case)g(where)h Fs(!)1510 4379 y Fq(n)1582 4367 y Fr(6)p Fv(=)e(0)h(is)h(the)g(Deb)n(y) n(e)f(mo)r(del.)h(In)g(more)f(sophisticated)523 4466 y(mo)r(dels)22 b(still,)h(di\013eren)n(t)g(geometries)e(ma)n(y)g(app)r (ear)h(\(hexagonal)e(lattices,)j(b)r(o)r(dy)f(or)g(face)523 4566 y(cen)n(tered)30 b(cubic)h(lattices)f Fp(etc.)p Fv(\),)h(and)g(the)g(spring)e(constan)n(ts)h(ma)n(y)f(v)-5 b(ary)30 b(from)g(site)h(to)523 4665 y(site)d(in)g(p)r(erio)r(dic,)f (quasi-p)r(erio)r(dic)g(or)f(random)h(w)n(a)n(ys.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free)h (b)r(ose)f(\014elds)228 b(9)523 282 y Fm(W)-8 b(a)m(v)m(e)33 b(and)f(Klein-Gordon)e(equations)523 451 y Fv(The)e(w)n(a)n(v)n(e)e (equation)1566 551 y Fs(@)1615 517 y Fq(2)1610 571 y Fn(t)1652 551 y Fs(q)s Fv(\()p Fs(x;)14 b(t)p Fv(\))24 b(=)f Fs(\001q)s Fv(\()p Fs(x;)14 b(t)p Fv(\))523 690 y(on)28 b(a)g(domain)g Fs(K)h Fr(\032)24 b Fo(R)1254 660 y Fn(d)1321 690 y Fv(with)k(Diric)n(hlet)h(b)r(oundary)e (conditions)h(is)g(another)f(example)523 790 y(of)40 b(a)f(free)g(oscillator)f(\014eld)i(where)f Fr(K)44 b Fv(=)f Fs(L)1966 759 y Fq(2)2003 790 y Fv(\()p Fs(K)q(;)14 b Fo(R)p Fv(\))40 b(and)f Fr(D)r Fv(\()p Fs(\012)t Fv(\))i(is)f(the)g (domain)f(of)523 889 y(the)e(square)f(ro)r(ot)g(of)g(the)h(Diric)n (hlet)g(Laplacian.)f(When)h Fs(K)42 b Fv(is)37 b(a)f(b)r(ounded)h(set,) g(the)523 989 y(sp)r(ectrum)k(of)f(the)h(Diric)n(hlet)f(Laplacian)f(is) i(discrete.)e(No)i(infrared)e(problem)h(then)523 1088 y(arises,)26 b(re\015ecting)g(the)i(fact)f(that)g(no)g(arbitrary)e (long)i(w)n(a)n(v)n(elengths)e(can)h(o)r(ccur)h(in)g(the)523 1188 y(system.)648 1288 y(The)h(case)g(where)g Fs(K)i Fv(=)24 b Fo(R)1487 1258 y Fn(d)1554 1288 y Fv(is)k(instructiv)n(e)g (and)h(easy)e(to)h(w)n(ork)g(out)g(thanks)g(to)g(its)523 1387 y(translational)i(in)n(v)-5 b(ariance.)29 b(The)i(situation)g(is)g (completely)g(analogous)e(with)j(the)f(one)523 1487 y(in)i(Sect.)g (2.2.)f(W)-7 b(riting)33 b Fs(!)s Fv(\()p Fs(k)s Fv(\))f(=)1598 1414 y Fr(p)p 1667 1414 84 4 v 73 x Fs(k)1713 1463 y Fq(2)1751 1487 y Fv(,)g(the)i(space)e Fr(K)2245 1499 y Fn(\025)2321 1487 y Fv(is)h(for)f(eac)n(h)g(real)g Fs(\025)i Fv(naturally)523 1587 y(isomorphic)23 b(to)g(the)h(real)f (subspace)g(of)g Fs(L)1824 1556 y Fq(2)1861 1587 y Fv(\()p Fo(R)1953 1556 y Fn(d)1992 1587 y Fs(;)14 b(!)s Fv(\()p Fs(k)s Fv(\))2194 1556 y Fq(2)p Fn(\025)2271 1587 y Fs(dk)s(;)g Fo(C)p Fv(\))24 b(giv)n(en)e(b)n(y)i(the)g(condition)p 523 1614 151 4 v 529 1686 a(^)-48 b Fs(q)s Fv(\()p Fs(k)s Fv(\))24 b(=)e Fs(q)s Fv(\()p Fr(\000)p Fs(k)s Fv(\).)28 b(If)g Fs(d)23 b Fr(\025)g Fv(2,)k(the)h(Sc)n(h)n(w)n(artz)e(space)h (is)h(a)f(subspace)g(of)g Fr(K)2754 1707 y Fl(\006)2816 1684 y Ff(1)p 2816 1693 29 3 v 2816 1727 a(2)2859 1686 y Fv(.)648 1786 y(One)35 b(can)h(also)f(consider)h(the)g(more)f (general)g(case)h(where)f Fs(K)42 b Fv(is)36 b(a)g(Riemannian)523 1885 y(manifold)21 b(with)h(metric)f Fs(\015)26 b Fv(and)21 b Fr(\000)p Fs(\001)g Fv(the)h(corresp)r(onding)d(Laplace-Beltrami)g (op)r(erator.)523 1985 y(Replacing)35 b Fr(\000)p Fs(\001)h Fv(b)n(y)g Fr(\000)p Fs(\001)24 b Fv(+)g Fs(m)1529 1955 y Fq(2)1602 1985 y Fv(\()p Fs(m)37 b(>)g Fv(0\))f(in)g(the)h(ab)r(o)n (v)n(e,)e(one)g(obtains)h(the)g(Klein-)523 2085 y(Gordon)27 b(equation.)g(It)i(pla)n(ys)e(an)g(imp)r(ortan)n(t)h(role)f(in)h(the)g (relativistic)g(quan)n(tum)f(\014eld)523 2184 y(theory)g(on)g(\015at)h (or)e(curv)n(ed)h(spacetimes.)523 2411 y Fm(2.3)41 b(A)32 b(preferred)g(complex)f(structure)i(on)e(the)h Fe(r)-5 b(e)g(al)32 b Fm(classical)f(phase)523 2511 y(space)h Fd(H)523 2680 y Fv(The)25 b(simple)f(linear)g(systems)g(w)n(e)g(are)g (dealing)f(with)i(here)f(ha)n(v)n(e)g(some)g(extra)f(structure)523 2780 y(that)41 b(is)f(enco)r(ded)g(in)h(the)g(matrix)f Fs(J)48 b Fv(de\014ned)41 b(in)g(\(4\).)g(Noticing)f(that)g Fs(J)2960 2749 y Fq(2)3042 2780 y Fv(=)k Fr(\000)p Fs(I)3252 2792 y Fq(2)3290 2780 y Fv(,)523 2879 y(one)32 b(sees)g Fs(J)41 b Fv(de\014nes)32 b(an)g Fs(s)p Fv(-compatible)g(\()p Fp(i.e.)i Fs(s)p Fv(\()p Fs(J)8 b(X)r(;)14 b(J)8 b(Y)19 b Fv(\))31 b(=)g Fs(s)p Fv(\()p Fs(X)r(;)14 b(Y)19 b Fv(\)\))33 b(and)f(p)r(ositiv)n(e)523 2979 y(de\014nite)42 b(\()p Fp(i.e.)h Fs(s)p Fv(\()p Fs(X)r(;)14 b(J)8 b(X)f Fv(\))45 b Fr(\025)g Fv(0)c(and)g Fs(s)p Fv(\()p Fs(X)r(;)14 b(J)8 b(Y)19 b Fv(\))46 b(=)f(0)p Fs(;)14 b Fr(8)p Fs(Y)64 b Fr(2)46 b(H)c Fv(implies)f Fs(X)52 b Fv(=)46 b(0\))523 3078 y(complex)34 b(structure)f(on)h Fr(H)q Fv(.)g(As)g(a)g(result,)f Fr(H)i Fv(can)f(\014rst)g(of)g(all)g(b)r(e)g(view)n(ed)g(as)f(a)h(real) 523 3178 y(Hilb)r(ert)28 b(space,)f(with)h(inner)g(pro)r(duct)868 3366 y Fs(g)908 3378 y Fn(\012)966 3366 y Fv(\()p Fs(X)r(;)14 b(Y)k Fv(\))1228 3319 y Fq(def)1241 3366 y Fv(=)36 b Fs(s)p Fv(\()p Fs(X)r(;)14 b(J)8 b(Y)19 b Fv(\))k(=)1785 3292 y Fr(p)p 1854 3292 69 4 v 74 x Fs(\012)t(q)f Fr(\001)2022 3292 y(p)p 2091 3292 V 74 x Fs(\012)5 b(q)2200 3332 y Fl(0)2241 3366 y Fv(+)2324 3292 y Fr(p)p 2394 3292 V 2394 3366 a Fs(\012)2462 3306 y Fl(\000)p Fq(1)2551 3366 y Fs(p)18 b Fr(\001)2653 3292 y(p)p 2722 3292 V 74 x Fs(\012)2790 3306 y Fl(\000)p Fq(1)2879 3366 y Fs(p)2921 3332 y Fl(0)2944 3366 y Fs(;)198 b Fv(\(11\))523 3532 y(where)26 b Fs(Y)42 b Fv(=)22 b(\()p Fs(q)1011 3502 y Fl(0)1035 3532 y Fs(;)14 b(p)1114 3502 y Fl(0)1137 3532 y Fv(\).)27 b(Of)g(course,)e(w)n(e)h(recognize)f(here)h(the)h (natural)f(inner)g(pro)r(duct)h(on)523 3631 y Fr(H)d Fv(=)f Fr(K)768 3646 y Fq(1)p Fn(=)p Fq(2)891 3631 y Fr(\010)18 b(K)1037 3646 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1193 3631 y Fv(,)28 b(written)g(in)g(terms)f(of)g(the)h(symplectic)g(form)f (and)h Fs(J)8 b Fv(.)648 3731 y(In)25 b(addition,)g Fs(J)33 b Fv(can)24 b(b)r(e)i(used)f(to)g(equip)g Fr(H)h Fv(with)f(a)g Fp(c)l(omplex)g Fv(Hilb)r(ert)h(space)e(struc-)523 3831 y(ture,)37 b(where)f(m)n(ultiplication)h(with)g(the)g(complex)f(n)n(um) n(b)r(er)g Fs(a)25 b Fv(+)f(i)p Fs(b)38 b Fr(2)g Fo(C)f Fv(is)f(de\014ned)523 3930 y(b)n(y)1285 4050 y(\()p Fs(a)19 b Fv(+)f(i)p Fs(b)p Fv(\))p Fs(X)1652 4003 y Fq(def)1665 4050 y Fv(=)37 b(\()p Fs(a)18 b Fv(+)g Fs(bJ)8 b Fv(\))p Fs(X)r(;)97 b Fr(8)p Fs(X)29 b Fr(2)24 b(H)523 4189 y Fv(and)j(with)i(the)f(inner)f(pro)r(duct)1272 4389 y Fr(h)p Fs(X)r(;)14 b(Y)k Fr(i)1510 4401 y Fq(+)1589 4389 y Fv(=)1686 4333 y(1)p 1686 4370 42 4 v 1686 4446 a(2)1738 4389 y(\()p Fs(g)1810 4401 y Fn(\012)1868 4389 y Fv(\()p Fs(X)r(;)c(Y)k Fv(\))h(+)f(i)p Fs(s)p Fv(\()p Fs(X)r(;)c(Y)k Fv(\)\))p Fs(:)602 b Fv(\(12\))523 4583 y(Note)30 b(that,)h(when)g Fr(H)g Fv(has)f(2)p Fs(n)g Fp(r)l(e)l(al)h Fv(dimensions,)f(the)g (complex)g(v)n(ector)g(space)f(\()p Fr(H)q Fs(;)14 b(J)8 b Fv(\))523 4682 y(has)27 b(only)g Fs(n)h Fv(complex)f(dimensions.)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 523 100 a Fu(10)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)648 282 y Fv(Since)27 b Fs(\010)919 294 y Fn(t)977 282 y Fv(is)g(symplectic)h(and)f(comm)n(utes)g(with)h Fs(J)8 b Fv(,)28 b(one)f(easily)g(c)n(hec)n(ks)g(that)846 465 y Fs(g)886 477 y Fn(\012)944 465 y Fv(\()p Fs(\010)1031 477 y Fn(t)1061 465 y Fs(X)r(;)14 b(\010)1224 477 y Fn(t)1253 465 y Fs(Y)k Fv(\))24 b(=)e Fs(g)1502 477 y Fn(\012)1560 465 y Fv(\()p Fs(X)r(;)14 b(Y)19 b Fv(\))83 b(and)g Fr(h)p Fs(\010)2186 477 y Fn(t)2216 465 y Fs(X)r(;)14 b(\010)2379 477 y Fn(t)2408 465 y Fs(Y)k Fr(i)2506 477 y Fq(+)2585 465 y Fv(=)23 b Fr(h)p Fs(X)r(;)14 b(Y)k Fr(i)2911 477 y Fq(+)2966 465 y Fs(;)523 648 y Fv(so)j(that)h Fs(\010)848 660 y Fn(t)899 648 y Fv(is)f(a)g(unitary)g(op)r(erator)f(on)h(the)h (complex)f(Hilb)r(ert)h(space)f(\()p Fr(H)q Fs(;)14 b(J)o(;)g Fr(h\001)p Fs(;)g Fr(\001i)3085 660 y Fq(+)3141 648 y Fv(\).)22 b(As)523 747 y(a)32 b(result,)g Fs(X)925 759 y Fn(H)1019 747 y Fv(=)f Fr(\000)p Fs(J)8 b(\012)t Fv(,)33 b(the)g(generating)e(Hamiltonian)h(v)n(ector)f(\014eld)i(is)f (necessarily)523 847 y(an)n(ti-self-adjoin)n(t)26 b(and)i(one)f(can)g (c)n(hec)n(k)g(that)h(in)g(addition)1514 1029 y(i)p Fr(h)p Fs(X)r(;)14 b(X)1746 1041 y Fn(H)1809 1029 y Fs(X)7 b Fr(i)1917 1041 y Fq(+)1995 1029 y Fv(=)23 b Fs(H)7 b Fv(\()p Fs(X)g Fv(\))p Fs(:)843 b Fv(\(13\))648 1212 y(It)28 b(is)h(natural)e(to)i(w)n(onder)e(if)i(there)f(exist)g(man)n(y) g(complex)g(structures)g(on)g Fr(H)h Fv(with)523 1312 y(these)22 b(prop)r(erties.)f(In)i(fact,)f Fs(J)30 b Fv(is)22 b(the)h(unique)f Fs(s)p Fv(-compatible,)g(p)r(ositiv)n(e)g (complex)f(struc-)523 1411 y(ture)30 b(on)g Fr(H)h Fv(so)e(that)i Fs(\010)1262 1423 y Fn(t)1321 1411 y Fv(is)f(unitary)g(on)g(the)g (corresp)r(onding)e(complex)i(Hilb)r(ert)g(space)523 1511 y([DB2].)36 b(In)h(other)e(w)n(ords,)g(the)i(phase)f(space)f Fr(H)i Fv(of)g(an)f(oscillator)e(\014eld,)j(whic)n(h)f(is)g(a)523 1611 y Fp(r)l(e)l(al)29 b Fv(symplectic)f(space,)g(carries)e(a)i (natural,)g(\015o)n(w-in)n(v)-5 b(arian)n(t)26 b(complex)i(Hilb)r(ert)g (space)523 1710 y(structure!)648 1810 y(The)38 b(ensuing)g(complex)g (Hilb)r(ert)h(space)f(seems)g(a)g(somewhat)f(abstract)h(ob)5 b(ject,)523 1910 y(but)24 b(it)g(can)g(b)r(e)g(naturally)f(iden)n (ti\014ed)h(with)g Fr(K)1969 1879 y Fg(C)2016 1910 y Fv(,)g(the)g(complexi\014cation)f(of)g Fr(K)q Fv(,)h(as)g(I)f(no)n(w) 523 2009 y(explain.)h(In)h(the)g(follo)n(wing,)f(whenev)n(er)f Fs(V)44 b Fv(is)24 b(a)g(real)g(v)n(ector)f(space,)h Fs(V)2756 1979 y Fg(C)2825 2009 y Fv(=)f Fs(V)31 b Fr(\010)12 b Fv(i)p Fs(V)43 b Fv(will)523 2109 y(denote)31 b(its)h (complexi\014cation.)e(In)i(the)g(concrete)e(examples)h(I)g(ha)n(v)n(e) g(in)g(mind,)h(where)523 2208 y Fs(V)48 b Fv(=)28 b Fr(K)i Fv(=)f Fo(R)959 2178 y Fn(n)1004 2208 y Fs(;)14 b(`)1076 2178 y Fq(2)1113 2208 y Fv(\()p Fo(Z)1200 2178 y Fn(d)1239 2208 y Fs(;)g Fo(R)p Fv(\))31 b(or)f Fs(L)1561 2178 y Fq(2)1598 2208 y Fv(\()p Fo(R)1690 2178 y Fn(d)1729 2208 y Fs(;)14 b Fo(R)p Fv(\),)31 b(one)g(\014nds)g Fs(V)2337 2178 y Fg(C)2412 2208 y Fv(=)d Fr(K)2569 2178 y Fg(C)2645 2208 y Fv(=)g Fo(C)2798 2178 y Fn(n)2843 2208 y Fs(;)14 b(`)2915 2178 y Fq(2)2952 2208 y Fv(\()p Fo(Z)3039 2178 y Fn(d)3078 2208 y Fs(;)g Fo(C)p Fv(\))32 b(or)523 2308 y Fs(L)580 2278 y Fq(2)617 2308 y Fv(\()p Fo(R)709 2278 y Fn(d)748 2308 y Fs(;)14 b Fo(C)p Fv(\),)28 b(resp)r(ectiv)n(ely)-7 b(.)26 b(The)i(iden)n(ti\014cation)g(go)r(es)e(as)h(follo)n(ws)g(:)860 2530 y Fs(z)899 2542 y Fn(\012)980 2530 y Fv(:)c Fs(X)29 b Fv(=)23 b(\()p Fs(q)s(;)14 b(p)p Fv(\))23 b Fr(2)h(H)g(7!)f Fs(z)1736 2542 y Fn(\012)1793 2530 y Fv(\()p Fs(X)7 b Fv(\))23 b(=)2089 2474 y(1)p 2054 2511 111 4 v 2054 2527 a Fr(p)p 2123 2527 42 4 v 69 x Fv(2)2175 2530 y(\()2207 2455 y Fr(p)p 2276 2455 69 4 v 75 x Fs(\012)t(q)f Fv(+)c(i)2567 2474 y(1)p 2519 2511 138 4 v 2519 2527 a Fr(p)p 2588 2527 69 4 v 71 x Fs(\012)2666 2530 y(p)p Fv(\))24 b Fr(2)f(K)2906 2495 y Fg(C)2953 2530 y Fs(:)189 b Fv(\(14\))523 2756 y(The)28 b(follo)n(wing)e(prop)r(osition)h(is)g(then)h(easily)f(pro)n (v)n(en.)523 2914 y Fm(Prop)s(osition)j(1.)24 b Fp(The)42 b(map)f Fs(z)1551 2926 y Fn(\012)1649 2914 y Fp(de\014nes)g(an)f (isomorphism)j(b)l(etwe)l(en)d(the)h(c)l(omplex)523 3013 y(Hilb)l(ert)27 b(sp)l(ac)l(es)g Fv(\()p Fr(H)q Fs(;)14 b(J)o(;)g Fr(h\001)p Fs(;)g Fr(\001i)1409 3025 y Fq(+)1465 3013 y Fv(\))26 b Fp(and)i Fr(K)1746 2983 y Fg(C)1792 3013 y Fp(,)f(intertwining)g(the)g(dynamics)h Fs(\010)2863 3025 y Fn(t)2919 3013 y Fp(with)f Fv(e)3133 2983 y Fl(\000)p Fq(i)p Fn(\012)s(t)3287 3013 y Fp(.)523 3113 y(Mor)l(e)k(pr)l(e)l (cisely,)1037 3296 y Fs(z)1076 3308 y Fn(\012)1134 3296 y Fv(\()p Fs(J)8 b(X)f Fv(\))23 b(=)f(i)p Fs(z)1500 3308 y Fn(\012)1558 3296 y Fv(\()p Fs(X)7 b Fv(\))p 1783 3223 237 4 v 85 w Fs(z)1822 3308 y Fn(\012)1879 3296 y Fv(\()p Fs(X)g Fv(\))19 b Fr(\001)f Fs(z)2118 3308 y Fn(\012)2176 3296 y Fv(\()p Fs(X)2284 3261 y Fl(0)2307 3296 y Fv(\))23 b(=)g Fr(h)p Fs(X)r(;)14 b(X)2666 3261 y Fl(0)2688 3296 y Fr(i)2720 3308 y Fq(+)2776 3296 y Fs(:)366 b Fv(\(15\))523 3478 y Fp(and)1477 3578 y Fs(z)1516 3590 y Fn(\012)1573 3578 y Fv(\()p Fs(\010)1660 3590 y Fn(t)1690 3578 y Fs(X)7 b Fv(\))23 b(=)g(e)1946 3544 y Fl(\000)p Fq(i)p Fn(\012)s(t)2099 3578 y Fs(z)2138 3590 y Fn(\012)2196 3578 y Fv(\()p Fs(X)7 b Fv(\))p Fs(:)806 b Fv(\(16\))523 3736 y(Note)37 b(that)g Fr(\001)g Fv(has)f(b)r(een)i(extended)f(to)f Fr(K)1881 3705 y Fg(C)1965 3736 y Fv(b)n(y)g(linearit)n(y)g(in)h(eac)n(h)f(v)-5 b(ariable)36 b(so)g(that)523 3835 y(the)c(inner)f(pro)r(duct)h(on)f Fr(K)1385 3805 y Fg(C)1463 3835 y Fv(is)h(giv)n(en)e(b)n(y)36 b(\026)-47 b Fs(z)25 b Fr(\001)c Fs(z)2041 3805 y Fl(0)2063 3835 y Fv(,)32 b(for)f Fs(z)t(;)14 b(z)2372 3805 y Fl(0)2423 3835 y Fr(2)30 b(K)2572 3805 y Fg(C)2619 3835 y Fv(.)i(The)f(c)n(hoice) g(of)g(the)523 3935 y(unnatural)k(lo)r(oking)g(factor)f(1)p Fs(=)1542 3866 y Fr(p)p 1611 3866 42 4 v 69 x Fv(2)h(in)h(the)f (de\014nition)h(of)g Fs(z)2462 3947 y Fn(\012)2555 3935 y Fv(and)g(of)f(the)h(matc)n(hing)523 4034 y(factor)i(1)p Fs(=)p Fv(2)f(in)i Fr(h)p Fs(X)r(;)14 b(X)1259 4004 y Fl(0)1282 4034 y Fr(i)1314 4046 y Fq(+)1408 4034 y Fv(are)38 b(con)n(v)n(en)n(tions)f(c)n(hosen)h(to)h(mak)n(e)f(comparison)f(to)h (the)523 4134 y(ph)n(ysics)18 b(literature)g(simple,)h(as)f(w)n(e)g (will)h(see)f(further)h(on.)f(Similarly)-7 b(,)18 b(for)g(later)g(purp) r(oses,)523 4234 y(w)n(e)27 b(de\014ne)860 4435 y Fs(z)903 4395 y Fl(y)899 4459 y Fn(\012)980 4435 y Fv(:)c Fs(X)29 b Fv(=)23 b(\()p Fs(q)s(;)14 b(p)p Fv(\))23 b Fr(2)h(H)g(7!)f Fs(z)1740 4395 y Fl(y)1736 4459 y Fn(\012)1793 4435 y Fv(\()p Fs(X)7 b Fv(\))23 b(=)2089 4379 y(1)p 2054 4416 111 4 v 2054 4432 a Fr(p)p 2123 4432 42 4 v 69 x Fv(2)2175 4435 y(\()2207 4360 y Fr(p)p 2276 4360 69 4 v 75 x Fs(\012)t(q)f Fr(\000)c Fv(i)2567 4379 y(1)p 2519 4416 138 4 v 2519 4432 a Fr(p)p 2588 4432 69 4 v 71 x Fs(\012)2666 4435 y(p)p Fv(\))24 b Fr(2)f(K)2906 4400 y Fg(C)2953 4435 y Fs(;)189 b Fv(\(17\))523 4661 y(whic)n(h)28 b(is)f(complex)g(an)n (ti-linear)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(11)1555 282 y Fs(z)1598 242 y Fl(y)1594 307 y Fn(\012)1652 282 y Fv(\()p Fs(J)8 b(X)f Fv(\))23 b(=)f Fr(\000)p Fv(i)p Fs(z)2087 242 y Fl(y)2083 307 y Fn(\012)2141 282 y Fv(\()p Fs(X)7 b Fv(\))884 b(\(18\))523 465 y(and)p 1410 474 237 4 v 1410 565 a Fs(z)1453 525 y Fl(y)1449 589 y Fn(\012)1507 565 y Fv(\()p Fs(X)7 b Fv(\))18 b Fr(\001)g Fs(z)1749 525 y Fl(y)1745 589 y Fn(\012)1803 565 y Fv(\()p Fs(X)1911 530 y Fl(0)1934 565 y Fv(\))23 b(=)p 2077 492 326 4 v 23 w Fr(h)p Fs(X)r(;)14 b(X)2293 541 y Fl(0)2315 565 y Fr(i)2347 577 y Fq(+)2403 565 y Fs(:)739 b Fv(\(19\))648 714 y(The)27 b(linear)g(map)h Fs(z)1273 726 y Fn(\012)1358 714 y Fv(is)f(readily)g(in)n(v)n(erted)g (and)g(one)h(has,)f(in)h(ob)n(vious)e(notations)843 955 y Fs(q)g Fv(=)1072 898 y(1)p 1003 935 179 4 v 1003 952 a Fr(p)p 1073 952 110 4 v 1073 1022 a Fv(2)p Fs(\012)1192 955 y Fv(\()p Fs(z)1263 967 y Fn(\012)1321 955 y Fv(\()p Fs(X)7 b Fv(\))19 b(+)f Fs(z)1606 915 y Fl(y)1602 979 y Fn(\012)1659 955 y Fv(\()p Fs(X)7 b Fv(\)\))28 b(and)g Fs(p)23 b Fv(=)2183 828 y Fr(p)p 2252 828 69 4 v 70 x Fs(\012)p 2183 935 138 4 v 2185 1021 a Fv(i)2208 952 y Fr(p)p 2277 952 42 4 v 69 x Fv(2)2331 955 y(\()p Fs(z)2402 967 y Fn(\012)2460 955 y Fv(\()p Fs(X)7 b Fv(\))18 b Fr(\000)g Fs(z)2744 915 y Fl(y)2740 979 y Fn(\012)2798 955 y Fv(\()p Fs(X)7 b Fv(\)\))p Fs(;)172 b Fv(\(20\))523 1181 y(and)1425 1280 y Fs(H)7 b Fv(\()p Fs(X)g Fv(\))23 b(=)f Fs(z)1790 1292 y Fn(\012)1848 1280 y Fv(\()p Fs(X)7 b Fv(\))1988 1246 y Fl(y)2041 1280 y Fr(\001)19 b Fs(\012)t(z)2190 1292 y Fn(\012)2248 1280 y Fv(\()p Fs(X)7 b Fv(\))p Fs(:)754 b Fv(\(21\))648 1430 y(In)22 b(conclusion,)g(w)n(e)f(established)h (that,)h(ha)n(ving)e(started)h(with)h(a)f Fp(r)l(e)l(al)g Fv(Hilb)r(ert)h(space)523 1529 y Fr(K)33 b Fv(and)f(a)f(p)r(ositiv)n(e) h(self-adjoin)n(t)f(op)r(erator)f Fs(\012)t Fv(,)j(the)f(classical)e (phase)i(space)f Fr(H)h Fv(of)g(the)523 1629 y(corresp)r(onding)c (oscillator)g(equation)36 b(\177)-49 b Fs(q)30 b Fv(=)d Fr(\000)p Fs(\012)2050 1599 y Fq(2)2087 1629 y Fs(q)33 b Fv(can)c(b)r(e)i(iden)n(ti\014ed)f(naturally)f(with)523 1729 y(the)j Fp(c)l(omplex)g Fv(Hilb)r(ert)g(space)e Fr(K)1565 1699 y Fg(C)1612 1729 y Fv(,)h(on)g(whic)n(h)g(the)h (dynamics)f(is)g(simply)g(the)h(unitary)523 1828 y(group)d(generated)f (b)n(y)i Fs(\012)t Fv(,)g(the)h(symplectic)f(structure)f(is)h(the)g (imaginary)e(part)h(of)h(the)523 1928 y(inner)h(pro)r(duct)f(and)h(the) g(Hamiltonian)g(is)g(giv)n(en)f(b)n(y)g Fs(H)7 b Fv(\()p Fs(z)t Fv(\))28 b(=)p 2574 1882 43 4 v 28 w Fs(z)c Fr(\001)d Fs(\012)t(z)t Fv(.)30 b(W)-7 b(e)32 b(therefore)523 2028 y(ended)24 b(up)f(with)h(a)f Fp(mathematic)l(al)t(ly)j Fv(completely)d(equiv)-5 b(alen)n(t)23 b(description)g(of)h(the)f (orig-)523 2127 y(inal)31 b(phase)g(space)f Fr(H)q Fv(,)h(its)g (symplectic)g(structure)g(and)g(the)g(dynamics)g Fs(\010)2901 2139 y Fn(t)2962 2127 y Fv(generated)523 2227 y(b)n(y)c(the)h (Hamiltonian)g Fs(H)34 b Fv(in)28 b(\(2\).)648 2326 y(It)h(is)g(ho)n(w) n(ev)n(er)f(imp)r(ortan)n(t)h(to)g(understand)g(that)g(the)h Fp(physic)l(al)h Fv(in)n(terpretation)e(of)523 2426 y(this)f(new)g (form)n(ulation)e(should)i(b)r(e)g(done)f(carefully)-7 b(,)27 b(as)g(I)h(explain)f(in)h(Sect.)g(2.4.)523 2667 y Fm(2.4)41 b(Ph)m(ysical)32 b(in)m(terpretation)523 2849 y Fv(It)c(is)g(instructiv)n(e)g(to)g(\014rst)g(lo)r(ok)f(at)h (what)g(the)h(formalism)e(of)h(the)h(Sect.)f(2.3)f(yields)h(for)523 2949 y Fp(\014nite)33 b(dimensional)h Fv(systems)d(of)h(coupled)g (oscillators,)d(suc)n(h)j(as)f(the)h(oscillator)e(ring.)523 3049 y(In)22 b(that)g(case)g Fr(K)i Fv(=)f Fo(R)1201 3019 y Fn(n)1268 3049 y Fv(and)f(hence)g Fr(K)1713 3019 y Fg(C)1782 3049 y Fv(=)h Fo(C)1930 3019 y Fn(n)1975 3049 y Fv(.)f(Note)g(ho)n(w)n(ev)n(er)e(that)i(the)h(iden)n (ti\014cation)523 3148 y(of)30 b Fo(R)680 3118 y Fq(2)p Fn(n)789 3148 y Fv(with)h Fo(C)1041 3118 y Fn(n)1117 3148 y Fv(dep)r(ends)g(in)g(a)f(non-trivial)f(w)n(a)n(y)g(on)i Fs(\012)j Fv(whic)n(h)d(mak)n(es)e(a)h(direct)h(in-)523 3248 y(terpretation)25 b(of)h(p)r(oin)n(ts)g(of)g Fo(C)1477 3218 y Fn(n)1548 3248 y Fv(di\016cult.)h(In)f(particular,)f(let)h Fs(X)j Fv(=)23 b(\()p Fs(q)s(;)14 b(p)p Fv(\))23 b Fr(2)h Fo(R)3030 3218 y Fq(2)p Fn(n)3131 3248 y Fv(=)f Fr(H)q Fv(.)523 3348 y(Then)29 b(the)h(comp)r(onen)n(ts)e(of)h Fs(q)j Fv(and)d Fs(p)g Fv(ha)n(v)n(e)f(a)h(direct)g(ph)n(ysical)f(in)n (terpretation)g(as)g(the)523 3447 y(displacemen)n(ts)23 b(and)h(momen)n(ta)g(of)g(the)g(di\013eren)n(t)g(oscillators.)e(The)i Fs(i)p Fv(th)g(comp)r(onen)n(t)g(of)523 3547 y(the)j(corresp)r(onding)e (v)n(ector)g Fs(z)h Fv(=)d Fs(z)1638 3559 y Fn(\012)1696 3547 y Fv(\()p Fs(X)7 b Fv(\))23 b Fr(2)g Fo(C)1997 3517 y Fn(n)2069 3547 y Fv(do)r(es)j(not)h(ha)n(v)n(e)e(suc)n(h)i(a)f (direct)g(simple)523 3646 y(in)n(terpretation)f(since)i(it)g(is)f(not)g (a)g(function)h(of)g(the)g(diplacemen)n(t)f Fs(q)2690 3658 y Fn(i)2744 3646 y Fv(and)g(momen)n(tum)523 3746 y Fs(p)565 3758 y Fn(i)623 3746 y Fv(of)31 b(the)f Fs(i)p Fv(th)h(oscillator)e(alone,)h(but)h(it)g(is)f(a)g(function)h(of)g(the)g (diplacemen)n(ts)f Fs(q)3113 3758 y Fn(j)3179 3746 y Fv(and)523 3846 y(momen)n(ta)k Fs(p)934 3858 y Fn(j)1003 3846 y Fv(of)g(all)g(the)g(oscillators.)e(This)i(is)g(so)g(b)r(ecause)g (in)g(general,)f(the)h(matrix)523 3945 y Fs(\012)591 3915 y Fq(1)p Fn(=)p Fq(2)729 3945 y Fv(has)e(no)h(\(or)f(few\))h(zero) f(o\013-diagonal)f(en)n(tries,)h(ev)n(en)h(if)g Fs(\012)2603 3915 y Fq(2)2674 3945 y Fv(is)f(tri-diagonal,)f(as)523 4045 y(in)g(the)h(oscillator)d(c)n(hain.)h(Indeed,)h(in)h(that)f(case,) f Fs(\012)2224 4015 y Fq(2)2293 4045 y Fv(is)g(a)h(di\013erence)g(op)r (erator,)e(but)523 4145 y Fs(\012)591 4115 y Fq(1)p Fn(=)p Fq(2)728 4145 y Fv(is)k(not.)f(Con)n(v)n(ersely)-7 b(,)31 b(as)h(is)g(clear)g(from)g(\(20\),)h Fs(q)2275 4157 y Fn(i)2335 4145 y Fv(and)f Fs(p)2543 4157 y Fn(i)2603 4145 y Fv(dep)r(end)i(on)e(all)g(com-)523 4244 y(p)r(onen)n(ts)26 b(of)h Fs(z)964 4256 y Fn(\012)1021 4244 y Fv(\()p Fs(X)7 b Fv(\),)27 b(not)f(only)g(on)g(the)h Fs(i)p Fv(th)f(one.)g(This)g (explains)g(wh)n(y)g(the)h(alternativ)n(e)523 4344 y(form)n(ulation)34 b(of)h(the)g(problem)f(in)h(terms)g(of)g(the)g(complex)f(space)g Fr(K)2780 4314 y Fg(C)2862 4344 y Fv(=)h Fo(C)3022 4314 y Fn(n)3102 4344 y Fv(is)g(not)523 4444 y(found)30 b(in)g(classical)e (mec)n(hanics)h(textb)r(o)r(oks.)h(Indeed,)g(one)f(is)h(t)n(ypically)f (in)n(terested)g(in)523 4543 y(questions)c(concerning)f(the)i (displacemen)n(ts)f(of)h(the)g(di\013eren)n(t)f(oscillators,)f(the)i (energy)523 4643 y(distribution)g(o)n(v)n(er)e(the)i(oscillators)d (when)j(the)g(system)g(is)f(in)h(a)f(normal)g(mo)r(de,)h(energy)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 523 100 a Fu(12)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(propagation)36 b(along)i(the)h(oscillators)d(when)j(originally)e(only)h(one)g (oscillator)f(is)h(ex-)523 382 y(cited,)27 b Fp(etc.)p Fv(.)g(Suc)n(h)f(questions)g(are)f(ob)n(viously)g(more)g(easily)h (addressed)f(in)i(the)f(original)523 482 y(form)n(ulation.)648 581 y(Another)33 b(w)n(a)n(y)f(to)i(see)f(wh)n(y)g(the)h(alternativ)n (e)e(form)n(ulation)h(leads)g(to)g(in)n(terpreta-)523 681 y(tional)25 b(problems)g(is)g(as)g(follo)n(ws.)f(Supp)r(ose)h(w)n (e)g(are)g(studying)g(t)n(w)n(o)f(oscillator)g(systems,)523 780 y(one)k(with)h(p)r(oten)n(tial)1227 748 y Fq(1)p 1227 762 34 4 v 1227 809 a(2)1270 780 y Fs(q)22 b Fr(\001)d Fs(\012)1439 750 y Fq(2)1477 780 y Fs(q)32 b Fv(and)c(another)f(with) 2213 748 y Fq(1)p 2213 762 V 2213 809 a(2)2256 780 y Fs(q)22 b Fr(\001)d Fs(\012)2425 750 y Fl(0)2449 739 y Fq(2)2486 780 y Fs(q)s Fv(,)29 b(where)f Fs(\012)2887 750 y Fq(2)2949 780 y Fr(6)p Fv(=)c Fs(\012)3106 750 y Fl(0)3129 739 y Fq(2)3166 780 y Fv(.)29 b(T)-7 b(o)523 897 y(\014x)24 b(ideas,)g(w)n(e)g(can)g(think)h(of)g Fs(\012)1509 867 y Fl(0)1532 855 y Fq(2)1594 897 y Fv(as)f(b)r(eing)g (a)g(p)r(erturbation)g(of)g Fs(\012)2622 867 y Fq(2)2684 897 y Fv(whic)n(h)h(is)f(obtained)523 996 y(b)n(y)j(c)n(hanging)f(just) i(one)f(spring)f(constan)n(t.)h(Supp)r(ose)g(no)n(w)g(that)g(the)h (state)f(of)g(the)h(\014rst)523 1096 y(system)j(is)f Fs(z)i Fr(2)d(K)1107 1066 y Fg(C)1153 1096 y Fv(,)i(and)g(of)g(the)g (second)f(is)h Fs(z)2017 1066 y Fl(0)2068 1096 y Fr(2)e(K)2216 1066 y Fg(C)2263 1096 y Fv(.)i(Supp)r(ose)f Fs(z)i Fv(=)c Fs(z)2852 1066 y Fl(0)2903 1096 y Fv(=)g Fs(z)3035 1108 y Fq(0)3101 1096 y Fr(2)g Fo(C)3244 1066 y Fn(n)3290 1096 y Fv(.)523 1196 y(W)-7 b(ould)30 b(y)n(ou)e(sa)n(y)h(the)h(t)n(w)n (o)e(systems)h(are)g(in)g(the)h(same)f(state?)g(Certainly)g(not)h(in)f (gen-)523 1295 y(eral!)d(Indeed,)i(as)e(a)h(result)g(of)g(what)g (precedes,)f(and)h(in)h(particular)e(of)h(\(20\),)g(the)g(same)523 1395 y(p)r(oin)n(t)i Fs(z)780 1407 y Fq(0)843 1395 y Fr(2)d(K)988 1365 y Fg(C)1064 1395 y Fv(yields)j(en)n(tirely)g (di\013eren)n(t)g(v)-5 b(alues)29 b(for)g(the)h(displacemen)n(ts)e Fs(q)3020 1407 y Fn(i)3048 1395 y Fs(;)14 b(q)3125 1365 y Fl(0)3122 1416 y Fn(i)3179 1395 y Fv(and)523 1494 y(the)30 b(momen)n(ta)f Fs(p)1074 1506 y Fn(i)1101 1494 y Fs(;)14 b(p)1180 1464 y Fl(0)1180 1516 y Fn(i)1237 1494 y Fv(of)30 b(the)g(t)n(w)n(o)f(oscillator)e(systems!)j(Indeed,)f(w)n(e)h(w)n(ould) f(normally)523 1594 y(sa)n(y)e(that)h(the)h(t)n(w)n(o)e(systems)g(are)g (in)i(the)f(same)g(state)f(if)i(the)f(p)r(ositions)g Fs(q)2855 1606 y Fn(i)2883 1594 y Fs(;)14 b(q)2960 1564 y Fl(0)2957 1616 y Fn(i)3012 1594 y Fv(and)28 b(mo-)523 1694 y(men)n(ta)37 b Fs(p)826 1706 y Fn(i)853 1694 y Fs(;)14 b(p)932 1664 y Fl(0)932 1715 y Fn(i)997 1694 y Fv(of)37 b(the)g(di\013eren)n(t)h(degrees)d(of)i(freedom)g(tak)n(e)g (the)g(same)g(v)-5 b(alues,)36 b(that)523 1793 y(is)h(to)g(sa)n(y)f(if) h Fs(X)45 b Fv(=)39 b Fs(X)1259 1763 y Fl(0)1281 1793 y Fv(.)e(But)h(that)f(is)g(not)g(the)g(same)g(as)f(sa)n(ying)g Fs(z)42 b Fv(=)c Fs(z)2927 1763 y Fl(0)2950 1793 y Fv(.)f(In)g(other) 523 1893 y(w)n(ords,)27 b(if)h(y)n(ou)f(decide)h(to)g(sa)n(y)f Fr(K)1582 1863 y Fg(C)1656 1893 y Fv(is)h(the)g(phase)f(space)h(of)f(y) n(our)g(system,)h(y)n(ou)f(should)523 1993 y(alw)n(a)n(ys)c(remem)n(b)r (er)h(that)h(the)g(ph)n(ysical)f(in)n(terpretation)g(of)g(its)h(p)r (oin)n(ts)g Fp(dep)l(ends)j(on)f(the)523 2092 y(dynamics,)37 b(i.e.)d Fv(on)f Fs(\012)t Fv(.)h(A)f(similar)g(phenomenon)g(pro)r (duces)f(itself)i(in)f(the)h(quan)n(tum)523 2192 y(mec)n(hanical)27 b(description)g(of)g(oscillator)f(systems)h(as)g(w)n(e)g(will)h(see)f (in)h(Sect.)g(3.3.)648 2291 y(Supp)r(ose)38 b(no)n(w)g(w)n(e)g(deal)g (with)h(an)f(in\014nite)h(dimensional)f(oscillator)e(\014eld,)j(suc)n (h)523 2391 y(as)31 b(an)h(oscillator)f(c)n(hain)g(or)g(a)h(w)n(a)n(v)n (e)f(equation.)g(As)h(in)h(the)f(\014nite)h(dimensional)e(case,)523 2491 y(the)40 b(elemen)n(ts)g(of)g Fr(H)h Fv(then)g(ha)n(v)n(e)e(a)g (direct)h(in)n(terpretation)f(in)i(terms)e(of)h(oscillator)523 2590 y(displacemen)n(ts,)35 b(w)n(a)n(v)n(e)f(propagation)f Fp(etc.)p Fv(,)j(whereas)e(those)h(of)h Fr(K)2665 2560 y Fg(C)2747 2590 y Fv(don't.)g(But)f(no)n(w)523 2690 y(an)29 b(additional)g(complicating)f(phenomenon)h(that)g(w)n(e)g (already)f(p)r(oin)n(ted)h(out)h(o)r(ccurs:)523 2790 y(starting)36 b(with)h(a)g(\014xed)g Fr(K)q Fv(,)g(di\013eren)n(t)g(c)n (hoices)f(of)h Fs(\012)k Fv(ma)n(y)36 b(lead)g(to)h(di\013eren)n(t)g (phase)523 2889 y(spaces)32 b Fr(H)q Fv(!)h(W)-7 b(e)33 b(ga)n(v)n(e)e(an)i(example)g(for)f(the)h(oscillator)e(c)n(hain)i(in)g (Sect.)g(2.2.)f(T)-7 b(alking)523 2989 y(ab)r(out)24 b(\\the)g(same)g(state")f(for)h(di\013eren)n(t)h(systems)e(no)n(w)h(b)r (ecomes)g(v)n(ery)f(di\016cult,)i(since)523 3088 y(the)32 b(state)f(space)f Fr(H)i Fv(dep)r(ends)g(on)f(the)g(system)g (considered.)g(It)g(is)g(then)h(tempting)g(to)523 3188 y(prefer)g(the)g(alternativ)n(e)f(form)n(ulation)g(where)h(the)g(phase) g(space)f Fr(K)2703 3158 y Fg(C)2782 3188 y Fv(is)h(indep)r(enden)n(t) 523 3288 y(of)d(the)h(dynamics,)e(but)i(at)f(that)h(p)r(oin)n(t)f(it)h (should)f(alw)n(a)n(ys)e(b)r(e)i(remem)n(b)r(ered)g(that)g(the)523 3387 y(same)j(p)r(oin)n(t)h(in)g Fr(K)1124 3357 y Fg(C)1204 3387 y Fv(has)f(a)g(di\013eren)n(t)h(in)n(terpretation)f(dep)r(ending)h (on)f(whic)n(h)h(system)523 3487 y(y)n(ou)27 b(consider.)648 3587 y(In)c(spite)h(of)g(those)f(in)n(terpretational)f(di\016culties,)j (the)f(alternativ)n(e)e(form)n(ulation)h(of)523 3686 y(the)34 b(classical)d(mec)n(hanics)i(of)g(oscillator)e(systems)i(will) g(turn)g(out)g(to)g(b)r(e)h(useful)g(\(and)523 3786 y(ev)n(en)g (crucial\))g(in)h(the)f(quan)n(tum)h(mec)n(hanical)e(description)h(of)h (oscillator)d(\014elds.)j(In-)523 3885 y(deed,)26 b(the)f(quan)n(tum)h (Hilb)r(ert)g(space)e(for)h(the)h(free)f(oscillator)f(\014eld)h(will)h (b)r(e)g(seen)f(to)g(b)r(e)523 3985 y(the)31 b(symmetric)f(F)-7 b(o)r(c)n(k)30 b(space)g(o)n(v)n(er)f(\()p Fr(H)q Fs(;)14 b(J)o(;)g Fr(h\001)p Fs(;)g Fr(\001i)2051 3997 y Fq(+)2106 3985 y Fv(\))31 b(\(see)g(Section)f(3\).)h(But)f(iden)n(tifying)523 4085 y(the)k(latter)e(with)i Fr(K)1162 4055 y Fg(C)1242 4085 y Fv(allo)n(ws)d(one)i(to)g(con)n(v)n(enien)n(tly)f(iden)n(tify)i (the)f(quan)n(tum)g(Hilb)r(ert)523 4184 y(space)j(as)f(the)i(symmetric) f(F)-7 b(o)r(c)n(k)36 b(space)f(o)n(v)n(er)g Fr(K)2116 4154 y Fg(C)2162 4184 y Fv(.)i(This)f(w)n(a)n(y)-7 b(,)35 b(one)h(can)g(w)n(ork)f(on)h(a)523 4284 y(\014xed)27 b(Hilb)r(ert)g(space,)f(while)h(c)n(hanging)e(the)i(dynamics)f(b)n(y)g (p)r(erturbing)g Fs(\012)t Fv(,)h(for)f(exam-)523 4384 y(ple.)34 b(This)g(is)f(v)n(ery)g(con)n(v)n(enien)n(t.)g(Still,)h(the)g (rather)f(ob)n(vious,)g(seemingly)g(trivial)g(and)523 4483 y(inno)r(cuous)g(remarks)e(ab)r(o)n(v)n(e)h(concerning)g(the)i(in) n(terpretation)e(of)h(the)g Fp(classic)l(al)i Fv(\014eld)523 4583 y(theory)i(are)f(at)h(the)h(origin)e(of)i(further,)f(more)g (subtle)g(in)n(terpretational)f(di\016culties)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(13)523 282 y Fv(with)28 b(the)g(quan)n(tum)f (\014eld)h(theory)f(of)g(in\014nite)i(dimensional)e(oscillator)e (\014elds)j(as)f(w)n(ell,)523 382 y(to)h(whic)n(h)f(I)h(shall)f(come)g (bac)n(k)g(in)h(Sects.)f(3.3)g(and)h(3.5.)523 618 y Fm(2.5)41 b(Creation)31 b(and)i(annihilation)d(functions)i(on)g Fd(H)523 795 y Fv(F)-7 b(or)28 b(the)i(purp)r(oses)e(of)h(quan)n(tum)g (mec)n(hanics,)f(it)i(will)f(turn)g(out)g(to)g(b)r(e)h(con)n(v)n(enien) n(t)e(to)523 895 y(dev)n(elop)f(the)h(previous)f(considerations)f (somewhat)h(further.)g(Ev)n(eryb)r(o)r(dy)g(is)g(familiar)523 995 y(with)20 b(creation)f(and)h(annihilation)f(op)r(erators)f(in)i (quan)n(tum)g(mec)n(hanics.)f(These)g(ob)5 b(jects)523 1094 y(are)35 b(usually)h(describ)r(ed)g(as)g(t)n(ypically)g(quan)n (tum)g(mec)n(hanical)f(in)i(nature,)f(but)g(they)523 1194 y(ha)n(v)n(e)j(a)g(p)r(erfectly)h(natural)f(classical)g(analog,)f (that)i(I)g(will)g(call)g(the)g(creation)f(and)523 1293 y(annihilation)27 b(functions,)h(and)g(that)g(are)e(de\014ned)i(as)f (follo)n(ws.)648 1393 y(F)-7 b(or)27 b(all)g Fs(\030)g Fr(2)c(K)1117 1363 y Fg(C)1164 1393 y Fv(,)1343 1569 y Fs(a)1387 1581 y Fq(c)1421 1569 y Fv(\()p Fs(\030)t Fv(\))g(:)h Fs(X)29 b Fr(2)23 b(H)h(7!)1980 1548 y Fv(\026)1971 1569 y Fs(\030)f Fr(\001)18 b Fs(z)2110 1581 y Fn(\012)2168 1569 y Fv(\()p Fs(X)7 b Fv(\))23 b Fr(2)h Fo(C)p Fs(;)523 1746 y Fv(and)1343 1846 y Fs(a)1387 1811 y Fl(y)1387 1866 y Fq(c)1421 1846 y Fv(\()p Fs(\030)t Fv(\))g(:)f Fs(X)29 b Fr(2)24 b(H)g(7!)f Fs(\030)f Fr(\001)d Fs(z)2115 1806 y Fl(y)2111 1870 y Fn(\012)2169 1846 y Fv(\()p Fs(X)7 b Fv(\))23 b Fr(2)g Fo(C)p Fs(:)523 1991 y Fv(Note)39 b(that)h Fs(a)971 2003 y Fq(c)1004 1991 y Fv(\()p Fs(\030)t Fv(\))h(is)e(an)n(ti-linear)f(in)h Fs(\030)t Fv(,)h(whereas)e Fs(a)2236 1961 y Fl(y)2236 2012 y Fq(c)2270 1991 y Fv(\()p Fs(\030)t Fv(\))i(is)f(linear.)g(The)g(index)h(\\c")523 2091 y(stands)e(for)f(\\classical",)f(so)i(that)g(the)h(notation)e (distinguishes)h(b)r(et)n(w)n(een)g(the)h(clas-)523 2190 y(sical)32 b(creation/annihilation)f(functions)j(and)e(the)i(quan)n (tum)f(creation/annihilation)523 2290 y(op)r(erators,)26 b(to)h(b)r(e)h(in)n(tro)r(duced)g(later.)f(A)g(direct)h(computation)f (no)n(w)g(yields)1439 2466 y Fr(f)p Fs(a)1525 2478 y Fq(c)1558 2466 y Fv(\()p Fs(\030)1626 2478 y Fq(1)1664 2466 y Fv(\))p Fs(;)14 b(a)1777 2432 y Fl(y)1777 2487 y Fq(c)1812 2466 y Fv(\()p Fs(\030)1880 2478 y Fq(2)1917 2466 y Fv(\))p Fr(g)23 b Fv(=)g Fr(\000)p Fv(i)2199 2445 y(\026)2190 2466 y Fs(\030)2226 2478 y Fq(1)2282 2466 y Fr(\001)18 b Fs(\030)2359 2478 y Fq(2)523 2643 y Fv(and)1068 2743 y Fs(a)1112 2755 y Fq(c)1146 2743 y Fv(\()p Fs(\030)t Fv(\))h Fr(\016)f Fs(\010)1384 2755 y Fn(t)1437 2743 y Fv(=)k Fs(a)1568 2755 y Fq(c)1602 2743 y Fv(\(e)1671 2708 y Fq(i)p Fn(\012)s(t)1773 2743 y Fs(\030)t Fv(\))p Fs(;)97 b(a)2009 2708 y Fl(y)2009 2763 y Fq(c)2044 2743 y Fv(\()p Fs(\030)t Fv(\))19 b Fr(\016)f Fs(\010)2282 2755 y Fn(t)2335 2743 y Fv(=)k Fs(a)2466 2708 y Fl(y)2466 2763 y Fq(c)2501 2743 y Fv(\(e)2570 2708 y Fq(i)p Fn(\012)s(t)2672 2743 y Fs(\030)t Fv(\))p Fs(:)523 2888 y Fv(Also,)27 b(for)g(all)h Fs(\021)e Fr(2)d(K)1185 2858 y Fg(C)1184 2915 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)581 3126 y Fs(\021)f Fr(\001)c Fs(q)26 b Fv(=)881 3070 y(1)p 846 3107 111 4 v 846 3124 a Fr(p)p 915 3124 42 4 v 68 x Fv(2)967 3126 y(\()p Fs(a)1043 3138 y Fq(c)1076 3126 y Fv(\()p Fs(\012)1176 3092 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1339 3126 y Fv(\026)-48 b Fs(\021)s Fv(\))19 b(+)f Fs(a)1555 3092 y Fl(y)1555 3147 y Fq(c)1590 3126 y Fv(\()p Fs(\012)1690 3092 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1847 3126 y Fs(\021)s Fv(\)\))23 b(=)2088 3070 y Fr(\000)p Fv(i)p 2076 3107 111 4 v 2076 3124 a Fr(p)p 2145 3124 42 4 v 68 x Fv(2)2197 3126 y(\()p Fs(a)2273 3092 y Fl(y)2273 3147 y Fq(c)2307 3126 y Fv(\(i)p Fs(\012)2430 3092 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2587 3126 y Fs(\021)s Fv(\))c Fr(\000)f Fs(a)2809 3138 y Fq(c)2843 3126 y Fv(\(i)p Fs(\012)2966 3092 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)3129 3126 y Fv(\026)-48 b Fs(\021)s Fv(\)\))p Fs(;)3165 3274 y Fv(\(22\))523 3374 y(and,)28 b(similarly)-7 b(,)26 b(for)h(all)h Fs(\021)e Fr(2)d(K)1515 3343 y Fg(C)1514 3400 y Fq(1)p Fn(=)p Fq(2)1270 3612 y Fs(\021)f Fr(\001)c Fs(p)23 b Fv(=)1581 3555 y(i)p 1537 3592 111 4 v 1537 3609 a Fr(p)p 1606 3609 42 4 v 69 x Fv(2)1657 3612 y(\()p Fs(a)1733 3577 y Fl(y)1733 3632 y Fq(c)1768 3612 y Fv(\()p Fs(\012)1868 3577 y Fq(1)p Fn(=)p Fq(2)1973 3612 y Fs(\021)s Fv(\))c Fr(\000)f Fs(a)2195 3624 y Fq(c)2229 3612 y Fv(\()p Fs(\012)2329 3577 y Fq(1)p Fn(=)p Fq(2)2440 3612 y Fv(\026)-48 b Fs(\021)s Fv(\)\))p Fs(:)600 b Fv(\(23\))523 3831 y(In)40 b(the)h(language)e(of)h(the)g(ph)n(ysics)g(literature,)f(these)h(t)n(w) n(o)g(equations)f(express)g(the)523 3931 y(oscilator)29 b(\014eld)j Fs(\021)24 b Fr(\001)d Fs(q)34 b Fv(and)d(its)g(conjugate)f (\014eld)i Fs(\021)24 b Fr(\001)d Fs(p)31 b Fp(viewe)l(d)j(as)f (functions)g(on)g(phase)523 4031 y(sp)l(ac)l(e)i Fv(in)28 b(terms)f(of)g(the)h(creation)f(and)g(annihilation)h(functions.)648 4130 y(It)k(is)g(\014nally)g(instructiv)n(e)f(to)h(write)g Fs(H)39 b Fv(explicitly)32 b(in)g(terms)g(of)g(the)g(annihilation)523 4230 y(and)c(creation)f(functions.)h(This)g(is)f(easily)g(done)h(when)g Fs(\012)k Fv(has)c(pure)f(p)r(oin)n(t)h(sp)r(ectrum,)523 4330 y Fp(i.e.)h Fv(when)f(there)f(exists)h(a)f(basis)g(of)g (normalized)g(eigen)n(v)n(ectors)e(for)i Fs(\012)32 b Fv(on)27 b Fr(K)2982 4300 y Fg(C)3029 4330 y Fv(:)1581 4506 y Fs(\012)t(\021)1690 4518 y Fn(i)1741 4506 y Fv(=)c Fs(!)1881 4518 y Fn(i)1908 4506 y Fs(\021)1949 4518 y Fn(i)1977 4506 y Fs(;)42 b(i)22 b Fr(2)i Fo(N)p Fs(:)523 4682 y Fv(Then,)k(from)f(\(21\))p eop end %%Page: 14 14 TeXDict begin 14 13 bop 523 100 a Fu(14)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)1143 309 y Fs(H)30 b Fv(=)1340 253 y(1)p 1340 290 42 4 v 1340 366 a(2)1405 230 y Fk(X)1453 407 y Fn(i)1539 309 y Fs(!)1591 321 y Fn(i)1632 242 y Fk(\000)1670 309 y Fs(a)1714 275 y Fl(y)1714 329 y Fq(c)1749 309 y Fv(\()p Fs(\021)1822 321 y Fn(i)1850 309 y Fv(\))p Fs(a)1926 321 y Fq(c)1960 309 y Fv(\()p Fs(\021)2033 321 y Fn(i)2061 309 y Fv(\))18 b(+)g Fs(a)2238 321 y Fq(c)2272 309 y Fv(\()6 b(\026)-48 b Fs(\021)2345 321 y Fn(i)2373 309 y Fv(\))p Fs(a)2449 275 y Fl(y)2449 329 y Fq(c)2484 309 y Fv(\()6 b(\026)-48 b Fs(\021)2557 321 y Fn(i)2585 309 y Fv(\))2617 242 y Fk(\001)2669 309 y Fs(:)473 b Fv(\(24\))523 569 y(Note)30 b(that)h(b)r(oth)g(sides)f(of)g (this)h(equation)e(are)h(functions)g(on)g(\(a)g(suitable)h(subset)f(of) 6 b(\))523 668 y Fr(H)q Fv(.)32 b(Corresp)r(ondingly)-7 b(,)30 b(in)i(quan)n(tum)g(mec)n(hanics,)f(b)r(oth)h(sides)g(will)g(b)r (e)g(op)r(erators)e(on)523 768 y(the)e(quan)n(tum)g(Hilb)r(ert)g(space) f(of)g(states.)523 1084 y Ft(3)42 b(The)37 b(quan)m(tum)i(theory)e(of)h (free)g(harmonic)g(systems)523 1283 y Fm(3.1)j(Finite)30 b(dimensional)g(harmonic)i(systems:)e(the)i(Sc)m(hr\177)-48 b(odinger)523 1382 y(represen)m(tation)523 1565 y Fv(Ho)n(w)28 b(to)h(giv)n(e)f(a)g(quan)n(tum)h(mec)n(hanical)f(description)g(of)g (the)i(classical)d(free)h(oscillator)523 1665 y(\014elds)h(studied)h (in)f(Section)g(2?)g(I)h(shall)e(pro)r(ceed)h(in)g(t)n(w)n(o)g(steps.)g (I)g(will)g(\014rst)g(recall)f(the)523 1764 y(quan)n(tum)38 b(description)f(of)h(a)g(system)f(of)h(a)g(\014nite)g(n)n(um)n(b)r(er)g (of)g(coupled)g(oscillators,)523 1864 y(and)27 b(then)g(rewrite)f(it)h (in)g(a)g(manner)f(suitable)h(for)f(immediate)h(adaptation)f(to)h (in\014nite)523 1964 y(dimension.)648 2063 y(The)f(quan)n(tum)g (Hamiltonian)g(for)g(a)g(system)g(with)h Fs(n)f Fv(degrees)f(of)i (freedom)f(ha)n(ving)523 2163 y(a)h(classical)f(Hamiltonian)i(giv)n(en) f(b)n(y)1606 2380 y Fs(H)j Fv(=)1803 2324 y(1)p 1803 2361 V 1803 2437 a(2)1854 2380 y Fs(p)1896 2346 y Fq(2)1952 2380 y Fv(+)18 b Fs(V)h Fv(\()p Fs(q)s Fv(\))p Fs(;)523 2590 y Fv(where)38 b(the)g(p)r(oten)n(tial)g Fs(V)57 b Fv(is)38 b(a)g(\(smo)r(oth\))g(real-v)-5 b(alued)37 b(function)i(on)f Fo(R)2890 2560 y Fn(n)2973 2590 y Fv(is,)g(in)g(the) 523 2690 y(so-called)26 b(p)r(osition)i(\(or)f(Sc)n(hr\177)-42 b(odinger\))26 b(represen)n(tation)g(giv)n(en)g(b)n(y)1582 2912 y Fs(H)k Fv(=)1778 2856 y(1)p 1778 2893 V 1778 2969 a(2)1830 2912 y Fs(P)1895 2878 y Fq(2)1950 2912 y Fv(+)18 b Fs(V)h Fv(\()p Fs(Q)p Fv(\))p Fs(;)523 3122 y Fv(where)31 b Fs(P)41 b Fv(=)28 b Fr(\000)p Fv(i)p Fs(@)5 b(=@)g(x)30 b Fv(and)h Fs(Q)e Fv(=)g Fs(x)i Fv(are)g(the)g(usual)g(momen)n(tum)h (and)f(p)r(osition)g(op)r(er-)523 3222 y(ators)d(whic)n(h)i(are)f (self-adjoin)n(t)g(on)h(there)f(natural)g(domains)g(in)h(the)h(\\quan)n (tum)e(state)523 3322 y(space")34 b Fs(L)851 3292 y Fq(2)888 3322 y Fv(\()p Fo(R)980 3292 y Fn(n)1025 3322 y Fs(;)14 b Fv(d)p Fs(x)p Fv(\).)37 b(Note)e(that,)h(just)g(as)f(in)g(the)h (classical)e(description,)h(the)h(state)523 3421 y(space)e(is)g(indep)r (enden)n(t)i(of)e(the)h(dynamics,)g(whic)n(h)f(mak)n(es)g(it)h(easy)e (to)i(compare)e(the)523 3521 y(dynamics)25 b(generated)f(b)n(y)g(t)n(w) n(o)h(di\013eren)n(t)g(Hamiltonians)f Fs(H)32 b Fv(and)25 b Fs(H)2699 3491 y Fl(0)2722 3521 y Fv(,)g(with)h(p)r(oten)n(tials)523 3621 y Fs(V)46 b Fv(and)27 b Fs(V)845 3590 y Fl(0)868 3621 y Fv(.)g(T)-7 b(o)27 b(put)g(it)h(di\013eren)n(tly)-7 b(,)27 b(just)g(as)g(a)f(giv)n(en)g(p)r(oin)n(t)i Fs(X)33 b Fv(in)27 b(the)h(classical)d(phase)523 3720 y(space)20 b Fo(R)798 3690 y Fq(2)p Fn(n)898 3720 y Fv(corresp)r(onds)f(to)i(the)g (same)f(state)h(of)g(the)g(system,)g(whatev)n(er)f(its)h(dynamics,)523 3820 y(so)k(a)g(giv)n(en)f Fs( )29 b Fv(in)c Fs(L)1138 3790 y Fq(2)1175 3820 y Fv(\()p Fo(R)1267 3790 y Fn(n)1312 3820 y Fv(\))h(yields)f(the)h(same)f(p)r(osition)g(and)g(momen)n(tum)h (distributions)523 3919 y(for)h(the)h(system,)g(whatev)n(er)e(the)i (dynamics)f(to)g(whic)n(h)h(it)g(is)g(sub)5 b(jected.)648 4019 y(Consequen)n(tly)-7 b(,)20 b(for)h(an)g Fs(n)p Fv(-dimensional)f(system)h(of)g(coupled)h(oscillators)d(with)j(clas-) 523 4119 y(sical)36 b(con\014guration)f(space)h Fr(K)k Fv(=)e Fo(R)1726 4089 y Fn(n)1808 4119 y Fv(and)f(phase)f(space)g Fr(H)j Fv(=)f Fo(R)2722 4089 y Fq(2)p Fn(n)2838 4119 y Fv(the)f(quan)n(tum)523 4218 y(Hamiltonian)27 b(reads)1500 4336 y Fs(H)j Fv(=)1696 4280 y(1)p 1696 4317 V 1696 4393 a(2)1748 4336 y(\()p Fs(P)1845 4302 y Fq(2)1901 4336 y Fv(+)18 b Fs(Q)g Fr(\001)g Fs(\012)2177 4302 y Fq(2)2215 4336 y Fs(Q)p Fv(\))p Fs(:)523 4509 y Fv(Unfortunately)-7 b(,)29 b(these)h(expressions)d(stop)i(making)g(sense)f(when)i Fr(K)h Fv(is)e(an)g(in\014nite)h(di-)523 4609 y(mensional)35 b(space,)f(in)i(particular)e(since)h(it)h(is)f(not)h(p)r(ossible)f(to)g (mak)n(e)f(sense)h(out)h(of)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(15)523 282 y Fs(L)580 252 y Fq(2)617 282 y Fv(\()p Fr(K)q Fv(\))30 b(in)e(that)h(case.)f(So)g(to) h(describ)r(e)f(the)h(quan)n(tum)g(mec)n(hanics)f(of)g(in\014nite)i (dimen-)523 382 y(sional)22 b(harmonic)g(systems,)g(I)h(will)g(\014rst) f(rewrite)g(the)i(ab)r(o)n(v)n(e)d(Hamiltonian)i(di\013eren)n(tly)-7 b(,)523 482 y(in)25 b(a)g(manner)f(allo)n(wing)g(for)h(immediate)g (generalization)e(to)i(in\014nite)h(dimension.)f(This)523 581 y(rewriting)31 b(is,)h(as)f(w)n(e)h(shall)f(see,)h(v)n(ery)f (analogous)f(to)h(the)i(rewriting)e(of)h(the)g(classical)523 681 y(mec)n(hanics)d(on)h Fr(K)1102 651 y Fg(C)1176 681 y Fv(=)d Fo(C)1328 651 y Fn(n)1373 681 y Fv(,)j(explained)g(in)g(Sect.) g(2.3,)g(and)f(is)h(therefore)g(also)f(a\013ected)523 780 y(b)n(y)35 b(the)g(in)n(terpretational)e(di\016culties)j(men)n (tioned)e(in)i(Sect.)f(2.4.)f(It)h(is)g(nev)n(ertheless)523 880 y(v)n(ery)26 b(e\016cien)n(t)i(and)g(essen)n(tial.)648 980 y(Let's)f(de\014ne,)h(for)f(an)n(y)g Fs(\030)h Fr(2)23 b(K)1605 950 y Fg(C)1675 980 y Fv(=)g Fo(C)1823 950 y Fn(n)1868 980 y Fv(,)28 b(the)g(so-called)e(creation)h(and)g (annihilation)523 1079 y(op)r(erators)566 1296 y(~)-43 b Fs(a)o Fv(\()p Fs(\030)t Fv(\))24 b(=)p 824 1228 41 4 v 23 w Fs(\030)18 b Fr(\001)960 1240 y Fv(1)p 926 1277 111 4 v 926 1293 a Fr(p)p 995 1293 42 4 v 69 x Fv(2)1047 1296 y(\()p Fs(\012)1147 1262 y Fq(1)p Fn(=)p Fq(2)1251 1296 y Fs(Q)d Fv(+)f(i)p Fs(\012)1502 1262 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1658 1296 y Fs(P)e Fv(\))p Fs(;)98 b Fv(~)-43 b Fs(a)1919 1262 y Fl(y)1954 1296 y Fv(\()p Fs(\030)t Fv(\))23 b(=)g Fs(\030)c Fr(\001)2306 1240 y Fv(1)p 2271 1277 111 4 v 2271 1293 a Fr(p)p 2340 1293 42 4 v 69 x Fv(2)2392 1296 y(\()p Fs(\012)2492 1262 y Fq(1)p Fn(=)p Fq(2)2597 1296 y Fs(Q)14 b Fr(\000)g Fv(i)p Fs(\012)2847 1262 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)3003 1296 y Fs(P)e Fv(\))p Fs(:)42 b Fv(\(25\))523 1521 y(Note)34 b(that)g(those)g(are)f(\014rst)h(order)e(di\013eren)n(tial)i(op)r (erators,)e(and)i(that)g(they)g(dep)r(end)523 1621 y(on)27 b Fs(\012)t Fv(,)h(although)f(the)i(notation)e(do)r(es)g(not)h(bring)f (this)h(dep)r(endence)g(out.)f(One)h(c)n(hec)n(ks)523 1721 y(easily)f(that)1539 1753 y Fk(\002)1574 1820 y Fv(~)-43 b Fs(a)p Fv(\()p Fs(\030)t Fv(\))p Fs(;)16 b Fv(~)-44 b Fs(a)1802 1786 y Fl(y)1837 1820 y Fv(\()p Fs(\030)1909 1786 y Fl(0)1933 1820 y Fv(\))1965 1753 y Fk(\003)2023 1820 y Fv(=)p 2110 1753 41 4 v 22 w Fs(\030)23 b Fr(\001)c Fs(\030)2251 1786 y Fl(0)2274 1820 y Fs(;)868 b Fv(\(26\))523 1969 y(all)27 b(other)g(comm)n(utators)f(v)-5 b(anishing.)28 b(In)f(addition,)h(for)f(an)n(y)g Fs(\021)f Fr(2)e Fo(C)2691 1939 y Fn(n)2736 1969 y Fv(,)1223 2186 y Fs(\021)e Fr(\001)c Fs(Q)23 b Fv(=)1548 2130 y(1)p 1514 2167 111 4 v 1514 2183 a Fr(p)p 1583 2183 42 4 v 69 x Fv(2)1634 2186 y(\()q(~)-43 b Fs(a)p Fv(\()p Fs(\012)1810 2152 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1973 2186 y Fv(\026)-48 b Fs(\021)t Fv(\))18 b(+)h(~)-43 b Fs(a)2189 2152 y Fl(y)2224 2186 y Fv(\()p Fs(\012)2324 2152 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2481 2186 y Fs(\021)s Fv(\)\))p Fs(;)553 b Fv(\(27\))523 2411 y(and,)28 b(similarly)-7 b(,)1275 2547 y Fs(\021)22 b Fr(\001)d Fs(P)35 b Fv(=)1609 2491 y(i)p 1565 2528 111 4 v 1565 2545 a Fr(p)p 1634 2545 42 4 v 69 x Fv(2)1686 2547 y(\()q(~)-43 b Fs(a)1762 2513 y Fl(y)1797 2547 y Fv(\()p Fs(\012)1897 2513 y Fq(1)p Fn(=)p Fq(2)2002 2547 y Fs(\021)s Fv(\))18 b Fr(\000)i Fv(~)-44 b Fs(a)p Fv(\()p Fs(\012)2323 2513 y Fq(1)p Fn(=)p Fq(2)2434 2547 y Fv(\026)c Fs(\021)s Fv(\)\))p Fs(:)606 b Fv(\(28\))523 2740 y(The)31 b(analogy)e(of)i(this)g(and)g(of)g(the)g(rest)g(of)g(this)g(section)g (with)g(the)h(dev)n(elopmen)n(ts)e(of)523 2840 y(Sect.)25 b(2.5)g(should)f(b)r(e)i(self-eviden)n(t.)f(In)g(particular,)f(it)h(is) g(clear)f(that)h(the)h(creation)e(and)523 2939 y(annihilation)f(op)r (erators)f(are)h(the)h(\\quan)n(tization")e(of)i(the)g(creation)e(and)i (annihilation)523 3039 y(functions)k Fs(a)925 3051 y Fq(c)959 3039 y Fs(;)14 b(a)1040 3009 y Fl(y)1040 3059 y Fq(c)1102 3039 y Fv(in)n(tro)r(duced)27 b(earlier.)648 3139 y(F)-7 b(urthermore,)35 b(let)i Fs(\021)1331 3151 y Fn(i)1396 3139 y Fr(2)h(K)1553 3108 y Fg(C)1638 3139 y Fv(=)f Fo(C)1800 3108 y Fn(n)1845 3139 y Fs(;)14 b(i)37 b Fv(=)g(1)14 b Fs(:)g(:)g(:)g(n)36 b Fv(b)r(e)h(an)f(orthonormal)e (basis)i(of)523 3238 y(eigen)n(v)n(ectors)c(of)j Fs(\012)1164 3208 y Fq(2)1236 3238 y Fv(with)g(eigen)n(v)-5 b(alues)34 b Fs(!)1924 3208 y Fq(2)1921 3259 y(1)1996 3238 y Fr(\024)g Fs(!)2150 3208 y Fq(2)2147 3259 y(2)2222 3238 y Fr(\024)g Fs(:)14 b(:)g(:)35 b Fr(\024)f Fs(!)2607 3208 y Fq(2)2604 3259 y Fn(n)2649 3238 y Fv(.)h(Then)g(it)g(is)f(easily)523 3338 y(c)n(hec)n(k)n(ed)26 b(that)633 3566 y Fs(H)j Fv(=)829 3509 y(1)p 829 3547 V 829 3623 a(2)934 3462 y Fn(n)894 3487 y Fk(X)900 3664 y Fn(i)p Fq(=1)1028 3566 y Fs(!)1080 3578 y Fn(i)1121 3498 y Fk(\000)1160 3566 y Fv(~)-43 b Fs(a)1203 3531 y Fl(y)1238 3566 y Fv(\()p Fs(\021)1311 3578 y Fn(i)1339 3566 y Fv(\))q(~)g Fs(a)p Fv(\()p Fs(\021)1488 3578 y Fn(i)1516 3566 y Fv(\))19 b(+)g(~)-43 b Fs(a)p Fv(\()6 b(\026)-48 b Fs(\021)1767 3578 y Fn(i)1795 3566 y Fv(\))q(~)-43 b Fs(a)1871 3531 y Fl(y)1906 3566 y Fv(\()6 b(\026)-48 b Fs(\021)1979 3578 y Fn(i)2007 3566 y Fv(\))2039 3498 y Fk(\001)2100 3566 y Fv(=)2227 3462 y Fn(n)2188 3487 y Fk(X)2194 3664 y Fn(i)p Fq(=1)2322 3566 y Fs(!)2374 3578 y Fn(i)2402 3566 y Fv(~)-43 b Fs(a)2445 3531 y Fl(y)2479 3566 y Fv(\()p Fs(\021)2552 3578 y Fn(i)2581 3566 y Fv(\))q(~)g Fs(a)p Fv(\()p Fs(\021)2730 3578 y Fn(i)2758 3566 y Fv(\))19 b(+)2902 3509 y(1)p 2902 3547 V 2902 3623 a(2)3006 3462 y Fn(n)2967 3487 y Fk(X)2973 3664 y Fn(i)p Fq(=1)3101 3566 y Fs(!)3153 3578 y Fn(i)3180 3566 y Fs(:)523 3820 y Fv(The)j(sp)r(ectral)f(analysis)f(of)h Fs(H)29 b Fv(is)21 b(no)n(w)g(straigh)n(tforw)n(ardly)d(w)n(ork)n(ed)i(out,)i(and)f (describ)r(ed)523 3920 y(in)28 b(an)n(y)f(textb)r(o)r(ok)g(on)h(quan)n (tum)f(mec)n(hanics.)g(Let)h(me)f(recall)g(the)h(essen)n(tials.)648 4020 y(It)36 b(is)g(\014rst)f(of)h(all)g(readily)f(c)n(hec)n(k)n(ed)g (that)h(there)g(exists)f(a)h(unit)h(v)n(ector)d Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)36 b Fv(in)523 4119 y Fs(L)580 4089 y Fq(2)617 4119 y Fv(\()p Fo(R)709 4089 y Fn(n)754 4119 y Fv(\))28 b(\(unique)g(up)g(to)g(a)f(global)f(phase\),)i(for)f (whic)n(h)1491 4301 y(~)-43 b Fs(a)p Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)24 b Fv(=)e(0)p Fs(;)14 b Fr(8)p Fs(\030)27 b Fr(2)c Fo(C)2277 4267 y Fn(n)2323 4301 y Fs(:)523 4483 y Fv(This)32 b(common)f(eigen)n(v)n (ector)e(of)j(all)f(the)h(annihilation)g(op)r(erators)e(~)-43 b Fs(a)p Fv(\()p Fs(\030)t Fv(\))32 b(is)g(called)f(the)523 4583 y(\\v)-5 b(acuum".)28 b(Remark)g(that,)g(as)g(a)g(v)n(ector)g(in)h Fs(L)2033 4553 y Fq(2)2069 4583 y Fv(\()p Fo(R)2161 4553 y Fn(n)2207 4583 y Fv(\),)g(the)g(v)-5 b(acuum)28 b Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)29 b Fv(ob)n(viously)523 4682 y(dep)r(ends)f(on)f Fs(\012)t Fv(.)h(One)g(has)f(indeed)h(v)n(ery) e(explicitly)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 523 100 a Fu(16)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)1272 329 y Fr(h)p Fs(x)p Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)24 b Fv(=)1674 273 y(\(det)q Fs(\012)t Fv(\))1922 243 y Fq(1)p Fn(=)p Fq(4)p 1674 310 353 4 v 1769 388 a Fs(\031)1819 364 y Fn(n=)p Fq(4)2051 329 y Fv(exp)13 b Fr(\000)2266 273 y Fv(1)p 2266 310 42 4 v 2266 386 a(2)2317 329 y Fs(x)19 b Fr(\001)g Fs(\012)t(x:)602 b Fv(\(29\))648 548 y(Clearly)30 b Fs(H)7 b Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)29 b Fv(=)1351 516 y Fq(1)p 1351 530 34 4 v 1351 577 a(2)1408 486 y Fk(P)1495 507 y Fn(n)1495 573 y(i)p Fq(=1)1621 548 y Fs(!)1673 560 y Fn(i)1700 548 y Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)p Fv(,)32 b(so)f(that)g(the)h(v)-5 b(acuum)32 b Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)31 b Fv(is)g(actually)523 666 y(the)h(ground)f(state)h(of)g Fs(H)7 b Fv(.)32 b(W)-7 b(riting)32 b(for)f(brevit)n(y)h(~)-43 b Fs(a)2166 678 y Fn(i)2224 666 y Fv(=)31 b(~)-43 b Fs(a)p Fv(\()p Fs(\021)2436 678 y Fn(i)2464 666 y Fv(\))p Fs(;)15 b Fv(~)-43 b Fs(a)2577 626 y Fl(y)2577 689 y Fn(i)2642 666 y Fv(=)31 b(~)-44 b Fs(a)2780 635 y Fl(y)2815 666 y Fv(\()p Fs(\021)2888 678 y Fn(i)2916 666 y Fv(\))32 b(it)h(follo)n(ws)523 765 y(\(after)28 b(some)f(w)n(ork\))f(that)i(the)g(v)n(ectors)1073 930 y(1)p 726 967 735 4 v 726 984 a Fr(p)p 795 984 666 4 v 64 x Fs(m)868 1060 y Fq(1)905 1048 y Fv(!)p Fs(m)1001 1060 y Fq(2)1039 1048 y Fv(!)p Fs(m)1135 1060 y Fq(3)1172 1048 y Fv(!)14 b Fs(:)g(:)g(:)f(m)1392 1060 y Fn(n)1438 1048 y Fv(!)1484 894 y Fk(\020)1535 986 y Fv(~)-43 b Fs(a)1578 946 y Fl(y)1578 1008 y Fq(1)1615 894 y Fk(\021)1665 911 y Fn(m)1724 919 y Ff(1)1774 894 y Fk(\020)1825 986 y Fv(~)g Fs(a)1868 946 y Fl(y)1868 1008 y Fq(2)1905 894 y Fk(\021)1954 911 y Fn(m)2013 919 y Ff(2)2064 894 y Fk(\020)2114 986 y Fv(~)g Fs(a)2157 946 y Fl(y)2157 1008 y Fq(3)2194 894 y Fk(\021)2244 911 y Fn(m)2303 919 y Ff(3)2353 986 y Fs(:)14 b(:)g(:)2464 919 y Fk(\000)2503 986 y Fv(~)-43 b Fs(a)2546 952 y Fl(y)2546 1007 y Fn(n)2591 919 y Fk(\001)2629 932 y Fn(m)2688 940 y Fc(n)2747 986 y Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)p Fs(;)193 b Fv(\(30\))523 1216 y(for)22 b(all)g(p)r(ossible)g(c)n(hoices)f(\()p Fs(m)1441 1228 y Fq(1)1478 1216 y Fs(;)14 b(:)g(:)g(:)g(;)g(m)1736 1228 y Fn(n)1781 1216 y Fv(\))23 b Fr(2)g Fo(N)1974 1186 y Fn(n)2042 1216 y Fv(form)f(an)g(orthonormal)e(basis)h(of)h(eigen-)523 1316 y(v)n(ectors)k(for)h Fs(H)7 b Fv(.)648 1416 y(Note)32 b(that)h(the)f(p)r(osition)g(and)h(momen)n(tum)f(distributions)g(of)h (the)g(ground)e(state)523 1515 y(eviden)n(tly)c(dep)r(end)h(on)f Fs(\012)32 b Fv(and)27 b(are)g(in)g(fact)h(not)f(totally)g(trivial)g (to)g(compute,)g(despite)523 1615 y(the)e(apparen)n(t)e(simplicit)n(y)i (of)f(the)h(Gaussian)f(expression)f(ab)r(o)n(v)n(e.)g(Indeed,)i(if)g(y) n(ou)e(w)n(an)n(t)523 1714 y(to)32 b(kno)n(w,)g(for)g(example,)g Fr(h)p Fv(0)p Fs(;)14 b(\012)t Fr(j)p Fs(Q)1627 1684 y Fq(2)1627 1735 y(7)1664 1714 y Fr(j)p Fv(0)p Fs(;)g(\012)t Fr(i)33 b Fv(y)n(ou)f(actually)g(need)g(to)h(b)r(e)g(able)f(to)g (diago-)523 1814 y(nalize)c Fs(\012)827 1784 y Fq(2)893 1814 y Fv(explicitly)-7 b(,)29 b(and)f(y)n(ou)g(need)g(in)h(particular) e(an)h(explicit)h(description)f(of)g(the)523 1914 y(normal)h(mo)r(des.) h(This)h(can)e(b)r(e)i(done)f(in)g(simple)h(cases,)e(suc)n(h)h(as)f (the)i(oscillator)d(ring,)523 2013 y(but)g(not)g(in)g(general.)648 2113 y(One)f(can)g(also)g(in)n(tro)r(duce)g(the)h(\\n)n(um)n(b)r(er)f (op)r(erator")1695 2336 y(~)1671 2357 y Fs(N)32 b Fv(=)1897 2253 y Fn(n)1858 2278 y Fk(X)1864 2455 y Fn(i)p Fq(=1)1993 2357 y Fv(~)-44 b Fs(a)2035 2317 y Fl(y)2035 2380 y Fn(i)2071 2357 y Fv(~)h Fs(a)2114 2369 y Fn(i)2141 2357 y Fs(;)523 2630 y Fv(whic)n(h)36 b(comm)n(utes)g(with)g Fs(H)7 b Fv(.)36 b(The)g(sp)r(ectrum)g(of)2177 2609 y(~)2153 2630 y Fs(N)45 b Fv(is)36 b(easily)f(seen)g(to)h(equal)g(to)g Fo(N)p Fv(.)523 2730 y(W)-7 b(riting)28 b Fr(E)872 2742 y Fn(m)962 2730 y Fv(for)f(the)h(eigenspace)f(of)1757 2709 y(~)1733 2730 y Fs(N)37 b Fv(with)28 b(eigen)n(v)-5 b(alue)26 b Fs(m)p Fv(,)i(one)f(has)g(eviden)n(tly)1586 2985 y Fs(L)1643 2950 y Fq(2)1680 2985 y Fv(\()p Fo(R)1772 2950 y Fn(n)1817 2985 y Fv(\))d(=)2007 2881 y Fl(\010)1973 2906 y Fk(X)1960 3084 y Fn(m)p Fl(2)p Fg(N)2120 2985 y Fr(E)2164 2997 y Fn(m)2227 2985 y Fs(:)915 b Fv(\(31\))523 3264 y(Eac)n(h)29 b(v)n(ector)g(in)i(\(30\))f(is)g(readily)g(c)n(hec)n (k)n(ed)f(to)h(b)r(e)h(an)f(eigen)n(v)n(ector)e(of)2826 3243 y(~)2802 3264 y Fs(N)40 b Fv(with)30 b(eigen-)523 3364 y(v)-5 b(alue)744 3301 y Fk(P)831 3322 y Fn(n)831 3388 y(k)q Fq(=1)970 3364 y Fs(m)1043 3376 y Fn(k)1084 3364 y Fv(.)33 b(The)h(preceding)f(considerations)f(will)i(b)r(e)g(the) g(starting)e(p)r(oin)n(t)i(for)523 3463 y(an)24 b(equiv)-5 b(alen)n(t)24 b(reform)n(ulation)e(of)i(the)h(quan)n(tum)f(theory)f(of) h(\014nite)h(dimensional)f(oscil-)523 3563 y(lator)30 b(systems)h(in)h(a)f(manner)f(suitable)i(for)e(generalization)g(to)h (in\014nite)h(dimensional)523 3662 y(systems.)c(This)h(reform)n (ulation)e(is)h(based)g(in)h(an)f(essen)n(tial)f(manner)h(on)h(the)g (notion)f(of)523 3762 y(F)-7 b(o)r(c)n(k)27 b(space,)g(whic)n(h)h(I)f (therefore)g(\014rst)g(brie\015y)h(recall)e(in)i(the)g(next)g(section.) 