Content-Type: multipart/mixed; boundary="-------------0106050727560" This is a multi-part message in MIME format. ---------------0106050727560 Content-Type: text/plain; name="01-210.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-210.keywords" Quantum Electrodynamics, Multipolar Hamiltonians, Moving ions and atoms. ---------------0106050727560 Content-Type: application/postscript; name="robert.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="robert.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: robert.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o robert.ps robert.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.06.05:1402 %%BeginProcSet: texc.pro %! /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 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v 718 4960 a(2)p Fl(m)852 4975 y Fj(1)919 4746 y Fh( )984 4892 y Fl(p)22 b Fm(+)1163 4824 y Fl(m)1248 4839 y Fj(1)p 1163 4868 125 4 v 1173 4960 a Fl(M)1298 4892 y(P)36 b Fg(\000)1506 4824 y Fm(1)p 1506 4868 49 4 v 1506 4960 a(2)1565 4892 y(\()p Fl(e)1648 4907 y Fj(1)1710 4892 y Fg(\000)1809 4770 y Fh(\022)1880 4824 y Fl(m)1965 4839 y Fj(1)p 1880 4868 125 4 v 1890 4960 a Fl(M)2015 4770 y Fh(\023)2076 4793 y Fj(2)2149 4892 y Fl(e)2194 4907 y Fk(T)2249 4892 y Fm(\))17 b Fl(B)2378 4907 y Fj(0)2439 4892 y Fg(^)23 b Fl(x)f Fg(\000)2715 4824 y Fl(m)2800 4839 y Fj(1)p 2715 4868 V 2725 4960 a Fl(M)2876 4824 y(e)2921 4839 y Fk(T)p 2876 4868 101 4 v 2902 4960 a Fm(2)3003 4892 y Fl(B)3077 4907 y Fj(0)3139 4892 y Fg(^)g Fl(R)718 5232 y Fg(\000)17 b Fl(e)857 5247 y Fk(T)939 5165 y Fl(m)1024 5180 y Fj(1)p 939 5209 125 4 v 949 5300 a Fl(M)1091 5232 y(A)p Fm(\()p Fl(R)q Fm(\))22 b(+)1435 5115 y Fh(Z)1534 5232 y Fl(d)1585 5191 y Fj(3)1625 5232 y Fl(r)d Fm(\002)1764 5247 y Fj(1)1803 5232 y Fm(\()p Fl(r)s Fm(\))j Fg(^)g Fl(B)5 b Fm(\()p Fl(r)s Fm(\))2238 5111 y Fh(\023)2299 5134 y Fj(2)708 5556 y Fm(+)873 5488 y(1)p 811 5532 174 4 v 811 5624 a(2)p Fl(m)945 5639 y Fj(2)1011 5410 y Fh( )1077 5556 y Fg(\000)p Fl(p)22 b Fm(+)1333 5488 y Fl(m)1418 5503 y Fj(2)p 1333 5532 125 4 v 1343 5624 a Fl(M)1468 5556 y(P)36 b Fm(+)1675 5488 y(1)p 1675 5532 49 4 v 1675 5624 a(2)1733 5556 y(\()p Fl(e)1816 5571 y Fj(2)1878 5556 y Fg(\000)1978 5435 y Fh(\022)2049 5488 y Fl(m)2134 5503 y Fj(2)p 2049 5532 125 4 v 2059 5624 a Fl(M)2183 5435 y Fh(\023)2245 5457 y Fj(2)2317 5556 y Fl(e)2362 5571 y Fk(T)2417 5556 y Fm(\))17 b Fl(B)2546 5571 y Fj(0)2608 5556 y Fg(^)22 b Fl(x)h Fg(\000)2883 5488 y Fl(m)2968 5503 y Fj(2)p 2883 5532 V 2893 5624 a Fl(M)3045 5488 y(e)3090 5503 y Fk(T)p 3045 5532 101 4 v 3070 5624 a Fm(2)3171 5556 y Fl(B)3245 5571 y Fj(0)3307 5556 y