523 4002 y Fm(3.2)41 b(F)-8 b(o)s(c)m(k)32 b(spaces)523 4184 y Fv(The)19 b(basic)g(theory)f(of)h(symmetric)g(and)g(an)n (ti-symmetric)f(F)-7 b(o)r(c)n(k)19 b(spaces)f(can)g(b)r(e)i(found)f (in)523 4284 y(man)n(y)g(places)f(\([RS2])i([BR1)o(])g(are)e(t)n(w)n(o) h(examples\))f(and)i(I)f(will)g(not)h(detail)f(it)h(here,)e(giving)523 4384 y(only)25 b(the)h(bare)f(essen)n(tials,)f(mostly)h(for)g (notational)g(purp)r(oses.)g(More)f(information)h(on)523 4483 y(this)34 b(sub)5 b(ject)33 b(can)g(also)f(b)r(e)i(found)g(in)f (the)h(con)n(tribution)f(of)g(Jan)g(Derezinski)g(in)g(this)523 4583 y(v)n(olume)27 b([D].)648 4682 y(Let)g Fr(V)35 b Fv(b)r(e)28 b(a)f(complex)g(Hilb)r(ert)h(space,)f(then)h(the)g(F)-7 b(o)r(c)n(k)27 b(space)g Fr(F)8 b Fv(\()p Fr(V)f Fv(\))28 b(o)n(v)n(er)e Fr(V)34 b Fv(is)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(17)1526 282 y Fr(F)8 b Fv(\()p Fr(V)f Fv(\))23 b(=)p 1827 210 460 4 v 23 w Fr(\010)1892 294 y Fn(m)p Fl(2)p Fg(N)2041 282 y Fr(F)2101 294 y Fn(m)2164 282 y Fv(\()p Fr(V)7 b Fv(\))p Fs(;)523 459 y Fv(where)24 b Fr(F)820 471 y Fn(m)882 459 y Fv(\()p Fr(V)7 b Fv(\))25 b(is)f(the)h Fs(m)p Fv(-fold)f(tensor)f(pro)r(duct)h(of)h Fr(V)31 b Fv(with)24 b(itself.)h(Moreo)n(v)n(er)d Fr(F)3066 471 y Fq(0)3103 459 y Fv(\()p Fr(V)7 b Fv(\))23 b(=)523 558 y Fo(C)p Fv(.)28 b(An)g(elemen)n(t)g Fs( )e Fr(2)d(F)8 b Fv(\()p Fr(V)f Fv(\))28 b(can)f(b)r(e)h(though)n(t)f(of)h(as)f(a)g (sequence)1463 735 y Fs( )f Fv(=)c(\()p Fs( )1716 747 y Fq(0)1754 735 y Fs(;)14 b( )1845 747 y Fq(1)1882 735 y Fs(;)g(:)g(:)g(:)g(;)g( )2121 747 y Fn(m)2184 735 y Fs(;)g(:)g(:)g(:)p Fv(\))p Fs(;)523 911 y Fv(where)20 b Fs( )810 923 y Fn(m)896 911 y Fr(2)k(F)1035 923 y Fn(m)1097 911 y Fv(\()p Fr(V)7 b Fv(\).)21 b(I)g(will)g(also)e(use)h(the)h (notation)f Fr(F)2288 881 y Fq(\014n)2367 911 y Fv(\()p Fr(V)7 b Fv(\))23 b(=)g Fr(\010)2665 923 y Fn(m)p Fl(2)p Fg(N)2814 911 y Fr(F)2874 923 y Fn(m)2937 911 y Fv(\()p Fr(V)7 b Fv(\),)21 b(whic)n(h)523 1011 y(is)28 b(the)g(dense)f (subspace)g(of)g Fr(F)8 b Fv(\()p Fr(V)f Fv(\))28 b(made)g(up)g(of)f (elemen)n(ts)h(of)f(the)h(t)n(yp)r(e)1396 1187 y Fs( )e Fv(=)d(\()p Fs( )1650 1199 y Fq(0)1687 1187 y Fs(;)14 b( )1778 1199 y Fq(1)1815 1187 y Fs(;)g(:)g(:)g(:)g(;)g( )2054 1199 y Fn(N)2117 1187 y Fs(;)g Fv(0)p Fs(;)g Fv(0)p Fs(;)g(:)g(:)g(:)o Fv(\))523 1364 y(for)42 b(some)f(in)n(teger)g Fs(N)56 b Fr(\025)47 b Fv(0.)41 b(Don't)i(confuse)f Fr(F)2134 1376 y Fq(0)2171 1364 y Fv(\()p Fr(V)7 b Fv(\))42 b(with)g Fr(F)2606 1334 y Fq(\014n)2685 1364 y Fv(\()p Fr(V)7 b Fv(\)!)42 b(Elemen)n(ts)g(of)523 1464 y Fr(F)591 1433 y Fq(\014n)669 1464 y Fv(\()p Fr(V)7 b Fv(\))38 b(will)f(b)r(e)g (referred)e(to)i(as)f(states)g Fp(with)j(a)g(\014nite)f(numb)l(er)f(of) i(quanta)p Fv(,)e(a)f(ter-)523 1563 y(minology)26 b(that)i(I)g(will)g (explain)f(later.)648 1663 y(I)e(will)g(freely)g(use)g(the)g(Dirac)g (notation)g(for)f(Hilb)r(ert)i(space)e(calculations.)g(So)h(I)g(will) 523 1762 y(write)30 b Fr(j)p Fs( )s Fr(i)f(2)f(F)8 b Fv(\()p Fr(V)f Fv(\))31 b(as)f(w)n(ell)h(as)f Fs( )h Fr(2)d(F)8 b Fv(\()p Fr(V)f Fv(\),)31 b(dep)r(ending)g(on)f(whic)n(h)h (one)f(seems)g(more)523 1862 y(con)n(v)n(enien)n(t)e(at)g(an)n(y)g(giv) n(en)g(time.)h(Also,)f(when)h(no)f(confusion)h(can)f(arise,)f(I)i(will) g(write)523 1962 y Fr(F)583 1974 y Fn(m)669 1962 y Fv(=)22 b Fr(F)816 1974 y Fn(m)879 1962 y Fv(\()p Fr(V)7 b Fv(\).)648 2061 y(Let)30 b Fr(P)857 2073 y Fn(m)951 2061 y Fv(b)r(e)h(the)f(p)r (erm)n(utation)h(group)e(of)h Fs(m)h Fv(elemen)n(ts,)f(then)h(for)f (eac)n(h)g Fs(\033)h Fr(2)e(P)3227 2073 y Fn(m)3290 2061 y Fv(,)523 2161 y(w)n(e)e(de\014ne)h(the)g(unitary)f(op)r(erator)k(^) -47 b Fs(\033)31 b Fv(on)c Fr(F)1909 2173 y Fn(m)1972 2161 y Fv(\()p Fr(V)7 b Fv(\))28 b(b)n(y)928 2337 y(^)-47 b Fs(\033)t(\030)1010 2349 y Fq(1)1066 2337 y Fr(\012)18 b Fs(\030)1185 2349 y Fq(2)1241 2337 y Fr(\012)g Fs(:)c(:)g(:)k Fr(\012)g Fs(\030)1558 2349 y Fn(m)1644 2337 y Fv(=)23 b Fs(\030)1768 2353 y Fn(\033)1808 2336 y Fb(\000)p Ff(1)1886 2353 y Fq(\(1\))1994 2337 y Fr(\012)18 b Fs(\030)2113 2353 y Fn(\033)2153 2336 y Fb(\000)p Ff(1)2231 2353 y Fq(\(2\))2339 2337 y Fr(\012)g Fs(:)c(:)g(:)k Fr(\012)g Fs(\030)2656 2353 y Fn(\033)2696 2336 y Fb(\000)p Ff(1)2774 2353 y Fq(\()p Fn(m)p Fq(\))2889 2337 y Fs(;)523 2514 y Fv(\()p Fs(\030)591 2526 y Fn(j)650 2514 y Fr(2)23 b Fo(C)788 2484 y Fn(n)833 2514 y Fs(;)14 b(j)28 b Fv(=)23 b(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(m)p Fv(\))28 b(and)f(the)h(pro)5 b(jectors)1007 2732 y Fs(P)1060 2744 y Fq(+)p Fn(;m)1218 2732 y Fv(=)1342 2676 y(1)p 1315 2713 96 4 v 1315 2789 a Fs(m)p Fv(!)1469 2654 y Fk(X)1435 2832 y Fn(\033)r Fl(2P)1567 2840 y Fc(m)1641 2732 y Fv(^)-47 b Fs(\033)t(;)180 b(P)1943 2744 y Fl(\000)p Fn(;m)2101 2732 y Fv(=)2225 2676 y(1)p 2198 2713 V 2198 2789 a Fs(m)p Fv(!)2352 2654 y Fk(X)2318 2832 y Fn(\033)r Fl(2P)2450 2840 y Fc(m)2519 2732 y Fv(sgn\()p Fs(\033)s Fv(\))t(^)-46 b Fs(\033)t(:)523 2996 y Fv(No)n(w)27 b(w)n(e)g(can)h(de\014ne)f(the)h(\(an)n (ti-\)symmetric)g(tensor)e(pro)r(duct)i(as)1523 3172 y Fr(F)1591 3138 y Fl(\006)1583 3193 y Fn(m)1647 3172 y Fv(\()p Fr(V)7 b Fv(\))23 b(=)g Fs(P)1933 3184 y Fl(\006)p Fn(;m)2068 3172 y Fr(F)2128 3184 y Fn(m)2190 3172 y Fv(\()p Fr(V)7 b Fv(\))523 3349 y(whereas)26 b(the)i(\(an)n(ti-\)symmetric)g(F) -7 b(o)r(c)n(k)27 b(space)g Fr(F)2105 3319 y Fl(\006)2161 3349 y Fv(\()p Fr(V)7 b Fv(\))27 b(o)n(v)n(er)f Fr(V)35 b Fv(is)1498 3543 y Fr(F)1566 3508 y Fl(\006)1621 3543 y Fv(\()p Fr(V)7 b Fv(\))24 b(=)p 1854 3461 461 4 v 22 w Fr(\010)1919 3555 y Fn(m)p Fl(2)p Fg(N)2069 3543 y Fr(F)2137 3507 y Fl(\006)2129 3552 y Fn(m)2193 3543 y Fv(\()p Fr(V)7 b Fv(\))p Fs(:)523 3719 y Fv(In)n(tro)r(ducing)27 b(the)h(pro)5 b(jector)26 b Fs(P)1527 3731 y Fl(\006)1606 3719 y Fv(=)1694 3657 y Fk(P)1782 3744 y Fn(m)p Fl(2)p Fg(N)1945 3719 y Fs(P)1998 3731 y Fl(\006)p Fn(;m)2133 3719 y Fv(,)i(w)n(e)f(also)g(ha)n(v)n(e)1006 3908 y Fr(F)1074 3874 y Fl(\006)1130 3908 y Fv(\()p Fr(V)7 b Fv(\))23 b(=)g Fs(P)1416 3920 y Fl(\006)1472 3908 y Fr(F)8 b Fv(\()p Fr(V)f Fv(\))83 b(and)g Fr(F)2030 3874 y Fq(\014n)o Fn(;)p Fl(\006)2180 3908 y Fv(\()p Fr(V)7 b Fv(\))24 b(=)e Fs(P)2466 3920 y Fl(\006)2522 3908 y Fr(F)2590 3874 y Fq(f)5 b Fn(in)2684 3908 y Fv(\()p Fr(V)i Fv(\))p Fs(:)523 4085 y Fv(One)23 b(refers)f(to)h Fr(F)1079 4055 y Fq(+)1134 4085 y Fv(\()p Fr(V)7 b Fv(\))23 b(as)g(the)g(symmetric)g(or)f(b)r (osonic)g(F)-7 b(o)r(c)n(k)23 b(space)f(and)h(to)g Fr(F)3037 4055 y Fl(\000)3093 4085 y Fv(\()p Fr(V)7 b Fv(\))23 b(as)523 4184 y(the)30 b(an)n(ti-symmetric)f(or)g(fermionic)g(F)-7 b(o)r(c)n(k)30 b(space.)f(I)g(will)h(only)g(deal)f(with)h(the)h(former) 523 4284 y(here.)648 4384 y(Computations)22 b(in)h(F)-7 b(o)r(c)n(k)22 b(space)g(are)g(greatly)f(simpli\014ed)j(through)e(the)h (use)f(of)h(\\cre-)523 4483 y(ation")k(and)g(\\annihilation")g(op)r (erators,)e(whic)n(h)j(are)e(abstract)h(v)n(ersions)f(of)i(the)g(op)r (er-)523 4583 y(ators)f(~)-43 b Fs(a)p Fv(\()p Fs(\030)t Fv(\))29 b(and)f(~)-43 b Fs(a)1113 4553 y Fl(y)1147 4583 y Fv(\()p Fs(\030)t Fv(\))29 b(in)n(tro)r(duced)e(in)h(Sect.)g(3.1.)648 4682 y(De\014ne,)g(for)f(an)n(y)g Fs(\030)g Fr(2)c(V)7 b Fv(,)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 523 100 a Fu(18)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)1128 282 y Fs(d)p Fv(\()p Fs(\030)t Fv(\))p Fs(\030)1311 294 y Fq(1)1367 282 y Fr(\012)19 b Fs(\030)1487 294 y Fq(2)1538 282 y Fs(:)14 b(:)g(:)k Fr(\012)g Fs(\030)1772 294 y Fn(m)1859 282 y Fv(=)k(\()p 1978 215 41 4 v Fs(\030)h Fr(\001)c Fs(\030)2115 294 y Fq(1)2152 282 y Fv(\))28 b Fs(\030)2248 294 y Fq(2)2304 282 y Fr(\012)18 b Fs(:)c(:)g(:)19 b Fr(\012)f Fs(\030)2622 294 y Fn(m)2685 282 y Fs(:)523 459 y Fv(This)k(extends)h(b)n(y)f (linearit)n(y)g(and)g(yields)g(a)g(w)n(ell-de\014ned)g(b)r(ounded)h(op) r(erator)e(from)h Fr(F)3250 471 y Fn(m)523 558 y Fv(to)i Fr(F)681 570 y Fn(m)p Fl(\000)p Fq(1)854 558 y Fv(whic)n(h)g(extends)h (to)f(a)h(b)r(ounded)g(op)r(erator)d(on)j(all)f(of)g Fr(F)8 b Fv(\()p Fr(V)f Fv(\),)25 b(denoted)g(b)n(y)f(the)523 658 y(same)j(sym)n(b)r(ol.)648 757 y(Note)33 b(that)h(I)g(use)g(the)g (notation)p 1741 690 V 33 w Fs(\030)27 b Fr(\001)22 b Fs(\021)37 b Fv(for)d(the)g(inner)f(pro)r(duct)h(on)g(the)g(abstract) 523 857 y(space)23 b Fr(V)31 b Fv(b)r(ecause)24 b(in)g(the)h (applications)e(in)i(these)f(notes)f Fr(V)31 b Fv(will)25 b(b)r(e)f Fr(K)2743 827 y Fg(C)2790 857 y Fv(,)g(in)g(whic)n(h)h(case) 523 957 y(this)30 b(notation)f(is)h(particularly)e(transparen)n(t.)g (Of)i(course,)e(on)i(a)f(general)f(abstract)h Fr(V)7 b Fv(,)523 1056 y(there)25 b(is)g(no)g(natural)f(de\014nition)h(of)g (\\the)g(complex)g(conjugate)p 2551 989 V 24 w Fs(\030)t Fv(",)g(but)h(that)f(do)r(es)g(not)523 1156 y(mean)i(w)n(e)h(can't)f (use)p 1218 1088 V 28 w Fs(\030)22 b Fr(\001)d Fs(\021)31 b Fv(as)c(a)g(notation)g(for)g(the)h(inner)f(pro)r(duct.)648 1256 y(One)g(has)g Fr(k)22 b Fs(d)p Fv(\()p Fs(\030)t Fv(\))i Fr(k)p Fv(=)p Fr(k)e Fs(\030)27 b Fr(k)p Fv(.)h(Similarly)-7 b(,)27 b(de\014ne)1106 1432 y Fs(c)p Fv(\()p Fs(\030)t Fv(\))p Fs(\030)1282 1444 y Fq(1)1338 1432 y Fr(\012)19 b Fs(\030)1458 1444 y Fq(2)1509 1432 y Fs(:)14 b(:)g(:)k Fr(\012)g Fs(\030)1743 1444 y Fn(m)1830 1432 y Fv(=)k Fs(\030)h Fr(\012)18 b Fs(\030)2095 1444 y Fq(1)2151 1432 y Fr(\012)g Fs(\030)2270 1444 y Fq(2)2326 1432 y Fr(\012)g Fs(:)c(:)g(:)k Fr(\012)g Fs(\030)2643 1444 y Fn(m)2707 1432 y Fs(:)523 1608 y Fv(This)33 b(again)e(yields)h(a)h(w) n(ell-de\014ned)f(b)r(ounded)h(op)r(erator)e(from)h Fr(F)2661 1620 y Fn(m)2757 1608 y Fv(to)g Fr(F)2923 1620 y Fn(m)p Fq(+1)3103 1608 y Fv(whic)n(h)523 1708 y(extends)d(to)g(a)f(b)r(ounded) h(op)r(erator)f(on)g(all)h(of)g Fr(F)8 b Fv(\()p Fr(V)f Fv(\),)29 b(denoted)g(b)n(y)f(the)i(same)e(sym)n(b)r(ol.)523 1807 y(One)f(has)g Fr(k)c Fs(c)p Fv(\()p Fs(\030)t Fv(\))h Fr(k)p Fv(=)p Fr(k)d Fs(\030)27 b Fr(k)h Fv(and)f Fs(d)p Fv(\()p Fs(\030)t Fv(\))1686 1777 y Fl(\003)1748 1807 y Fv(=)c Fs(c)p Fv(\()p Fs(\030)t Fv(\).)648 1907 y(In)n(tro)r(ducing)k (the)g(self-adjoin)n(t)h(\\n)n(um)n(b)r(er)e(op)r(erator")g Fs(N)36 b Fv(b)n(y)1358 2083 y Fs(N)9 b( )26 b Fv(=)d(\(0)p Fs(;)14 b( )1767 2095 y Fq(1)1804 2083 y Fs(;)g Fv(2)p Fs( )1937 2095 y Fq(2)1974 2083 y Fs(;)g(:)g(:)g(:)f(m )2248 2095 y Fn(m)2325 2083 y Fs(:)h(:)g(:)p Fv(\))p Fs(;)523 2260 y Fv(w)n(e)27 b(can)g(then)i(de\014ne,)e(on)h Fr(F)1433 2229 y Fq(\014n)1511 2260 y Fv(,)760 2436 y Fs(a)804 2448 y Fl(\006)860 2436 y Fv(\()p Fs(\030)t Fv(\))c(=)f Fs(P)1129 2448 y Fl(\006)1185 2365 y Fr(p)p 1254 2365 219 4 v 71 x Fs(N)k Fv(+)18 b(1)p Fs(d)p Fv(\()p Fs(\030)t Fv(\))p Fs(P)1673 2448 y Fl(\006)1730 2436 y Fs(;)180 b Fv(and)166 b Fs(a)2277 2396 y Fl(y)2277 2456 y(\006)2333 2436 y Fv(\()p Fs(\030)t Fv(\))23 b(=)g Fs(P)2601 2448 y Fl(\006)2657 2361 y Fr(p)p 2727 2361 76 4 v 2727 2436 a Fs(N)8 b(c)p Fv(\()p Fs(\030)t Fv(\))p Fs(P)2995 2448 y Fl(\006)3052 2436 y Fs(:)523 2629 y Fv(The)29 b Fs(a)739 2641 y Fl(\006)795 2629 y Fv(\()p Fs(\030)t Fv(\))h(are)f(called)g (\\annihilation)f(op)r(erators")f(and)i(the)h Fs(a)2574 2589 y Fl(y)2574 2649 y(\006)2630 2629 y Fv(\()p Fs(\030)t Fv(\))g(creation)e(op)r(era-)523 2728 y(tors.)f(I)i(will)f(think)h(of)f Fs(a)1288 2740 y Fl(\000)1344 2728 y Fv(\()p Fs(\030)t Fv(\))g(as)g(an)g(op)r(erator)e(on)i Fr(F)2214 2698 y Fl(\000)2298 2728 y Fv(and)g(of)g Fs(a)2599 2740 y Fq(+)2654 2728 y Fv(\()p Fs(\030)t Fv(\))h(as)f(an)f(op)r(erator)523 2828 y(on)c Fr(F)702 2798 y Fq(+)757 2828 y Fv(.)g(Direct)h (computation)f(\(on)g Fr(F)1743 2798 y Fq(f)5 b Fn(in)1837 2828 y Fv(,)23 b(for)g(example\))g(yields)g(the)g(follo)n(wing)g (crucial)523 2927 y(comm)n(utation)k(and)h(an)n(ti-comm)n(utation)e (relations)g(b)r(et)n(w)n(een)i(those)f(op)r(erators:)605 3104 y([)p Fs(a)672 3116 y Fq(+)727 3104 y Fv(\()p Fs(\030)795 3116 y Fq(1)833 3104 y Fv(\))p Fs(;)14 b(a)946 3116 y Fq(+)1001 3104 y Fv(\()p Fs(\030)1069 3116 y Fq(2)1107 3104 y Fv(\)])23 b(=)g(0)f(=)h([)p Fs(a)1492 3064 y Fl(y)1492 3124 y Fq(+)1547 3104 y Fv(\()p Fs(\030)1615 3116 y Fq(1)1653 3104 y Fv(\))p Fs(;)14 b(a)1766 3064 y Fl(y)1766 3124 y Fq(+)1821 3104 y Fv(\()p Fs(\030)1889 3116 y Fq(2)1927 3104 y Fv(\)])p Fs(;)180 b Fv([)p Fs(a)2252 3116 y Fq(+)2307 3104 y Fv(\()p Fs(\030)2375 3116 y Fq(1)2413 3104 y Fv(\))p Fs(;)14 b(a)2526 3064 y Fl(y)2526 3124 y Fq(+)2581 3104 y Fv(\()p Fs(\030)2649 3116 y Fq(2)2687 3104 y Fv(\)])24 b(=)p 2853 3036 74 4 v 22 w Fs(\030)2889 3116 y Fq(1)2945 3104 y Fr(\001)19 b Fs(\030)3023 3116 y Fq(2)3060 3104 y Fs(;)82 b Fv(\(32\))523 3280 y(Those)33 b(are)g(referred)g(to)h(as)f (the)h(canonical)f(comm)n(utation)g(relations)g(or)g(CCR.)h(Y)-7 b(ou)523 3380 y(should)27 b(compare)g(\(32\))g(to)g(\(26\))h(and)f(b)r (e)h(amazed.)648 3479 y(W)-7 b(orking)27 b(in)h(the)g(b)r(osonic)f(F)-7 b(o)r(c)n(k)28 b(space)f Fr(F)2009 3449 y Fq(+)2092 3479 y Fv(and)h(using)f(the)i(ab)r(o)n(v)n(e)d(relations)h(one)523 3579 y(establishes)g(through)g(direct)g(computation)h(that)911 3680 y Fr(p)p 980 3680 96 4 v 75 x Fs(m)p Fv(!)p Fs(P)1129 3767 y Fq(+)1184 3755 y Fs(\030)1220 3767 y Fq(1)1277 3755 y Fr(\012)18 b Fs(\030)1396 3767 y Fq(2)1452 3755 y Fr(\012)g Fs(:)c(:)g(:)k Fr(\012)g Fs(\030)1769 3767 y Fn(m)1855 3755 y Fv(=)23 b Fs(a)1987 3715 y Fl(y)1987 3776 y Fq(+)2042 3755 y Fv(\()p Fs(\030)2110 3767 y Fq(1)2148 3755 y Fv(\))p Fs(a)2224 3715 y Fl(y)2224 3776 y Fq(+)2279 3755 y Fv(\()p Fs(\030)2347 3767 y Fq(2)2385 3755 y Fv(\))14 b Fs(:)g(:)g(:)g(a)2586 3715 y Fl(y)2586 3776 y Fq(+)2641 3755 y Fv(\()p Fs(\030)2709 3767 y Fn(m)2772 3755 y Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fs(:)523 3931 y Fv(Here)34 b(I)g(in)n(tro)r(duced) g(the)h(notation)e Fr(j)p Fv(0)p Fr(i)h Fv(=)g(\(1)p Fs(;)14 b Fv(0)p Fs(;)g Fv(0)p Fs(;)g(:)g(:)g(:)n Fv(\))35 b Fr(2)f(F)2510 3943 y Fq(0)2581 3931 y Fr(\032)f(F)8 b Fv(.)35 b(This)f(v)n(ector)f(is)523 4031 y(usually)g(referred)f(to)h (as)f(the)i(F)-7 b(o)r(c)n(k)33 b(v)-5 b(acuum)33 b(or)f(simply)h(as)g (the)h(v)-5 b(acuum.)33 b(It)g(can)g(b)r(e)523 4131 y(c)n(haracterized) 26 b(as)g(b)r(eing)i(the)g(unique)g(v)n(ector)e(in)i Fr(F)2181 4101 y Fq(+)2264 4131 y Fv(for)f(whic)n(h)1448 4307 y Fs(a)1492 4319 y Fq(+)1547 4307 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fs( )s Fr(i)d Fv(=)f(0)p Fs(;)179 b Fr(8)p Fs(\030)28 b Fr(2)23 b(V)7 b Fs(:)523 4483 y Fv(F)-7 b(or)22 b(explicit)h(computations)f(and)h(in)g(order)e(to)i(understand) f(the)h(ph)n(ysics)f(literature,)g(it)523 4583 y(is)27 b(a)g(Go)r(o)r(d)g(Thing)g(to)g(ha)n(v)n(e)f(a)h(con)n(v)n(enien)n(t)f (basis)h(at)g(hand.)g(So)g(supp)r(ose)g(y)n(ou)f(ha)n(v)n(e)g(an)523 4682 y(orthonormal)h(basis)h Fs(\021)1246 4694 y Fn(j)1311 4682 y Fv(of)h Fr(V)35 b Fv(\(with)30 b Fs(j)h Fv(=)25 b(1)p Fs(;)14 b Fv(2)p Fs(;)g(:)g(:)g(:)e Fv(dim)q Fr(V)7 b Fv(\).)29 b(Then)g(y)n(ou)f(can)h(de\014ne,)g(for)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(19)523 282 y Fv(an)n(y)30 b(p)r(ositiv)n(e)g(in)n(teger)f Fs(k)h Fr(\024)e Fv(dim)p Fr(V)37 b Fv(and)30 b(for)g(an)n(y)g(c)n(hoice)f(of)i(\()p Fs(m)2568 294 y Fq(1)2605 282 y Fs(;)14 b(m)2715 294 y Fq(2)2752 282 y Fs(;)g(m)2862 294 y Fq(3)2899 282 y Fs(;)g(:)g(:)g(:)g(;)g(m)3157 294 y Fn(k)3197 282 y Fv(\))28 b Fr(2)523 382 y Fo(N)583 352 y Fn(k)624 382 y Fv(,)g(the)g(v)n(ector) 567 561 y Fr(j)p Fs(m)663 573 y Fq(1)701 561 y Fs(;)14 b(m)811 573 y Fq(2)848 561 y Fs(;)g(m)958 573 y Fq(3)995 561 y Fs(;)g(:)g(:)g(:)f(;)h(m)1252 573 y Fn(k)1293 561 y Fr(i)23 b Fv(:=)g(\()p Fs(m)1564 573 y Fq(1)1601 561 y Fv(!)p Fs(m)1697 573 y Fq(2)1735 561 y Fv(!)14 b Fs(:)g(:)g(:)f(m) 1955 573 y Fn(k)1996 561 y Fv(!\))2051 527 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2207 561 y Fr(\002)3165 660 y Fv(\(33\))898 697 y Fk(\020)947 790 y Fs(a)991 750 y Fl(y)991 810 y Fq(+)1046 790 y Fv(\()p Fs(\021)1119 802 y Fq(1)1157 790 y Fv(\))1189 697 y Fk(\021)1239 715 y Fn(m)1298 723 y Ff(1)1348 697 y Fk(\020)1398 790 y Fs(a)1442 750 y Fl(y)1442 810 y Fq(+)1497 790 y Fv(\()p Fs(\021)1570 802 y Fq(2)1608 790 y Fv(\))1640 697 y Fk(\021)1689 715 y Fn(m)1748 723 y Ff(2)1799 697 y Fk(\020)1848 790 y Fs(a)1892 750 y Fl(y)1892 810 y Fq(+)1947 790 y Fv(\()p Fs(\021)2020 802 y Fq(3)2058 790 y Fv(\))2090 697 y Fk(\021)2140 715 y Fn(m)2199 723 y Ff(3)2249 790 y Fs(:)h(:)g(:)2360 697 y Fk(\020)2410 790 y Fs(a)2454 750 y Fl(y)2454 810 y Fq(+)2509 790 y Fv(\()p Fs(\021)2582 802 y Fn(k)2623 790 y Fv(\))2655 697 y Fk(\021)2705 715 y Fn(m)2764 724 y Fc(k)2818 790 y Fr(j)p Fv(0)p Fr(i)p Fs(:)523 965 y Fv(Those)34 b(v)n(ectors)f(are)h(no)n(w)g(easily)g(c)n(hec)n(k)n(ed)f (to)i(form)f(an)g(orthonormal)f(basis)h(of)g Fr(F)3234 935 y Fq(+)3290 965 y Fv(.)523 1065 y(The)i(n)n(um)n(b)r(ers)f Fs(m)1118 1077 y Fn(j)1188 1065 y Fv(are)g(often)h(referred)e(to)i(as)f (the)h(\\o)r(ccupation)e(n)n(um)n(b)r(ers")h(of)g(the)523 1165 y(states)e Fs(\021)806 1177 y Fn(j)841 1165 y Fv(.)h(Note)f(that)h (eac)n(h)e(of)i(them)g(is)f(an)g(eigen)n(v)n(ector)e(of)j(the)f(n)n(um) n(b)r(er)g(op)r(erator)523 1279 y(with)28 b(eigen)n(v)-5 b(alue)27 b(giv)n(en)g(b)n(y)1441 1217 y Fk(P)1529 1237 y Fn(k)1529 1304 y(j)s Fq(=1)1661 1279 y Fs(m)1734 1291 y Fn(j)1769 1279 y Fv(.)648 1378 y(It)33 b(is)f(a)h(go)r(o)r(d)f (exercise)g(to)h(pro)n(v)n(e)e(that)i Fs(N)2015 1390 y Fq(+)2101 1378 y Fv(=)f Fs(P)2251 1390 y Fq(+)2306 1378 y Fs(N)9 b(P)2435 1390 y Fq(+)2490 1378 y Fv(,)33 b(the)h(restriction)d(of)i(the)523 1478 y(n)n(um)n(b)r(er)27 b(op)r(erator)f(to)i Fr(F)1330 1448 y Fq(+)1412 1478 y Fv(can)g(b)r(e)g(written)1483 1669 y Fs(N)1550 1681 y Fq(+)1628 1669 y Fv(=)1716 1590 y Fk(X)1760 1767 y Fn(j)1850 1669 y Fs(a)1894 1629 y Fl(y)1894 1690 y Fq(+)1949 1669 y Fv(\()p Fs(\021)2022 1681 y Fn(j)2057 1669 y Fv(\))p Fs(a)2133 1681 y Fq(+)2188 1669 y Fv(\()p Fs(\021)2261 1681 y Fn(j)2297 1669 y Fv(\))p Fs(:)648 1940 y Fv(If)40 b Fs(U)48 b Fv(is)40 b(a)f(unitary)h(op)r(erator)e(on)h Fr(V)7 b Fv(,)40 b(the)g(unitary)f(op)r(erator)g Fs(\000)12 b Fv(\()p Fs(U)d Fv(\))39 b(on)h Fr(F)3162 1910 y Fq(+)3257 1940 y Fv(is)523 2040 y(de\014ned)31 b(as)f Fr(\012)982 2010 y Fn(m)982 2063 y(k)q Fq(=1)1107 2040 y Fs(U)39 b Fv(when)31 b(restricted)f(to)h Fr(F)1969 2010 y Fq(+)1961 2060 y Fn(m)2024 2040 y Fv(.)g(When)g Fs(A)g Fv(is)g(a)f(self-adjoin)n (t)g(op)r(erator)523 2140 y(on)d Fr(V)7 b Fv(,)28 b(d)p Fs(\000)12 b Fv(\()p Fs(A)p Fv(\))28 b(is)g(the)f(self-adjoin)n(t)h(op) r(erator)d(on)j Fr(F)2182 2109 y Fq(+)2264 2140 y Fv(de\014ned)g(as)883 2319 y Fs(A)19 b Fr(\012)f Fv(1)-18 b(l)18 b Fr(\012)g Fs(:)c(:)g(:)k Fr(\012)g Fv(1)-18 b(l)18 b(+)g(1)-18 b(l)17 b Fr(\012)h Fs(A)h Fr(\012)f Fs(:)c(:)g(:)k Fr(\012)g Fv(1)-18 b(l)18 b(+)g Fs(:)c(:)g(:)k Fv(+)g(1)-18 b(l)18 b Fr(\012)g Fs(:)c(:)g(:)k Fr(\012)g Fv(1)-18 b(l)18 b Fr(\012)g Fs(A)523 2497 y Fv(on)33 b(\(a)g(suitable)g(domain\))h(in)f Fr(F)1570 2467 y Fq(+)1562 2518 y Fn(m)1625 2497 y Fv(,)g(for)g(eac)n (h)g Fs(m)f(>)g Fv(0.)h(Also)g(d)p Fs(\000)12 b Fv(\()p Fs(A)p Fv(\))p Fr(F)2803 2462 y Fq(+)2795 2520 y(0)2891 2497 y Fv(=)32 b(0.)h(It)h(is)f(a)523 2597 y(go)r(o)r(d)27 b(exercise)f(to)i(c)n(hec)n(k)f(that,)h(if)g Fs(A)g Fv(has)f(a)g(basis) g(of)g(eigen)n(v)n(ectors)1711 2776 y Fs(A\021)1814 2788 y Fn(j)1873 2776 y Fv(=)22 b Fs(\013)2013 2788 y Fn(j)2048 2776 y Fs(\021)2089 2788 y Fn(j)523 2955 y Fv(then)1380 3055 y(d)p Fs(\000)12 b Fv(\()p Fs(A)p Fv(\))23 b(=)1726 2976 y Fk(X)1774 3153 y Fn(i)1860 3055 y Fs(\013)1913 3067 y Fn(i)1968 3055 y Fs(a)2012 3015 y Fl(y)2012 3075 y Fq(+)2067 3055 y Fv(\()p Fs(\021)2140 3067 y Fn(i)2168 3055 y Fv(\))p Fs(a)2244 3067 y Fq(+)2300 3055 y Fv(\()p Fs(\021)2373 3067 y Fn(i)2401 3055 y Fv(\))p Fs(:)523 3365 y Fm(3.3)41 b(The)32 b(F)-8 b(o)s(c)m(k)32 b(represen)m(tation:)f (\014nite)h(dimensional)d(\014elds)523 3545 y Fv(It)k(is)g(no)n(w)f (straigh)n(tforw)n(ard)d(to)k(reform)n(ulate)e(the)i(quan)n(tum)g (description)f(of)h(the)g(os-)523 3644 y(cillator)i(system)h(in)h (Sect.)g(3.1)e(as)h(follo)n(ws.)f(First)i(of)f(all,)g(in)h(view)f(of)g (\(31\))g(and)g(the)523 3744 y(considerations)31 b(of)h(the)h(previous) f(section,)g(it)h(is)f(clear)g(that)h(there)f(exists)g(a)g(unitary)523 3844 y(map)c Fs(T)757 3856 y Fn(\012)1487 3943 y Fs(T)1536 3955 y Fn(\012)1617 3943 y Fv(:)23 b Fs(L)1720 3909 y Fq(2)1757 3943 y Fv(\()p Fo(R)1849 3909 y Fn(n)1894 3943 y Fv(\))h Fr(!)f(F)2124 3909 y Fq(+)2179 3943 y Fv(\()p Fo(C)2271 3909 y Fn(n)2316 3943 y Fv(\))523 4090 y(satisfying)1001 4331 y Fs(T)1050 4343 y Fn(\012)1108 4331 y Fr(E)1152 4343 y Fn(m)1238 4331 y Fv(=)f Fr(F)1393 4297 y Fq(+)1385 4352 y Fn(m)1448 4331 y Fv(\()p Fo(C)1540 4297 y Fn(n)1586 4331 y Fv(\))p Fs(;)97 b(T)1787 4343 y Fn(\012)1845 4331 y Fs(H)7 b(T)1982 4296 y Fl(\000)p Fq(1)1970 4356 y Fn(\012)2093 4331 y Fv(=)22 b(d)p Fs(\000)12 b Fv(\()p Fs(\012)t Fv(\))19 b(+)2533 4275 y(1)p 2533 4312 42 4 v 2533 4388 a(2)2638 4228 y Fn(n)2598 4252 y Fk(X)2605 4429 y Fn(i)p Fq(=1)2732 4331 y Fs(!)2784 4343 y Fn(i)2812 4331 y Fs(;)523 4583 y Fv(and)1547 4682 y Fs(T)1596 4694 y Fn(\012)1655 4682 y Fv(~)-43 b Fs(a)o Fv(\()p Fs(\030)t Fv(\))p Fs(T)1862 4647 y Fl(\000)p Fq(1)1850 4707 y Fn(\012)1974 4682 y Fv(=)23 b Fs(a)2106 4694 y Fq(+)2161 4682 y Fv(\()p Fs(\030)t Fv(\))p Fs(;)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 523 100 a Fu(20)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(for)27 b(all)g Fs(\030)h Fr(2)23 b Fo(C)967 252 y Fn(n)1012 282 y Fv(.)28 b(In)g(fact,)f(quite)h(explicitly)-7 b(,)28 b(one)f(has,)h(for)f(all)g Fs(\030)2541 294 y Fq(1)2579 282 y Fs(;)14 b(\030)2652 294 y Fq(2)2689 282 y Fs(;)g(:)g(:)g(:)g(\030)2873 294 y Fn(m)2959 282 y Fr(2)23 b Fo(C)3097 252 y Fn(n)3143 282 y Fv(,)600 565 y Fs(T)649 577 y Fn(\012)729 565 y Fv(:)i(~)-44 b Fs(a)819 530 y Fl(y)854 565 y Fv(\()p Fs(\030)922 577 y Fq(1)960 565 y Fv(\))14 b Fs(:)g(:)g(:)h Fv(~)-43 b Fs(a)1161 530 y Fl(y)1195 565 y Fv(\()p Fs(\030)1263 577 y Fn(m)1327 565 y Fv(\))p Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)39 b(2)g(E)1738 577 y Fn(m)1824 565 y Fr(\032)22 b Fs(L)1968 530 y Fq(2)2005 565 y Fv(\()p Fo(R)2097 530 y Fn(n)2143 565 y Fv(\))1586 699 y Fr(7!)j Fs(a)1738 659 y Fl(y)1738 719 y Fq(+)1793 699 y Fv(\()p Fs(\030)1861 711 y Fq(1)1899 699 y Fv(\))14 b Fs(:)g(:)g(:)g(a)2100 659 y Fl(y)2100 719 y Fq(+)2155 699 y Fv(\()p Fs(\030)2223 711 y Fn(m)2286 699 y Fv(\))p Fr(j)p Fv(0)p Fr(i)23 b(2)h(F)2585 664 y Fq(+)2577 719 y Fn(m)2640 699 y Fv(\()p Fo(C)2732 664 y Fn(n)2777 699 y Fv(\))g Fr(\032)e(F)2988 664 y Fq(+)3043 699 y Fv(\()p Fo(C)3135 664 y Fn(n)3181 699 y Fv(\))p Fs(:)523 881 y Fv(The)37 b(unitary)f(map)h Fs(T)1248 893 y Fn(\012)1342 881 y Fv(transp)r(orts)e(eac)n(h)h(ob)5 b(ject)37 b(of)f(the)h(theory)f(from)h Fs(L)2996 851 y Fq(2)3032 881 y Fv(\()p Fo(R)3124 851 y Fn(n)3170 881 y Fv(\))g(to)523 981 y(the)27 b(symmetric)e(F)-7 b(o)r(c)n(k)26 b(space)f(o)n(v)n(er)g Fo(C)1720 951 y Fn(n)1791 981 y Fv(and)h(pro)n(vides)f(in)h(this)h(manner)e(an)h(equiv)-5 b(alen)n(t)523 1081 y(quan)n(tum)26 b(mec)n(hanical)f(description)g(of) h(the)g(oscillator)e(system,)i(that)g(go)r(es)f(under)g(the)523 1180 y(name)i(of)h(F)-7 b(o)r(c)n(k)27 b(represen)n(tation.)648 1280 y(Note)g(that)h(in)g(the)f(left)i(hand)e(side)g(of)h(the)g(ab)r(o) n(v)n(e)e(equations,)h(the)g(op)r(erators)g(~)-43 b Fs(a)p Fv(\()p Fs(\030)t Fv(\))523 1380 y(or)32 b(~)-43 b Fs(a)673 1349 y Fl(y)708 1380 y Fv(\()p Fs(\030)t Fv(\))32 b(are)f(the)i (concrete)e(di\013eren)n(tial)h(op)r(erators)e(on)h Fs(L)2435 1349 y Fq(2)2472 1380 y Fv(\()p Fo(R)2564 1349 y Fn(n)2610 1380 y Fv(\))h(that)g(w)n(ere)f(de\014ned)523 1479 y(in)e(\(25\))f(and) g(that)g(dep)r(end)h(explicitly)g(on)f Fs(\012)t Fv(.)h(In)f(the)h (righ)n(t)e(hand)i(side,)f(y)n(ou)g(\014nd)g(the)523 1579 y(abstract)33 b(creation)f(and)i(annihilation)f(op)r(erators)f (de\014ned)i(in)g(Sect.)h(3.2.)e(Note)g(that)523 1678 y(those)c(do)g(not)h(dep)r(end)g(on)f Fs(\012)34 b Fv(at)c(all.)f (Similarly)-7 b(,)29 b(the)h(ground)f(state)g(v)n(ector)f Fr(j)p Fv(0)p Fs(;)14 b(\012)t Fr(i)30 b Fv(of)523 1778 y Fs(H)i Fv(app)r(earing)23 b(in)i(the)g(left)g(hand)g(side)g(is)f(of)h (course)e Fs(\012)t Fv(-dep)r(enden)n(t,)j(whereas)d(the)i(F)-7 b(o)r(c)n(k)523 1878 y(v)i(acuum)20 b Fr(j)p Fv(0)p Fr(i)f Fv(in)h(the)g(righ)n(t)f(hand)h(side)g(is)f(not.)h(This)g(is)f (somewhat)g(parado)n(xical.)e(Indeed,)523 1977 y(since)22 b(the)h(v)-5 b(acuum)23 b(is)f(the)h(ground)e(state)h(of)h(the)f (Hamiltonian,)h(should)f(it)h(not)f(dep)r(end)523 2077 y(on)32 b(this)g(Hamiltonian?)g(The)g(answ)n(er)f(to)h(this)g(con)n (undrum)f(go)r(es)g(as)h(follo)n(ws,)f(and)h(is)523 2177 y(v)n(ery)20 b(similar)g(to)g(the)i(discussion)e(in)h(Sect.)g(2.4)f(in) h(the)g(classical)e(con)n(text.)i(Recall)f(that)h(it)523 2276 y(is)26 b(customary)e(to)i(sa)n(y)e(that)i(eac)n(h)f(ph)n(ysical)g (state)g(of)h(the)g(system)g(is)f(represen)n(ted)g(b)n(y)g(a)523 2376 y(v)n(ector)e(in)h(a)g(Hilb)r(ert)h(space.)e(Consider)g(for)h (example)f(the)i(v)-5 b(acuum)24 b(v)n(ector)f Fr(j)p Fv(0)p Fr(i)h Fv(in)g(F)-7 b(o)r(c)n(k)523 2475 y(space.)31 b(T)-7 b(o)31 b(\014nd)g(out)h(to)f(whic)n(h)g(ph)n(ysical)g(state)g (of)g(the)h(system)f(it)h(corresp)r(onds,)d(one)523 2575 y(has)35 b(to)h(compute)g(the)g(exp)r(ectation)g(v)-5 b(alue)36 b(of)f(ph)n(ysical)g(observ)-5 b(ables)35 b(in)h(this)g (state.)523 2675 y(No)n(w,)f(for)g(a)g(system)g(of)h(coupled)f (oscillators,)f(the)i(most)f(relev)-5 b(an)n(t)35 b(observ)-5 b(ables)34 b(are)523 2774 y(arguably)22 b(the)j(co)r(ordinates)d(of)i (p)r(osition)g(and)g(momen)n(tum.)g(In)g(view)g(of)g(\(27\))f(and)h (\(28\))523 2874 y(it)k(is)g(no)n(w)f(clear)f(that)1057 3075 y Fs(T)1106 3087 y Fn(\012)1164 3075 y Fs(\021)c Fr(\001)c Fs(QT)1395 3039 y Fl(\000)p Fq(1)1383 3099 y Fn(\012)1506 3075 y Fv(=)1638 3019 y(1)p 1604 3056 111 4 v 1604 3072 a Fr(p)p 1673 3072 42 4 v 69 x Fv(2)1724 3075 y(\()p Fs(a)1800 3087 y Fq(+)1856 3075 y Fv(\()p Fs(\012)1956 3041 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)p 2113 3029 45 4 v 2113 3075 a Fs(\021)s Fv(\))g(+)g Fs(a)2334 3035 y Fl(y)2334 3096 y Fq(+)2389 3075 y Fv(\()p Fs(\012)2489 3041 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2646 3075 y Fs(\021)s Fv(\)\))p Fs(;)388 b Fv(\(34\))523 3301 y(and,)28 b(similarly)-7 b(,)1110 3520 y Fs(T)1159 3532 y Fn(\012)1216 3520 y Fs(\021)22 b Fr(\001)d Fs(P)12 b(T)1447 3484 y Fl(\000)p Fq(1)1435 3544 y Fn(\012)1558 3520 y Fv(=)1696 3463 y Fs(i)p 1655 3500 111 4 v 1655 3517 a Fr(p)p 1724 3517 42 4 v 69 x Fv(2)1776 3520 y(\()p Fs(a)1852 3480 y Fl(y)1852 3540 y Fq(+)1907 3520 y Fv(\()p Fs(\012)2007 3485 y Fq(1)p Fn(=)p Fq(2)2112 3520 y Fs(\021)s Fv(\))19 b Fr(\000)f Fs(a)2334 3532 y Fq(+)2389 3520 y Fv(\()p Fs(\012)2489 3485 y Fq(1)p Fn(=)p Fq(2)p 2594 3474 45 4 v 2594 3520 a Fs(\021)s Fv(\)\))p Fs(:)440 b Fv(\(35\))523 3761 y(I)35 b(will)g(in)g(the)g(follo)n(wing)f(not)g(hesitate)h(to)g(write)f Fs(T)2212 3773 y Fn(\012)2270 3761 y Fs(\021)26 b Fr(\001)d Fs(QT)2510 3725 y Fl(\000)p Fq(1)2498 3785 y Fn(\012)2633 3761 y Fv(=)35 b Fs(\021)26 b Fr(\001)d Fs(Q)35 b Fv(and)f Fs(T)3164 3773 y Fn(\012)3222 3761 y Fs(\021)27 b Fr(\001)523 3861 y Fs(P)12 b(T)649 3825 y Fl(\000)p Fq(1)637 3885 y Fn(\012)770 3861 y Fv(=)32 b Fs(\021)25 b Fr(\001)e Fs(P)12 b Fv(,)33 b(in)h(agreemen)n(t)e(with)i(the)f(usual)g(con)n(v)n (en)n(tion)f(that)i(consists)e(of)i(not)523 3960 y(making)40 b(the)g(iden)n(ti\014cation)h(op)r(erator)d Fs(T)1894 3972 y Fn(\012)1992 3960 y Fv(notationally)i(explicit.)h(But)f(it)h(is) f(no)n(w)523 4060 y(clear)35 b(that,)h(con)n(trary)d(to)j(what)f(happ)r (ens)h(on)f Fs(L)2120 4030 y Fq(2)2157 4060 y Fv(\()p Fo(R)2249 4030 y Fn(n)2295 4060 y Fv(\),)h(the)g(explicit)g(expression) e(of)523 4160 y(the)28 b(p)r(osition)f(and)g(momen)n(tum)h(observ)-5 b(ables)26 b(as)h(op)r(erators)e(on)i(F)-7 b(o)r(c)n(k)28 b(space)e(dep)r(ends)523 4259 y(on)39 b(the)g(dynamics,)g(via)g Fs(\012)t Fv(!)g(Hence)h(the)f(exp)r(ectation)g(v)-5 b(alues)39 b(of)g(those)g(op)r(erators,)523 4359 y(and)34 b(of)f(p)r(olynomial)g(expressions)f(in)i(these)g(op)r(erators)d(will)j (also)f(dep)r(end)h(on)f Fs(\012)t Fv(.)h(In)523 4458 y(this)j(sense,)e Fp(the)j(same)g(mathematic)l(al)h(obje)l(ct)p Fv(,)e(namely)f(the)h(v)n(ector)e Fr(j)p Fv(0)p Fr(i)i(2)h(F)3088 4428 y Fq(+)3143 4458 y Fv(\()p Fo(C)3235 4428 y Fn(n)3280 4458 y Fv(\))523 4558 y Fp(c)l(orr)l(esp)l(onds)30 b(to)e(a)h(di\013er) l(ent)g(physic)l(al)i(state)d(of)i(the)e(system)f Fv(of)f Fs(n)g Fv(coupled)h(oscillators)523 4658 y(for)e(di\013eren)n(t)h(c)n (hoices)e(of)i Fs(\012)t Fv(,)g Fp(i.e.)h Fv(of)f(the)f(spring)g (constan)n(ts.)g(Also,)g(the)h Fp(same)i(physic)l(al)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(21)523 282 y Fp(quantity)p Fv(,)27 b(suc)n(h)f(as)g(the)h(displacemen)n(t)g(of)f(the)i(sev)n(en)n (th)e(oscillator,)f(is)h(represen)n(ted)g(b)n(y)523 382 y(a)e(di\013eren)n(t)g(mathematical)f(op)r(erator,)g(namely)g(the)i(op) r(erator)d(in)i(the)h(righ)n(t)e(hand)h(side)523 482 y(of)f(\(34\),)f(with)h Fs(\021)s Fv(\()p Fs(j)5 b Fv(\))24 b(=)f Fs(\016)1286 494 y Fn(j)s Fq(7)1354 482 y Fv(.)f(In)h (particular,)f(if)h(y)n(ou)f(are)f(in)n(terested)i(in)f(the)i(mean)e (square)523 581 y(displacemen)n(t)i(of)h(the)g(sev)n(en)n(th)f (oscillator)e(when)j(the)g(system)f(is)h(in)f(the)h(ground)f(state,)523 681 y Fp(i.e.)k Fr(h)p Fv(0)p Fr(j)p Fs(Q)827 651 y Fq(2)827 701 y(7)864 681 y Fr(j)p Fv(0)p Fr(i)p Fv(,)e(y)n(ou)f(will)h(need)h(a) e(detailed)i(sp)r(ectral)e(analysis)g(of)h Fs(\012)31 b Fv(and)26 b(in)g(particular)523 780 y(a)f(go)r(o)r(d)g(understanding) f(of)i(the)f(spatial)g(distribution)g(of)h(its)f(normal)g(mo)r(des)g(o) n(v)n(er)e(the)523 880 y Fs(n)29 b Fv(degrees)f(of)h(freedom)f(of)h (the)h(system,)e(as)h(I)g(already)e(p)r(oin)n(ted)j(out.)f(The)g (result)f(y)n(ou)523 980 y(\014nd)g(will)g(of)f(course)g(dep)r(end)h (on)f Fs(\012)t Fv(.)648 1079 y(In)h(the)g(same)g(manner,)f(an)n(y)g (other)h(giv)n(en)f(\014xed)h(v)n(ector)e(in)j(the)f(F)-7 b(o)r(c)n(k)28 b(space,)f(suc)n(h)523 1179 y(as)i(for)g(example)g(a)g (state)g(of)g(the)h(form)f Fs(a)1842 1139 y Fl(y)1842 1200 y Fq(+)1897 1179 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fv(,)h(for)f(some)g(\014xed)g(c)n(hoice)f(of)i Fs(\030)g Fr(2)c Fo(C)3244 1149 y Fn(n)3290 1179 y Fv(,)523 1279 y(represen)n(ts)g(a)h Fp(di\013er)l(ent)h Fv(ph)n(ysical)f(state)g (dep)r(ending)h(on)g Fs(\012)t Fv(.)648 1378 y(In)k(short,)g(the)h(in)n (terpretation)e(of)h(a)g(giv)n(en)g(v)n(ector)f(in)h(F)-7 b(o)r(c)n(k)32 b(space)g(as)g(a)g(state)g(of)523 1478 y(a)h(ph)n(ysical)g(system)g(dep)r(ends)h(on)g(the)g(dynamics)f(of)h (the)g(system)f(under)h(considera-)523 1577 y(tion)d(b)r(ecause)f(the)h (represen)n(tation)d(of)j(the)g(ph)n(ysical)e(observ)-5 b(ables)29 b(of)i(the)g(system)f(b)n(y)523 1677 y(op)r(erators)c(on)h (F)-7 b(o)r(c)n(k)27 b(space)g(is)g(dynamics)h(dep)r(enden)n(t.)648 1777 y(T)-7 b(o)35 b(a)n(v)n(oid)e(confusion,)i(these)h(simple)f (remarks)f(need)h(to)g(b)r(e)h(remem)n(b)r(ered)e(when)523 1876 y(dealing)g(with)g(the)h(in\014nite)g(dimensional)e(theory)-7 b(,)34 b(where)f(only)h(the)g(F)-7 b(o)r(c)n(k)34 b(represen-)523 1976 y(tation)c(surviv)n(es.)e(In)j(particular,)d(the)j(name)f(\\v)-5 b(acuum)29 b(v)n(ector")g(or)g(\\v)-5 b(acuum)29 b(state")523 2076 y(giv)n(en)h(to)h(the)g(F)-7 b(o)r(c)n(k)31 b(v)-5 b(acuum)31 b(con)n(v)n(eys)e(the)j(wrong)d(idea)i(that,)g(someho)n(w,)f (when)h(the)523 2175 y(system)g(state)h(is)f(represen)n(ted)f(b)n(y)i (this)f(v)n(ector,)g(space)f(is)i(empt)n(y)-7 b(,)32 b(there)f(is)g(\\nothing)523 2275 y(there")c(and)g(therefore)g(this)h (state)f(should)h(ha)n(v)n(e)e(trivial)h(ph)n(ysical)g(prop)r(erties)g (that)g(in)523 2374 y(fact)19 b(should)h(b)r(e)f(indep)r(enden)n(t)h (of)g(the)f(system)g(under)h(consideration)d(and)i(in)h(particular)523 2474 y(of)28 b(the)g(dynamics.)523 2702 y Fm(3.4)41 b(The)32 b(F)-8 b(o)s(c)m(k)32 b(represen)m(tation:)f(general)h(free)g(\014elds) 523 2873 y Fv(Summing)26 b(up,)g(w)n(e)g(ha)n(v)n(e)f(no)n(w)g(reform)n (ulated)f(the)j(quan)n(tum)e(mec)n(hanical)g(description)523 2972 y(of)34 b(a)g(\014nite)g(dimensional)g(coupled)g(oscillator)e (system)i(in)g(a)g(w)n(a)n(y)f(that)h(will)g(b)r(e)h(seen)523 3072 y(to)29 b(carry)f(o)n(v)n(er)f(immediately)j({)e(with)i(only)f (one)g(mo)r(derate)f(c)n(hange)g({)h(to)g(the)h(in\014nite)523 3171 y(dimensional)35 b(case.)f(Indeed,)i(giv)n(en)e(a)h(free)g (oscillator)f(\014eld)h(determined)h(b)n(y)f Fr(K)i Fv(and)523 3271 y Fs(\012)t Fv(,)j(it)h(is)f(no)n(w)f(p)r(erfectly)h(natural)g(to) f(c)n(ho)r(ose)g(as)h(the)g(quan)n(tum)g(Hilb)r(ert)g(space)g(of)523 3371 y(suc)n(h)35 b(a)f(system)h(the)g(F)-7 b(o)r(c)n(k)34 b(space)g Fr(F)1728 3341 y Fq(+)1784 3371 y Fv(\()p Fr(K)1880 3341 y Fg(C)1926 3371 y Fv(\),)i(and)e(as)h(quan)n(tum)f(Hamiltonian)h Fs(H)42 b Fv(=)523 3470 y(d)p Fs(\000)12 b Fv(\()p Fs(\012)t Fv(\).)28 b(Note)g(that)g(this)g(is)f(a)h(p)r(ositiv)n(e)f(op)r(erator) f(and)h(that)h(the)g(F)-7 b(o)r(c)n(k)27 b(v)-5 b(acuum)28 b(is)f(its)523 3570 y(ground)18 b(state,)h(with)g(eigen)n(v)-5 b(alue)18 b(0.)h(Pro)r(ceeding)e(in)i(complete)g(analogy)e(with)j(the)f (\014nite)523 3670 y(dimensional)36 b(case,)g(the)h(quan)n(tization)f (of)h(the)g(classical)e(creation)h(and)h(annhilition)523 3769 y(functions)21 b Fs(a)918 3781 y Fq(c)951 3769 y Fv(\()p Fs(\030)t Fv(\))p Fs(;)14 b(a)1136 3739 y Fl(y)1136 3790 y Fq(c)1172 3769 y Fv(\()p Fs(\030)t Fv(\))21 b(are)f(the)h (creation)e(and)h(annihilation)h(op)r(erators)d Fs(a)2889 3781 y Fq(+)2944 3769 y Fv(\()p Fs(\030)t Fv(\))p Fs(;)c(a)3129 3729 y Fl(y)3129 3790 y Fq(+)3185 3769 y Fv(\()p Fs(\030)t Fv(\).)523 3869 y(In)33 b(terms)f(of)h(those)g(the)g(quan)n(tized)f (\014elds)h(and)g(their)f(conjugates)g(are)g(then)h Fp(de\014ne)l(d)523 3968 y Fv(precisely)27 b(as)g(b)r(efore)g(\()p Fs(\021)g Fr(2)c(K)1456 3938 y Fg(C)1455 3995 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1611 3968 y Fv(\):)1174 4195 y Fs(\021)f Fr(\001)c Fs(Q)23 b Fv(:=)1522 4139 y(1)p 1487 4176 111 4 v 1487 4193 a Fr(p)p 1557 4193 42 4 v 1557 4261 a Fv(2)1608 4195 y(\()p Fs(a)1684 4207 y Fq(+)1739 4195 y Fv(\()p Fs(\012)1839 4161 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1996 4195 y Fs(\021)s Fv(\))c(+)f Fs(a)2218 4155 y Fl(y)2218 4216 y Fq(+)2273 4195 y Fv(\()p Fs(\012)2373 4161 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2530 4195 y Fs(\021)s Fv(\)\))p Fs(;)504 b Fv(\(36\))523 4418 y(and,)28 b(similarly)e(\()p Fs(\021)h Fr(2)c(K)1289 4388 y Fg(C)1288 4445 y Fq(1)p Fn(=)p Fq(2)1393 4418 y Fv(\),)1226 4647 y Fs(\021)f Fr(\001)c Fs(P)35 b Fv(:=)1583 4591 y(i)p 1539 4628 111 4 v 1539 4644 a Fr(p)p 1608 4644 42 4 v 69 x Fv(2)1660 4647 y(\()p Fs(a)1736 4607 y Fl(y)1736 4667 y Fq(+)1791 4647 y Fv(\()p Fs(\012)1891 4612 y Fq(1)p Fn(=)p Fq(2)1996 4647 y Fs(\021)s Fv(\))19 b Fr(\000)f Fs(a)2218 4659 y Fq(+)2273 4647 y Fv(\()p Fs(\012)2373 4612 y Fq(1)p Fn(=)p Fq(2)p 2478 4601 45 4 v 2478 4647 a Fs(\021)s Fv(\)\))p Fs(:)556 b Fv(\(37\))p eop end %%Page: 22 22 TeXDict begin 22 21 bop 523 100 a Fu(22)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(It)29 b(is)g(often)g(con)n(v)n (enien)n(t)e(to)i(think)g(of)g(\\the)f(\014eld)h Fs(Q)p Fv(")f(as)g(the)h(map)g(that)g(asso)r(ciates)e(to)523 382 y(eac)n(h)k Fs(\021)i Fr(2)d(K)936 397 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1125 382 y Fv(the)i(self-adjoin)n(t)f(op)r(erator)f (in)i(the)g(righ)n(t)f(hand)h(side)g(of)f(\(36\),)h(and)523 482 y(similarly)22 b(for)h(\\the)g(conjugate)f(\014eld)h Fs(P)12 b Fv(",)23 b(de\014ned)g(on)g Fr(K)2315 497 y Fq(1)p Fn(=)p Fq(2)2419 482 y Fv(.)h(With)g(this)f(language,)e(the)523 581 y(\014eld)30 b(op)r(erator)d Fs(\021)c Fr(\001)c Fs(Q)29 b Fv(is)h(the)f(v)-5 b(alue)29 b(of)g(the)h(\014eld)g Fs(Q)f Fv(at)g Fs(\021)g Fr(2)d(K)2523 596 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2679 581 y Fv(.)k(This)f(notation)g(is)523 681 y(reasonable)d(since)h(the)h(\014eld)g(is)f(a)h(linear)e(function)j (of)e(its)h(argumen)n(t.)648 780 y(The)35 b(mo)r(derate)g(c)n(hange)g (to)g(whic)n(h)h(I)g(referred)e(to)i(ab)r(o)n(v)n(e)e(is)i(the)g(fact)f (that,)h(if)h(I)523 880 y(compare)i(the)i(ab)r(o)n(v)n(e)e(quan)n (tization)h(prescription)f(for)h(the)h(case)e Fr(K)46 b Fv(=)e Fo(R)2997 850 y Fn(n)3083 880 y Fv(to)c(the)523 980 y(one)c(of)h(Sect.)g(3.1)f(and)h(Sect.)g(3.3,)f(then)h(it)g(is)g (clear)f(that)h(I)g(substracted)f(from)g(the)523 1079 y(Hamiltonian)g(the)g(\\zero-p)r(oin)n(t)e(energy",)1940 1017 y Fk(P)2027 1038 y Fn(n)2027 1104 y(i)p Fq(=1)2153 1079 y Fs(!)2205 1091 y Fn(i)2232 1079 y Fv(.)i(It)g(is)g(argued)e(in)i (all)g(quan)n(tum)523 1179 y(\014eld)30 b(theory)f(texts)g(that)h(this) g(constitutes)f(an)h(inno)r(cuous)f(c)n(hange,)f(for)h(t)n(w)n(o)g (distinct)523 1279 y(reasons.)37 b(First,)i(adding)g(a)g(constan)n(t)f (to)h(the)h(Hamiltonian)e(do)r(es)h(not)g(c)n(hange)f(the)523 1378 y(dynamics)d(in)g(an)n(y)f(fundamen)n(tal)h(w)n(a)n(y)-7 b(.)34 b(Second)h(only)f(energy)g(di\013erences)h(coun)n(t)f(in)523 1478 y(ph)n(ysics,)h(so)h(tossing)f(out)g(an)h(additiv)n(e)g(constan)n (t)f(in)h(the)g(de\014nition)h(of)f(the)g(energy)523 1577 y(should)30 b(not)h(c)n(hange)f(an)n(ything)g(fundamen)n(tally)-7 b(.)30 b(As)h(a)f(result,)h(since)f(the)h(expression)523 1615 y Fk(P)611 1635 y Fn(n)611 1702 y(i)p Fq(=1)736 1677 y Fs(!)788 1689 y Fn(i)841 1677 y Fv(mak)n(es)25 b(no)g(sense)g(in)h(general)e(in)i(in\014nite)g(dimensions,)f(where)g (it)h(is)g(formally)523 1777 y(t)n(ypically)i(equal)h(to)f(+)p Fr(1)p Fv(,)h(it)g(seems)g(lik)n(e)f(a)g(go)r(o)r(d)h(idea)f(to)h(toss) f(it)h(out)g(from)g(the)g(v)n(ery)523 1876 y(b)r(eginning!)k(This)f (means)g(y)n(ou)g(calibrate)f(the)i(energy)e(so)h(that)h(the)g(ground)e (state)i(of)523 1976 y(the)e(system,)f(whic)n(h)g(is)h(represen)n(ted)e (b)n(y)h(the)h(F)-7 b(o)r(c)n(k)30 b(v)-5 b(acuum,)31 b(has)f(zero)f Fp(total)i Fv(energy)-7 b(,)523 2076 y(indep)r(enden)n (tly)21 b(of)f Fs(\012)t Fv(,)g(and)g(leads)g(to)g(the)g(c)n(hoice)g (of)g Fs(H)29 b Fv(=)23 b(d)p Fs(\000)12 b Fv(\()p Fs(\012)t Fv(\))21 b(as)e(the)i(Hamiltonian.)648 2175 y(While)27 b(this)g(is)g(the)g(reasoning)e(found)i(in)g(all)f(ph)n(ysics)g(and)h (mathematical)f(ph)n(ysics)523 2275 y(texts)21 b(the)f(tossing)g(out)g (of)g(the)h(zero-p)r(oin)n(t)e(energy)g(is)i(not)f(suc)n(h)g(an)g(inno) r(cen)n(t)h(op)r(eration)523 2374 y(after)f(all.)g(F)-7 b(or)20 b(the)h(ph)n(ysics)f(of)h(the)f(zero-p)r(oin)n(t)g(energy)-7 b(,)19 b(I)i(refer)e(to)i([Mi].)g(See)f(also)g([DB2])523 2474 y(for)27 b(further)h(commen)n(ts.)648 2574 y(It)33 b(is)h(instructiv)n(e)f(to)g(compute)g(the)h(ev)n(olution)f(of)g(the)h (\014eld)f(and)h(the)f(conjugate)523 2673 y(\014eld)28 b(under)f(the)h(dynamics.)g(Since)1594 2852 y(e)1631 2817 y Fl(\000)p Fq(i)p Fn(H)t(t)1813 2852 y Fv(=)22 b Fs(\000)12 b Fv(\(e)2032 2817 y Fl(\000)p Fq(i)p Fn(\012)s(t)2186 2852 y Fv(\))p Fs(;)523 3030 y Fv(it)28 b(is)g(easy)e(to)i(c)n(hec)n(k) e(that)1408 3208 y(e)1445 3174 y Fq(i)p Fn(H)t(t)1552 3208 y Fs(a)1596 3220 y Fq(+)1651 3208 y Fv(\()p Fs(\030)t Fv(\)e)1792 3174 y Fl(\000)p Fq(i)p Fn(H)t(t)1974 3208 y Fv(=)d Fs(a)2106 3220 y Fq(+)2161 3208 y Fv(\(e)2230 3174 y Fq(i)p Fn(\012)s(t)2332 3208 y Fs(\030)t Fv(\))p Fs(:)523 3386 y Fv(De\014ne)d(then)g(the)g(ev)n(olv)n(ed)d(\014eld)j Fs(Q)p Fv(\()p Fs(t)p Fv(\))g(as)e(the)i(map)f(that)h(asso)r(ciates)e (to)h(eac)n(h)f Fs(\021)27 b Fr(2)c(K)3156 3401 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)523 3486 y Fv(the)28 b(self-adjoin)n(t)f(op)r (erator)f Fs(\021)c Fr(\001)c Fs(Q)p Fv(\()p Fs(t)p Fv(\))28 b(de\014ned)g(as)f(follo)n(ws:)1432 3664 y Fs(\021)22 b Fr(\001)c Fs(Q)p Fv(\()p Fs(t)p Fv(\))23 b Fr(\021)g Fv(e)1844 3630 y Fq(i)p Fn(H)t(t)1951 3664 y Fv(\()p Fs(\021)f Fr(\001)c Fs(Q)p Fv(\)e)2222 3630 y Fl(\000)p Fq(i)p Fn(H)t(t)2381 3664 y Fs(:)523 3842 y Fv(A)28 b(simple)g (computation)f(then)h(yields)999 4055 y Fs(\021)22 b Fr(\001)c Fs(Q)p Fv(\()p Fs(t)p Fv(\))24 b(=)1419 3999 y(1)p 1384 4036 111 4 v 1384 4053 a Fr(p)p 1453 4053 42 4 v 68 x Fv(2)1505 4055 y(\()p Fs(a)1581 4067 y Fq(+)1636 4055 y Fv(\()p Fs(\012)1736 4021 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1893 4055 y Fv(e)1930 4021 y Fq(i)p Fn(\012)s(t)p 2032 4009 45 4 v 2032 4055 a Fs(\021)s Fv(\))19 b(+)f Fs(a)2254 4015 y Fl(y)2254 4076 y Fq(+)2309 4055 y Fv(\()p Fs(\012)2409 4021 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2566 4055 y Fv(e)2603 4021 y Fq(i)p Fn(\012)s(t)2705 4055 y Fs(\021)s Fv(\)\))p Fs(:)523 4276 y Fv(Hence)1460 4352 y(d)1506 4322 y Fq(2)p 1445 4389 114 4 v 1445 4465 a Fv(d)p Fs(t)1521 4441 y Fq(2)1568 4408 y Fs(\021)k Fr(\001)c Fs(Q)p Fv(\()p Fs(t)p Fv(\))24 b(=)e Fr(\000)p Fs(\012)2076 4374 y Fq(2)2113 4408 y Fs(\021)g Fr(\001)c Fs(Q)p Fv(\()p Fs(t)p Fv(\))p Fs(:)523 4583 y Fv(One)25 b(de\014nes)g(similarly)f Fs(\021)16 b Fr(\001)d Fs(P)f Fv(\()p Fs(t)p Fv(\),)26 b(whic)n(h)f(ob)r(eys)g(the)g(same)g(equation.)f(In)h(fact,)h Fs(\021)16 b Fr(\001)d Fs(Q)p Fv(\()p Fs(t)p Fv(\))523 4682 y(and)30 b Fs(\021)23 b Fr(\001)e Fs(P)12 b Fv(\()p Fs(t)p Fv(\))30 b(are)g(op)r(erator-v)-5 b(alued)28 b(solutions)h(of)h (this)h(with)g Fs(\021)23 b Fr(\001)d Fs(Q)p Fv(\()p Fs(t)p Fv(\))31 b(satisfying)e(the)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(23)523 282 y Fp(e)l(qual)36 b(time)g(c)l(ommutation)g(r)l(elations)p Fv(.)f(They)f(are)f(called)h (the)h(Heisen)n(b)r(erg)e(\014eld)i(and)523 382 y(conjugate)27 b(\014eld)h(in)g(the)g(ph)n(ysics)f(literature.)648 482 y(W)-7 b(e)36 b(are)e(no)n(w)h(in)h(a)f(p)r(osition)h(to)f(further)h (study)g(these)f(systems,)h(a)f(task)g(I)h(turn)523 581 y(to)28 b(next.)g(First,)f(a)h(w)n(ord)e(on)i(the)g(\\particle)e(in)n (terpretation)h(of)h(the)g(\014eld)g(states")f(is)g(in)523 681 y(order.)523 913 y Fm(3.5)41 b(P)m(article)32 b(in)m(terpretation)g (of)g(the)g(\014eld)f(states)523 1088 y Fv(Ph)n(ysicists)17 b(refer)h(to)h Fr(F)1235 1100 y Fn(m)1316 1088 y Fv(as)f(the)h Fs(m)g Fv(particle)f(sector)f(of)i(the)g(F)-7 b(o)r(c)n(k)18 b(space)g(\()p Fs(m)23 b Fr(\025)g Fv(1\))18 b(and)h(to)523 1188 y Fr(F)583 1200 y Fq(0)646 1188 y Fv(as)26 b(the)h(v)-5 b(acuum)26 b(sector.)g(This)g(terminology)f(comes)h(from)g(the)h(follo) n(wing)e(remark.)523 1287 y(As)40 b(an)n(y)g(b)r(eginners')g(text)h(in) f(quan)n(tum)g(mec)n(hanics)g(will)h(tell)f(y)n(ou,)g(whenev)n(er)f (the)523 1387 y(quan)n(tum)20 b(Hilb)r(ert)i(space)d(of)i(a)f(single)g (particle)g(\(or)g(a)g(single)g(system\))g(is)h Fr(V)7 b Fv(,)20 b(the)h(Hilb)r(ert)523 1487 y(space)34 b(of)i(states)e(for)h Fs(m)g Fv(\(iden)n(tical\))g(particles)g(\(or)f(systems\))h(is)g(the)h Fs(m)p Fv(-fold)e(tensor)523 1586 y(pro)r(duct)d(of)f Fr(V)7 b Fv(.)31 b(The)f(simplest)h(case)f(is)h(the)g(one)f(where)g Fr(V)35 b Fv(=)27 b Fs(L)2592 1556 y Fq(2)2629 1586 y Fv(\()p Fo(R)2721 1556 y Fn(d)2760 1586 y Fv(\).)k(Then)g(the)g Fs(m)p Fv(-)523 1686 y(fold)24 b(tensor)f(pro)r(duct)i(can)e(b)r(e)i (naturally)e(iden)n(ti\014ed)h(with)h Fs(L)2451 1656 y Fq(2)2488 1686 y Fv(\()p Fo(R)2580 1656 y Fn(d)2630 1686 y Fr(\002)11 b Fs(:)j(:)g(:)d Fr(\002)g Fo(R)2950 1656 y Fn(d)3012 1686 y Fv(=)23 b Fo(R)3160 1656 y Fn(dm)3257 1686 y Fv(\),)523 1785 y(whic)n(h)j(is)g(isomorphic)f(to)h Fr(\012)1425 1797 y Fn(m)1488 1785 y Fr(V)7 b Fv(.)26 b(The)g(same)f(quan)n(tum)i(mec)n(hanics)e(course)g(will)h(teac)n(h)523 1885 y(y)n(ou)d(that,)i(when)f(the)g(particles)f(are)g (indistinguishable,)h(the)h(state)e(space)h(needs)f(to)h(b)r(e)523 1985 y(restricted)33 b(either)g(to)g(the)h(symmetric)f(or)f(an)n (ti-symmetric)h(tensor)f(pro)r(duct.)i(In)f(the)523 2084 y(\014rst)28 b(case,)f(whic)n(h)h(is)f(the)i(one)e(w)n(e)h(are)f (dealing)g(with)i(here,)e(the)h(particles)f(are)g(said)h(to)523 2184 y(b)r(e)33 b(b)r(osons,)f(otherwise)g(they)h(are)f(fermions.)h(In) g(the)g(case)f(where)g Fr(V)39 b Fv(=)31 b Fs(L)2941 2154 y Fq(2)2978 2184 y Fv(\()p Fo(R)3070 2154 y Fn(d)3109 2184 y Fv(\),)i(the)523 2284 y Fs(m)p Fv(-fold)28 b(symmetric)g(tensor) g(pro)r(duct)g(of)h Fr(V)35 b Fv(consists)27 b(of)i(all)f(symmetric)g Fs(L)2918 2253 y Fq(2)2955 2284 y Fv(-functions)523 2383 y(of)g Fs(m)f Fv(v)-5 b(ariables.)648 2483 y(The)24 b(ab)r(o)n(v)n(e)f (considerations)g(suggest)g(that,)i Fp(c)l(onversely)p Fv(,)g(whenev)n(er)f(the)h(quan)n(tum)523 2582 y(state)40 b(space)g(of)h(a)f(ph)n(ysical)f(system)h(turns)h(out)f(to)h(b)r(e)g(a) f(F)-7 b(o)r(c)n(k)40 b(space)g(o)n(v)n(er)e(some)523 2682 y(Hilb)r(ert)33 b(space)f Fr(V)7 b Fv(,)33 b(one)f(ma)n(y)h(think) g(of)g Fr(V)39 b Fv(as)32 b(a)h(one-particle)e(space,)h(and)h(of)g Fr(F)3128 2694 y Fn(m)3190 2682 y Fv(\()p Fr(V)7 b Fv(\))523 2782 y(as)30 b(the)h(corresp)r(onding)e Fs(m)p Fv(-particle)h(space.)g (An)h(arbitrary)e(state)i(of)f(the)i(system)e(can)523 2881 y(then)23 b(b)r(e)g(though)n(t)f(of)g(as)g(a)g(sup)r(erp)r (osition)g(of)g(states)g(with)h(0,)f(1,)g(2,)h(.)13 b(.)h(.)g Fs(m)p Fv(,)23 b(.)14 b(.)g(.)f(particles.)523 2981 y(These)33 b(ideas)f(emerged)h(v)n(ery)f(quic)n(kly)g(after)h(the)g(birth)h(of)f (quan)n(tum)g(mec)n(hanics,)f(as)523 3081 y(so)r(on)23 b(as)g(ph)n(ysicists)g(attac)n(k)n(ed)g(the)h(problem)f(of)h(analyzing) f(the)h(quan)n(tum)g(mec)n(hanical)523 3180 y(b)r(eha)n(viour)39 b(of)g(systems)h(with)g(an)f(in\014nite)i(n)n(um)n(b)r(er)f(of)f (degrees)g(of)h(freedom,)f(suc)n(h)523 3280 y(as)34 b(the)i (electromagnetic)e(\014eld.)h(The)g(F)-7 b(o)r(c)n(k)35 b(space)f(structure)h(of)g(the)h(Hilb)r(ert)f(space)523 3379 y(of)h(states)g(describing)g(the)h Fp(\014eld)g Fv(immediately)f(lead)g(to)h(suc)n(h)f(an)g(in)n(terpretation)f(in)523 3479 y(terms)d(of)g Fp(p)l(articles)p Fv(.)i(F)-7 b(or)31 b(the)i(electromagnetic)e(\014eld,)h(the)h(particles)e(w)n(ere)h (baptized)523 3579 y(\\photons",)d(and)h(in)h(complete)f(analogy)-7 b(,)29 b(the)i(quan)n(tum)f(mec)n(hanical)g(description)g(of)523 3678 y(lattice)f(vibrations)f(in)h(solid)f(state)h(ph)n(ysics)f(lead)g (to)h(the)g(notion)g(of)f(\\phonons".)g(The)523 3778 y(idea)33 b(that)g(one)f(can)h(asso)r(ciate)e(a)i(particle)f(in)n (terpretation)g(to)h(the)g(states)g(of)g(a)f(F)-7 b(o)r(c)n(k)523 3878 y(space)20 b(is)h(further)g(corrob)r(orated)d(b)n(y)j(the)g (observ)-5 b(ation)20 b(that)h(those)g(states)f(carry)g(energy)523 3977 y(and)40 b(momen)n(tum)g(in)g(\\lumps".)f(This)h(can)f(b)r(e)h (seen)g(as)f(follo)n(ws.)g(Supp)r(ose,)g(in)h(our)523 4077 y(notations,)27 b(that)h Fs(\012)k Fv(has)27 b(a)g(pure)g(p)r(oin) n(t)h(sp)r(ectrum:)1565 4249 y Fs(\012)t(\021)1674 4261 y Fn(j)1732 4249 y Fv(=)23 b Fs(!)1872 4261 y Fn(j)1907 4249 y Fs(\021)1948 4261 y Fn(j)1983 4249 y Fs(;)41 b(j)28 b Fr(2)c Fo(N)p Fs(:)523 4422 y Fv(Then)k(the)g(quan)n(tum)f (Hamiltonian)h(is)1448 4606 y Fs(H)i Fv(=)22 b(d)p Fs(\000)12 b Fv(\()p Fs(\012)t Fv(\))24 b(=)1987 4528 y Fk(X)2031 4704 y Fn(j)2120 4606 y Fs(!)2172 4618 y Fn(j)2207 4606 y Fs(a)2251 4566 y Fl(y)2251 4630 y Fn(j)2286 4606 y Fs(a)2330 4618 y Fn(j)2365 4606 y Fs(;)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 523 100 a Fu(24)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(where)37 b(I)g(wrote)f Fs(a)1124 242 y Fl(y)1124 305 y Fn(j)1198 282 y Fv(=)j Fs(a)1346 252 y Fl(y)1380 282 y Fv(\()p Fs(\021)1453 294 y Fn(j)1489 282 y Fv(\).)f(Note)f(that)g(I)h(ha)n(v)n(e)e(dropp)r (ed)h(the)g(index)h(+)e(on)h(the)523 382 y(creation)32 b(and)i(annihilation)f(op)r(erators,)e(a)i(practice)g(that)h(I)f(shall) h(stic)n(k)f(to)g(in)h(what)523 482 y(follo)n(ws)e(since)h(I)f(will)i (at)e(an)n(y)g(rate)h(b)r(e)g(w)n(orking)e(on)i(the)g(symmetric)f(F)-7 b(o)r(c)n(k)33 b(space)f(all)523 581 y(the)c(time.)g(No)n(w)f(consider) g(for)g(example)g(the)h(state)1669 739 y Fs(a)1713 699 y Fl(y)1713 761 y Fq(1)1750 739 y Fv(\()p Fs(a)1826 699 y Fl(y)1826 761 y Fq(5)1863 739 y Fv(\))1895 705 y Fq(3)1933 739 y Fs(a)1977 699 y Fl(y)1977 761 y Fq(10)2047 739 y Fr(j)p Fv(0)p Fr(i)p Fs(:)523 897 y Fv(This)i(is)g(a)f(5-particle)g (state,)g(and)h(an)g(eigen)n(v)n(ector)d(of)j(the)g(Hamiltonian)g(with) g(eigen-)523 996 y(v)-5 b(alue)31 b Fs(!)793 1008 y Fq(1)850 996 y Fv(+)20 b(3)p Fs(!)1029 1008 y Fq(5)1086 996 y Fv(+)h Fs(!)1224 1008 y Fq(10)1294 996 y Fv(.)31 b(It)g(is)f(natural)g (to)h(think)g(of)g(it)g(in)n(tuitiv)n(ely)g(as)g(b)r(eing)f(a)h(state) 523 1096 y(\\con)n(taining")d(3)h(particles)g(of)h(energy)e Fs(!)1834 1108 y Fq(5)1871 1096 y Fv(,)i(and)g(one)f(particle)g(of)h (energy)e Fs(!)2960 1108 y Fq(1)3027 1096 y Fv(and)i Fs(!)3243 1108 y Fq(10)523 1196 y Fv(eac)n(h.)h(Similarly)-7 b(,)31 b(in)h(translationally)e(in)n(v)-5 b(arian)n(t)31 b(systems,)g(suc)n(h)g(states)h(can)f(b)r(e)h(seen)523 1295 y(to)25 b(carry)f(a)g(total)h(momen)n(tum)h(whic)n(h)f(is)g(the)h (sum)f(of)g(\\lumps")g(of)g(momen)n(tum)h(corre-)523 1395 y(sp)r(onding)i(to)f(its)i(individual)f(constituen)n(ts.)f(Of)h (course,)f(the)h(particle)g(in)n(terpretation)523 1494 y(of)j(the)g(states)f(of)h(the)g(\014eld)g(is)g(a)f(v)n(ery)g(imp)r (ortan)n(t)h(feature)f(of)h(the)g(theory)f(since)h(it)g(is)523 1594 y(essen)n(tial)23 b(for)g(the)h(in)n(terpretation)e(of)i(high)f (energy)g(exp)r(erimen)n(ts,)g(and)g(so)g(it)h(has)f(quite)523 1694 y(naturally)k(receiv)n(ed)f(a)h(lot)h(of)f(atten)n(tion.)648 1793 y(Despite)37 b(its)g(undeniable)g(v)-5 b(alue,)36 b(the)h(suggestiv)n(e)f(in)n(terpretation)f(of)i(the)g(states)523 1893 y(of)30 b(F)-7 b(o)r(c)n(k)29 b(space)g(in)i(terms)e(of)h (particles)f(ma)n(y)g(lead)h(\(and)g(has)f(lead\))h(to)g(some)f(amoun)n (t)523 1993 y(of)j(confusion)f(and)g(has)g(to)g(b)r(e)h(tak)n(en)f (with)h(a)f(\(large\))g(grain)f(of)i(salt.)f(Some)h(of)f(those)523 2092 y(problems)26 b(seem)g(to)h(ha)n(v)n(e)e(b)r(een)i(brough)n(t)e (out)i(clearly)e(only)i(when)f(ph)n(ysicists)g(started)523 2192 y(to)k(in)n(v)n(estigate)e(quan)n(tum)i(\014eld)g(theory)f(on)g (curv)n(ed)g(space-times.)g(A)h(critical)f(discus-)523 2291 y(sion)e(of)h(this)g(issue)f(can)h(b)r(e)g(found)g(throughout)f ([F)-7 b(u].)28 b(Although)g(F)-7 b(ulling)28 b(do)r(es)f(adopt)523 2391 y(the)i(second)f(quan)n(tization)g(viewp)r(oin)n(t,)g(he)h (stresses)e(rep)r(eatedly)h(the)h(need)g(to)f(escap)r(e)523 2491 y(\\from)e(the)h(t)n(yrann)n(y)e(of)h(the)h(particle)f(concept")g (in)h(order)e(to)i(\\come)e(to)i(a)f(completely)523 2590 y(\014eld)k(theoretic)f(understanding)g(of)h(quan)n(tum)f(\014eld)h (theory)-7 b(.")29 b(Similarly)-7 b(,)29 b(W)-7 b(ald,)30 b(who)523 2690 y(do)r(es)24 b(indeed)g(adopt)f(a)h(\014eld)g(theoretic) g(viewp)r(oin)n(t)f(throughout)h(in)g([W)-7 b(a],)24 b(giv)n(es)f(a)g(crit-)523 2790 y(ical)k(analysis)f(of)h(the)g(merits)g (and)g(limitations)g(of)g(the)h(particle)e(concept)h(in)h(quan)n(tum) 523 2889 y(\014eld)k(theory)-7 b(.)31 b(He)g(actually)g(stresses)f(the) i(need)g(to)f(\\unlearn")f(some)h(of)g(the)h(familiar)523 2989 y(concepts)22 b(of)h(quan)n(tum)g(\014eld)g(theory)e(on)i(\015at)g (space)f(times)h(to)f(understand)g(the)i(curv)n(ed)523 3088 y(space)j(time)h(v)n(ersion)e(of)i(the)g(theory)-7 b(.)648 3188 y(There)26 b(are)h(in)g(fact)h(sev)n(eral)e(sources)g(of)h (problems)f(with)i(the)g(particle)f(in)n(terpreta-)523 3288 y(tion)33 b(of)f(the)h(states)g(in)g(quan)n(tum)f(\014eld)h (theory)-7 b(.)32 b(The)h(\014rst)f(one)g(w)n(as)g(already)f(hin)n(ted) 523 3387 y(at)c(in)h(Sect.)f(3.3:)g(the)g(use)g(of)h(the)f(w)n(ord)f (\\v)-5 b(acuum")27 b(to)g(describ)r(e)g(the)g(ground)g(state)g(of)523 3487 y(the)33 b(system)f(in)n(vites)g(one)g(to)g(think)h(that)g(when)f (the)h(system)f(is)g(in)h(this)g(state,)f(there)523 3587 y(is)f(\\nothing)g(there".)g(Actually)-7 b(,)31 b(one)g(ma)n(y)g(b)r(e) h(tempted)g(to)f(think)h(the)g(system)f(itself)523 3686 y(is)36 b(simply)h(not)f(there!)g(But)h(to)f(see)g(that)g(mak)n(es)g (no)g(sense,)f(it)i(is)f(enough)g(to)g(think)523 3786 y(of)29 b(an)g(oscillator)e(lattice.)i(Certainly)-7 b(,)29 b(when)g(this)g(system)g(is)g(in)g(its)g(ground)f(state,)h(all)523 3885 y(oscillators)d(are)i(there!)g(It)h(is)f(just)h(that)g(the)g (system)f(is)g(not)h(excited,)f(so)g(there)g(are)g(no)523 3985 y(\\particles")20 b(in)h(the)h(ab)r(o)n(v)n(e)e(\(F)-7 b(o)r(c)n(k)21 b(space\))g(sense)g(of)h(the)f(w)n(ord,)g(and)g(this)h (in)f(spite)h(of)f(the)523 4085 y(fact)k(that)g(the)g(mec)n(hanical)e (particles)h(making)g(up)h(the)g(lattice)g(are)e(certainly)h(presen)n (t.)523 4184 y(Also,)g(if)h(one)g(thinks)f(of)h(the)g(v)-5 b(acuum)25 b(state)f(as)g(empt)n(y)h(space,)f(it)h(b)r(ecomes)f(imp)r (ossible)523 4284 y(to)h(understand)f(ho)n(w)g(its)h(prop)r(erties)f (can)h(dep)r(end)g(on)f(the)i(system)e(considered)g(via)g Fs(\012)t Fv(.)523 4384 y(In)29 b(fact,)g(it)g(is)g(quite)g(ba\017ing)f (to)h(think)g(\\empt)n(y)g(space")e(could)i(ha)n(v)n(e)f(an)n(y)g(prop) r(erties)523 4483 y(at)i(all.)g(In)g(particular,)f(the)h(mean)g(square) f(displacemen)n(t)h(of)g(the)g(\014eld,)g(for)g(example,)523 4583 y(giv)n(en)d(b)n(y)1685 4682 y Fr(h)p Fv(0)p Fr(j)p Fv(\()p Fs(\021)22 b Fr(\001)c Fs(Q)p Fv(\))2016 4648 y Fq(2)2054 4682 y Fr(j)p Fv(0)p Fr(i)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(25)523 282 y Fv(is)20 b(a)f(function)h(of)g Fs(\012)t Fv(,)g(as)f(is)g(easily)g(seen)h(ev)n(en)f(in)h(\014nite)g (dimensional)f(oscillator)f(systems.)523 382 y(This)j(quan)n(tit)n(y)f (is)g(an)h(example)f(of)g(a)g(so-called)g(\\v)-5 b(acuum)20 b(\015uctuation".)g(Of)h(course,)e(for)523 482 y(systems)26 b(with)g(a)g(\014nite)h(n)n(um)n(b)r(er)f(of)g(degrees)e(of)i(freedom,) g(w)n(e)g(\014nd)h(this)f(phenomenon)523 581 y(p)r(erfectly)32 b(natural,)e(but)i(if)g(y)n(ou)e(study)i(the)g(Klein-Gordon)d(\014eld,) j(for)f(example,)g(and)523 681 y(call)d(the)g(ground)g(state)g(the)g(v) -5 b(acuum,)29 b(y)n(ou)e(end)i(up)f(b)r(eing)g(surprised)g(to)g(see)g (v)-5 b(acuum)523 780 y(exp)r(ectation)27 b(v)-5 b(alues)28 b(dep)r(end)g(on)f(the)h(mass)f(of)h(particles)e(that)i(are)f(not)g (there!)648 880 y(A)22 b(second)f(source)g(of)g(confusion)h(is)f(that)h (the)h(notion)e(of)h(\\particle")e(ev)n(ok)n(es)g(a)i(lo)r(cal-)523 980 y(ized)h(en)n(tit)n(y)-7 b(,)23 b(carrying)e(not)i(only)g(momen)n (tum)g(and)g(energy)-7 b(,)22 b(but)h(that)g(one)g(should)g(also)523 1079 y(b)r(e)32 b(able)f(to)g(lo)r(calize)g(in)h(space,)f(preferably)f (with)i(the)g(help)g(of)f(a)g(p)r(osition)g(op)r(erator.)523 1179 y(I)d(will)h(sho)n(w)e(in)i(Section)f(4)g(that)h(there)f(is)g(no)g (reasonable)f(notion)h(of)g(\\p)r(osition")f(that)523 1279 y(can)k(b)r(e)h(asso)r(ciated)e(to)h(the)h(one-particle)e(states)h (of)g(F)-7 b(o)r(c)n(k)31 b(space,)g(con)n(trary)e(to)j(what)523 1378 y(happ)r(ens)h(in)h(the)g(usual)f(non-relativistic)f(quan)n(tum)h (mec)n(hanics)g(of)g(systems)g(with)h(a)523 1478 y(\014nite)24 b(n)n(um)n(b)r(er)f(of)g(particles.)f(In)i(particular,)e(there)h(is)g (no)g(reasonable)e(\\p)r(osition)i(op)r(er-)523 1577 y(ator".)g(This)i(has)g(nothing)f(to)h(do)g(with)g(relativit)n(y)-7 b(,)24 b(but)i(is)e(true)h(for)f(large)g(classes)f(of)i