Fg(^)g Fl(R)1865 5871 y Fm(6)p eop %%Page: 7 7 7 6 bop 718 330 a Fg(\000)17 b Fl(e)857 345 y Fk(T)939 263 y Fl(m)1024 278 y Fj(2)p 939 307 125 4 v 949 398 a Fl(M)1091 330 y(A)p Fm(\()p Fl(R)q Fm(\))22 b(+)1435 213 y Fh(Z)1534 330 y Fl(d)1585 289 y Fj(3)1625 330 y Fl(r)d Fm(\002)1764 345 y Fj(2)1803 330 y Fm(\()p Fl(r)s Fm(\))j Fg(^)g Fl(B)5 b Fm(\()p Fl(r)s Fm(\))2238 209 y Fh(\023)2299 232 y Fj(2)3606 330 y Fm(\(16\))708 615 y(+)874 547 y Fl(e)919 562 y Fj(1)975 547 y Fl(e)1020 562 y Fj(2)p 811 592 312 4 v 811 683 a Fm(4)p Fl(\031)20 b(")981 698 y Fj(0)1020 683 y Fg(j)p Fl(r)s Fg(j)1154 615 y(\000)j Fl(M)1348 630 y Fj(1)1410 615 y Fg(\001)f Fl(B)5 b Fm(\()p Fl(r)1621 630 y Fj(1)1660 615 y Fm(\))22 b Fg(\000)h Fl(M)1914 630 y Fj(2)1976 615 y Fg(\001)f Fl(B)5 b Fm(\()p Fl(r)2187 630 y Fj(2)2226 615 y Fm(\))708 957 y(+)811 890 y(1)p 811 934 49 4 v 811 1025 a(2)886 840 y Fh(Z)986 957 y Fl(d)1037 916 y Fj(3)1076 957 y Fl(r)1139 811 y Fh( )1215 890 y Fm(\005\()p Fl(r)s Fm(\))1411 854 y Fj(2)p 1215 934 236 4 v 1290 1025 a Fl(")1336 1040 y Fj(0)1482 957 y Fm(+)1590 890 y Fl(B)g Fm(\()p Fl(r)s Fm(\))1792 854 y Fj(2)p 1590 934 242 4 v 1662 1025 a Fl(\026)1721 1040 y Fj(0)1841 811 y Fh(!)1929 957 y Fm(+)2027 840 y Fh(Z)2127 957 y Fl(d)2178 916 y Fj(3)2217 957 y Fl(r)2290 890 y Fm(\005\()p Fl(r)s Fm(\))22 b Fg(\001)g Fl(P)14 b Fm(\()p Fl(r)s Fm(\))p 2290 934 467 4 v 2481 1025 a Fl(")2527 1040 y Fj(0)2789 957 y Fm(+)2939 890 y(1)p 2897 934 134 4 v 2897 1025 a(2)p Fl(")2992 1040 y Fj(0)3057 840 y Fh(Z)3157 957 y Fl(d)3208 916 y Fj(3)3247 957 y Fl(r)19 b(P)14 b Fm(\()p Fl(r)s Fm(\))3510 916 y Fj(2)0 1239 y Fm(In)26 b(order)f(to)g(get)g(the)h(electric-dip)s(ole) d(appro)m(ximation)g(of)i(the)g(Hamiltonians)d(\(2.14\))j(and)g (\(2.16\),)h(w)m(e)0 1360 y(ha)m(v)m(e)i(to)f(in)m(tro)s(duce)g(the)g (electric-dip)s(ole)e(appro)m(ximation)f(of)j Fl(P)14 b Fm(\()p Fl(r)s Fm(\))26 b(and)h(\002)2780 1375 y Fk(\013)2829 1360 y Fm(\()p Fl(r)s Fm(\).)41 b(These)29 b(appro)m(xima-)0 1480 y(tions)h(are)h(obtained)g(from)f(\(2.6\))g(and)h(\(2.10\))g(b)m (y)h(retaining)d(the)i(leading)f(terms)h(in)f(the)i(expansions)0 1600 y(of)e(the)g(Dirac)f Fl(\016)34 b Fm(distributions)29 b(app)s(earing)g(in)g(the)i(v)m(ectors)h Fl(P)14 b Fm(\()p Fl(r)s Fm(\))29 b(and)h(\002)2729 1615 y Fk(\013)2779 1600 y Fm(\()p Fl(r)s Fm(\))f(in)h(p)s(o)m(w)m(ers)h(of)f Fl(r)3516 1615 y Fk(\013)3583 1600 y Fg(\000)17 b Fl(R)q Fm(.)0 1754 y(Let)28 b Fl(P)233 1769 y Fk(dip)333 1754 y Fm(\()p Fl(r)s Fm(\))g(and)g(\002)745 1769 y Fk(\013;dip)910 1754 y Fm(\()p Fl(r)s Fm(\))g(denote)h(the)g(dip)s(ole)d(appro)m (ximation)g(of)i Fl(P)14 b Fm(\()p Fl(r)s Fm(\))27 