Fs(\012)523 1677 y Fv(and)h(in)g(particular)f(for)g(all)h(examples)f (giv)n(en)g(so)h(far.)f(So)h(ev)n(en)f(if)i(the)f(particles)f(of)h (\014eld)523 1777 y(theory)37 b(share)f(a)h(certain)f(n)n(um)n(b)r(er)h (of)g(prop)r(erties)g(with)g(the)h(usual)f(p)r(oin)n(t)g(particles)523 1876 y(of)f(classical)g(and)g(quan)n(tum)g(mec)n(hanical)g(textb)r(o)r (oks,)g(they)h(ha)n(v)n(e)e(some)h(imp)r(ortan)n(t)523 1976 y(features)26 b(that)i(mak)n(e)e(them)h(quite)g(di\013eren)n(t.)g (They)g(are)f(analogous)f(ob)5 b(jects,)26 b(but)i(not)523 2076 y(totally)i(similar)f(ones.)g(This,)h(I)g(will)g(argue,)f(has)g (nothing)h(to)g(do)g(either)g(with)g(sp)r(ecial)523 2175 y(or)25 b(general)g(relativit)n(y)-7 b(,)25 b(but)h(is)g(clear)f(if)i (one)e(remem)n(b)r(ers)g(systematically)g(the)i(analogy)523 2275 y(with)h(\014nite)g(dimensional)f(oscillator)f(systems.)648 2374 y(As)f(a)g(constan)n(t)g(reminder)g(of)h(the)g(fact)f(that)h(the)g (so-called)e(particles)h(of)h(quan)n(tum)523 2474 y(\014eld)k(theory)f (are)g(nothing)h(but)g(excitations)f(of)h(its)g(ground)f(state,)g(it)i (is)e(a)h(go)r(o)r(d)f(idea)523 2574 y(to)39 b(use)g(the)g(older)f(ph)n (ysics)g(terminology)g(and)h(to)g(talk)f(systematically)g(of)h (\\quasi-)523 2673 y(particles",)22 b(\\quan)n(ta",)g(\\\014eld)h(quan) n(ta")f(or)h(of)g(\\elemen)n(tary)f(excitations)h(of)g(the)h(\014eld") 523 2773 y(rather)e(than)h(simply)g(of)g(particles)g(when)g(describing) f(the)i(states)e(of)h(F)-7 b(o)r(c)n(k)23 b(space.)f(I)h(will)523 2873 y(adhere)k(as)g(m)n(uc)n(h)g(as)g(p)r(ossible)g(to)h(this)g (pruden)n(t)f(practice.)648 2972 y(Moreo)n(v)n(er,)h(when)j(testing)f (y)n(our)g(understanding)g(of)h(a)f(notion)h(in)g(quan)n(tum)f(\014eld) 523 3072 y(theory)-7 b(,)34 b(try)g(to)h(see)f(what)h(it)g(giv)n(es)e (for)h(a)h(\014nite)g(system)f(of)h(oscillators.)d(If)k(it)f(lo)r(oks) 523 3171 y(funn)n(y)30 b(there,)g(it)g(is)g(lik)n(ely)f(to)h(b)r(e)h(a) e(bad)h(idea)f(to)h(use)g(it)g(in)g(the)h(in\014nite)f(dimensional)523 3271 y(case.)648 3371 y(The)g(remaining)g(parts)g(of)h(this)g(section)f (dev)n(elop)g(material)g(that)h(will)f(b)r(e)i(needed)523 3470 y(in)24 b(Section)f(4.)g(It)h(is)f(p)r(erhaps)g(a)g(go)r(o)r(d)g (idea)g(to)g(start)g(reading)f(the)i(latter,)f(coming)g(bac)n(k)523 3570 y(to)28 b(this)f(material)g(only)g(as)g(I)h(refer)f(to)g(it.)523 3811 y Fm(3.6)41 b(W)-8 b(eyl)31 b(op)s(erators)g(and)i(coheren)m(t)f (states)523 3993 y Fv(Giv)n(en)24 b(a)f(Hilb)r(ert)h(space)f Fr(V)30 b Fv(and)24 b(the)g(corresp)r(onding)e(symmetric)h(F)-7 b(o)r(c)n(k)23 b(space)g Fr(F)3112 3963 y Fq(+)3167 3993 y Fv(\()p Fr(V)7 b Fv(\),)523 4093 y(w)n(e)27 b(can)g(\014rst)h (de\014ne,)g(for)f(an)n(y)g Fs(\030)g Fr(2)c(V)7 b Fv(,)28 b(the)g Fp(Weyl)i(op)l(er)l(ator)1553 4289 y Fs(W)1631 4301 y Fq(F)1679 4289 y Fv(\()p Fs(\030)t Fv(\))24 b(=)e(e)1931 4254 y Fn(a)1967 4229 y Fb(y)1999 4254 y Fq(\()p Fn(\030)r Fq(\))p Fl(\000)p Fn(a)p Fq(\()p Fn(\030)r Fq(\))2259 4289 y Fs(:)523 4471 y Fv(A)28 b(coheren)n(t)f(state)g(is)g(then)i (de\014ned)f(as)e(a)i(v)n(ector)e(of)i Fr(F)2293 4441 y Fq(+)2348 4471 y Fv(\()p Fr(V)7 b Fv(\))28 b(of)f(the)h(form)1626 4679 y Fr(j)p Fs(\030)t Fr(i)1745 4632 y Fq(def)1758 4679 y Fv(=)36 b Fs(W)1937 4691 y Fq(F)1985 4679 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fs(;)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 523 100 a Fu(26)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(for)27 b(some)g Fs(\030)g Fr(2)d(V)7 b Fv(.)27 b(Note)h(that)g(the)g(map)1532 460 y Fs(\030)g Fr(2)23 b(V)30 b(7!)23 b(j)p Fs(\030)t Fr(i)g(2)h(F)2126 426 y Fq(+)2181 460 y Fv(\()p Fr(V)7 b Fv(\))523 638 y(pro)n(vides)26 b(a)i Fp(nonline)l(ar)g Fv(im)n(b)r(edding)g(of)g Fr(V)34 b Fv(in)n(to)28 b Fr(F)2112 608 y Fq(+)2167 638 y Fv(\()p Fr(V)7 b Fv(\))28 b(whic)n(h)g(is)f(not)h (to)g(b)r(e)g(confused)523 737 y(with)35 b(the)f(trivial)g(linear)g(im) n(b)r(edding)g Fr(V)1864 715 y(\030)1864 742 y Fv(=)1963 737 y Fr(F)2023 749 y Fq(1)2060 737 y Fv(\()p Fr(V)7 b Fv(\))34 b Fr(\032)g(F)2383 707 y Fq(+)2438 737 y Fv(\()p Fr(V)7 b Fv(\).)35 b(Giv)n(en)f(an)g(arbitrary)523 837 y(0)c Fr(6)p Fv(=)f Fs( )k Fr(2)e(V)7 b Fv(,)31 b(one)h(can)f(lik)n (ewise)g(consider)g(the)h(family)g Fs(W)2407 849 y Fq(F)2454 837 y Fv(\()p Fs(\030)t Fv(\))p Fs( )s Fv(,)h(and)f(those)f(v)n(ectors) 523 937 y(are)c(also)f(referred)h(to)g(as)g(a)g(family)h(of)f(coheren)n (t)g(states.)648 1036 y(Coheren)n(t)36 b(states)g(pla)n(y)h(an)f(imp)r (ortan)n(t)h(role)f(in)h(the)h(semi-classical)d(analysis)h(of)523 1136 y(quan)n(tum)e(systems)f(and)h(in)g(v)-5 b(arious)33 b(branc)n(hes)g(of)g(theoretical)g(ph)n(ysics)h([KSk)o(])g([P)n(e)o(].) 523 1236 y(W)-7 b(e)40 b(describ)r(e)f(them)g(here)g(in)h(the)f (abstract)g(con)n(text)g(of)g(symmetric)g(F)-7 b(o)r(c)n(k)39 b(spaces.)523 1335 y(They)d(are)f(v)n(ery)h(simple)g(ob)5 b(jects)36 b(to)g(de\014ne)g(but)h(nev)n(ertheless)e(ha)n(v)n(e)g(an)h (seemingly)523 1435 y(inexhaustable)27 b(set)h(of)f(in)n(teresting)g (prop)r(erties.)g(I)g(will)h(only)f(men)n(tion)h(those)f(I)h(need.)648 1534 y(T)-7 b(o)28 b(compute)g(with)h(the)f(coheren)n(t)g(states,)g(w)n (e)g(need)g(a)g(n)n(um)n(b)r(er)g(of)g(form)n(ulas)f(that)523 1634 y(are)g(listed)g(b)r(elo)n(w)g(and)h(that)g(can)f(all)g(b)r(e)h (obtained)f(easily)-7 b(,)27 b(if)h(one)f(remem)n(b)r(ers)f(\014rst)i (of)523 1734 y(all)c(that,)g(if)h Fs(A)f Fv(and)g Fs(B)k Fv(are)c(b)r(ounded)g(op)r(erators)e(so)i(that)g Fs(C)29 b Fv(=)23 b([)p Fs(A;)14 b(B)t Fv(])24 b(comm)n(utes)g(with)523 1833 y(b)r(oth)k Fs(A)g Fv(and)f Fs(B)t Fv(,)h(then)1370 2021 y(e)1407 1987 y Fn(A)p Fq(+)p Fn(B)1588 2021 y Fv(=)22 b(e)1712 1987 y Fn(A)1766 2021 y Fv(e)1803 1987 y Fn(B)1860 2021 y Fv(e)1897 1987 y Fl(\000)1959 1964 y Fc(C)p 1959 1973 45 3 v 1967 2007 a Ff(2)2017 2021 y Fs(;)14 b(C)30 b Fv(=)22 b([)p Fs(A;)14 b(B)t Fv(])p Fs(:)523 2199 y Fv(Computing)20 b(with)g Fs(a)1174 2169 y Fl(y)1209 2199 y Fv(\()p Fs(\030)t Fv(\))g(and)g Fs(a)p Fv(\()p Fs(\030)t Fv(\))h(as)e(if)h(they)g(w)n(ere)f(b)r(ounded)h(op)r(erators,)e(all)i (form)n(ulas)523 2298 y(b)r(elo)n(w)g(follo)n(w)f(from)g(this)h(and)g (some)f(p)r(ersev)n(erance)f(in)i(computing.)g(T)-7 b(aking)19 b(care)f(of)i(the)523 2398 y(domain)29 b(problems)f(to)h(mak)n(e)f (them)i(completely)f(rigorous)d(is)j(tedious)g(but)h(c)n(haracter)523 2498 y(building)j(and)f(can)g(b)r(e)g(done)g(using)g(the)h(tec)n (hniques)f(describ)r(ed)g(in)g([BR2])g(or)g([RS2)o(].)523 2597 y(First)c(of)f(all,)g(w)n(e)h(ha)n(v)n(e,)e(for)h(all)h Fs(\030)1574 2609 y Fq(1)1611 2597 y Fs(;)14 b(\030)1684 2609 y Fq(2)1745 2597 y Fr(2)23 b(V)7 b Fv(,)1174 2789 y Fs(W)1252 2801 y Fq(F)1299 2789 y Fv(\()p Fs(\030)1367 2801 y Fq(1)1405 2789 y Fv(\))p Fs(W)1515 2801 y Fq(F)1563 2789 y Fv(\()p Fs(\030)1631 2801 y Fq(2)1668 2789 y Fv(\))24 b(=)e Fs(W)1889 2801 y Fq(F)1937 2789 y Fv(\()p Fs(\030)2005 2801 y Fq(1)2061 2789 y Fv(+)c Fs(\030)2180 2801 y Fq(2)2218 2789 y Fv(\)e)2287 2754 y Fl(\000)p Fq(iIm)o(\()p 2462 2705 33 3 v Fn(\030)2494 2770 y Ff(1)2527 2754 y Fl(\001)p Fn(\030)2577 2762 y Ff(2)2609 2754 y Fq(\))2639 2789 y Fs(:)523 2966 y Fv(As)28 b(a)f(result)1113 3066 y Fs(W)1191 3078 y Fq(F)1238 3066 y Fv(\()p Fs(\030)1306 3078 y Fq(1)1344 3066 y Fv(\))p Fs(W)1454 3078 y Fq(F)1502 3066 y Fv(\()p Fs(\030)1570 3078 y Fq(2)1608 3066 y Fv(\))c(=)g Fs(W)1829 3078 y Fq(F)1876 3066 y Fv(\()p Fs(\030)1944 3078 y Fq(2)1982 3066 y Fv(\))p Fs(W)2092 3078 y Fq(F)2139 3066 y Fv(\()p Fs(\030)2207 3078 y Fq(1)2245 3066 y Fv(\)e)2314 3032 y Fl(\000)p Fq(2iIm\()p 2523 2983 V Fn(\030)2555 3047 y Ff(1)2587 3032 y Fl(\001)p Fn(\030)2637 3040 y Ff(2)2670 3032 y Fq(\))2700 3066 y Fs(;)523 3213 y Fv(and)1026 3312 y([)p Fs(W)1127 3324 y Fq(F)1174 3312 y Fv(\()p Fs(\030)t Fv(\))p Fs(;)14 b(W)1393 3324 y Fq(F)1442 3312 y Fv(\()p Fs(\030)1514 3278 y Fl(0)1537 3312 y Fv(\)])24 b(=)e Fs(W)1781 3324 y Fq(F)1829 3312 y Fv(\()p Fs(\030)1901 3278 y Fl(0)1924 3312 y Fv(\))p Fs(W)2034 3324 y Fq(F)2082 3312 y Fv(\()p Fs(\030)t Fv(\))2200 3220 y Fk(\020)2250 3312 y Fv(e)2287 3278 y Fq(i2Im\()p 2444 3229 V Fn(\030)2476 3244 y Fb(0)2498 3278 y Fl(\001)p Fn(\030)r 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3934 y Fs(;)523 4081 y Fv(or)1077 4180 y Fs(W)1155 4192 y Fq(F)1202 4180 y Fv(\()p Fr(\000)p Fs(\020)6 b Fv(\))p Fs(W)1451 4192 y Fq(F)1499 4180 y Fv(\()p Fs(\030)t Fv(\))p Fs(W)1681 4192 y Fq(F)1729 4180 y Fv(\()p Fs(\020)g Fv(\))24 b(=)f(e)1984 4151 y([)2007 4144 y Fq(\()p Fn(a)2069 4119 y Fb(y)2101 4144 y Fq(\()p Fn(\030)r Fq(\)+)p Fn(\030)r Fl(\001)p 2288 4096 V Fn(\020)t Fq(\))p Fl(\000)p Fq(\()p Fn(a)p Fq(\()p Fn(\030)r Fq(\)+)p 2597 4096 33 3 v Fn(\030)r Fl(\001)p Fn(\020)t Fq(\))2709 4150 y Fv(])2736 4180 y Fs(:)523 4327 y Fv(One)k(then)h(\014nds)523 4505 y Fs(W)601 4517 y Fq(F)648 4505 y Fv(\()p Fr(\000)p Fs(\020)6 b Fv(\))833 4437 y Fk(\002)868 4505 y Fs(a)912 4470 y Fl(y)947 4505 y Fv(\()p Fs(\030)t Fv(\))1051 4437 y Fk(\003)1086 4451 y Fn(n)1145 4505 y Fs(W)1223 4517 y Fq(F)1271 4505 y Fv(\()p Fs(\020)g Fv(\))24 b(=)e(\()p Fs(a)1564 4470 y Fl(y)1599 4505 y Fv(\()p Fs(\030)t Fv(\))r(+)r Fs(\030)6 b Fr(\001)p 1839 4437 43 4 v 2 w Fs(\020)j Fv(\))1916 4470 y Fn(n)1961 4505 y Fs(;)34 b(W)2096 4517 y Fq(F)2143 4505 y Fv(\()p Fr(\000)p Fs(\020)6 b Fv(\))p Fs(a)2358 4470 y Fn(n)2404 4505 y Fv(\()p Fs(\030)t Fv(\))p Fs(W)2586 4517 y Fq(F)2634 4505 y Fv(\()p Fs(\020)g Fv(\))24 b(=)e(\()p Fs(a)p Fv(\()p Fs(\030)t Fv(\))r(+)p 3100 4437 41 4 v 2 w Fs(\030)8 b Fr(\001)r Fs(\020)e Fv(\))3243 4470 y Fn(n)3290 4505 y Fs(:)523 4682 y Fv(It)28 b(is)f(often)h(con)n(v)n (enien)n(t)f(to)g(write)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(27)1027 287 y Fs(W)1105 299 y Fq(F)1152 287 y Fv(\()p Fs(\030)t Fv(\))24 b(=)f(e)1405 253 y Fn(a)1441 228 y Fb(y)1472 253 y Fq(\()p Fn(\030)r Fq(\))1561 287 y Fv(e)1598 253 y Fl(\000)p Fn(a)p Fq(\()p Fn(\030)r Fq(\))1774 287 y Fv(e)1811 253 y Fl(\000)1872 231 y Ff(1)p 1872 240 29 3 v 1872 273 a(2)1911 253 y Fl(k)p Fn(\030)r Fl(k)2011 228 y Ff(2)2070 287 y Fv(=)g(e)2195 253 y Fl(\000)p Fn(a)p Fq(\()p Fn(\030)r Fq(\))2371 287 y Fv(e)2408 253 y Fn(a)2444 228 y 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Fr(j)p Fs(a)1502 1190 y Fn(n)1547 1224 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fs(\030)1714 1190 y Fl(0)1739 1224 y Fr(i)23 b Fv(=)g(e)1919 1190 y Fl(\000)1980 1167 y Ff(1)p 1980 1176 V 1980 1210 a(2)2019 1190 y Fl(k)p Fn(\030)2085 1165 y Fb(0)2107 1190 y Fl(k)2141 1165 y Ff(2)2178 1224 y Fv(\()p 2210 1156 41 4 v Fs(\030)g Fr(\001)18 b Fs(\030)2350 1190 y Fl(0)2374 1224 y Fv(\))2406 1190 y Fn(n)2451 1224 y Fs(;)691 b Fv(\(39\))523 1373 y(and)1347 1473 y Fr(h)p Fs(\020)6 b Fr(j)p Fs(W)1522 1485 y Fq(F)1570 1473 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fs(\020)g Fr(i)24 b Fv(=)f(e)1920 1439 y Fl(\000)1982 1416 y Ff(1)p 1982 1425 29 3 v 1982 1459 a(2)2020 1439 y Fl(k)p Fn(\030)r Fl(k)2120 1414 y Ff(2)2157 1473 y Fv(e)2194 1439 y Fq(i2Im)o(\()p 2350 1390 34 3 v Fn(\020)t Fl(\001)p Fn(\030)r Fq(\))2466 1473 y Fs(:)676 b Fv(\(40\))523 1713 y Fm(3.7)41 b(Observ)-5 b(ables)30 b(and)j(observ)-5 b(able)31 b(algebras)523 1896 y Fv(Ph)n(ysically)d(measurable)h(quan)n(tities)g(of)h(a)g(system) f(are,)g(in)i(its)f(classical)e(description,)523 1995 y(represen)n(ted)22 b(b)n(y)i(functions)f(on)h(phase)f(space.)g (Consider)f(\014rst)h(\014nite)i(dimensional)e(sys-)523 2095 y(tems.)34 b(An)g(example,)f(in)h(the)g(case)e(of)i(an)f (oscillator)f(ring,)h(is)g(\\the)g(displacemen)n(t)h(of)523 2195 y(the)29 b(nin)n(th)h(oscillator",)c(represen)n(ted)i(b)n(y)h Fs(q)1906 2207 y Fq(9)1968 2195 y Fv(:)d Fs(X)31 b Fv(=)25 b(\()p Fs(q)s(;)14 b(p)p Fv(\))26 b Fr(2)f(H)i(!)e Fs(q)2738 2207 y Fq(9)2800 2195 y Fr(2)h Fo(R)p Fv(.)j(Some)g(in-)523 2294 y(teresting)23 b(observ)-5 b(ables)21 b(are)h(represen)n(ted)g(b)n (y)h(linear)f(functions)h(\(suc)n(h)h(as)e(p)r(osition)h(and)523 2394 y(momen)n(tum\))d(or)f(b)n(y)h(quadratic)f(functions)h(\(suc)n(h)g (as)f(energy)g(or)g(angular)f(momen)n(tum\).)523 2493 y(More)33 b(generally)-7 b(,)31 b(they)j(ma)n(y)f(b)r(e)g(p)r (olynomial.)g(T)-7 b(o)33 b(discuss)g(the)h(linear)e(functions,)i(it) 523 2593 y(is)d(helpful)i(to)e(notice)g(that)h(the)g Fp(top)l(olo)l(gic)l(al)h Fv(dual)f(space)e(of)i Fr(H)g Fv(can)f(con)n(v)n(enien)n(tly)f(b)r(e)523 2693 y(iden)n(ti\014ed)22 b(with)g Fr(H)h Fv(itself)f(using)f(the)h(symplectic)g(form:)g(to)f (eac)n(h)g Fs(Y)42 b Fr(2)23 b(H)q Fv(,)f(w)n(e)f(asso)r(ciate)523 2792 y(the)28 b(linear)f(map)1503 2892 y Fs(X)i Fr(2)24 b(H)g(7!)f Fs(s)p Fv(\()p Fs(Y)5 b(;)14 b(X)7 b Fv(\))22 b Fr(2)i Fo(R)p Fs(:)523 3041 y Fv(One)j(has,)g(from)h(\(8\),)g(for)f (ev)n(ery)f Fs(Y)1617 3053 y Fq(1)1654 3041 y Fs(;)14 b(Y)1739 3053 y Fq(2)1800 3041 y Fr(2)23 b(H)q Fv(,)1373 3223 y Fr(f)p Fs(s)p Fv(\()p Fs(Y)1534 3235 y Fq(1)1571 3223 y Fs(;)14 b Fr(\001)p Fv(\))p Fs(;)42 b(s)p Fv(\()p Fs(Y)1847 3235 y Fq(2)1884 3223 y Fs(;)14 b Fr(\001)p Fv(\))p Fr(g)23 b Fv(=)g Fs(s)p Fv(\()p Fs(Y)2248 3235 y Fq(1)2285 3223 y Fs(;)14 b(Y)2370 3235 y Fq(2)2408 3223 y Fv(\))p Fs(:)702 b Fv(\(41\))523 3406 y(It)28 b(is)f(then)i(con)n(v)n(enien)n(t)d(to)i(in)n(tro)r(duce)1617 3588 y Fs(V)1665 3600 y Fq(c)1699 3588 y Fv(\()p Fs(Y)19 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Fs(Y)42 b Fv(=)23 b(\()p Fs(a;)14 b(b)p Fv(\))23 b Fr(2)g(H)q Fs(:)452 b Fv(\(43\))523 465 y(The)32 b Fs(V)19 b Fv(\()p Fs(Y)g Fv(\))32 b(are)f(clearly)g(unitary)h(op)r(erators)e(on)i Fs(L)2188 435 y Fq(2)2224 465 y Fv(\()p Fo(R)2316 435 y Fn(n)2362 465 y Fv(\))g(and)g(satisfy)g(the)g(so-called)523 565 y(W)-7 b(eyl)28 b(relations)910 757 y Fs(V)19 b Fv(\()p Fs(Y)1057 769 y Fq(1)1095 757 y Fv(\))p Fs(V)g Fv(\()p Fs(Y)1274 769 y Fq(2)1312 757 y Fv(\))k(=)g(e)1492 723 y Fl(\000)1559 700 y Ff(i)p 1553 709 29 3 v 1553 743 a(2)1592 723 y Fn(s)p Fq(\()p Fn(Y)1688 731 y Ff(1)1720 723 y Fn(;Y)1779 731 y Ff(2)1812 723 y Fq(\))1842 757 y Fs(V)18 b Fv(\()p Fs(Y)1988 769 y Fq(1)2045 757 y Fv(+)g Fs(Y)2176 769 y Fq(2)2213 757 y Fv(\))p Fs(;)180 b Fr(8)28 b Fs(Y)2570 769 y Fq(1)2608 757 y Fs(;)14 b(Y)2693 769 y Fq(2)2753 757 y Fr(2)23 b(H)q Fs(:)523 939 y Fv(In)36 b(a)f(F)-7 b(o)r(c)n(k)36 b(represen)n(tation)e(\(determined)i(b)n(y)f (a)g(c)n(hoice)g(of)h Fs(\012)t 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b(\014nite)h(sums)g(of)f Fs(V)19 b Fv(\()p Fs(Y)g Fv(\).)28 b(One)f(therefore)g(considers)f(the)i(algebra)629 1986 y(CCR)810 1998 y Fq(0)847 1986 y Fv(\()p Fo(R)939 1951 y Fq(2)p Fn(n)1018 1986 y Fv(\))23 b(=)g(span)k Fr(f)p Fs(W)1475 1998 y Fq(F)1522 1986 y Fv(\()p Fs(z)1593 1998 y Fn(\012)1651 1986 y Fv(\()p Fs(Y)19 b Fv(\)\))28 b Fr(j)g Fs(Y)41 b Fr(2)24 b Fo(R)2121 1951 y Fq(2)p Fn(n)2199 1986 y Fr(g)f Fv(=)f(span)27 b Fr(f)p Fs(W)2665 1998 y Fq(F)2712 1986 y Fv(\()p Fs(\030)t Fv(\))i Fr(j)e Fs(\030)h Fr(2)23 b Fo(C)3097 1951 y Fn(n)3142 1986 y Fr(g)p Fs(:)523 2168 y Fv(This)41 b(algebra)f(is)h(irreducible.)g(This) g(means)g(that)h(the)g(only)f(closed)f(subspaces)h(of)523 2268 y Fs(L)580 2238 y Fq(2)617 2268 y Fv(\()p Fo(R)709 2238 y Fn(n)754 2268 y Fv(\))810 2246 y Fr(\030)810 2272 y Fv(=)897 2268 y Fr(F)965 2238 y Fq(+)1020 2268 y Fv(\()p Fo(C)1112 2238 y Fn(n)1158 2268 y Fv(\))27 b(in)n(v)-5 b(arian)n(t)26 b(under)h(the)g(ab)r(o)n(v)n(e)f(algebra)f(are)h(the)i (trivial)e(ones)h(and)523 2367 y(is)37 b(equiv)-5 b(alen)n(t,)36 b(via)g(Sc)n(h)n(ur's)g(Lemma,)h(to)f(the)h(statemen)n(t)g(that)g(the)g (only)f(b)r(ounded)523 2467 y(op)r(erators)e(that)i(comm)n(ute)g(with)g (all)g Fs(F)48 b Fv(in)36 b(the)g(algebra)e(are)h(the)h(m)n(ultiples)g (of)g(the)523 2567 y(iden)n(tit)n(y)-7 b(.)24 b(F)-7 b(or)23 b(a)g(simple)g(pro)r(of)g(of)h(these)f(facts)g(one)h(ma)n(y)e (consult)i([DB1)o(].)g(This)f(implies)523 2666 y(via)33 b(a)g(w)n(ell)g(kno)n(wn)f(result)h(in)g(the)h(theory)f(of)g(v)n(on)f (Neumann)i(algebras)d(\(see)i([BR1)o(],)523 2766 y(for)i(example\))g (that)h(its)f(w)n(eak)f(closure)h(is)g(all)g(of)g Fr(B)s Fv(\()p Fr(F)2309 2736 y Fq(+)2363 2766 y Fv(\()p Fo(C)2455 2736 y Fn(n)2501 2766 y Fv(\)\):)h(in)f(this)h(sense,)f(\\an)n(y)523 2866 y(b)r(ounded)i(op)r(erator)d(on)i(F)-7 b(o)r(c)n(k)36 b(space)g(can)g(b)r(e)g(appro)n(ximated)f(\(in)i(the)g(w)n(eak)e(top)r (ol-)523 2965 y(ogy!\))h(b)n(y)g(a)g(function)h(of)g Fs(Q)f Fv(and)h Fs(P)12 b Fv(.")36 b(This)g(is)h(clearly)e(a)h(w)n(a)n (y)g(of)g(sa)n(ying)f(that)i(the)523 3065 y(original)30 b(algebra)f(is)i(quite)h(large.)d(Note)j(nev)n(ertheless)e(that)h(its)g (op)r(erator)f(norm)g(clo-)523 3164 y(sure)k(\(called)h(the)h (CCR-algebra)c(o)n(v)n(er)i Fo(R)1884 3134 y Fq(2)p Fn(n)1997 3164 y Fv(and)h(denoted)g(b)n(y)f(CCR\()p Fo(R)2882 3134 y Fq(2)p Fn(n)2961 3164 y Fv(\))h(is)g(m)n(uc)n(h)523 3264 y(smaller,)i(since)h(it)g(con)n(tains)f(no)h(compact)f(op)r (erators.)f(F)-7 b(or)37 b(the)i(purp)r(oses)e(of)h(these)523 3364 y(notes,)29 b(I)h(will)g(consider)f(CCR\()p Fo(R)1583 3334 y Fq(2)p Fn(n)1662 3364 y Fv(\)\))h(or)f(CCR)2041 3376 y Fq(0)2078 3364 y Fv(\()p Fo(R)2170 3334 y Fq(2)p Fn(n)2249 3364 y Fv(\))h(as)f(\\the")g(observ)-5 b(able)28 b(algebra)523 3463 y(of)g(the)g(systems)f(considered.)648 3563 y(Remark)k(that)i(these)f(algebras)e(do,)i(as)g(sets,)g(not)h(dep) r(end)g(on)f Fs(\012)t Fv(.)h(But)f(again,)f(in)523 3663 y(close)k(analogy)e(to)j(what)f(w)n(e)g(observ)n(ed)f(in)i(Sect.)g (3.3,)e(giv)n(en)h(an)g(op)r(erator)f(on)h(F)-7 b(o)r(c)n(k)523 3762 y(space)36 b(b)r(elonging)g(to)h(one)f(of)h(these)f(algebras,)f (its)i(expression)e(in)i(terms)g(of)f Fs(Q)h Fv(and)523 3862 y Fs(P)42 b Fv(do)r(es)30 b(dep)r(end)h(on)f Fs(\012)t Fv(,)h(and)f(so)g(do)r(es)g(therefore)f(its)i(ph)n(ysical)e(in)n (terpretation)g(as)h(an)523 3961 y(observ)-5 b(able.)30 b(So)i(it)g(is)g(not)f(only)h(the)g(iden)n(ti\014cation)f(of)h(the)g (appropriate)e(observ)-5 b(able)523 4061 y(algebra)32 b(whic)n(h)i(is)f(imp)r(ortan)n(t,)h(but)g(the)g(lab)r(eling,)g(within) g(this)g(algebra,)e(of)i(the)g(el-)523 4161 y(emen)n(ts)h(that)g (describ)r(e)g(the)g(relev)-5 b(an)n(t)34 b(ph)n(ysical)h(observ)-5 b(ables.)33 b(This)i(will)g(b)r(e)h(crucial)523 4260 y(once)28 b(w)n(e)h(discuss)f(lo)r(cal)h(observ)-5 b(ables)27 b(in)i(Section)g(4,)f(and)h(b)r(ecome)g(hop)r(efully)g(quite)g(a)523 4360 y(bit)f(clearer)e(then)i(to)r(o.)648 4460 y(It)k(is)g(ob)n (viously)f(not)h(of)g(m)n(uc)n(h)g(in)n(terest)g(to)g(discuss)f(observ) -5 b(able)31 b(algebras)f(if)j(one)523 4559 y(is)d(not)h(going)e(to)h (sa)n(y)f(ho)n(w)h(the)h(observ)-5 b(ables)28 b(ev)n(olv)n(e)h(in)i (time.)g(In)f(\014nite)h(dimensional)523 4659 y(systems,)k(one)g(is)g (giv)n(en)g(a)g(Hamiltonian)g Fs(H)7 b Fv(,)35 b(whic)n(h)g(is)h(a)e (self-adjoin)n(t)h(op)r(erator)f(on)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(29)523 282 y Fs(L)580 252 y Fq(2)617 282 y Fv(\()p Fo(R)709 252 y Fn(n)754 282 y Fv(\))824 260 y Fr(\030)824 286 y Fv(=)927 282 y Fr(F)995 252 y Fq(+)1050 282 y Fv(\()p Fo(C)1142 252 y Fn(n)1187 282 y Fv(\).)37 b(It)g(generates)e(the)i(so-called)e(Heisen)n(b)r(erg)g (ev)n(olution)h(of)g(eac)n(h)523 382 y(observ)-5 b(able)28 b Fs(F)12 b Fv(,)29 b(whic)n(h)g(is)g(de\014ned)h(b)n(y)e Fs(\013)1827 394 y Fn(t)1857 382 y Fv(\()p Fs(F)12 b Fv(\))26 b(=)f(e)2139 352 y Fq(ii)p Fn(H)t(t)2264 382 y Fs(F)12 b Fv(e)2366 352 y Fl(\000)p Fq(i)p Fn(H)t(t)2525 382 y Fs(:)29 b Fv(It)h(has)e(to)h(b)r(e)h(c)n(hec)n(k)n(ed)523 482 y(that)d(the)h(algebra)d(of)i(observ)-5 b(ables)25 b(and)i Fs(H)34 b Fv(are)26 b(suc)n(h)g(that)h(this)h(de\014nes)f(an)f (automor-)523 581 y(phism)h(of)g(the)h(algebra)d(\()p Fp(i.e.)k Fv(so)d(that)i(e)1786 551 y Fq(i)p Fn(H)t(t)1892 581 y Fs(F)12 b Fv(e)1994 551 y Fl(\000)p Fq(i)p Fn(H)t(t)2180 581 y Fv(still)27 b(b)r(elongs)g(to)g(the)g(algebra)e(if)j Fs(F)523 681 y Fv(do)r(es\).)648 780 y(That)33 b Fs(\013)914 792 y Fn(t)977 780 y Fv(is)g(an)h(automorphism)e(of)i(the)g(CCR)f (algebra)f(is)h(not)h(true)f(in)h(general.)523 880 y(F)-7 b(or)33 b(example,)g(it)i(is)e(pro)n(v)n(en)f(in)i([FV)q(])g(that,)g (when)g Fs(H)7 b Fv(\()p Fs(\025)p Fv(\))34 b(=)2548 847 y Fq(1)p 2548 861 34 4 v 2548 909 a(2)2591 880 y Fs(P)2656 850 y Fq(2)2716 880 y Fv(+)22 b Fs(\025V)d Fv(,)34 b(with)g Fs(V)53 b Fv(a)523 980 y(b)r(ounded)29 b Fs(L)920 950 y Fq(1)986 980 y Fv(function,)h(then)f(the)h(Heisen)n(b) r(erg)e(ev)n(olution)g(lea)n(v)n(es)f(the)i(CCR)g(algebra)523 1079 y(in)n(v)-5 b(arian)n(t)27 b(for)g(all)g(v)-5 b(alues)28 b(of)g Fs(t)g Fv(and)f(of)h Fs(\025)g Fv(if)h(and)e(only)h(if)g Fs(V)42 b Fv(=)23 b(0.)28 b(In)g(other)f(w)n(ords,)f(the)523 1179 y(CCR)f(algebra)e(cannot)i(p)r(ossibly)f(b)r(e)i(a)e(suitable)h (algebra)e(to)i(describ)r(e)g(most)g(standard)523 1279 y(quan)n(tum)j(mec)n(hanical)e(systems)h(with)h(a)g(\014nite)g(n)n(um)n (b)r(er)f(of)h(degrees)e(of)h(freedom.)648 1378 y(An)21 b(exception)g(to)h(this)f(rule)g(are)g(systems)g(describ)r(ed)g(b)n(y)g (quadratic)f(hamiltonians,)523 1478 y(whic)n(h)36 b(are)e(precisely)h (the)h(ones)f(w)n(e)g(are)g(in)n(terested)g(in)h(here.)f(An)h(easy)f (example)g(is)523 1577 y(pro)n(vided)23 b(b)n(y)h(quadratic)g (Hamiltonians)g(of)g(the)h(t)n(yp)r(e)f Fs(H)30 b Fv(=)2460 1545 y Fq(1)p 2460 1559 V 2460 1606 a(2)2503 1577 y Fs(P)2568 1547 y Fq(2)2617 1577 y Fv(+)2704 1545 y Fq(1)p 2704 1559 V 2704 1606 a(2)2747 1577 y Fs(Q)12 b Fr(\001)g Fs(\012)2928 1547 y Fq(2)2965 1577 y Fs(Q)24 b Fv(in)h(view)523 1677 y(of)1041 1777 y(e)1078 1742 y Fq(id)p Fn(\000)9 b Fq(\()p Fn(\012)s Fq(\))p Fn(t)1318 1777 y Fs(W)1396 1789 y Fq(F)1443 1777 y Fv(\()p Fs(\030)t Fv(\)e)1584 1742 y Fl(\000)p Fq(id)p Fn(\000)g Fq(\()p Fn(\012)s Fq(\))p Fn(t)1900 1777 y Fv(=)22 b Fs(W)2065 1789 y Fq(F)2113 1777 y Fv(\(e)2182 1742 y Fl(\000)p Fq(i)p Fn(\012)s(t)2336 1777 y Fs(\030)t Fv(\))p Fs(;)42 b Fr(8)p Fs(\030)27 b Fr(2)d(K)2725 1742 y Fg(C)2771 1777 y Fs(;)523 1926 y Fv(whic)n(h)c(follo)n(ws)e(immediately)i(from)f(the)i(discussion)d (in)i(Sect.)g(3.4.)f(This)h(clearly)e(implies)523 2026 y(that)j(the)g(dynamics)f(lea)n(v)n(es)e(the)j(CCR)g(algebra)d(CCR\()p Fo(R)2316 1996 y Fq(2)p Fn(n)2395 2026 y Fv(\))j(in)n(v)-5 b(arian)n(t.)19 b(Note)i(that)f(this)523 2125 y(will)29 b(w)n(ork)e(in)i(in\014nite)g(dimensional)f(systems)g(just)h(as)f(w)n (ell)g(as)g(in)h(\014nite)g(dimensional)523 2225 y(ones.)648 2325 y(The)20 b(discussion)f(carries)g(o)n(v)n(er)f(to)i(the)h (in\014nite)g(dimensional)f(case)f(without)i(c)n(hange.)523 2424 y(One)27 b(de\014nes)h(the)g(algebra)e(of)h(observ)-5 b(ables)26 b(in)i(the)g(quan)n(tum)g(theory)e(to)i(b)r(e)693 2607 y(CCR)874 2619 y Fq(0)912 2607 y Fv(\()p Fr(H)q Fv(\))23 b(=)g(span)k Fr(f)p Fs(W)1472 2619 y Fq(F)1519 2607 y Fv(\()p Fs(z)1590 2619 y Fn(\012)1648 2607 y Fv(\()p Fs(Y)19 b Fv(\)\))28 b Fr(j)g Fs(Y)42 b Fr(2)23 b(H)q(g)g Fv(=)f(span)27 b Fr(f)p Fs(W)2595 2619 y Fq(F)2642 2607 y Fv(\()p Fs(\030)t Fv(\))i Fr(j)e Fs(\030)h Fr(2)23 b(K)3031 2573 y Fg(C)3078 2607 y Fr(g)p Fs(:)523 2790 y Fv(Again,)g(this)h(algebra)d(is)i(indep)r(enden)n(t)i(of)e Fs(\012)28 b Fv(and)23 b(turns)g(out)g(to)h(b)r(e)f(irreducible)g([BR1) o(],)523 2889 y(so)g(that)h(its)g(w)n(eak)e(closure)g(is)i(the)g (algebra)e(of)h(all)g(b)r(ounded)h(op)r(erators)e(on)h(F)-7 b(o)r(c)n(k)23 b(space.)523 2989 y(Its)35 b(norm)g(closure,)f(whic)n(h) h(is)g(m)n(uc)n(h)g(smaller,)g(is)g(the)h(so-called)d(CCR-algebra)g(o)n (v)n(er)523 3088 y Fr(H)q Fv(,)i(for)f(whic)n(h)h(I)g(will)g(write)g (CCR\()p Fr(H)q Fv(\).)h(Since)f(w)n(e)f(will)h(only)g(w)n(ork)e(with)j (quadratic)523 3188 y(Hamiltonians,)i(this)h(algebra)e(is)h(adequate)g (for)g(the)h(description)f(of)h(suc)n(h)f(systems)523 3288 y(since)29 b(it)h(is)f(then)g(in)n(v)-5 b(arian)n(t)28 b(under)h(the)h(dynamics.)f(Here)f(also,)g(to)h(no)g(one's)g(surprise) 523 3387 y(b)n(y)34 b(no)n(w,)f(I)h(hop)r(e,)g(the)h(in)n(terpretation) e(of)h(a)f(giv)n(en)g(op)r(erator)g(in)h(the)g(algebra)f(as)g(an)523 3487 y(observ)-5 b(able)26 b Fp(wil)t(l)38 b Fv(dep)r(end)28 b(on)f Fs(\012)t Fv(,)h(as)f(w)n(e)h(will)f(see)h(in)g(more)e(detail)i (in)g(Section)f(4.)648 3587 y(F)-7 b(or)27 b(further)g(reference,)g (let)h(me)f(de\014ne)h(also)f(the)h(algebra)1167 3769 y(CCR)1348 3781 y Fq(0)1385 3769 y Fv(\()p Fr(M)p Fv(\))23 b(=)g(span)k Fr(f)p Fs(W)1974 3781 y Fq(F)2021 3769 y Fv(\()p Fs(z)2092 3781 y Fn(\012)2150 3769 y Fv(\()p Fs(Y)19 b Fv(\)\))p Fr(j)p Fs(Y)42 b Fr(2)24 b(Mg)p Fs(;)523 3952 y Fv(whenev)n(er)e Fr(M)h Fv(is)g(a)g(v)n(ector)f(subspace)g(of)h Fr(H)i Fv(\(ev)n(en)d(if)i Fr(M)f Fv(is)g(not)g(symplectic\).)h(In)f (man)n(y)523 4051 y(situations)29 b(it)g(is)g(natural)f(and)h(elegan)n (t)f(not)h(to)g(w)n(ork)e(with)j(the)f(norm)g(closure)e(of)i(the)523 4151 y(CCR)704 4163 y Fq(0)741 4151 y Fv(\()p Fr(M)p Fv(\),)34 b(but)f(with)h(their)f(w)n(eak)e(closure,)h(for)h(whic)n(h)g (I)g(shall)f(write)h(CCR)3073 4163 y Fq(w)3125 4151 y Fv(\()p Fr(M)p Fv(\).)523 4251 y(F)-7 b(urther)30 b(dev)n(elopmen)n(ts) f(concerning)f(the)j(CCR)f(can)f(b)r(e)i(found)f(in)g(the)g(con)n (tribution)523 4350 y(of)e(J.)f(Derezinski)g(in)h(this)g(v)n(olume)f ([D].)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 523 100 a Fu(30)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Ft(4)42 b(Lo)s(cal)c(observ)-6 b(ables)39 b(and)f(lo)s(cal)g(states)523 482 y Fm(4.1)j(In)m(tro)s (duction)523 664 y Fv(The)30 b(issue)f(of)h(what)f(are)g(lo)r(cal)g (observ)-5 b(ables,)28 b(lo)r(cal)h(states)g(and)h(lo)r(cal)f (measuremen)n(ts)523 764 y(has)22 b(attracted)f(a)h(fair)g(amoun)n(t)g (of)g(atten)n(tion)g(and)g(has)f(generated)g(some)h(surprises)f(and)523 863 y(ev)n(en)32 b(some)f(con)n(tro)n(v)n(ersy)e(in)k(the)g (mathematical)e(ph)n(ysics)h(literature)f(on)h(relativistic)523 963 y(quan)n(tum)22 b(\014eld)g(theory)-7 b(.)21 b(The)h(con)n(tro)n(v) n(ersy)d(has)i(cen)n(tered)g(on)h(the)g(question)f(of)h(particle)523 1063 y(lo)r(calization,)36 b(of)h(p)r(ossible)g(causalit)n(y)f (violations)g(and)h(of)g(relativistic)g(in)n(v)-5 b(ariance.)36 b(I)523 1162 y(will)c(address)f(these)h(issues)f(in)h(the)g(presen)n(t) g(section)f(within)i(the)f(restricted)f(con)n(text)523 1262 y(of)25 b(the)g(free)f(oscillator)f(\014elds)i(under)g(study)f (here,)h(some)f(of)h(whic)n(h)f(are)g(relativistically)523 1362 y(in)n(v)-5 b(arian)n(t,)28 b(while)i(others)e(are)h(not.)g(I)g (will)h(argue)e(that)h(there)g(is)h(not)f(m)n(uc)n(h)g(reason)f(to)523 1461 y(b)r(e)g(surprised)f(and)g(certainly)g(no)g(ground)g(for)g(con)n (tro)n(v)n(ersy)-7 b(.)648 1561 y(After)30 b(de\014ning)f(what)h(is)g (mean)n(t)f(b)n(y)h(a)f(lo)r(cal)g(observ)-5 b(able)28 b(\(Sect.)j(4.2\))e(and)g(giving)523 1660 y(some)f(examples)g(\(Sect.)i (4.3\),)e(the)h(notion)f(of)h(\\strictly)f(lo)r(cal)g(excitation)g(of)h (the)g(v)-5 b(ac-)523 1760 y(uum")23 b(is)g(in)n(tro)r(duced)g(in)g (Sect.)h(4.4.)e(I)i(will)f(then)h(state)e(a)h(generalization)e(of)i(a)g (theorem)523 1860 y(of)i(Knigh)n(t)f(asserting)f(that,)i(if)g Fs(\012)k Fv(is)24 b(a)g(non-lo)r(cal)g(op)r(erator,)f(then)i(states)f (with)h(a)f(\014nite)523 1959 y(n)n(um)n(b)r(er)j(of)g(\014eld)g (excitations)g(cannot)g(b)r(e)g(strictly)g(lo)r(cal)g(excitations)f(of) h(the)h(v)-5 b(acuum)523 2059 y(\(Sect.)26 b(4.5\).)f(It)g(will)h(b)r (e)g(sho)n(wn)e(through)h(examples)f(\(Sect.)i(4.6\))f(that)h(the)f(ab) r(o)n(v)n(e)f(con-)523 2159 y(dition)31 b(on)g Fs(\012)k Fv(is)c(t)n(ypically)g(satis\014ed)f(in)i(mo)r(dels)f(of)g(in)n(terest) f(and)h(I)g(will)h(explain)e(the)523 2258 y(link)g(b)r(et)n(w)n(een)g (the)h(ab)r(o)n(v)n(e)d(notion)i(of)g(lo)r(calized)f(excitation)h(of)g (the)g(v)-5 b(acuum)30 b(and)g(the)523 2358 y(so-called)i (Newton-Wigner)h(lo)r(calization)g(\(Sect.)h(4.7\).)f(It)h(will)g(b)r (e)g(argued)e(that)i(the)523 2457 y(latter)d(is)f(not)h(a)g(suitable)f (notion)h(to)g(discuss)f(the)h(lo)r(cal)f(prop)r(erties)g(of)h(the)g (states)g(of)523 2557 y(oscillator)23 b(\014elds.)j(The)f(actual)g(pro) r(of)f(of)h(Knigh)n(t's)g(theorem)f(is)h(deferred)g(to)g(Sect.)g(4.8.) 523 2798 y Fm(4.2)41 b(De\014nition)30 b(of)i(a)g(lo)s(cal)f(structure) 523 2980 y Fv(Among)g(the)h(in)n(teresting)f(observ)-5 b(ables)30 b(of)h(the)h(oscillator)d(systems)i(w)n(e)g(are)g(studying) 523 3080 y(are)j(certainly)h(the)g(\\lo)r(cal")f(ones.)h(I)g(will)g (giv)n(e)g(a)f(precise)h(de\014nition)g(in)h(a)f(momen)n(t,)523 3180 y(but)25 b(thinking)f(again)f(of)i(the)f(oscillator)f(c)n(hain,)h (\\the)g(displacemen)n(t)g Fs(q)2750 3192 y Fq(7)2811 3180 y Fv(of)h(the)f(sev)n(en)n(th)523 3279 y(oscillator")k(is)h (certainly)g(a)h(\\lo)r(cal")e(observ)-5 b(able.)29 b(In)h(the)g(same)f (w)n(a)n(y)-7 b(,)29 b(if)h(dealing)f(with)523 3379 y(a)e(w)n(a)n(v)n (e)e(equation,)i(\\the)g(v)-5 b(alue)27 b Fs(q)s Fv(\()p Fs(x)p Fv(\))i(of)e(the)h(\014eld)f(at)g Fs(x)h Fv(")f(is)g(a)g(lo)r (cal)g(observ)-5 b(able.)26 b(The)523 3479 y(Hamiltonian)c(is)g(on)f (the)i(other)e(hand)h(not)g(a)f(lo)r(cal)h(observ)-5 b(able,)20 b(since)i(it)h(in)n(v)n(olv)n(es)d(sums)523 3578 y(or)30 b(in)n(tegrals)e(o)n(v)n(er)h(all)h(oscillator)f (displacemen)n(ts)h(and)g(momen)n(ta.)g(Generally)-7 b(,)30 b(\\lo)r(cal)523 3678 y(observ)-5 b(ables")38 b(are)h(functions)h(of)g(the)h(\014elds)f(and)g(conjugate)f(\014elds)h (in)h(a)e(b)r(ounded)523 3778 y(region)30 b(of)g(space.)g(Of)h(course,) f(this)h(notion)f(do)r(es)h(not)g(mak)n(e)f(sense)g(for)g(all)h (harmonic)523 3877 y(systems,)i(de\014ned)g(b)n(y)g(giving)f(a)h(p)r (ositiv)n(e)g(op)r(erator)e Fs(\012)2320 3847 y Fq(2)2391 3877 y Fv(on)h(some)h(abstract)f(Hilb)r(ert)523 3977 y(space)23 b Fr(K)q Fv(.)i(So)e(let)h(me)g(reduce)g(the)g(lev)n(el)f (of)h(abstractness)e(of)i(the)g(discussion,)g(therefore)523 4076 y(hop)r(efully)c(increasing)e(its)h(lev)n(el)g(of)h(p)r (ertinence,)f(and)g(de\014ne)h(what)f(I)h(mean)f(b)n(y)g(a)g(system)523 4176 y(with)28 b(a)f(lo)r(cal)g(structure.)648 4276 y(In)g(view)h(of)g (what)f(precedes,)g(I)h(will)g(limit)h(m)n(y)e(atten)n(tion)h(to)f (free)h(oscillator)e(\014elds)523 4375 y(o)n(v)n(er)d(a)h(real)f(Hilb)r (ert)i(space)f Fr(K)h Fv(of)g(the)f(form)g Fr(K)h Fv(=)e Fs(L)2169 4345 y Fq(2)2169 4398 y Fg(R)2214 4375 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026)p Fv(\),)25 b(where)f Fs(K)30 b Fv(is)24 b(a)g(top)r(olog-)523 4475 y(ical)g(space)g(and)g Fs(\026)h Fv(a)f(Borel)f(measure)g(on)i Fs(K)6 b Fv(.)24 b(Here)g(the)g(subscript)h(\\)p Fo(R)p Fv(")e(indicates)h(that)523 4575 y(w)n(e)h(are)e(dealing)i(with)g(the)g(real)f(Hilb)r(ert)i(space)e (of)h(real-v)-5 b(alued)23 b(functions.)j(In)f(fact,)g(all)523 4674 y(examples)i(I)h(ha)n(v)n(e)e(giv)n(en)h(so)g(far)g(are)f(of)i (the)g(ab)r(o)n(v)n(e)e(t)n(yp)r(e.)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(31)523 282 y Fm(De\014nition)31 b(1.)24 b Fp(A)33 b(lo)l(c)l(al)i(structur)l(e)d(for)i(the)g(oscil)t (lator)i(\014eld)e(determine)l(d)g(by)g Fs(\012)k Fp(and)523 382 y Fr(K)25 b Fv(=)d Fs(L)755 352 y Fq(2)755 405 y Fg(R)801 382 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026)p Fv(\))30 b Fp(is)g(a)g(subsp)l(ac)l(e)g Fr(S)37 b Fp(of)30 b Fr(K)h Fp(with)g(the)f(fol)t(lowing)i(pr)l(op)l(erties:)571 531 y(1.)26 b Fr(S)k(\032)22 b(K)894 546 y Fq(1)p Fn(=)p Fq(2)1017 531 y Fr(\\)d(K)1154 546 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1311 531 y Fp(;)571 731 y(2.)26 b(L)l(et)41 b Fs(B)46 b Fp(b)l(e)41 b(a)h(Bor)l(el)g(subset)f(of)h Fs(K)6 b Fp(,)42 b(then)f Fr(S)2114 743 y Fn(B)2216 731 y Fr(\021)i(S)34 b(\\)27 b Fs(L)2547 700 y Fq(2)2547 754 y Fg(R)2593 731 y Fv(\()p Fs(B)t(;)14 b Fv(d)p Fs(\026)p Fv(\))42 b Fp(is)g(dense)f(in)664 830 y Fs(L)721 800 y Fq(2)721 853 y Fg(R)767 830 y Fv(\()p Fs(B)t(;)14 b Fv(d)p Fs(\026)p Fv(\))p Fp(.)523 988 y Fv(This)26 b(is)f(a)g(prett)n (y)h(strange)e(de\014nition,)i(and)g(I)f(will)h(giv)n(e)f(some)g (examples)g(in)h(a)f(second,)523 1088 y(but)34 b(let)g(me)f(\014rst)g (sho)n(w)g(ho)n(w)g(to)g(use)g(this)h(de\014nition)f(to)h(de\014ne)f (what)h(is)f(mean)n(t)g(b)n(y)523 1187 y(\\lo)r(cal)27 b(observ)-5 b(ables".)25 b(Note)j(that,)g(thanks)f(to)h(the)f(densit)n (y)h(condition)f(ab)r(o)n(v)n(e,)1537 1390 y Fr(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\))1868 1343 y Fq(def)1881 1390 y Fv(=)36 b Fr(S)2032 1402 y Fn(B)2108 1390 y Fr(\002)18 b(S)2241 1402 y Fn(B)523 1573 y Fv(is)30 b(a)f(symplectic)h(subspace)g (of)g Fr(H)g Fv(so)g(that)g(the)g(restriction)f(of)h Fs(W)2645 1585 y Fq(F)2713 1573 y Fr(\016)19 b Fs(z)2813 1585 y Fn(\012)2901 1573 y Fv(to)30 b Fr(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\))523 1673 y(is)28 b(a)f(represen)n(tation)f (of)h(the)h(CCR)g(o)n(v)n(er)e Fr(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\).)523 1830 y Fm(De\014nition)31 b(2.)24 b Fp(L)l(et)41 b Fr(K)j Fv(=)f Fs(L)1495 1800 y Fq(2)1495 1854 y Fg(R)1541 1830 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026)p Fv(\))p Fs(;)g(\012)t(;)g Fr(S)48 b Fp(b)l(e)41 b(as)g(ab)l(ove)i(and)e(let)g Fs(B)k Fp(b)l(e)c(a)h(Bor)l(el)523 1930 y(subset)29 b(of)i Fs(K)6 b Fp(.)29 b(The)i(algebr)l(a)g(of)g(lo)l (c)l(al)g(observables)g(over)g Fs(B)j Fp(is)c(the)g(algebr)l(a)924 2113 y Fv(CCR)1105 2125 y Fq(0)1142 2113 y Fv(\()p Fr(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\)\))24 b(=)f(span)29 b Fr(f)p Fs(W)1941 2125 y Fq(F)1988 2113 y Fv(\()p Fs(z)2059 2125 y Fn(\012)2117 2113 y Fv(\()p Fs(Y)19 b Fv(\)\))30 b Fr(j)g Fs(Y)42 b Fr(2)23 b(S)2581 2125 y Fn(B)2657 2113 y Fr(\002)18 b(S)2790 2125 y Fn(B)2847 2113 y Fr(g)p Fs(:)523 2295 y Fv(Note)26 b(that)h Fs(\012)j Fv(pla)n(ys)c(a)f(role)h (in)g(the)h(de\014nition)f(of)h Fr(S)32 b Fv(through)26 b(the)g(app)r(earance)f(of)h(the)523 2395 y(spaces)31 b Fr(K)845 2407 y Fn(\025)889 2395 y Fv(.)i(The)f(\014rst)g(condition)h (on)f Fr(S)39 b Fv(guaran)n(tees)30 b(that)j Fr(S)28 b(\002)21 b(S)38 b(\032)30 b(H)k Fv(so)e(that,)g(in)523 2495 y(particular,)22 b(for)h(all)g Fs(Y)42 b Fr(2)23 b(S)16 b(\002)10 b(S)c Fv(,)24 b Fs(s)p Fv(\()p Fs(Y)5 b(;)14 b Fr(\001)p Fv(\))23 b(is)g(w)n(ell)g(de\014ned)h(as)f(a)f (function)i(on)f Fr(H)i Fv(whic)n(h)e(is)523 2594 y(imp)r(ortan)n(t)e (for)f(the)h(de\014nition)h(of)f(the)g(lo)r(cal)f(observ)-5 b(ables)20 b(to)h(mak)n(e)f(sense.)g(In)h(practice,)523 2694 y(one)j(w)n(an)n(ts)f(to)h(b)r(e)h(able)f(to)g(use)g(the)h(same)f (spatial)f(structure)h Fr(S)31 b Fv(for)24 b(v)-5 b(arious)23 b(c)n(hoices)g(of)523 2793 y Fs(\012)t Fv(,)j(in)g(order)f(to)h(b)r(e)g (able)g(to)f(compare)g(di\013eren)n(t)h(systems)f(built)i(o)n(v)n(er)d (the)i(same)f(space)523 2893 y Fr(K)g Fv(=)d Fs(L)755 2863 y Fq(2)755 2916 y Fg(R)801 2893 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026)p Fv(\).)25 b(Note)f(nev)n(ertheless)f(that)h(ev)n(en) g(then,)g(the)h(algebras)d(of)i(lo)r(cal)g(and)g(of)523 2993 y(quasi-lo)r(cal)h(observ)-5 b(ables,)25 b(whic)n(h)i(are)e (algebras)g(of)h(b)r(ounded)h(op)r(erators)e(on)h(the)h(F)-7 b(o)r(c)n(k)523 3092 y(space)27 b Fr(F)813 3062 y Fq(+)868 3092 y Fv(\()p Fr(K)964 3062 y Fg(C)1011 3092 y Fv(\))g(do,)h Fp(as)h(sets)p Fv(,)f(dep)r(end)g(on)f Fs(\012)t Fv(.)h(This)f(is)g(in) h(con)n(trast)e(to)h(the)h(algebra)e(of)523 3192 y(\\all")g(observ)-5 b(ables,)693 3375 y(CCR)874 3387 y Fq(0)912 3375 y Fv(\()p Fr(H)q Fv(\))23 b(=)g(span)k Fr(f)p Fs(W)1472 3387 y Fq(F)1519 3375 y Fv(\()p Fs(z)1590 3387 y Fn(\012)1648 3375 y Fv(\()p Fs(Y)19 b Fv(\)\))28 b Fr(j)g Fs(Y)42 b Fr(2)23 b(H)q(g)g Fv(=)f(span)27 b Fr(f)p Fs(W)2595 3387 y Fq(F)2642 3375 y Fv(\()p Fs(\030)t Fv(\))i Fr(j)e Fs(\030)h Fr(2)23 b(K)3031 3340 y Fg(C)3078 3375 y Fr(g)p Fs(;)523 3557 y Fv(whic)n(h)32 b(is,)g(as)f(a)h(set,)g(indep)r(enden)n (t)h(of)f Fs(\012)t Fv(,)g(as)f(p)r(oin)n(ted)i(out)f(b)r(efore.)f(In)i (other)e(w)n(ords,)523 3657 y(some)f(of)h(the)g(ph)n(ysics)f(is)h (hidden)g(in)g(the)g(w)n(a)n(y)f(the)h(lo)r(cal)f(algebras)f(are)h(im)n (b)r(edded)h(in)523 3757 y(the)d(CCR)g(algebra)e(o)n(v)n(er)f Fr(H)q Fv(.)523 3997 y Fm(4.3)41 b(Examples)30 b(of)i(lo)s(cal)f (structures)523 4180 y(Oscillator)g(lattices)g({)h(Klein-Gordon)e (equations)523 4363 y Fv(In)24 b(the)g(case)f(of)h(the)g (translationally)e(in)n(v)-5 b(arian)n(t)23 b(oscillator)f(lattices)h (in)h(dimension)g(2)g(or)523 4462 y(higher)30 b(presen)n(ted)f(in)i (Sect.)g(2.2,)e Fr(S)37 b Fv(can)30 b(b)r(e)h(tak)n(en)e(to)i(b)r(e)f (the)h(space)f(of)g(sequences)g Fs(q)523 4562 y Fv(of)24 b(\014nite)h(supp)r(ort,)g(ev)n(en)f(in)g(the)h(massless)e(case,)h(as)g (is)g(easily)g(c)n(hec)n(k)n(ed.)f(Alternativ)n(ely)-7 b(,)523 4661 y(y)n(ou)23 b(could)h(tak)n(e)f Fr(S)31 b Fv(to)24 b(b)r(e)g(the)g(larger)e(space)i(of)g(sequences)f(of)h(fast) g(decrease.)e(This)i(has)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 523 100 a Fu(32)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(the)28 b(adv)-5 b(an)n(tage)25 b(that)j(then)g Fr(S)c(\002)18 b(S)34 b Fv(is)27 b(dynamics)g(in)n(v)-5 b(arian)n(t.)26 b(Note)h(that)h(in)f (neither)h(of)523 382 y(these)20 b(examples)f Fr(S)9 b(\002)s(S)26 b Fv(is)20 b Fs(J)27 b Fv(in)n(v)-5 b(arian)n(t,)19 b(though,)h(so)f(that)h Fr(S)9 b(\002)s(S)26 b Fv(will)20 b(not)g(b)r(e)g(a)f(complex)523 482 y(v)n(ector)28 b(subspace)h(of)h (\()p Fr(H)q Fs(;)14 b(J)8 b Fv(\),)30 b(just)g(a)g(real)e(one.)i(This) f(is)h(also)e(true)i(for)f Fr(S)2880 494 y Fn(B)2957 482 y Fr(\002)20 b(S)3092 494 y Fn(B)3179 482 y Fv(and)523 581 y(will)27 b(b)r(e)f(crucial)g(when)g(discussing)g(\\lo)r(cal)f (excitations)g(of)i(the)f(v)-5 b(acuum")26 b(in)h(quan)n(tum)523 681 y(\014eld)h(theory)-7 b(.)523 855 y Fm(Exercise)31 b(1.)25 b Fv(Chec)n(k)i(all)g(of)h(the)g(ab)r(o)n(v)n(e)e(statemen)n (ts)h(in)h(detail.)648 1029 y(As)37 b(an)h(example)f(of)g(a)g(lo)r(cal) g(observ)-5 b(able,)37 b(w)n(e)g(ha)n(v)n(e,)f(with)j Fs(\021)j Fr(2)f(S)j Fv(of)37 b(b)r(ounded)523 1129 y(supp)r(ort)27 b(in)h(some)f(set)h Fs(B)f Fr(\032)c Fo(Z)1498 1099 y Fn(d)1537 1129 y Fv(,)1472 1348 y(e)1509 1314 y Fq(i)p Fn(\021)r Fl(\001)p Fn(Q)1663 1348 y Fv(=)g Fs(W)1829 1360 y Fq(F)1876 1348 y Fv(\()1962 1292 y(i)p 1918 1329 111 4 v 1918 1346 a Fr(p)p 1988 1346 42 4 v 1988 1414 a Fv(2)2039 1348 y Fs(\012)2107 1314 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2264 1348 y Fs(\021)s Fv(\))p Fs(:)523 1590 y Fv(V)-7 b(ery)33 b(explicitly)-7 b(,)33 b(one)g(ma)n(y)g(think)g(of)g (taking)g Fs(\021)s Fv(\()p Fs(j)5 b Fv(\))33 b(=)f Fs(\016)2350 1602 y Fn(j;k)2471 1590 y Fv(and)h(then)g(this)h(is)f(e)3126 1560 y Fq(i)p Fn(Q)p Fq(\()p Fn(k)q Fq(\))3290 1590 y Fv(,)523 1690 y(a)c(simple)g(function)h(of)f(the)h(displacemen)n(t)f (of)g(the)h(oscillator)e(at)h(site)g Fs(k)g Fr(2)d Fo(Z)2984 1660 y Fn(d)3023 1690 y Fv(.)k(A)n(t)f(the)523 1790 y(risk)h(of)h(b)r (oring)f(the)h(wits)g(out)g(of)g(y)n(ou,)f(let)h(me)g(p)r(oin)n(t)g (out)g(y)n(et)g(again)f(that)h(this)g Fp(\014xe)l(d)523 1889 y(observable)41 b Fv(is)33 b(represen)n(ted)e(on)h(F)-7 b(o)r(c)n(k)33 b(space)f(b)n(y)g(a)g Fp(di\013er)l(ent)j(op)l(er)l (ator)42 b Fv(for)33 b(di\013eren)n(t)523 1989 y(c)n(hoices)27 b(of)g Fs(\012)t Fv(.)648 2089 y(Similarly)1478 2223 y(e)1515 2188 y Fq(i)p Fn(\021)r Fl(\001)p Fn(P)1668 2223 y Fv(=)c Fs(W)1834 2235 y Fq(F)1881 2223 y Fv(\()p Fr(\000)2022 2167 y Fv(1)p 1988 2204 111 4 v 1988 2220 a Fr(p)p 2057 2220 42 4 v 69 x Fv(2)2109 2223 y Fs(\012)2177 2188 y Fq(1)p Fn(=)p Fq(2)2281 2223 y Fs(\021)s Fv(\))523 2416 y(is)28 b(a)f(function)h(of)f(the)h(momen)n(ta)f(of)h(the)g (oscillators)e(in)h(the)h(supp)r(ort)g(of)f Fs(\021)s Fv(.)648 2515 y(In)f(the)h(one-dimensional)e(translationally)f(in)n(v) -5 b(arian)n(t)25 b(lattice)i(a)f(spatial)f(structure)523 2615 y(do)r(es)c(not)g(exist)g(when)g Fs(\027)29 b Fv(=)22 b(1)p Fs(=)p Fv(2)e(b)r(ecause)h(of)g(the)g(strong)f(infrared)h (singularit)n(y)-7 b(.)19 b(Indeed,)523 2715 y(due)26 b(to)g(the)h(densit)n(y)f(condition)f(in)i(the)f(de\014nition)h(of)f (the)g(lo)r(cal)g(structure,)f(it)i(is)f(clear)523 2814 y(that)j Fr(S)36 b Fv(m)n(ust)29 b(con)n(tain)f(all)h(sequences)f(of)h (\014nite)h(supp)r(ort,)f(and)g(those)f(do)h(not)g(b)r(elong)523 2914 y(to)f Fr(K)688 2929 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)872 2914 y Fv(in)g(dimension)f(1,)g(as)g(w)n(e)g(already)g(p)r(oin)n(ted)g (out)h(in)g(Sect.)g(2.2.)648 3022 y(Similarly)-7 b(,)26 b(the)g(w)n(a)n(v)n(e)f(and)i(Klein-Gordon)d(equations)i(on)g Fo(R)2585 2992 y Fn(d)2650 3022 y Fv(admit)h(for)f(example)523 3122 y Fs(C)582 3134 y Fq(0)620 3122 y Fv(\()p Fo(R)712 3092 y Fn(d)751 3122 y Fv(\))f(or)g(the)g(space)g(of)g(Sc)n(h)n(w)n (artz)f(functions)i(as)f(a)g(spatial)f(structure)h(in)h(dimension)523 3221 y(2)h(or)g(higher,)g(as)g(follo)n(ws)f(from)i(the)g(discussion)e (in)i(Sect.)g(2.2.)523 3462 y Fm(The)k(\014nite)f(dimensional)f(case) 523 3645 y Fv(I)e(\014nd)h(this)f(example)f(p)r(ersonally)g(most)h (instructiv)n(e.)g(It)g(forces)f(one)h(in)n(to)f(an)h(un)n(usual)523 3744 y(p)r(oin)n(t)k(of)h(view)e(on)h(a)g(system)g(of)g Fs(n)g Fv(coupled)g(oscillators)e(that)j(is)f(w)n(ell)g(suited)g(to)g (the)523 3844 y(in\014nite)c(dimensional)f(case.)f(Think)i(therefore)e (of)h(a)g(system)g(of)g Fs(n)g Fv(oscillators)f(c)n(harac-)523 3944 y(terized)i(b)n(y)g(a)g(p)r(ositiv)n(e)g Fs(n)h Fv(b)n(y)f Fs(n)g Fv(matrix)g Fs(\012)1902 3914 y Fq(2)1940 3944 y Fv(,)g(as)g(in)h(Sect.)f(2.2.)g(A)h(lo)r(cal)f(observ)-5 b(able)27 b(of)523 4043 y(suc)n(h)j(a)g(system)g(should)h(b)r(e)g(a)f (function)h(of)f(the)h(p)r(ositions)f(and)g(momen)n(ta)g(of)h(a)f (\014xed)523 4143 y(\014nite)g(set)f(of)g(oscillators.)e(Do)r(es)i(the) h(de\014nition)g(giv)n(en)e(ab)r(o)n(v)n(e)g(correctly)f(incorp)r (orate)523 4243 y(this)h(in)n(tuition?)g(Let's)g(c)n(hec)n(k.)648 4342 y(In)i(this)h(case,)e Fr(K)h Fv(=)d Fo(R)1365 4312 y Fn(n)1410 4342 y Fv(,)k(whic)n(h)f(I)g(view)g(as)g Fs(L)2120 4312 y Fq(2)2120 4365 y Fg(R)2166 4342 y Fv(\()p Fs(K)6 b Fv(\),)30 b(where)g Fs(K)36 b Fv(is)30 b(simply)h(the)g(set) 523 4442 y(of)g Fs(n)f Fv(elemen)n(ts.)h(Indeed,)g Fs(q)g Fr(2)d Fo(R)1574 4412 y Fn(n)1650 4442 y Fv(can)i(b)r(e)h(seen)g(as)e (a)i(function)g Fs(q)g Fv(:)d Fs(j)33 b Fr(2)c(f)p Fv(1)p Fs(;)14 b(:)g(:)g(:)e(n)p Fr(g)28 b(7!)523 4541 y Fs(q)s Fv(\()p Fs(j)5 b Fv(\))43 b Fr(2)f Fo(R)p Fv(,)d(ob)n(viously)f(square) g(in)n(tegrable)g(for)g(the)i(coun)n(ting)e(measure.)g(I)h(already)523 4641 y(explained)22 b(in)g(detail)g(the)h(iden)n(ti\014cation)f(b)r(et) n(w)n(een)g(the)g(quan)n(tum)g(state)g(space)f Fs(L)3106 4611 y Fq(2)3143 4641 y Fv(\()p Fo(R)3235 4611 y Fn(n)3280 4641 y Fv(\))p eop end %%Page: 33 33 TeXDict begin 33 32 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(33)523 282 y Fv(and)30 b Fr(F)755 252 y Fq(+)810 282 y Fv(\()p Fo(C)902 252 y Fn(n)948 282 y Fv(\))h(\(Sect.)g(3.3\).)f(Here)g Fo(C)1700 252 y Fn(n)1776 282 y Fv(is)h(the)g(complexi\014cation)e(of)i Fo(R)2785 252 y Fn(n)2830 282 y Fv(,)g(and)f(as)g(suc)n(h)523 382 y(naturally)d(iden)n(ti\014ed)h(with)g Fs(L)1489 352 y Fq(2)1526 382 y Fv(\()p Fs(K)q(;)14 b Fo(C)p Fv(\).)28 b(So,)f(\014nally)1462 565 y Fs(L)1519 530 y Fq(2)1556 565 y Fv(\()p Fo(R)1648 530 y Fn(n)1693 565 y Fv(\))1748 542 y Fr(\030)1748 569 y Fv(=)1836 565 y Fr(F)1904 530 y Fq(+)1959 565 y Fv(\()p Fs(L)2048 530 y Fq(2)2085 565 y Fv(\()p Fs(K)q(;)14 b Fo(C)p Fv(\)\))p Fs(:)523 747 y Fv(Consider)32 b(no)n(w)h(a)g(subset)g Fs(B)38 b Fv(of)33 b Fs(K)6 b Fv(,)33 b(sa)n(y)f Fs(B)37 b Fv(=)32 b Fr(f)p Fv(1)p Fs(;)14 b Fv(6)p Fs(;)g Fv(9)p Fr(g)p Fv(\()p Fs(n)30 b Fr(\025)j Fv(9\).)g(It)h(is)f(an)g(excellen)n(t)523 847 y(exercise)21 b(to)h(con)n(vince)f(oneself)h(that,)h(unra)n(v)n (eling)d(the)j(v)-5 b(arious)21 b(iden)n(ti\014cations,)g(a)h(lo)r(cal) 523 946 y(observ)-5 b(able)24 b(o)n(v)n(er)f Fs(B)29 b Fv(is)c(a)f(\014nite)i(linear)e(com)n(bination)g(of)h(op)r(erators)e (on)i Fs(L)2874 916 y Fq(2)2911 946 y Fv(\()p Fo(R)3003 916 y Fn(n)3048 946 y Fv(\))g(of)g(the)523 1046 y(form)i(\()p Fs(a)795 1058 y Fn(j)830 1046 y Fs(;)14 b(b)903 1058 y Fn(j)961 1046 y Fr(2)23 b Fo(R)p Fs(;)14 b(j)28 b Fr(2)c Fs(B)t Fv(\):)1396 1331 y(exp)14 b Fr(\000)p Fv(i)1639 1164 y Fk(0)1639 1314 y(@)1715 1252 y(X)1711 1431 y Fn(j)s Fl(2)p Fn(B)1839 1331 y Fv(\()p Fs(a)1915 1343 y Fn(j)1951 1331 y Fs(P)2004 1343 y Fn(j)2057 1331 y Fr(\000)k Fs(b)2176 1343 y Fn(j)2211 1331 y Fs(Q)2277 1343 y Fn(j)2312 1331 y Fv(\))2344 1164 y Fk(1)2344 1314 y(A)2416 1331 y Fs(:)523 1614 y Fv(Better)27 b(y)n(et,)h(if)g(y)n(ou)f(write)g(\(with)i Fs(]B)j Fv(denoting)27 b(the)h(cardinalit)n(y)e(of)i(the)g(set)f Fs(B)t Fv(\))1023 1813 y Fs(L)1080 1779 y Fq(2)1117 1813 y Fv(\()p Fo(R)1209 1779 y Fn(n)1255 1813 y Fv(\))1310 1791 y Fr(\030)1310 1817 y Fv(=)1398 1813 y Fs(L)1455 1779 y Fq(2)1491 1813 y Fv(\()p Fo(R)1583 1779 y Fn(]B)1668 1813 y Fs(;)1716 1734 y Fk(Y)1705 1912 y Fn(j)s Fl(2)p Fn(B)1847 1813 y Fv(d)p Fs(x)1940 1825 y Fn(j)1975 1813 y Fv(\))19 b Fr(\012)f Fs(L)2166 1779 y Fq(2)2203 1813 y Fv(\()p Fo(R)2295 1779 y Fn(n)p Fl(\000)p Fn(]B)2497 1734 y Fk(Y)2486 1913 y Fn(j)s Fl(62)p Fn(B)2628 1813 y Fv(d)p Fs(x)2721 1825 y Fn(j)2757 1813 y Fv(\))p Fs(;)523 2081 y Fv(then)28 b(it)g(is)g(clear)e(that)i(the)g(w)n(eak)f(closure)f (of)i(the)g(ab)r(o)n(v)n(e)e(algebra)g(is)1467 2276 y Fr(B)s Fv(\()p Fs(L)1614 2242 y Fq(2)1650 2276 y Fv(\()p Fo(R)1742 2242 y Fn(]B)1826 2276 y Fs(;)1874 2197 y Fk(Y)1863 2375 y Fn(j)s Fl(2)p Fn(B)2005 2276 y Fv(d)p Fs(x)2098 2288 y Fn(j)2134 2276 y Fv(\)\))19 b Fr(\012)f Fv(1)-18 b(l)o Fs(:)523 2544 y Fv(So,)29 b(indeed,)g(a)g(lo)r(cal)g(observ)-5 b(able)27 b(is)i(clearly)f(one)h(that)g(in)n(v)n(olv)n(es)f(only)g(the) i(degrees)e(of)523 2643 y(freedom)f(indexed)h(b)n(y)f(elemen)n(ts)h(of) f Fs(B)t Fv(.)523 2818 y Fm(Exercise)k(2.)25 b Fv(Con)n(vince)h(y)n (ourself)h(all)g(of)h(this)g(is)f(true.)523 3058 y Fm(Un)m(b)s(ounded) 32 b(lo)s(cal)f(observ)-5 b(ables)523 3241 y Fv(T)e(o)25 b(mak)n(e)f(con)n(tact)g(with)i(the)f(ph)n(ysics)g(literature,)f(it)i (will)f(b)r(e)g(con)n(v)n(enien)n(t)f(on)h(o)r(ccasion)523 3341 y(in)20 b(the)g(follo)n(wing)f(to)h(refer)f(to)h(p)r(olynomials)f (in)2027 3308 y Fq(d)p 2015 3322 62 4 v 2015 3369 a(d)p Fn(t)2087 3341 y Fs(W)12 b Fv(\()p Fs(z)2248 3353 y Fn(\012)2306 3341 y Fv(\()p Fs(tY)18 b Fv(\)\))p Fr(j)2521 3353 y Fn(t)p Fq(=0)2655 3341 y Fv(with)j Fs(Y)41 b Fr(2)24 b(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\))523 3440 y(as)25 b(lo)r(cal)f(observ)-5 b(ables)24 b(o)n(v)n(er)g Fs(B)29 b Fv(as)c(w)n(ell.)g(These)g(are)f(sums)h(of)h(expressions)d(of)i(the)h (form)1209 3623 y Fs(\005)1278 3635 y Fn(S)1326 3623 y Fv(\()p Fs(z)1397 3635 y Fn(\012)1455 3623 y Fv(\()p Fs(Y)1535 3635 y Fq(1)1573 3623 y Fv(\)\))p Fs(\005)1706 3635 y Fn(S)1754 3623 y Fv(\()p Fs(z)1825 3635 y Fn(\012)1883 3623 y Fv(\()p Fs(Y)1963 3635 y Fq(2)2001 3623 y Fv(\)\))14 b Fs(:)g(:)g(:)g(\005)2259 3635 y Fn(S)2307 3623 y Fv(\()p Fs(z)2378 3635 y Fn(\012)2436 3623 y Fv(\()p Fs(Y)2516 3635 y Fn(n)2562 3623 y Fv(\)\))523 3806 y(where)27 b(eac)n(h)f Fs(Y)997 3818 y Fn(j)1055 3806 y Fr(2)e(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\))28 b(Alternativ)n(ely)f(and)g(p)r(erhaps)g(more)f (suggestiv)n(ely)-7 b(,)26 b(these)523 3905 y(are)h(sums)g(of)h (expressions)e(of)h(the)h(form)1070 4088 y(\()p Fs(\021)1143 4100 y Fq(1)1199 4088 y Fr(\001)18 b Fs(Q)p Fv(\))c Fs(:)g(:)g(:)g Fv(\()p Fs(\021)1536 4100 y Fn(m)1618 4088 y Fr(\001)k Fs(Q)p Fv(\))83 b(and)g(\()p Fs(\021)2130 4100 y Fq(1)2186 4088 y Fr(\001)19 b Fs(P)12 b Fv(\))i Fs(:)g(:)g(:)f Fv(\()p Fs(\021)2522 4100 y Fn(m)2604 4088 y Fr(\001)19 b Fs(P)12 b Fv(\))p Fs(;)523 4271 y Fv(or)38 b(of)g(pro)r(ducts)h (thereof,)f(where)g(eac)n(h)g Fs(\021)1900 4283 y Fn(j)1977 4271 y Fr(2)k(S)2124 4283 y Fn(B)2181 4271 y Fv(.)d(Again,)f(for)g (lattices,)h(these)g(are)523 4370 y(p)r(olynomials)h(in)h(the)g(p)r (ositions)f(and)g(momen)n(ta)g(of)h(the)g(individual)g(oscillators)d (in)523 4470 y(some)27 b(subset)h Fs(B)j Fv(of)d(the)g(lattice)g Fo(Z)1627 4440 y Fn(d)1666 4470 y Fv(.)p eop end %%Page: 34 34 TeXDict begin 34 33 bop 523 100 a Fu(34)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fm(4.4)41 b(Strictly)32 b(lo)s(calized)f(v)-5 b(acuum)32 b(excitations)523 457 y Fv(I)26 b(no)n(w)f(w)n(an)n(t)g(to)h(giv)n(e)e(meaning)i(to)f(the)h (notion)g(of)f(\\lo)r(cal)g(excitation)g(of)h(the)g(v)-5 b(acuum")523 556 y(for)35 b(general)g(free)g(oscillator)f(\014elds)i (with)h(a)e(lo)r(cal)g(structure)h Fr(S)6 b Fv(.)36 b(So)g(in)g(this)g (section)523 656 y Fr(K)25 b Fv(=)d Fs(L)755 626 y Fq(2)755 676 y(r)792 656 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026)p Fv(\),)28 b(and)g Fr(S)34 b Fv(satis\014es)27 b(the)h(conditions)f(of)g (De\014nition)h(1.)648 756 y(The)33 b(equiv)-5 b(alen)n(t)32 b(classical)g(notion)g(is)h(readily)f(describ)r(ed)h(and)g(w)n(as)f (already)f(dis-)523 855 y(cussed)37 b(in)h(Sect.)g(2.2.)f(The)g(v)-5 b(acuum,)38 b(b)r(eing)g(the)g(ground)e(state)i(of)f(the)h(system,)g (is)523 955 y(the)33 b(quan)n(tum)g(mec)n(hanical)f(equiv)-5 b(alen)n(t)32 b(of)h(the)g(global)f(equilibrium)h Fs(X)38 b Fv(=)31 b(0,)i(whic)n(h)523 1054 y(b)r(elongs)25 b(of)h(course)f(to)h (the)g(phase)f(space)g Fr(H)q Fv(,)h(and)g(a)f(lo)r(cal)h(p)r (erturbation)f(of)h(this)g(equi-)523 1154 y(librium)k(is)g(an)g (initial)g(condition)g Fs(X)k Fv(=)27 b(\()p Fs(q)s(;)14 b(p)p Fv(\))27 b Fr(2)h(S)e(\002)20 b(S)37 b Fv(with)30 b(the)h(supp)r(ort)f(of)g Fs(q)j Fv(and)523 1254 y(of)26 b Fs(p)g Fv(con)n(tained)f(in)i(a)e(\(t)n(ypically)h(b)r(ounded\))h (subset)f Fs(B)k Fv(of)c Fs(K)6 b Fv(.)26 b(An)g(example)g(of)g(a)f(lo) r(cal)523 1353 y(p)r(erturbation)32 b(of)h(an)g(oscillator)e(lattice)i (is)g(a)f(state)h Fs(X)38 b Fr(2)33 b(H)h Fv(where)e(only)g Fs(q)2996 1365 y Fq(0)3067 1353 y Fv(and)h Fs(p)3276 1365 y Fq(0)523 1453 y Fv(di\013er)23 b(from)f(0.)g(In)g(the)h (classical)e(theory)-7 b(,)22 b(lo)r(cal)g(p)r(erturbations)g(of)g(the) h(equilibrium)g(are)523 1553 y(therefore)29 b(states)h(that)g(di\013er) h(from)f(the)g(equilibrium)g(state)g(only)g(inside)g(a)g(b)r(ounded)523 1652 y(subset)c Fs(B)k Fv(of)25 b Fs(K)6 b Fv(.)26 b(It)g(is)f(this)i (last)e(form)n(ulation)g(that)h(is)f(readily)g(adapted)h(to)f(the)h (quan-)523 1752 y(tum)31 b(con)n(text,)g(through)f(the)h(use)f(of)h (the)g(notion)g(of)f(\\lo)r(cal)g(observ)-5 b(able")29 b(in)n(tro)r(duced)523 1851 y(previously)-7 b(.)648 1951 y(F)g(or)28 b(that)g(purp)r(ose,)g(w)n(e)h(\014rst)f(in)n(tro)r(duce)g (the)h(follo)n(wing)f(notion,)g(whic)n(h)h(is)f(due)h(to)523 2051 y(Knigh)n(t)e([Kn].)523 2200 y Fm(De\014nition)k(3.)24 b Fp(L)l(et)33 b Fs( )s(;)14 b( )1366 2170 y Fl(0)1418 2200 y Fr(2)30 b(F)1571 2170 y Fq(+)1626 2200 y Fv(\()p Fr(K)1722 2170 y Fg(C)1768 2200 y Fv(\))p Fp(.)k(We)f(wil)t(l)i(say)f (that)f Fs( )j Fp(and)d Fs( )2808 2170 y Fl(0)2865 2200 y Fp(ar)l(e)g(indistin-)523 2300 y(guishable)e(inside)g(a)f(Bor)l(el)h (set)f Fs(B)d Fr(\032)22 b Fs(K)36 b Fp(if,)31 b(for)f(al)t(l)h Fs(X)e Fr(2)24 b(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\))p Fp(,)1176 2472 y Fr(h)p Fs( )s Fr(j)p Fs(W)1366 2484 y Fq(F)1414 2472 y Fv(\()p Fs(z)1485 2484 y Fn(\012)1543 2472 y Fv(\()p Fs(X)7 b Fv(\)\))p Fr(j)p Fs( )s Fr(i)24 b Fv(=)e Fr(h)p Fs( )2027 2438 y Fl(0)2051 2472 y Fr(j)p Fs(W)2152 2484 y Fq(F)2199 2472 y Fv(\()p Fs(z)2270 2484 y Fn(\012)2328 2472 y Fv(\()p Fs(X)7 b Fv(\)\))p Fr(j)p Fs( )2580 2438 y Fl(0)2604 2472 y Fr(i)p Fs(:)506 b Fv(\(44\))523 2644 y(Note)26 b(that,)g(giv)n(en)f Fs( )k Fv(and)d Fs(B)t Fv(,)g(it)g(is)g(easy)f(to)g(construct)h(man)n(y)f(states)g(that)i(are) d(lo)r(cally)523 2744 y(indistinguishable)29 b(from)g Fs( )k Fv(in)d Fs(B)t Fv(.)g(Indeed,)f(one)g(ma)n(y)g(consider)g Fs(W)2700 2756 y Fq(F)2747 2744 y Fv(\()p Fs(z)2818 2756 y Fn(\012)2876 2744 y Fv(\()p Fs(X)7 b Fv(\)\))p Fr(j)p Fs( )s Fr(i)p Fv(,)30 b(for)523 2844 y(an)n(y)d Fs(X)i Fr(2)24 b(H)q Fv(\()p Fs(B)1027 2813 y Fn(c)1061 2844 y Fs(;)14 b(\012)t Fv(\).)648 2943 y(W)-7 b(e)26 b(are)e(no)n(w)h (ready)g(to)g(de\014ne)h(what)g(w)n(e)f(mean)h(b)n(y)f(a)g(strictly)g (lo)r(cal)h(excitation)f(of)523 3043 y(the)j(v)-5 b(acuum.)523 3192 y Fm(De\014nition)31 b(4.)24 b Fp(If)36 b Fs(B)k Fp(is)c(a)g(Bor)l(el)h(subset)e(of)h Fs(K)6 b Fp(,)36 b(a)g(strictly)g(lo)l(c)l(al)h(excitation)f(of)g(the)523 3292 y(vacuum)27 b(with)h(supp)l(ort)g(in)g Fs(B)j Fp(is)d(a)g (normalize)l(d)h(ve)l(ctor)f Fs( )e Fr(2)d(F)2535 3262 y Fq(+)2590 3292 y Fv(\()p Fr(K)2686 3262 y Fg(C)2733 3292 y Fv(\))p Fp(,)28 b(di\013er)l(ent)g(fr)l(om)523 3391 y(the)35 b(vacuum)f(itself,)i(which)h(is)e(indistinguishable)i(fr) l(om)e(the)g(vacuum)f(outside)h(of)h Fs(B)t Fp(.)523 3491 y(In)29 b(other)i(wor)l(ds,)1235 3663 y Fr(h)p Fs( )s Fr(j)p Fs(W)1425 3675 y Fq(F)1473 3663 y Fv(\()p Fs(z)1544 3675 y Fn(\012)1602 3663 y Fv(\()p Fs(Y)19 b Fv(\)\))p Fr(j)p Fs( )s Fr(i)24 b Fv(=)f Fr(h)p Fv(0)p Fr(j)p Fs(W)2164 3675 y Fq(F)2211 3663 y Fv(\()p Fs(z)2282 3675 y Fn(\012)2340 3663 y Fv(\()p Fs(Y)c Fv(\)\))p Fr(j)p Fv(0)p Fr(i)565 b Fv(\(45\))523 3836 y Fp(for)31 b(al)t(l)f Fs(Y)42 b Fv(=)23 b(\()p Fs(q)s(;)14 b(p)p Fv(\))23 b Fr(2)h(H)q Fv(\()p Fs(B)1407 3805 y Fn(c)1441 3836 y Fs(;)14 b(\012)t Fv(\))p Fp(.)648 3985 y Fv(F)-7 b(or)37 b(brevit)n(y)-7 b(,)38 b(I)h(will)f(o)r(ccasionally)f(call)h(suc)n(h)g(states)g(\\lo)r (cal)f(states",)g(although)523 4085 y(this)j(terminology)d(conjures)i (up)g(images)g(that)g(are)f(misleading.)h(In)g(view)h(of)f(what)523 4184 y(precedes,)22 b(the)h(coheren)n(t)e(states)h Fs(W)1650 4196 y Fq(F)1698 4184 y Fv(\()p Fs(z)1769 4196 y Fn(\012)1827 4184 y Fv(\()p Fs(X)7 b Fv(\)\))p Fr(j)p Fv(0)p Fr(i)p Fv(,)22 b(for)g(an)n(y)g Fs(X)30 b Fr(2)23 b(H)q Fv(\()p Fs(B)t(;)14 b(\012)t Fv(\))23 b(are)f(strictly)523 4284 y(lo)r(cal)34 b(excitations)g(of)h(the)g(v)-5 b(acuum)35 b(in)g Fs(B)t Fv(.)g(The)g(use)g(of)g(the)g(adjectiv)n(e)f(\\strictly") g(is)523 4384 y(motiv)-5 b(ated)41 b(b)n(y)f(the)h(p)r(ossibilit)n(y)g (of)g(relaxing)e(condition)h(\(45\))h(to)f(allo)n(w)g(for)g(states)523 4483 y(that)e(are)e(only)h(appro)n(ximately)f(lo)r(calized)h(in)h Fs(B)t Fv(,)f(but)i(for)e(whic)n(h)g(the)h(exp)r(ectation)523 4583 y(v)-5 b(alues)27 b(of)h(observ)-5 b(ables)26 b(lo)r(cated)i(far)f (from)g Fs(B)32 b Fv(con)n(v)n(erge)26 b(more)h(or)f(less)i(rapidly)f (to)g(the)523 4682 y(corresp)r(onding)f(v)-5 b(acuum)27 b(exp)r(ectation)h(v)-5 b(alues.)27 b(I)h(refer)f(to)g([DB2])h(for)f (details.)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(35)523 282 y Fm(4.5)41 b(Knigh)m(t's)30 b(theorem)h(revisited)523 458 y Fv(Recall)g(that)h(states)g(with)g(a)f (\014nite)i(n)n(um)n(b)r(er)e(of)h(\014eld)g(quan)n(ta,)f Fp(i.e.)i Fv(states)e(b)r(elonging)523 558 y(to)i Fr(F)698 528 y Fq(\014n)o Fn(;)p Fq(+)847 558 y Fv(\()p Fr(K)943 528 y Fg(C)989 558 y Fv(\),)g(are)f(in)n(terpreted)g(as)g(states)g (describing)g(a)g(\014nite)i(n)n(um)n(b)r(er)e(of)h(quasi-)523 658 y(particles)d(\(see)i(Sect.)f(3.5\).)g(Hence)g(one)g(natural)g (question)g(is)g(whether)g(suc)n(h)g(a)g(state)523 757 y(can)d(b)r(e)g(a)f(strictly)h(lo)r(cal)f(excitation)h(of)g(the)g(v)-5 b(acuum)28 b(in)g(a)f(set)h Fs(B)t Fv(.)g(Theorem)g(1)f(b)r(elo)n(w)523 857 y(giv)n(es)f(a)i(necessary)d(and)j(su\016cien)n(t)g(condition)f (for)g(this)h(to)f(happ)r(en.)648 956 y(First,)g(I)h(need)f(a)h (de\014nition:)523 1108 y Fm(De\014nition)j(5.)24 b Fs(\012)31 b Fp(is)26 b(said)i(to)e(b)l(e)g(str)l(ongly)h(non-lo)l(c)l(al)f(on)h Fs(B)j Fp(if)d(ther)l(e)f(do)l(es)h(not)f(exist)g(a)523 1207 y(non-vanishing)32 b Fs(h)26 b Fr(2)g(K)1277 1222 y Fq(1)p Fn(=)p Fq(2)1413 1207 y Fp(with)31 b(the)h(pr)l(op)l(erty)g (that)f(b)l(oth)h Fs(h)f Fp(and)g Fs(\012)t(h)h Fp(vanish)g(outside)523 1307 y Fs(B)t Fp(.)523 1458 y Fv(Here)27 b(I)h(used)g(the)f(further)h (de\014nition:)523 1609 y Fm(De\014nition)j(6.)24 b Fp(L)l(et)33 b Fs(h)c Fr(2)h(K)1440 1624 y Fl(\006)p Fq(1)p Fn(=)p Fq(2)1629 1609 y Fp(and)k Fs(B)g Fr(\032)29 b Fs(K)6 b Fp(.)33 b(Then)g Fs(h)g Fp(is)h(said)h(to)e(vanish)h(in)f Fs(B)38 b Fp(if)523 1709 y(for)c(al)t(l)g Fs(\021)f Fr(2)c(S)989 1721 y Fn(B)1046 1709 y Fp(,)34 b Fs(\021)24 b Fr(\001)d Fs(h)29 b Fv(=)g(0)p Fp(.)k(Similarly,)i(it)e(is)h(said)g(to)f(vanish)h (outside)g Fs(B)t Fp(,)g(if)g(for)f(al)t(l)523 1808 y Fs(\021)26 b Fr(2)e(S)719 1820 y Fn(B)772 1804 y Ff(c)806 1808 y Fp(,)30 b Fs(\021)21 b Fr(\001)e Fs(h)k Fv(=)g(0)p Fp(.)523 1959 y Fv(Note)30 b(that)g(this)g(de\014nition)g(uses)f(the)h (densit)n(y)g(of)f Fs(S)2202 1971 y Fn(B)2289 1959 y Fv(in)h Fs(L)2445 1929 y Fq(2)2482 1959 y Fv(\()p Fs(B)t Fv(\))h(implicitly)-7 b(,)30 b(b)r(ecause)523 2059 y(without)h(this)g (prop)r(ert)n(y)-7 b(,)30 b(it)h(w)n(ould)f(not)h(mak)n(e)e(m)n(uc)n(h) i(sense.)f(In)n(tuitiv)n(ely)-7 b(,)31 b(a)f(strongly)523 2159 y(non-lo)r(cal)i(op)r(erator)f(is)h(one)h(that)g(do)r(es)f(not)h (lea)n(v)n(e)e(the)i(supp)r(ort)g(of)f(an)n(y)g(function)i Fs(h)523 2258 y Fv(in)n(v)-5 b(arian)n(t.)523 2424 y Fm(Theorem)31 b(1.)25 b Fp(L)l(et)f Fs(B)30 b Fp(b)l(e)25 b(a)h(Bor)l(el)g(subset)e(of)i Fs(K)6 b Fp(.)25 b(Then)h(the)g(fol)t (lowing)h(ar)l(e)f(e)l(quivalent:)648 2524 y(\(i\))k Fs(\012)k Fp(is)c(str)l(ongly)g(non-lo)l(c)l(al)g(on)g Fs(B)t Fp(;)648 2623 y(\(ii\))25 b(Ther)l(e)i(do)e Fv(not)g Fp(exist)g(states)g(in)g Fr(F)1863 2593 y Fn(f)7 b(in)q(;)p Fq(+)2041 2623 y Fv(\()p Fr(K)2137 2593 y Fg(C)2184 2623 y Fv(\))26 b Fp(which)g(ar)l(e)g(strictly)f(strictly)h(lo)l(c)l(al)523 2723 y(excitations)k(of)h(the)f(vacuum)f(with)i(supp)l(ort)e(in)h Fs(B)d Fr(\032)c Fs(K)6 b Fp(;)648 2889 y Fv(I)27 b(will)h(giv)n(e)f (the)h(pro)r(of)f(of)g(this)h(result)f(in)h(Sect.)g(4.8.)648 2989 y(Statemen)n(t)23 b(\(i\))h(of)f(the)g(theorem)f(giv)n(es)g(a)h (more)f(or)g(less)h(easily)f(c)n(hec)n(k)n(ed)g(neccessary)523 3088 y(and)32 b(su\016cien)n(t)h(condition)f(for)g(the)h(non-existence) f(of)h(states)f(with)h(a)f(\014nite)h(n)n(um)n(b)r(er)523 3188 y(of)e(\014eld)h(quan)n(ta)f(that)g(are)g(lo)r(calized)g(in)g(a)g (region)f Fs(B)t Fv(.)i(I)g(will)f(sho)n(w)g(in)h(the)f(examples)523 3288 y(dev)n(elop)r(ed)g(in)g(the)h(follo)n(wing)e(sections)g(that)h (this)h(condition)f(is)g(so)f(to)h(sp)r(eak)g(alw)n(a)n(ys)523 3387 y(satis\014ed)24 b(when)g Fs(B)29 b Fv(is)24 b(a)g(b)r(ounded)h (set:)f(I)g(mean,)h(it)f(is)h(satis\014ed)e(in)i(the)g(v)-5 b(arious)23 b(mo)r(dels)523 3487 y(that)30 b(are)g(t)n(ypically)f (studied)i(in)f(solid)g(state)g(ph)n(ysics,)f(in)i(relativistic)e(quan) n(tum)h(\014eld)523 3587 y(theory)-7 b(,)25 b(or)g(in)h(the)g(theory)f (of)h(free)g(quan)n(tum)f(\014elds)h(on)g(curv)n(ed)e(space-times.)h (Indeed,)523 3686 y(in)c(these)g(examples,)f Fs(\012)1261 3656 y Fq(2)1319 3686 y Fv(is)h(a)f(\014nite)i(di\013erence)f(or)e (\(second)i(order)e(elliptic\))j(di\013eren)n(tial)523 3786 y(op)r(erator,)28 b(so)h(that)i(it)f(is)g(lo)r(cal:)f(it)h (preserv)n(es)e(the)i(supp)r(ort.)g(But)g(its)g(p)r(ositiv)n(e)g (square)523 3885 y(ro)r(ot,)35 b Fs(\012)t Fv(,)h(is)g(more)f(lik)n(e)g (a)g(pseudo-di\013eren)n(tial)g(op)r(erator,)f(and)h(therefore)g(do)r (es)h(not)523 3985 y(preserv)n(e)30 b(supp)r(orts.)i(This)h(will)f(b)r (e)h(sho)n(wn)e(in)i(sev)n(eral)d(cases)h(b)r(elo)n(w.)h(The)h(upshot)f (is)523 4085 y(that)22 b(states)g(with)h(a)e(\014nite)i(n)n(um)n(b)r (er)f(of)g(particles,)f(and)h(a)f(fortiori,)g(one-particle)g(states,) 523 4184 y(are)29 b(nev)n(er)g(strictly)h(lo)r(calized)g(in)g(a)g(b)r (ounded)h(set)f Fs(B)t Fv(.)g(This)h(giv)n(es)e(a)g(precise)h(sense)f (in)523 4284 y(whic)n(h)41 b(the)g(elemen)n(tary)f(excitations)g(of)h (the)g(v)-5 b(acuum)41 b(in)g(a)f(b)r(osonic)g(\014eld)h(theory)523 4384 y(\(relativistic)26 b(or)g(not\))h(di\013er)g(from)f(the)h (ordinary)e(p)r(oin)n(t)i(particles)f(of)h(non-relativistic)523 4483 y(mec)n(hanics:)i(their)h(Hilb)r(ert)g(space)f(of)h(states)f(con)n (tains)g(no)g(states)g(in)h(whic)n(h)g(they)g(are)523 4583 y(p)r(erfectly)e(lo)r(calized.)648 4682 y(So,)f(to)g(sum)h(it)g (all)f(up,)h(y)n(ou)f(could)h(put)g(it)g(this)g(w)n(a)n(y)-7 b(.)26 b(T)-7 b(o)28 b(the)f(question)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 523 100 a Fu(36)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)664 282 y Fv(Wh)n(y)28 b(is)f(there)h(no)f (sharp)g(p)r(osition)g(observ)-5 b(able)26 b(for)h(particles?)523 431 y(the)h(answ)n(er)e(is)664 580 y(It)i(is)g(the)g(non-lo)r(calit)n (y)e(of)h Fs(\012)t Fv(,)h(stupid!)648 729 y(Should)36 b(all)f(this)h(mak)n(e)g(y)n(ou)f(feel)h(uncomfortable,)f(I)h(hop)r(e)g (the)g(further)g(discus-)523 829 y(sion)f(in)h(Sects.)f(4.7)f(of)i(the) g(history)e(of)h(the)h(quest)f(for)g(a)g(\\p)r(osition)g(observ)-5 b(able")33 b(in)523 929 y(relativistic)27 b(\014eld)h(theory)f(will)g (b)r(e)h(of)g(some)f(help.)523 1169 y Fm(4.6)41 b(Examples)523 1351 y Fv(As)28 b(a)f(w)n(arm-up,)f(here)i(is)f(m)n(y)g(fa)n(v)n (ourite)g(example.)523 1524 y Fm(Exercise)k(3.)25 b Fv(Let)g Fr(K)f Fv(=)f Fo(R)1382 1494 y Fq(2)1444 1524 y Fv(so)i(that)g Fs(\012)1789 1494 y Fq(2)1852 1524 y Fv(is)f(a)h(t)n(w)n(o)f(b)n(y)h(t) n(w)n(o)f(matrix)h(and,)g(as)f(explained)523 1624 y(in)f(Sect.)g(4.3,)e Fs(K)29 b Fv(=)22 b Fr(f)p Fv(1)p Fs(;)14 b Fv(2)p Fr(g)p Fv(.)21 b(Sho)n(w)h(that)h(in)g(this)f(case,)g(a)g(state)g(with)h(a)f (\014nite)h(n)n(um)n(b)r(er)g(of)523 1723 y(quan)n(ta)28 b(can)h(b)r(e)h(a)f(strictly)g(lo)r(calized)g(excitation)f(of)i(the)f (v)-5 b(acuum)30 b(on)f Fs(B)h Fv(=)25 b Fr(f)p Fv(1)p Fr(g)j Fv(only)523 1823 y(if)j Fs(\012)670 1793 y Fq(2)737 1823 y Fv(is)f(diagonal.)f(In)h(other)f(w)n(ords,)g(this)h(can)g(happ)r (en)g(only)g(if)g(the)g(t)n(w)n(o)g(oscillators)523 1923 y(are)d(not)g(coupled.)648 2096 y(F)-7 b(or)23 b(t)n(ypical)g (translationally)f(in)n(v)-5 b(arian)n(t)23 b(systems,)g(it)h(is)g (easy)f(to)h(see)f Fs(\012)28 b Fv(is)c(strongly)523 2196 y(non-lo)r(cal)36 b(o)n(v)n(er)f(b)r(ounded)i(sets,)f(so)g(that)h (w)n(e)g(can)f(conclude)g(there)h(are)e(no)i(strictly)523 2295 y(lo)r(calized)27 b(\014nite)h(particle)f(states.)g(This)h(is)g (the)f(con)n(ten)n(t)h(of)f(the)h(follo)n(wing)f(results.)523 2469 y Fm(Theorem)k(2.)25 b Fp(L)l(et)f Fr(K)g Fv(=)f Fs(L)1398 2438 y Fq(2)1398 2492 y Fg(R)1443 2469 y Fv(\()p Fo(R)1535 2438 y Fn(d)1574 2469 y Fs(;)14 b Fv(d)p Fs(x)p Fv(\))26 b Fp(and)f(let)f Fs(!)j Fp(b)l(e)e(a)g(p)l(ositive)h(function) e(b)l(elonging)i(to)523 2568 y Fs(L)580 2538 y Fl(1)580 2592 y Fq(lo)r(c)667 2568 y Fv(\()p Fo(R)759 2538 y Fn(d)798 2568 y Fs(;)14 b Fv(d)p Fs(k)s Fv(\))30 b Fp(with)g Fs(!)1224 2538 y Fl(\000)p Fq(1)1336 2568 y Fr(2)23 b Fs(L)1471 2538 y Fq(1)1471 2592 y(lo)r(c)1558 2568 y Fv(\()p Fo(R)1650 2538 y Fn(d)1689 2568 y Fs(;)14 b Fv(d)p Fs(k)s Fv(\))p Fp(.)31 b(Supp)l(ose)f(b)l(oth)g Fs(!)i Fp(and)f Fs(!)2699 2538 y Fl(\000)p Fq(1)2817 2568 y Fp(ar)l(e)f(p)l(olynomi-)523 2668 y(al)t(ly)g(b)l(ounde)l(d)f(at)g(in\014nity.)f(L)l(et)g Fs(\012)g Fv(=)22 b Fs(!)s Fv(\()p Fr(jrj)p Fv(\))p Fp(.)30 b(Then)f Fr(S)h Fv(=)22 b Fr(S)6 b Fv(\()p Fo(R)2547 2638 y Fn(d)2587 2668 y Fv(\))29 b Fp(is)g(a)g(lo)l(c)l(al)g(structur)l (e)523 2767 y(for)c(this)g(system.)g(If)g Fs(!)i Fp(do)l(es)f(not)e (extend)g(to)g(a)h(holomorphic)j(function)d(on)f(the)h(c)l(omplex)523 2867 y(plane,)k(then)e Fs(\012)k Fp(is)d(str)l(ongly)f(non-lo)l(c)l(al) g(on)h(any)f(b)l(ounde)l(d)h(op)l(en)f(set)g Fs(B)t Fp(.)h(Conse)l (quently,)523 2967 y(ther)l(e)36 b(exist)f(no)h(states)g(with)g(a)g (\014nite)g(numb)l(er)f(of)h(quasi-p)l(articles)h(that)f(ar)l(e)g (strictly)523 3066 y(lo)l(c)l(alize)l(d)c(excitations)e(of)g(the)g (vacuum)g(in)g(such)f(a)i(set)e Fs(B)t Fp(.)648 3240 y Fv(The)k(pro)r(of)g(is)h(a)f(simple)h(application)f(of)h(the)g(P)n (aley-Wiener)d(theorem)j(together)523 3339 y(with)22 b(Theorem)f(1.)g(Note)g(that)h(the)g(theorem)f(applies)g(to)g(the)h (Klein-Gordon)e(equation:)523 3439 y(so)f(w)n(e)h(reco)n(v)n(er)e(in)i (this)g(w)n(a)n(y)f(Knigh)n(t's)g(original)g(result.)g(Pushing)h(the)g (use)g(of)g(the)g(P)n(aley-)523 3539 y(Wiener)28 b(theorem)f(a)g (little)h(further,)g(one)f(can)g(also)g(pro)n(v)n(e:)523 3712 y Fm(Theorem)k(3.)25 b Fp(L)l(et)j Fr(K)d Fv(=)d Fs(L)1402 3682 y Fq(2)1402 3735 y Fg(R)1448 3712 y Fv(\()p Fo(R)1540 3682 y Fn(d)1579 3712 y Fs(;)14 b Fv(d)p Fs(x)p Fv(\))30 b Fp(and)f Fs(\012)1999 3682 y Fq(2)2060 3712 y Fv(=)22 b Fr(\000)p Fs(\001)17 b Fv(+)f Fs(m)2452 3682 y Fq(2)2489 3712 y Fp(,)30 b(with)f Fs(d)24 b Fr(\025)e Fv(1)p Fs(;)14 b(m)23 b(>)f Fv(0)p Fp(,)29 b(or)523 3812 y Fs(d)e Fr(\025)f Fv(2)p Fs(;)14 b(m)26 b Fr(\025)h Fv(0)p Fp(.)k(Then)i Fr(S)g Fv(=)26 b Fr(S)6 b Fv(\()p Fo(R)1593 3782 y Fn(d)1633 3812 y Fv(\))32 b Fp(is)g(a)g(lo)l(c)l(al)h (structur)l(e)d(for)i(this)h(system)e(and)h(ther)l(e)523 3911 y(exist)26 b(no)g(states)g(with)h(a)f(\014nite)g(numb)l(er)f(of)i (quasi-p)l(articles)h(that)e(ar)l(e)g(strictly)h(lo)l(c)l(alize)l(d)523 4011 y(excitations)j(of)h(the)f(vacuum)f(in)h(any)g(set)g Fs(B)j Fp(with)e(non-empty)e(op)l(en)i(c)l(omplement.)648 4184 y Fv(The)26 b(result)g(one)g(needs)h(here)f(is)g(pro)n(v)n(en)f (in)i([SeGo]:)f(for)g Fs(\012)i Fv(=)2640 4115 y Fr(p)p 2709 4115 346 4 v 69 x(\000)p Fs(\001)19 b Fv(+)f Fs(m)3018 4160 y Fq(2)3055 4184 y Fv(,)26 b Fs(h)h Fv(and)523 4284 y Fs(\012)t(h)f Fv(cannot)e(b)r(oth)i(v)-5 b(anish)25 b(on)g(the)h(same)f(op)r(en)g(set.)h(Via)f(Theorem)f(1)h(this)h (implies)f(the)523 4384 y(ab)r(o)n(v)n(e)h(result.)648 4483 y(An)40 b(analogous)e(result)h(holds)h(for)f(the)h (translationally)e(in)n(v)-5 b(arian)n(t)39 b(lattices)h(dis-)523 4583 y(cussed)23 b(in)g(Sect.)h(2.2.)e(In)h(particular,)f(with)i Fs(\012)1975 4553 y Fq(2)2036 4583 y Fv(as)e(in)i(\(10\),)f(it)g(is)g (v)n(ery)f(easy)h(to)g(see)f(that)523 4682 y(there)j(are)f(no)h(states) g(with)h(a)f(\014nite)g(n)n(um)n(b)r(er)g(of)h(quan)n(ta)e(that)i(are)e (p)r(erfectly)h(lo)r(calized)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(37)523 282 y Fv(p)r(erturbations)32 b(of)g(the)h(v)-5 b(acuum)33 b(on)f(a)g(\014nite)h(n)n(um)n(b)r(er)f (of)h(lattice)f(sites.)h(The)f(spatial)523 382 y(structure)23 b(is)g(giv)n(en)g(here)g(b)n(y)g(the)h(sequences)e(of)i(\014nite)g (supp)r(ort,)f(as)g(discussed)g(in)g(Sect.)523 482 y(4.3.)648 581 y(Similarly)-7 b(,)41 b(for)f(the)i(w)n(a)n(v)n(e)e(and)h (Klein-Gordon)f(equations)h(the)g(op)r(erator)f Fs(\012)46 b Fv(is)523 681 y(t)n(ypically)27 b(also)g(strictly)g(non-lo)r(cal,)f (but)j(I)e(will)h(not)g(go)e(in)n(to)i(this)g(here.)648 780 y(It)g(is)g(clear)g(from)g(these)g(examples)g(that)g(Knigh)n(t's)g (theorem)f(has)h(less)g(to)g(do)g(with)523 880 y(relativit)n(y)g(than)h (with)h(coupled)f(oscillators,)e(whic)n(h)i(is)g(the)h(p)r(oin)n(t)f(I) g(w)n(an)n(ted)g(to)f(mak)n(e)523 980 y(all)f(along.)523 1220 y Fm(4.7)41 b(Newton-Wigner)30 b(lo)s(calization)523 1403 y Fv(Knigh)n(t's)21 b(result)h(app)r(ears)e(coun)n(terin)n(tuitiv) n(e.)h(Indeed,)h(w)n(e)g(argued)e(\014rst)i(that)g(the)g(F)-7 b(o)r(c)n(k)523 1503 y(space)19 b(structure)g(of)h(the)g(Hilb)r(ert)g (space)f(of)h(states)f(of)g(the)i(\014eld)f(in)n(vites)f(a)g(particle)g (in)n(ter-)523 1602 y(pretation)g(\(Sect.)i(3.5\),)e(w)n(e)g(then)i(in) n(tro)r(duced)e(what)h(lo)r(oks)f(lik)n(e)g(a)h(p)r(erfectly)g (reasonable)523 1702 y(notion)25 b(of)g(\\strictly)f(lo)r(calized)g (excitation)h(of)f(the)i(v)-5 b(acuum",)24 b(only)h(to)g(end)g(up)g (disco)n(v-)523 1802 y(ering)f(that)g(states)g(with)h(a)f(\014nite)h(n) n(um)n(b)r(er)f(of)h(particles)e(cannot)h(b)r(e)h(strictly)f(lo)r (calized.)523 1901 y(Since)29 b(the)g(notion)g(of)g(a)f(particle)g(ev)n (ok)n(es)f(an)i(en)n(tit)n(y)g(that)g(is)g(lo)r(calized)f(in)h(space,)f (this)523 2001 y(ma)n(y)21 b(seem)h(parado)n(xical.)d(My)j(p)r(oin)n(t) g(of)f(view)h(is)g(simple:)g(the)g(w)n(a)n(y)e(out)i(of)g(this)g (parado)n(x)523 2100 y(is,)g(as)f(I)h(ha)n(v)n(e)f(suggested)g(b)r (efore)g(\(Sect.)i(3.5\),)e(that)h(one)g(has)f(to)h(k)n(eep)f(in)i (mind)f(that)g(the)523 2200 y(particles)31 b(under)h(discussion)f(here) h(are)f(just)h Fp(excite)l(d)i(states)g(of)g(an)g(extende)l(d)g(system) 523 2300 y Fv(and)25 b(that,)h(just)h(lik)n(e)e(in)h(an)f(oscillator)f (ring,)h(c)n(hain,)g(or)g(lattice,)g(the)h(analogy)e(with)i(the)523 2399 y(p)r(oin)n(t)31 b(particles)g(of)g(elemen)n(tary)f(classical)g (or)g(quan)n(tum)h(mec)n(hanics)f(courses)g(should)523 2499 y(not)k(b)r(e)g(pushed)g(to)r(o)f(far.)h(Calling)f(those)g (excitations)g(particles)g(amoun)n(ts)g(to)h(noth-)523 2599 y(ing)28 b(more)g(than)h(an)f(o)r(ccasionally)f(confusing)h(abuse) h(of)f(language.)f(The)i(lesson)f(to)g(b)r(e)523 2698 y(learned)22 b(from)h(Knigh)n(t's)g(result)g(is)g(therefore)f(that)i (suc)n(h)e(\014eld)i(quan)n(ta)e(ma)n(y)h(carry)e(mo-)523 2798 y(men)n(tum)27 b(and)g(energy)-7 b(,)25 b(but)i(they)g(cannot)f(b) r(e)h(p)r(erfectly)g(lo)r(calized.)f(View)n(ed)h(from)f(the)523 2897 y(angle)g(I)i(ha)n(v)n(e)e(c)n(hosen,)g(this)i(is)f(not)g(ev)n(en) g(surprising.)f(The)h(examples)f(sho)n(w)n(ed)g(indeed)523 2997 y(this)31 b(statemen)n(t)h(is)e(true)h(in)h(a)f(system)f(with)i(t) n(w)n(o)e(oscillators,)f(and)i(in)h(oscillator)d(lat-)523 3097 y(tices.)d(One)f(should)h(in)g(particular)e(not)i(hop)r(e)f(to)h (asso)r(ciate)e(a)i(p)r(osition)f(op)r(erator)f(with)523 3196 y(those)j(quan)n(ta,)g(ha)n(ving)g(all)g(the)h(usual)g(prop)r (erties)e(familiar)h(from)h(the)g(description)f(of)523 3296 y(p)r(oin)n(t)h(particles)f(in)g(ordinary)f(Sc)n(hr\177)-42 b(odinger)26 b(quan)n(tum)i(mec)n(hanics.)648 3396 y(I)f(could)g(end)h (the)f(story)f(there.)i(But)f(a)g(v)n(ery)f(di\013eren)n(t)i(p)r(oin)n (t)f(of)g(view,)h(based)e(pre-)523 3495 y(cisely)c(on)h(the)g(use)g(of) g(a)f(p)r(osition)h(op)r(erator)e(\(the)i(so-called)f(Newton-Wigner)g (p)r(osition)523 3595 y(op)r(erator\))j(to)h(lo)r(cate)g(the)h (particles,)f(w)n(as)f(dev)n(elop)r(ed)h(w)n(ell)g(b)r(efore)h(Knigh)n (t's)e(w)n(ork,)g(in)523 3694 y(the)40 b(con)n(text)g(of)g(\(free\))g (relativistic)g(quan)n(tum)f(\014eld)i(theory)e(of)h(whic)n(h)g(the)g (Klein-)523 3794 y(Gordon)27 b(\014eld)i(is)f(a)g(particular)f (example.)h(Since)g(this)h(alternativ)n(e)e(p)r(oin)n(t)h(of)g(view)g (has)523 3894 y(met)k(with)f(a)g(certain)f(amoun)n(t)h(of)g(p)r (opularit)n(y)-7 b(,)30 b(it)h(cannot)g(b)r(e)g(dismissed)g(to)r(o)g (ligh)n(tly)-7 b(.)523 3993 y(Belo)n(w)26 b(I)h(will)g(explain)f(it)i (has)e(an)g(ob)n(vious)g(analog)f(for)h(the)h(oscillator)e(systems)i (under)523 4093 y(study)f(here)g(and)f(I)h(will)h(sho)n(w)e(wh)n(y)-7 b(,)26 b(although)f(it)h(seems)g(at)g(\014rst)f(sigh)n(t)h(p)r (erfectly)g(nat-)523 4193 y(ural,)i(it)i(is)e(clearly)g(ill-conceiv)n (ed.)g(The)h(implication)g(of)g(this)g(remark)e(for)h(the)i(debate)523 4292 y(ab)r(out)d(supp)r(osed)f(causalit)n(y)g(problems)g(in)h (relativistic)f(quan)n(tum)g(\014eld)i(theory)d(and)i(a)523 4392 y(further)d(o)n(v)n(erview)f(of)h(some)g(other)g(issues)g(related) g(to)g(\\particle)g(lo)r(calization")f(in)h(that)523 4491 y(con)n(text)j(will)h(also)f(b)r(e)h(giv)n(en.)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 523 100 a Fu(38)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fm(Newton-Wigner)30 b(lo)s(calization:)h(the)g(de\014nition)523 461 y Fv(Let)39 b(us)h(therefore)e(turn)h(again)f(to)h(an)g(oscillator)f(\014eld)h (with)h(spatial)f(structure)f(so)523 561 y(that)32 b Fr(K)771 531 y Fg(C)847 561 y Fv(=)d Fs(L)998 531 y Fq(2)1035 561 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026;)g Fo(C)p Fv(\).)32 b(The)f(state)h(space)e(of)i(this)g(system)f(is)g(the)h(b)r (osonic)f(F)-7 b(o)r(c)n(k)523 660 y(space)24 b Fr(F)810 630 y Fq(+)865 660 y Fv(\()p Fr(K)961 630 y Fg(C)1008 660 y Fv(\))h(of)g(whic)n(h)g Fr(K)1456 630 y Fg(C)1526 660 y Fv(=)d Fs(L)1670 630 y Fq(2)1707 660 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026;)g Fo(C)p Fv(\))25 b(represen)n(ts)f(the)h (one-particle)f(sector.)523 760 y(No)n(w,)35 b(if)h(the)g(system)f(is)h (in)f(the)h(state)f Fs( )40 b Fr(2)c(K)2068 730 y Fg(C)2151 760 y Fr(\032)g(F)2320 730 y Fq(+)2375 760 y Fv(\()p Fr(K)2471 730 y Fg(C)2518 760 y Fv(\),)g(it)f(is)h(in)g(view)f(of)g (the)523 860 y(particle)26 b(in)n(terpretation)g(of)h(the)g(\014eld)h (states)e(explained)h(in)g(Sect.)g(3.5)f(v)n(ery)g(tempting)523 959 y(to)38 b(in)n(terpret)f Fr(j)j Fs( )k Fr(j)1173 929 y Fq(2)1250 959 y Fv(\()p Fs(y)s Fv(\)d)p Fs(\026)39 b Fv(as)e(the)h(probabilit)n(y)f(for)g(\014nding)h(the)h(\\particle")d (in)i(a)523 1059 y(v)n(olume)28 b(d)p Fs(\026)h Fv(around)f Fs(y)s Fv(,)g(or)g(in)h(a)f(preciser)g(manner,)g(to)h(sa)n(y)e(that)i (the)g(probabilit)n(y)f(for)523 1158 y(\014nding)g(the)g(particle)f(in) h Fs(B)f Fr(\032)22 b Fs(K)34 b Fv(is)27 b(giv)n(en)g(b)n(y)1636 1261 y Fk(Z)1682 1450 y Fn(B)1762 1374 y Fr(j)d Fs( )i Fr(j)1912 1340 y Fq(2)1972 1374 y Fv(\()p Fs(y)s Fv(\)d)p Fs(\026:)523 1594 y Fv(This)j(seems)g(lik)n(e)g(a)g(quite)g(reasonable) e(thing)j(to)f(do)g(b)r(ecause)g(it)g(is)h(completely)f(anal-)523 1694 y(ogous)i(to)i(what)f(is)h(done)f(in)h(the)g(non-relativistic)e (Sc)n(hr\177)-42 b(odinger)31 b(quan)n(tum)h(mec)n(han-)523 1793 y(ics)k(of)h(particle)e(systems.)h(I)h(will)g(call)f(the)g(pro)5 b(jection)36 b(v)-5 b(alued)36 b(measure)g Fs(B)42 b Fr(7!)37 b Fs(\037)3232 1805 y Fn(B)3290 1793 y Fv(,)523 1893 y(where)c Fs(\037)821 1905 y Fn(B)911 1893 y Fv(is)h(the)f(op)r (erator)f(of)h(m)n(ultiplication)h(b)n(y)f(the)g(c)n(haracteristic)f (function)i(of)523 1993 y Fs(B)h Fv(the)c(Newton-Wigner)e(p)r(osition)i (observ)-5 b(able.)29 b(If)i Fs( )g Fr(2)d Fs(L)2430 1962 y Fq(2)2467 1993 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026;)g Fo(C)p Fv(\))30 b(is)h(supp)r(orted)523 2092 y(in)k Fs(B)j Fr(\032)c Fs(K)6 b Fv(,)34 b(w)n(e)g(sa)n(y)f Fs( )k Fv(is)d(\\Newton-Wigner)f(lo)r(calized)h(in)h Fs(B)t Fv(".)f(This)g(terminology)523 2192 y(is)29 b(inspired)g(b)n(y)g (the)g(observ)-5 b(ation)28 b(that,)h(when)g(considering)f(the)h (particular)f(example)523 2291 y(of)37 b(an)f(oscillator)f(\014eld)h (giv)n(en)g(b)n(y)h(the)f(w)n(a)n(v)n(e)f(or)h(Klein-Gordon)f (equation,)h(one)g(has)523 2391 y Fr(K)k Fv(=)e Fs(L)786 2361 y Fq(2)823 2391 y Fv(\()p Fo(R)915 2361 y Fq(3)952 2391 y Fs(;)14 b Fv(d)p Fs(x;)g Fo(C)p Fv(\))38 b(and)f(in)g(that)g (case)f(the)h(ab)r(o)n(v)n(e)f(measure)f(is)i(indeed)g(the)h(join)n(t) 523 2491 y(sp)r(ectral)24 b(measure)f(of)i(the)g(usual)f(Newton-Wigner) g(p)r(osition)g(op)r(erator)f(of)h(relativistic)523 2590 y(quan)n(tum)i(\014eld)g(theory)f([NW)q(].)h(This)f(c)n(hoice)g(of)h(p) r(osition)g(observ)-5 b(able)24 b(ma)n(y)h(seem)h(rea-)523 2690 y(sonable,)i(but)h(it)g(is)g(only)f(based)h(on)f(an)g(analogy)-7 b(,)27 b(and)i(as)f(I)h(will)g(no)n(w)f(sho)n(w,)g(it)h(is)g(not)523 2790 y(reasonable)d(at)h(all.)648 2889 y(F)-7 b(or)29 b(that)h(purp)r(ose,)f(let)i(us)f(go)f(bac)n(k)g(to)g(the)i(particular) d(example)i(of)g(the)g(oscilla-)523 2989 y(tor)d(ring)g(treated)h(b)r (efore)f(\(Sects.)h(2.2)f(and)h(4.3\))f(and)g(see)h(what)f(the)h (Newton-Wigner)523 3088 y(p)r(osition)40 b(op)r(erator)e(means)h(in)i (that)f(case.)f(Remem)n(b)r(er,)h(this)g(is)g(just)g(a)g(system)g(of) 523 3188 y Fs(n)f Fv(coupled)g(oscillators.)e(So)h(the)i(quan)n(tum)f (Hilb)r(ert)g(space)f(can)h(on)g(the)g(one)g(hand)523 3288 y(b)r(e)e(seen)g(as)f Fs(L)1003 3258 y Fq(2)1040 3288 y Fv(\()p Fo(R)1132 3258 y Fn(n)1177 3288 y Fs(;)14 b Fv(d)p Fs(x)p Fv(\))38 b(\(Sc)n(hr\177)-42 b(odinger)36 b(represen)n(tation\))f(and)i(the)g(system)g(can)f(b)r(e)523 3387 y(studied)k(through)e(the)i(displacemen)n(ts)e(and)h(momen)n(ta)g (of)g(those)g(oscillators.)e(This)523 3487 y(is)i(the)h(usual)f(p)r (oin)n(t)g(of)h(view.)f(Alternativ)n(ely)-7 b(,)39 b(it)g(b)r(e)h(iden) n(ti\014ed)g(with)g(the)f(b)r(osonic)523 3587 y(F)-7 b(o)r(c)n(k)33 b(space)h Fr(F)1023 3556 y Fq(+)1078 3587 y Fv(\()p Fr(K)1174 3556 y Fg(C)1220 3587 y Fv(\))h(\(F)-7 b(o)r(c)n(k)33 b(represen)n(tation\),)g(where)g(no)n(w)g(the)h (one-particle)f(sub-)523 3686 y(space)28 b(is)g Fr(K)894 3656 y Fg(C)965 3686 y Fv(=)23 b Fo(C)1113 3656 y Fn(n)1158 3686 y Fv(.)29 b(The)f(latter,)g(as)g(explained)g(in)g(Sect.)h(4.3,)e (can)h(b)r(e)h(though)n(t)f(of)g(as)523 3786 y Fs(L)580 3756 y Fq(2)617 3786 y Fv(\()p Fo(Z)p Fs(=n)p Fo(Z)p Fs(;)14 b Fo(C)p Fv(\).)27 b(In)f(other)g(w)n(ords,)f(it)i(is)f (tempting)h(to)f(in)n(terpret)g Fs( )g Fr(2)e(K)2792 3756 y Fg(C)2865 3786 y Fv(as)i(the)g(quan-)523 3885 y(tum)i(mec)n(hanical)f(state)g(of)g(a)g(\\particle")f(hopping)h(along) f Fs(n)i Fv(sites!)f(Its)g(probabilit)n(y)g(of)523 3985 y(b)r(eing)c(at)f(site)h Fs(i)g Fv(is)f(then)i(giv)n(en)e(b)n(y)g Fr(j)p Fs( )s Fv(\()p Fs(i)p Fv(\))p Fr(j)1819 3955 y Fq(2)1857 3985 y Fv(.)g(More)g(generally)-7 b(,)22 b(an)n(y)g(state)g (of)h(the)g Fs(n)g Fv(oscil-)523 4085 y(lators)f(can)h(b)r(e)g(seen)g (as)g(a)f(sup)r(erp)r(osition)h(of)g(0)p Fs(;)14 b Fv(1)p Fs(;)g Fv(2)p Fs(;)g(:)g(:)g(:)21 b Fv(\\particle")h(states,)h(where)f (no)n(w)523 4184 y(\\particle")30 b(refers)h(to)h(an)g(imagined)f(en)n (tit)n(y)h(hopping)g(along)f(the)h(sites)g(of)g(the)g(c)n(hain.)523 4284 y(Sp)r(eaking)26 b(lik)n(e)g(this,)h(w)n(e)g(are)e(pushing)i(the)g (particle)f(in)n(terpretation)g(maximally)-7 b(.)26 b(The)523 4384 y(state)h Fs(a)770 4353 y Fl(y)805 4384 y Fv(\()p Fs(\016)874 4396 y Fn(i)902 4384 y Fv(\))p Fr(j)p Fv(0)p Fr(i)h Fv(is)f(then)h(though)n(t)f(of)h(as)f(particle)g(p)r(erfectly)h (lo)r(calized)f(on)g(the)h(site)g Fs(i)p Fv(.)648 4483 y(But)39 b(do)r(es)f(this)h(mak)n(e)f(sense?)h(Certainly)-7 b(,)38 b(whatev)n(er)g(picture)h(used,)g(the)g(mean)523 4583 y(square)23 b(displacemen)n(t)g(of)h(the)g(oscillator)f(at)g(site) h Fs(j)29 b Fv(is)24 b(a)g(relev)-5 b(an)n(t)23 b(ph)n(ysical)g(observ) -5 b(able)523 4682 y(in)27 b(this)g(system.)g(The)g(problem)f(is)h (that)g(this)g(mean)g(square)e(displacemen)n(t)i(will)g(di\013er)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(39)523 282 y Fv(from)27 b(its)h(v)-5 b(acuum)28 b(v)-5 b(alue)27 b(if)h Fs(j)g Fr(6)p Fv(=)23 b Fs(i)p Fv(:)1364 465 y Fr(h)p Fv(0)p Fr(j)p Fs(a)p Fv(\()p Fs(\016)1574 477 y Fn(i)1602 465 y Fv(\))p Fs(Q)1700 431 y Fq(2)1700 485 y Fn(j)1737 465 y Fs(a)1781 431 y Fl(y)1815 465 y Fv(\()p Fs(\016)1884 477 y Fn(i)1912 465 y Fv(\))p Fr(j)p Fv(0)p Fr(i)g(6)p Fv(=)g Fr(h)p Fv(0)p Fr(j)p Fs(Q)2315 431 y Fq(2)2315 485 y Fn(j)2352 465 y Fr(j)p Fv(0)p Fr(i)p Fs(:)523 648 y Fv(So)34 b(the)h(idea)g(that) g(the)g(system)f(con)n(tains)g(only)g(one)g(particle,)h(and)f(that)h (the)g(latter)523 747 y(is)29 b(lo)r(calized)f(p)r(erfectly)g(at)h Fs(i)p Fv(,)f(the)h(rest)f(of)h(the)g(sites)f(b)r(eing)h(\\empt)n(y",)f (is)g(not)h(tenable.)523 847 y(Indeed,)37 b(if)h(the)f(particle)f(is)h (at)g(site)g Fs(i)p Fv(,)f(and)h(if)g(space)g(\(here)f(represen)n(ted)g (b)n(y)g(the)i Fs(n)523 946 y Fv(sites\))f(is)f(otherwise)g(\\empt)n (y",)f(ho)n(w)h(can)g(an)n(y)g(observ)-5 b(able)35 b(at)i(site)f Fs(j)42 b Fv(tak)n(e)36 b(a)g(v)-5 b(alue)523 1046 y(di\013eren)n(t)35 b(from)g(its)g(v)-5 b(acuum)36 b(v)-5 b(alue?)35 b(The)g(problem)g(is)g (of)g(course)f(readily)g(solv)n(ed)g(if)523 1146 y(one)27 b(stops)g(trying)g(to)h(in)n(terpret)f(the)h(quan)n(tit)n(y)f Fr(j)p Fs( )s Fv(\()p Fs(i)p Fv(\))p Fr(j)2252 1116 y Fq(2)2317 1146 y Fv(as)g(a)g(probabilit)n(y)g(of)g(presence)523 1245 y(for)g(a)g(particle.)648 1345 y(The)f(same)f(analysis)f(carries)h (immediately)h(o)n(v)n(er)e(to)h(the)i(oscillator)d(c)n(hains)h(or)g (lat-)523 1445 y(tices)39 b(discussed)g(b)r(efore.)g(It)g(is)g(p)r (erhaps)g(ev)n(en)f(more)h(telling)g(there.)g(No)n(w)g(the)g(one)523 1544 y(particle)30 b(space)g(is)g Fs(`)1172 1514 y Fq(2)1209 1544 y Fv(\()p Fo(Z)1296 1514 y Fn(d)1335 1544 y Fs(;)14 b Fo(C)p Fv(\))31 b(and)f(so)g(the)h(idea)f(of)g(thinking)h(of)f (states)g(in)h(this)f(space)523 1644 y(as)d(describing)h(a)f(particle)h (hopping)f(on)h(the)g(sites)g(of)g(the)h(lattice)f Fo(Z)2700 1614 y Fn(d)2767 1644 y Fv(ma)n(y)f(seem)h(ev)n(en)523 1743 y(more)35 b(reasonable.)e(Mo)r(dels)i(of)g(this)h(t)n(yp)r(e)g (are)e(used)h(in)h(solid)f(state)g(ph)n(ysics)g(to)g(de-)523 1843 y(scrib)r(e)e(lattice)h(vibrations,)e(and)h(the)h(quan)n(ta)e(are) h(then)h(called)f(phonons.)f(They)i(are)523 1943 y(excitations)22 b(of)h(the)g(oscillator)e(lattice)i(and)g({)f(as)g(Knigh)n(t's)h (theorem)f(tells)h(us)g({)f(cannot)523 2042 y(b)r(e)k(p)r(erfectly)g (lo)r(calized)f(in)g(the)h(sense)f(that,)h(if)g(the)g(system)f(is)h(in) f(a)h(one-phonon)e(state)523 2142 y Fs( )i Fr(2)e Fs(`)717 2112 y Fq(2)753 2142 y Fv(\()p Fo(Z)840 2112 y Fn(d)880 2142 y Fv(\),)g(then)h(it)g(cannot)e(coincide)h(with)h(the)g(v)-5 b(acuum)24 b(outside)g(a)g(\014nite)g(subset)h(of)523 2242 y(the)31 b(lattice.)g(This)g(do)r(es)f(not)h(lead)g(to)f(an)n(y)g (in)n(terpretational)g(di\016culties,)h Fp(as)i(long)g(as)523 2341 y(one)h(do)l(es)g(not)f(try)g(to)h(interpr)l(et)d Fr(j)p Fs( )s Fv(\()p Fs(i)p Fv(\))p Fr(j)1799 2311 y Fq(2)1869 2341 y Fp(as)j(the)f(pr)l(ob)l(ability)j(of)e(\014nding)g (the)f(p)l(article)523 2441 y(at)d(site)e Fs(i)f Fp(of)j(the)g(lattic)l (e)p Fv(.)648 2540 y(Finally)-7 b(,)38 b(without)g(an)n(y)f(c)n(hange)g (whatso)r(ev)n(er,)g(the)h(same)g(analysis)e(carries)g(o)n(v)n(er)523 2640 y(to)g(the)h(Klein-Gordon)d(equation.)i(Let)g Fs(B)k Fv(b)r(e)d(a)f(b)r(ounded)g(subset)g(of)g Fo(R)2919 2610 y Fq(3)2993 2640 y Fv(and)g Fs( )k Fr(2)523 2740 y Fs(L)580 2710 y Fq(2)617 2740 y Fv(\()p Fo(R)709 2710 y Fq(3)746 2740 y Fs(;)14 b Fv(d)p Fs(x;)g Fo(C)p Fv(\))40 b(b)r(e)f(supp)r(orted) g(in)g Fs(B)t Fv(.)g(As)g(Knigh)n(t's)f(theorem)g(tells)h(us,)g(the)h (corre-)523 2839 y(sp)r(onding)d(one-particle)f(state)h(of)g(the)g (\014eld)h(is)f(not)g(an)g(excitation)g(of)g(the)h(v)-5 b(acuum)523 2939 y(lo)r(calized)27 b(inside)h Fs(B)t Fv(.)648 3039 y(The)33 b(conclusion)f(I)h(dra)n(w)f(from)h(all)g(this)g (is)g(that)h Fp(the)h(Newton-Wigner)g(op)l(er)l(ator)523 3138 y(do)l(es)h(not)f(pr)l(ovide)j(an)d(appr)l(opriate)j(to)l(ol)e(to) g(describ)l(e)g(the)g(strict)f(lo)l(c)l(alization)j(pr)l(op-)523 3238 y(erties)e(of)h(the)f(states)g(of)h(extende)l(d)f(systems)g(of)h (the)f(typ)l(e)g(discusse)l(d)h(her)l(e)p Fv(.)e(It)g(sho)n(ws)523 3337 y(up)e(only)e(b)r(ecause)h(of)h(an)e(understandable)h(but)h (ill-fated)f(desire)g(to)g(force)f(a)h Fp(p)l(article)523 3437 y(interpr)l(etation)g Fv(with)f(all)g(its)g(usual)f(attributes)h (on)g(the)g(states)f(of)h(a)g Fp(\014eld)p Fv(.)g(The)g(righ)n(t)523 3537 y(notion)k(of)h(a)f(\(strictly\))h(lo)r(calized)f(state)g(is)h (the)g(one)f(giv)n(en)f(b)n(y)i(Knigh)n(t)f(\(De\014nition)523 3636 y(4\).)30 b(This)g(has)g(lead)g(to)g(some)g(debate)g(in)g(the)h (con)n(text)f(of)g(relativistic)f(quan)n(tum)h(\014eld)523 3736 y(theory)-7 b(,)37 b(up)r(on)h(whic)n(h)f(I)g(shall)g(commen)n(t)h (b)r(elo)n(w.)f(An)n(ticipating)g(on)h(the)f(discussion)523 3836 y(there,)29 b(I)f(w)n(ould)g(lik)n(e)h(to)f(stress)g(that)h(m)n(y) f(line)h(of)g(argumen)n(t)e(here,)i(and)f(in)h(particular)523 3935 y(m)n(y)37 b(criticism)h(of)f(the)h(use)f(of)h(the)g (Newton-Wigner)e(op)r(erator)g(has)h(nothing)h(to)f(do)523 4035 y(with)25 b(relativit)n(y)-7 b(,)23 b(or)h(with)g(causalit)n(y)-7 b(,)24 b(but)g(is)g(related)g(instead)g(to)g(the)h(fact)f(that)h(w)n(e) f(are)523 4134 y(dealing)j(with)h(extended)g(systems.)523 4375 y Fm(Causalit)m(y)k(problems)523 4558 y Fv(In)23 b(the)g(early)e(da)n(ys)h(of)g(relativistic)g(quan)n(tum)h(\014eld)f (theory)-7 b(,)22 b(and)h(w)n(ell)f(b)r(efore)g(an)n(ything)523 4658 y(lik)n(e)34 b(Knigh)n(t's)g(theorem)g(w)n(as)f(form)n(ulated)h (or)f(pro)n(v)n(en,)g(the)i(particle)f(in)n(terpretation)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 523 100 a Fu(40)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(of)29 b(the)h(\014eld)g (states)f(made)g(it)h(p)r(erfectly)g(natural)e(to)h(searc)n(h)f(for)h (a)g(p)r(osition)g(op)r(erator)523 382 y(with)k(the)f(usual)g(prop)r (erties)f(familiar)g(from)h(non-relativistic)e(quan)n(tum)i(mec)n (hanics.)523 482 y(In)24 b(other)f(w)n(ords,)g(if)h(the)h(\014eld)f (quan)n(ta)f(are)g(particles,)g(one)g(w)n(ould)h(w)n(an)n(t)f(to)h (answ)n(er)e(the)523 581 y(question:)28 b(\\Where)f(is)h(the)h (particle?")e(It)i(should)f(therefore)f(not)h(come)g(as)f(a)h(surprise) 523 681 y(that)40 b(a)g(fair)f(amoun)n(t)g(of)h(literature)f(w)n(as)g (dev)n(oted)g(to)h(this)g(problem.)g(The)f(theory)523 780 y(receiv)n(ed)25 b(its)h(de\014nite)g(form)g(in)g([NW)q(])g(and)g (a)f(sligh)n(tly)g(more)h(rigorous)d(treatmen)n(t)j(w)n(as)523 880 y(subsequen)n(tly)j(giv)n(en)g(in)h([Wi)q(].)f(References)h(to)f (earlier)f(w)n(ork)h(can)g(b)r(e)h(found)g(in)g(those)523 980 y(t)n(w)n(o)35 b(pap)r(ers)h(and)g(in)g([ScWi)q(].)g(The)g (discussion)f(in)i([NW])f(cen)n(ters)g(on)g(the)g(question)523 1079 y(ho)n(w)30 b(to)h(iden)n(tify)-7 b(,)32 b(inside)f(a)g (relativistic)f(elemen)n(tary)g(system)h(\()p Fp(i.e.)i Fv(inside)e(a)g(unitary)523 1179 y(irreducible)j(represen)n(tation)e (of)i(the)h(P)n(oincar)n(\023)-39 b(e)31 b(group\),)i(a)h(\\p)r (osition)f(op)r(erator)38 b(^)-47 b Fs(x)34 b Fv(=)523 1279 y(\()5 b(^)-47 b Fs(x)602 1291 y Fq(1)640 1279 y Fs(;)19 b Fv(^)-47 b Fs(x)724 1291 y Fq(2)762 1279 y Fs(;)19 b Fv(^)-47 b Fs(x)846 1291 y Fq(3)883 1279 y Fv(\)",)23 b(using)g(only)f(natural)h(requiremen)n(ts)f({)g(form)n (ulated)h(as)f(axioms)g({)h(on)f(the)523 1378 y(transformation)32 b(prop)r(erties)g(of)h(this)h(op)r(erator)d(under)i(rotations)f(and)h (translations.)523 1478 y(The)26 b(upshot)f(of)h(this)f(analysis)g(is)g (that)h(suc)n(h)f(an)g(op)r(erator)f(exists)h(\(for)g(most)h(v)-5 b(alues)25 b(of)523 1577 y(spin)h(and)f(mass\))h(and)f(that)h(it)g(is)g (unique.)g(It)g(is)f(called)h(the)g(Newton-Wigner)f(p)r(osition)523 1677 y(op)r(erator)d(in)i(the)g(literature.)f(As)h(an)f(example,)h (there)f(exists)h(suc)n(h)f(an)g(op)r(erator)f(in)i(the)523 1777 y(one)d(\014eld)i(quan)n(tum)e(sector)g(of)h(the)g(quan)n(tized)g (Klein-Gordon)e(\014eld,)i(whic)n(h)g(carries)e(an)523 1876 y(irreducible)i(represen)n(tation)e(of)j(the)f(P)n(oincar)n(\023) -39 b(e)19 b(group)j(of)g(zero)f(spin)h(and)h(it)f(is)h(precisely)523 1976 y(the)j(one)g(discussed)f(in)h(the)h(previous)d(subsection.)i(No)n (w,)f(the)h(join)n(t)g(sp)r(ectral)g(measure)523 2076 y(of)j(the)h(three)f(comp)r(onen)n(ts)f(of)35 b(^)-48 b Fs(x)30 b Fv(de\014nes)f(a)g(pro)5 b(jection)28 b(v)-5 b(alued)29 b(measure)f Fs(P)2990 2088 y Fn(B)3048 2076 y Fv(,)h(where)523 2175 y Fs(B)39 b Fv(is)34 b(a)h(Borel)e(subset)i(of) f Fo(R)1442 2145 y Fq(3)1480 2175 y Fv(.)g(If)h(the)g(in)n (terpretation)f(of)40 b(^)-47 b Fs(x)35 b Fv(as)f(a)g(p)r(osition)g(op) r(erator)523 2275 y(along)26 b(the)h(lines)g(of)g(the)g(usual)f(in)n (terpretational)g(rules)g(of)h(quan)n(tum)g(mec)n(hanics)f(is)h(to)523 2374 y(mak)n(e)g(sense,)h(then)g(eigenstates)f(of)h Fs(P)1735 2386 y Fn(B)1821 2374 y Fv(with)h(eigen)n(v)-5 b(alue)27 b(1)g(are)g(to)h(b)r(e)h(though)n(t)e(of)h(as)523 2474 y(states)j(\\p)r(erfectly)g(lo)r(calized)g(inside)h Fs(B)t Fv(".)f(This)h(is)f(referred)f(to)i(as)e(NW-lo)r(calization.)523 2574 y(This)i(is)f(precisely)g(the)h(in)n(terpretation)e(giv)n(en)h(to) h(the)g(Newton-Wigner)f(op)r(erator)e(in)523 2673 y(the)j(literature)e (whic)n(h)i(is,)f(as)g(explained)g(b)r(efore,)g(at)g(o)r(dds)g(with)h (Knigh)n(t's)f(notion)g(of)523 2773 y(lo)r(cal)e(excitation)f(of)i(the) f(v)-5 b(acuum.)30 b(Nev)n(ertheless,)e(the)h(axiomatic)g(deriv)-5 b(ation)28 b(of)h(the)523 2873 y(Newton-Wigner)f(op)r(erator,)g(and)h (its)g(p)r(erfect)g(analogy)e(with)j(the)f(familiar)g(situation)523 2972 y(in)c(the)g(quan)n(tum)f(mec)n(hanics)g(of)g(non-relativistic)f (particles)h(giv)n(es)f(it)i(something)f(v)n(ery)523 3072 y(comp)r(elling,)37 b(whic)n(h)g(probably)f(explains)g(its)h (success.)f(As)h(a)g(result,)f(some)h(authors)523 3171 y(ha)n(v)n(e)30 b(written)i(that)g(the)g(Newton-Wigner)f(op)r(erator)f (is)h(the)h(only)g(p)r(ossible)f(p)r(osition)523 3271 y(op)r(erator)i(for)h(relativistic)f(quan)n(tum)i(particles.)e(In)i ([Wi],)g(one)f(reads)f(the)i(follo)n(wing)523 3371 y(claim:)26 b(\\I)h(v)n(en)n(ture)e(to)i(sa)n(y)e(that)i(an)n(y)f(notion)g(of)g(lo) r(calizabilit)n(y)g(in)h(three-dimensional)523 3470 y(space)32 b(whic)n(h)h(do)r(es)g(not)g(satisfy)g([the)g(axioms])f(will)h (represen)n(t)f(a)h(radical)f(departure)523 3570 y(from)26 b(presen)n(t)g(ph)n(ysical)g(ideas.")g(Newton)h(and)f(Wigner)g(sa)n(y)g (something)g(similar,)g(but)523 3670 y(do)34 b(not)g(put)g(it)h(so)e (forcefully:)g(\\It)h(seems)g(to)g(us)f(that)i(the)f(ab)r(o)n(v)n(e)f (p)r(ostulates)g(are)g(a)523 3769 y(reasonable)22 b(expression)h(for)g (the)h(lo)r(calization)f(of)h(the)h(system)e(to)h(the)h(exten)n(t)f (that)g(one)523 3869 y(w)n(ould)g(naturally)f(call)h(a)g(system)g(unlo) r(calizable)f(if)i(it)g(should)e(pro)n(v)n(e)g(to)h(b)r(e)h(imp)r (ossible)523 3968 y(to)f(satisfy)g(these)g(requiremen)n(ts.)f(In)i ([ScWi])f(one)g(can)g(read:)f(\\One)h(either)g(accepts)f(the)523 4068 y(Newton-Wigner)h(p)r(osition)g(op)r(erator)e(when)j(it)g(exists,) f(or)f(abandons)h(his)g(axioms.)f(W)-7 b(e)523 4168 y(b)r(eliev)n(e)24 b(the)h(\014rst)g(alternativ)n(e)e(is)h(w)n(ell)g(w)n(orth)g(in)n(v)n (estigation)f(and)h(adopt)h(it)g(here.")e(I)i(of)523 4267 y(course)30 b(ha)n(v)n(e)g(argued)f(ab)r(o)n(v)n(e)h(that)h(one)g (should)g(abandon)f(it,)h(and)g(that)g(this)h(neither)523 4367 y(constitutes)40 b(a)f(departure)g(from)h(standard)e(ph)n(ysical)h (ideas,)h(nor)f(means)g(that)h(one)523 4467 y(abandons)26 b(the)i(notion)g(of)f(lo)r(calizabilit)n(y)-7 b(.)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(41)648 282 y Fv(Still,)35 b(ev)n(en)g(among)f(those)g(that)i(ha)n(v)n(e)d(adv)n(o)r(cated)h(the)i (use)e(of)h(Newton-Wigner)523 382 y(lo)r(calization,)19 b(this)i(notion)f(has)g(stirred)g(up)g(a)g(fair)g(amoun)n(t)g(of)g (debate,)h(since)f(it)h(violates)523 482 y(causalit)n(y)-7 b(,)27 b(as)g(I)g(no)n(w)g(brie\015y)g(explain.)648 581 y(Indeed,)f(\014rst)h(of)f(all,)g(a)g(one-particle)f(state)i(of)f(the)h (Klein-Gordon)e(\014eld)h(p)r(erfectly)523 681 y(NW-lo)r(calized)g(in)g (some)g(b)r(ounded)h(set)f Fs(B)k Fv(at)c(an)g(initial)h(time,)f(is)g (easily)g(seen)g(to)g(ha)n(v)n(e)523 780 y(a)k(non-zero)f(probabilit)n (y)h(to)g(b)r(e)h(found)g(arbitrarily)d(far)j(a)n(w)n(a)n(y)d(from)i Fs(B)t Fv(,)h(at)g(an)n(y)e(later)523 880 y(time,)i(violating)e (causalit)n(y)-7 b(.)29 b(Since)h(the)h(theory)e(is)h(supp)r(osed)g(to) g(b)r(e)h(relativistic,)e(this)523 980 y(is)35 b(a)f(real)g(problem)h (that)g(has)f(receiv)n(ed)g(m)n(uc)n(h)h(atten)n(tion.)f(Actually)-7 b(,)36 b(replacing)d(the)523 1079 y(pro)5 b(jection)32 b(op)r(erators)g Fs(P)1350 1091 y Fn(B)1440 1079 y Fv(of)i(the)f(NW-p)r (osition)g(op)r(erator)f(b)n(y)h(an)n(y)f(other)h(p)r(ositiv)n(e)523 1179 y(op)r(erators)27 b(transforming)g(correctly)g(under)i(space)f (translations,)g(Hegerfeldt)g(pro)n(v)n(ed)523 1279 y(that)37 b(the)g(causalit)n(y)e(problem)h(remains)g(\(see)g([He1])h([He2)o(])g (and)f(for)g(a)h(more)e(recen)n(t)523 1378 y(o)n(v)n(erview,)24 b([He3)o(]\).)j(In)f(addition,)f(and)h(directly)f(link)n(ed)h(to)f(the) h(previous)f(observ)-5 b(ation,)523 1478 y(a)29 b(state)h(p)r(erfectly) g(lo)r(calized)f(in)h(one)f(Loren)n(tz)g(frame)g(is)h(not)g(in)g (another)f(one.)g(These)523 1577 y(di\016culties,)h(while)f(w)n(ell)g (kno)n(wn)f(and)h(widely)g(stressed,)f(are)g(often)i(dismissed)e(with)i (a)523 1677 y(v)-5 b(ague)33 b(app)r(eal)h(to)g(one)g(of)g(the)h(follo) n(wing)e(somewhat)g(related)g(ideas.)h(Although)g(the)523 1777 y(Newton-Wigner)23 b(deriv)-5 b(ation)23 b(do)r(es)g(not)h(refer)f (to)g(an)n(y)g(underlying)g(\014eld)h(theory)-7 b(,)23 b(these)523 1876 y(argumen)n(ts)37 b(all)h(in)n(v)n(olv)n(e)f(remem)n (b)r(ering)g(that)h(the)h(\\particles")d(in)j(relativistic)e(\014eld) 523 1976 y(theory)27 b(are)f(excitations)h(of)h(the)g(\014eld.)648 2076 y(The)g(\014rst)g(suc)n(h)g(argumen)n(t)f(go)r(es)h(as)f(follo)n (ws.)h(In)g(a)g(\014eld)h(theory)e(a)h(p)r(osition)g(mea-)523 2175 y(suremen)n(t)36 b(of)h(a)f(particle)g(w)n(ould)g(lead)g(to)h (pair)f(creation)f(\(see)h([ScWi)q(]\))h(and)f(so)g(the)523 2275 y(app)r(earance)28 b(of)i(particles)f(far)h(a)n(w)n(a)n(y)e(is)i (not)f(parado)n(xical.)f(This)i(line)g(of)g(reasoning)e(is)523 2374 y(not)20 b(v)n(ery)g(satisfactory)e(\(as)i(already)f(p)r(oin)n (ted)i(out)f(in)h([NW]\),)g(since)f(it)h(seems)f(to)g(app)r(eal)523 2474 y(to)j(a)f(\(non-sp)r(eci\014ed\))h(theory)f(of)h(in)n(teracting)e (\014elds)i(to)g(deal)f(with)h(the)h(a)e(priori)f(simple)523 2574 y(non-in)n(teracting)i(\014eld.)i(An)g(alternativ)n(e)f(argumen)n (t)f(stresses)h(that)h(in)g(a)f(theory)g(whic)n(h)523 2673 y(allo)n(ws)j(for)i(m)n(ulti-particle)f(states,)g(the)i(observ)-5 b(ation)27 b(of)i(exactly)f(one)g(particle)h(inside)523 2773 y(a)22 b(b)r(ounded)h(set)f Fs(B)27 b Fv(en)n(tails)22 b(the)h(observ)-5 b(ation)21 b(of)h(the)h(absence)f(of)g(particles)g (ev)n(erywhere)523 2873 y(else,)i(and)g(is)h(therefore)e(not)h(really)g (a)g(lo)r(cal)f(measuremen)n(t.)h(As)g(suc)n(h,)g(the)h(app)r(earance) 523 2972 y(later)i(on)g(of)h(particles)e(far)h(a)n(w)n(a)n(y)f(do)r(es) h(not)h(violate)e(causalit)n(y)h(\(see)g([F)-7 b(u)q(]\).)28 b(This)f(argu-)523 3072 y(men)n(t)k(is)g(certainly)f(correct.)f(But)i (it)g(is)g(again)f(qualitativ)n(e)g(and)g(nothing)h(guaran)n(tees)523 3171 y(that)22 b(it)g(can)f(correctly)f(accoun)n(t)h(for)g(the)g (\\amoun)n(t")f(of)i(causalit)n(y)e(violation)h(generated)523 3271 y(b)n(y)27 b(the)h(Newton-Wigner)f(p)r(osition.)648 3371 y(All)34 b(in)g(all,)f(it)i(seems)e(considerably)f(simpler)i(to)g (adopt)f(the)h(notion)g(of)f(\\strictly)523 3470 y(lo)r(calized)19 b(v)-5 b(acuum)20 b(excitation")f(in)n(tro)r(duced)g(b)n(y)h(Knigh)n (t,)f(whic)n(h)g(is)h(p)r(erfectly)g(adapted)523 3570 y(to)28 b(the)g(study)f(of)h(the)g(extended)g(systems)f(under)h (consideration)e(here)h(and)g(to)h(accept)523 3670 y(once)c(and)g(for)g (all)g(that)h(the)f(particles)g(of)g(\014eld)h(theory)e(are)h(elemen)n (tary)f(excitations)h(of)523 3769 y(the)31 b(system)g(\(or)f(\014eld)h (quan)n(ta\))f(that)i(do)e(not)h(ha)n(v)n(e)f(all)g(the)h(usual)g (attributes)f(of)h(the)523 3869 y(p)r(oin)n(t)i(particles)f(of)h(our)f (\014rst)h(mec)n(hanics)f(and)g(quan)n(tum)h(mec)n(hanics)f(courses.)g (This)523 3968 y(seems)25 b(to)h(b)r(e)g(the)g(p)r(oin)n(t)f(of)h(view) f(implicitly)i(prev)-5 b(alen)n(t)25 b(among)f(ph)n(ysicists,)h (although)523 4068 y(it)e(is)g(nev)n(er)f(clearly)g(sp)r(elled)i(out)f (in)g(the)g(theoretical)f(ph)n(ysics)h(textb)r(o)r(oks)g(for)f (example,)523 4168 y(as)33 b(I)h(will)g(discuss)g(in)g(more)f(detail)h (in)g([DB2].)g(It)h(also)e(has)g(the)h(adv)-5 b(an)n(tage)33 b(that)h(no)523 4267 y(causalit)n(y)20 b(problems)g(arise.)g(Although)i (traces)e(of)h(this)g(argumen)n(t)f(can)h(o)r(ccasionally)e(b)r(e)523 4367 y(found)25 b(in)g(the)g(more)g(mathematically)f(orien)n(ted)g (literature,)g(Knigh)n(t's)g(de\014nition)h(of)g(a)523 4467 y(strictly)g(lo)r(cal)h(excitation)f(of)g(the)h(v)-5 b(acuum)26 b(and)g(his)f(result)h(on)f(the)h(non-lo)r(calizabilit)n(y) 523 4566 y(of)k(\014nite)h(particle)f(states)g(seem)g(to)g(b)r(e)h (mostly)f(ignored)f(in)i(discussions)e(of)h(the)h(issue)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 523 100 a Fu(42)191 b(Stephan)24 b(De)i(Bi)n(\022)-36 b(evre)523 282 y Fv(of)29 b(the)h(lo)r (calizabilit)n(y)e(of)h(particles)g(in)g(\014eld)h(theory)-7 b(,)28 b(of)h(whic)n(h)g(there)g(con)n(tin)n(ue)g(to)g(b)r(e)523 382 y(man)n(y)e([Ba)o(])h([Ha])g([T)-7 b(el])28 b([Stre)o(])g([Fl])g ([FlBu])648 482 y(Ha)n(ving)d(adv)n(o)r(cated)h(Knigh)n(t's)g (de\014nition)h(of)g(\\lo)r(cal)f(state",)g(it)i(remains)e(to)g(pro)n (v)n(e)523 581 y(the)i(extension)f(of)h(his)f(theorem)g(giv)n(en)g(ab)r (o)n(v)n(e.)523 822 y Fm(4.8)41 b(Pro)s(of)31 b(of)h(Theorem)f(1)523 1005 y Fv(The)36 b(theorem)g(is)g(reduced)g(to)g(abstract)f(nonsense)h (through)f(the)i(follo)n(wing)e(prop)r(o-)523 1104 y(sition.)h(Note)g (that,)h(for)e(an)n(y)g(subset)h Fr(M)g Fv(of)g(a)g(Hilb)r(ert)h(space) e Fr(V)7 b Fv(,)36 b Fr(M)2824 1074 y Fl(?)2916 1104 y Fv(denotes)f(its)523 1204 y(orthogonal)25 b(complemen)n(t,)j(whic)n (h)g(is)f(a)g(complex)g(subspace)g(of)h Fr(V)7 b Fv(.)523 1378 y Fm(Prop)s(osition)30 b(2.)24 b Fp(L)l(et)j Fr(K)d Fv(=)f Fs(L)1511 1348 y Fq(2)1511 1401 y Fg(R)1557 1378 y Fv(\()p Fs(K)q(;)14 b Fv(d)p Fs(\026)p Fv(\))p Fp(,)27 b Fs(\012)32 b Fp(and)27 b Fr(S)33 b Fp(b)l(e)27 b(as)h(b)l(efor)l(e.)g (L)l(et)e Fs(B)h Fr(\032)c Fs(K)6 b Fp(.)27 b(Then)523 1478 y(the)j(fol)t(lowing)i(statements)d(ar)l(e)h(e)l(quivalent.)648 1577 y(\(i\))g Fs(\012)k Fp(is)c(str)l(ongly)g(non-lo)l(c)l(al)g(over)g Fs(B)t Fp(.)648 1689 y(\(ii\))g Fv(\()p Fs(z)867 1701 y Fn(\012)925 1689 y Fv(\()p Fr(H)q Fv(\()p Fs(B)1127 1658 y Fn(c)1162 1689 y Fs(;)14 b(\012)t Fv(\)\)\))1364 1647 y Fl(?)1443 1689 y Fv(=)22 b Fr(f)p Fv(0)p Fr(g)29 b Fp(or,)h(e)l(quivalently,)1371 1883 y Fv(\()p 1403 1837 167 4 v(span)1569 1903 y Fg(C)1615 1883 y Fs(z)1654 1895 y Fn(\012)1712 1883 y Fv(\()p Fr(H)q Fv(\()p Fs(B)1914 1848 y Fn(c)1949 1883 y Fs(;)14 b(\012)t Fv(\)\)\))2151 1841 y Fl(?)2230 1883 y Fv(=)22 b Fr(f)p Fv(0)p Fr(g)p Fs(:)699 b Fv(\(46\))648 2065 y(Indeed,)28 b(that)h(Theorem)f(1)g (\(i\))h(and)g(\(ii\))g(are)f(equiv)-5 b(alen)n(t)28 b(no)n(w)g(follo)n(ws)g(from)g(The-)523 2165 y(orem)f(4)g(b)r(elo)n(w.) 523 2339 y Fp(Pr)l(o)l(of.)43 b Fv(\(Prop)r(osition)26 b(2\))g(It)i(is)e(easy)g(to)h(see)g(that)g Fs(\030)g Fr(2)c Fv(\()q Fs(z)2343 2351 y Fn(\012)2400 2339 y Fv(\()p Fr(H)q Fv(\()p Fs(B)2602 2309 y Fn(c)2637 2339 y Fs(;)14 b(\012)t Fv(\)\)\))2839 2298 y Fl(?)2922 2339 y Fv(if)27 b(and)g(only)523 2439 y(if)p 1432 2471 41 4 v 1432 2539 a Fs(\030)c Fr(\001)18 b Fs(\012)1600 2504 y Fq(1)p Fn(=)p Fq(2)1705 2539 y Fs(\021)26 b Fv(=)d(0)f(=)p 2012 2471 V 23 w Fs(\030)g Fr(\001)d Fs(\012)2180 2504 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2336 2539 y Fs(\021)s(;)523 2688 y Fv(for)24 b(all)h Fs(\021)h Fr(2)d(S)955 2700 y Fn(B)1008 2684 y Fc(c)1044 2688 y Fv(.)i(W)-7 b(e)25 b(can)f(supp)r(ose)h (without)g(loss)f(of)g(generalit)n(y)g(that)h Fs(\030)k Fv(is)24 b(real.)g(No)n(w,)523 2788 y(if)k Fs(\030)f Fr(2)d(K)q Fv(,)k(then)g Fs(\012)1113 2758 y Fq(1)p Fn(=)p Fq(2)1218 2788 y Fs(\030)f Fr(2)c(K)1422 2803 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1606 2788 y Fv(and)28 b(hence)1469 2986 y(0)23 b(=)g Fs(\030)f Fr(\001)d Fs(\012)1790 2952 y Fq(1)p Fn(=)p Fq(2)1894 2986 y Fs(\021)27 b Fv(=)22 b Fs(\012)2117 2952 y Fq(1)p Fn(=)p Fq(2)2222 2986 y Fs(\030)g Fr(\001)d Fs(\021)523 3169 y Fv(whic)n(h)25 b(pro)n(v)n(es)f Fs(\012)1081 3139 y Fq(1)p Fn(=)p Fq(2)1185 3169 y Fs(\030)30 b Fv(v)-5 b(anishes)24 b(outside)h Fs(B)t Fv(.)h(Similarly)f Fs(\012)2393 3139 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)2549 3169 y Fs(\030)30 b Fv(v)-5 b(anishes)24 b(outside)h Fs(B)t Fv(.)523 3269 y(Setting)j Fs(h)23 b Fv(=)g Fs(\012)1036 3239 y Fl(\000)p Fq(1)p Fn(=)p Fq(2)1192 3269 y Fs(\030)32 b Fv(the)c(result)f(follo)n(ws.)523 3443 y Fm(Theorem)k(4.)25 b Fp(L)l(et)k Fr(W)36 b Fp(b)l(e)30 b(a)g(r)l(e)l(al)h(subsp)l(ac)l(e)f(of)g Fr(V)7 b Fp(.)648 3543 y(\(i\))30 b(If)g Fs( )c Fr(2)d(F)1084 3513 y Fq(+)1139 3543 y Fv(\(\()p 1203 3497 167 4 v(span)1370 3563 y Fg(C)1417 3543 y Fr(W)7 b Fv(\))1538 3513 y Fl(?)1594 3543 y Fv(\))23 b Fr(\032)g(F)1805 3513 y Fq(+)1860 3543 y Fv(\()p Fr(V)7 b Fv(\))p Fp(,)30 b Fr(k)23 b Fs( )j Fr(k)p Fv(=)c(1)p Fp(,)30 b(then)1170 3725 y Fr(h)p Fs( )s Fr(j)p Fs(W)1360 3737 y Fq(F)1408 3725 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fs( )s Fr(i)24 b Fv(=)f Fr(h)p Fv(0)p Fr(j)p Fs(W)1911 3737 y Fq(F)1958 3725 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fs(;)184 b Fr(8)p Fs(\030)27 b Fr(2)d(W)7 b Fs(:)499 b Fv(\(47\))648 3908 y Fp(\(ii\))29 b(If)g Fv(span)1047 3928 y Fg(C)1094 3908 y Fr(W)35 b Fp(is)29 b(dense)g(in)g Fr(V)35 b Fp(then)28 b(ther)l(e)h(exist)f(no)h Fs( )d Fr(2)d(F)2639 3878 y Fq(\014n)p Fn(;)p Fq(+)2788 3908 y Fv(\()p Fr(V)7 b Fv(\))29 b Fp(other)g(than)523 4008 y Fr(j)p Fv(0)p Fr(i)h Fp(itself)g(so)g(that)g(\(47\))g(holds.)648 4182 y Fv(Clearly)-7 b(,)25 b(the)i(equiv)-5 b(alence)26 b(of)g(\(i\))h(and)f(\(ii\))i(in)e(Theorem)g(1)g(is)g(obtained)g(b)n(y) g(taking)523 4282 y Fr(W)k Fv(=)23 b Fs(z)762 4294 y Fn(\012)819 4282 y Fv(\()p Fr(H)q Fv(\()p Fs(B)1021 4251 y Fq(c)1056 4282 y Fs(;)14 b(\012)t Fv(\)\))28 b(in)g(the)g(ab)r(o)n(v) n(e)e(theorem)h(and)h(applying)f(Prop)r(osition)f(2.)523 4456 y Fp(Pr)l(o)l(of.)43 b Fv(As)32 b(a)g(w)n(arm-up,)f(let)h(us)g (pro)n(v)n(e)f(that,)h(if)h Fr(j)p Fs( )s Fr(i)e Fv(=)f Fs(a)2394 4426 y Fl(y)2428 4456 y Fv(\()p Fs(\030)2500 4426 y Fl(0)2524 4456 y Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fv(,)i(for)g(some)f Fs(\030)3092 4426 y Fl(0)3146 4456 y Fr(2)g(V)7 b Fv(,)523 4556 y(then)31 b(\(47\))f(holds)g(if)h(and)f (only)g(if)h Fs(\030)1660 4525 y Fl(0)1711 4556 y Fr(2)d Fv(\()p 1826 4510 V(span)1993 4576 y Fg(C)2039 4556 y Fr(W)7 b Fv(\))2160 4525 y Fl(?)2246 4556 y Fv(and)31 b Fr(k)c Fs(\030)2520 4525 y Fl(0)2571 4556 y Fr(k)p Fv(=)f(1.)k(Indeed,)h(for)f(all)523 4655 y Fs(\030)d Fr(2)d(W)7 b Fv(,)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 2040 100 a Fu(Lo)r(cal)27 b(states)g(of)f(free) h(b)r(ose)f(\014elds)190 b(43)662 285 y Fr(h)p Fs( )s Fr(j)p Fs(W)852 297 y Fq(F)900 285 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fs( )s Fr(i)26 b Fv(=)e Fr(h)p Fv(0)p Fr(j)p Fs(a)p Fv(\()p Fs(\030)1444 251 y Fl(0)1468 285 y Fv(\)[1)-18 b(l)18 b(+)g Fs(a)1715 251 y Fl(y)1749 285 y Fv(\()p Fs(\030)t Fv(\)][1)-18 b(l)19 b Fr(\000)f Fs(a)p Fv(\()p Fs(\030)t Fv(\)])p Fs(a)2263 251 y Fl(y)2298 285 y Fv(\()p Fs(\030)2370 251 y Fl(0)2394 285 y Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fv(e)2560 251 y Fl(\000)2622 229 y Ff(1)p 2622 238 29 3 v 2622 271 a(2)2660 251 y Fl(k)p Fn(\030)r Fl(k)2760 226 y Ff(2)1142 425 y Fv(=)24 b Fr(h)p Fv(0)p Fr(j)p Fs(W)1406 437 y Fq(F)1454 425 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fv(0)p Fr(i)f(k)g Fs(\030)1783 391 y Fl(0)1829 425 y Fr(k)1871 391 y Fq(2)1931 425 y Fr(\000h)p Fv(0)p Fr(j)p Fs(a)p Fv(\()p Fs(\030)2209 391 y Fl(0)2232 425 y Fv(\))p Fs(a)2308 391 y Fl(y)2343 425 y Fv(\()p Fs(\030)t Fv(\))p Fs(a)p Fv(\()p Fs(\030)t Fv(\))p Fs(a)2639 391 y Fl(y)2675 425 y Fv(\()p Fs(\030)2747 391 y Fl(0)2771 425 y Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fv(e)2937 391 y Fl(\000)2998 369 y Ff(1)p 2998 378 V 2998 411 a(2)3037 391 y Fl(k)p Fn(\030)r Fl(k)3137 366 y Ff(2)1142 575 y Fv(=)h Fr(h)p Fv(0)p Fr(j)p Fs(W)1406 587 y Fq(F)1454 575 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fv(0)p Fr(i)1669 483 y Fk(h)1708 575 y Fr(k)f Fs(\030)1813 541 y Fl(0)1859 575 y Fr(k)1901 541 y Fq(2)1961 575 y Fr(\000)p Fv(\()p 2058 507 41 4 v Fs(\030)2098 521 y Fl(0)2140 575 y Fr(\001)18 b Fs(\030)t Fv(\)\()p 2285 507 V Fs(\030)24 b Fr(\001)18 b Fs(\030)2426 541 y Fl(0)2450 575 y Fv(\))2482 483 y Fk(i)2535 575 y Fs(:)523 791 y Fv(Supp)r(osing)25 b(\(47\))f(holds,)g (this)h(clearly)f(implies)h Fr(k)d Fs(\030)2134 760 y Fl(0)2181 791 y Fr(k)p Fv(=)g(1)i(and)h Fs(\030)2575 760 y Fl(0)2621 791 y Fr(2)f Fv(\()p 2732 745 167 4 v(span)2898 811 y Fg(C)2945 791 y Fr(W)7 b Fv(\))3066 760 y Fl(?)3122 791 y Fv(.)25 b(The)523 890 y(con)n(v)n(erse)30 b(is)h(equally)g(ob)n (vious.)g(This)h(pro)n(v)n(es)e(the)i(theorem)f(for)g(the)h(v)n(ery)f (particular)523 990 y(case)40 b(of)h(states)g(con)n(taining)f(exactly)h (one)g(quan)n(tum.)g(Note)g(that)g(this)h(completely)523 1089 y(c)n(haracterizes)25 b(the)j(states)f(with)h(exactly)f(one)h (\014eld)f(quan)n(tum)h(that)g(are)e(\\lo)r(calized".)648 1189 y(T)-7 b(o)27 b(pro)n(v)n(e)f(part)h(\(i\),)h(w)n(e)f(can)h(no)n (w)f(pro)r(ceed)g(as)f(follo)n(ws.)h(Recall)g(that)1396 1389 y Fs(W)1474 1401 y Fq(F)1521 1389 y Fv(\()p Fs(\030)t Fv(\))d(=)f(e)1774 1355 y Fl(\000)1836 1333 y Ff(1)p 1836 1342 29 3 v 1836 1375 a(2)1874 1355 y Fl(k)p Fn(\030)r Fl(k)1974 1330 y Ff(2)2011 1389 y Fv(e)2048 1355 y Fn(a)2084 1330 y Fb(y)2115 1355 y Fq(\()p Fn(\030)r Fq(\))2204 1389 y Fv(e)2241 1355 y Fl(\000)p Fn(a)p Fq(\()p Fn(\030)r Fq(\))2417 1389 y Fs(:)523 1572 y Fv(Let)30 b Fs(\030)i Fr(2)c(W)7 b Fv(.)30 b(Supp)r(ose)g Fs( )g Fv(=)d(\()p Fs( )1557 1584 y Fq(0)1595 1572 y Fs(;)14 b( )1686 1584 y Fq(1)1723 1572 y Fs(;)g( )1814 1584 y Fq(2)1851 1572 y Fs(;)g(:)g(:)g(:)g(;)g( )2090 1584 y Fn(N)2153 1572 y Fs(;)g Fv(0)p Fs(;)g Fv(0)p Fs(;)g(:)g(:)g(:)n Fv(\))28 b Fr(2)g(F)2654 1542 y Fq(\014n)o Fn(;)p Fq(+)2803 1572 y Fv(\(\()p 2867 1526 167 4 v(span)3034 1592 y Fg(C)3080 1572 y Fr(W)7 b Fv(\))3201 1542 y Fl(?)3257 1572 y Fv(\).)523 1672 y(Then)1112 1771 y Fr(h)p Fs( )s Fr(j)p Fs(W)1302 1783 y Fq(F)1349 1771 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fs( )s Fr(i)25 b Fv(=)d(e)1714 1737 y Fl(\000)1776 1715 y Ff(1)p 1776 1724 29 3 v 1776 1757 a(2)1814 1737 y Fl(k)p Fn(\030)r Fl(k)1914 1712 y Ff(2)1951 1771 y Fr(h)p Fs( )s(;)14 b( )s Fr(i)24 b Fv(=)e Fr(h)p Fv(0)p Fr(j)p Fs(W)2452 1783 y Fq(F)2499 1771 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fv(0)p Fr(i)p Fs(:)523 1921 y Fv(Indeed,)35 b(as)f(a)g(result)h(of)f (the)h(fact)g(that)g Fs(\030)k Fr(2)c(W)42 b Fv(and)35 b Fs( )j Fr(2)d(F)2564 1891 y Fq(\014n)o Fn(;)p Fq(+)2713 1921 y Fv(\(\()p 2777 1875 167 4 v(span)2944 1941 y Fg(C)2990 1921 y Fr(W)7 b Fv(\))3111 1891 y Fl(?)3167 1921 y Fv(\),)35 b(it)523 2020 y(follo)n(ws)c(that)i Fs(a)p Fv(\()p Fs(\030)t Fv(\))p Fs( )h Fv(=)c(0)i(so)g(that)g(e)1718 1990 y Fl(\000)p Fn(a)p Fq(\()p Fn(\030)r Fq(\))1894 2020 y Fs( )i Fv(=)c Fs( )s Fv(.)j(F)-7 b(rom)31 b(this)i(one)f(can)f(conclude)h(as)523 2120 y(follo)n(ws.)27 b(F)-7 b(or)27 b(an)n(y)g Fs( )f Fv(=)d(\()p Fs( )1379 2132 y Fq(0)1417 2120 y Fs(;)14 b(:)g(:)g(:)f(;)h( )1655 2132 y Fn(n)1701 2120 y Fs(;)g(:)g(:)g(:)o Fv(\))24 b Fr(2)g(F)2037 2090 y Fq(+)2092 2120 y Fv(\(\()p 2156 2074 V(span)2323 2140 y Fg(C)2369 2120 y Fr(W)7 b Fv(\))2490 2090 y Fl(?)2546 2120 y Fv(\))28 b(and)g(for)f(an)n(y)g Fs(N)32 b Fr(2)24 b Fo(N)p Fv(,)523 2220 y(w)n(e)j(can)g(write)1614 2319 y Fs( )f Fv(=)d Fs( )1836 2331 y Fn(N)523 2469 y Fv(where)29 b Fs( )819 2481 y Fn()g Fv(0,)j(there)h (exists)f Fs(N)3113 2481 y Fn(\017)3171 2469 y Fr(2)e Fo(N)523 2568 y Fv(so)g(that)725 2751 y Fr(h)p Fs( )s Fr(j)p Fs(W)915 2763 y Fq(F)963 2751 y Fv(\()p Fs(\030)t Fv(\))p Fr(j)p Fs( )s Fr(i)d Fv(=)f Fr(h)p Fs( )1377 2763 y Fn(