b(and)h(\002)3061 1769 y Fk(\013)3111 1754 y Fm(\()p Fl(r)s Fm(\))f(resp)s(ectiv)m(ely)-8 b(.)0 1874 y(W)g(e)33 b(ha)m(v)m(e,)h(for)e(an)m(y)h(system)h(of)e Fl(N)43 b Fm(c)m(harges,)1091 2163 y Fl(P)1154 2178 y Fk(dip)1253 2163 y Fm(\()p Fl(r)s Fm(\))28 b(=)1507 2017 y Fh( )1609 2055 y Fk(N)1581 2080 y Fh(X)1573 2262 y Fk(\013)p Fj(=1)1725 2163 y Fl(e)1770 2178 y Fk(\013)1820 2163 y Fm(\()p Fl(r)1902 2178 y Fk(\013)1973 2163 y Fg(\000)23 b Fl(R)q Fm(\)\))2224 2017 y Fh(!)2323 2163 y Fl(\016)t Fm(\()p Fl(r)h Fg(\000)f Fl(R)q Fm(\))917 b(\(17\))492 2552 y(\002)568 2567 y Fk(\013;dip)733 2552 y Fm(\()p Fl(r)s Fm(\))27 b(=)987 2381 y Fh(0)987 2531 y(@)1095 2444 y Fk(N)1066 2469 y Fh(X)1060 2654 y Fk(\014)s Fj(=1)1209 2552 y Fl(e)1254 2567 y Fk(\014)1302 2552 y Fm(\()p Fl(r)1384 2567 y Fk(\014)1453 2552 y Fg(\000)c Fl(R)q Fm(\)\))1721 2435 y Fh(Z)1803 2461 y Fj(1)1766 2624 y(0)1859 2552 y Fl(d\025)p Fm(\()p Fl(\025\016)2105 2567 y Fk(\013\014)2219 2552 y Fg(\000)2329 2485 y Fl(m)2414 2500 y Fk(\013)p 2329 2529 135 4 v 2344 2620 a Fl(M)2474 2552 y Fm(\()p Fl(\025)f Fg(\000)g Fm(1\)\))2815 2381 y Fh(1)2815 2531 y(A)2921 2552 y Fl(\016)t Fm(\()p Fl(r)j Fg(\000)d Fl(R)q Fm(\))319 b(\(18\))0 2807 y(Let)1492 2977 y Fl(d)28 b Fm(=)1710 2869 y Fk(N)1682 2894 y Fh(X)1674 3075 y Fk(\013)p Fj(=1)1826 2977 y Fl(e)1871 2992 y Fk(\013)1921 2977 y Fm(\()p Fl(r)2003 2992 y Fk(\013)2075 2977 y Fg(\000)22 b Fl(R)q Fm(\))1319 b(\(19\))0 3215 y Fl(d)32 b Fm(is)g(the)h (electric-dip)s(ole)d(momen)m(t)i(v)m(ector)i(of)e(the)h(system)g(of)f Fl(N)43 b Fm(c)m(harges.)0 3401 y(Th)m(us,)34 b(for)e Fl(N)38 b Fm(=)28 b(2,)k(w)m(e)i(get)f Fl(d)27 b Fm(=)1239 3280 y Fh(\022)1311 3334 y Fl(e)1356 3349 y Fj(1)1395 3334 y Fl(m)1480 3349 y Fj(2)1542 3334 y Fg(\000)c Fl(e)1687 3349 y Fj(2)1726 3334 y Fl(m)1811 3349 y Fj(1)p 1311 3378 541 4 v 1528 3470 a Fl(M)1861 3280 y Fh(\023)1955 3401 y Fl(x)p Fm(.)0 3587 y(W)-8 b(e)33 b(then)g(ha)m(v)m(e)616 3807 y Fl(P)679 3822 y Fk(dip)778 3807 y Fm(\()p Fl(r)s Fm(\))28 b(=)f Fl(d)17 b(\016)t Fm(\()p Fl(r)24 b Fg(\000)f Fl(R)q Fm(\))2140 b(\(20\))616 4066 y(\002)692 4081 y Fk(\013;dip)856 4066 y Fm(\()p Fl(r)s Fm(\))28 b(=)1120 3998 y(1)p 1120 4042 49 4 v 1120 4134 a(2)1195 3945 y Fh(\022)1257 4066 y Fl(e)1302 4081 y Fk(\013)1351 4066 y Fm(\()p Fl(r)1433 4081 y Fk(\013)1505 4066 y Fg(\000)22 b Fl(R)q Fm(\))h(+)1848 3998 y Fl(m)1933 4013 y Fk(\013)p 1848 4042 135 4 v 1863 4134 a Fl(M)2009 4066 y(d)2060 3945 y Fh(\023)2154 4066 y Fl(\016)t Fm(\()p Fl(r)i Fg(\000)d Fl(R)q Fm(\))1086 b(\(21\))644 4304 y(1)p 626 4348 86 4 v 626 4440 a Fl(")672 4455 y Fj(0)737 4255 y Fh(Z)837 4372 y Fl(d)888 4331 y Fj(3)927 4372 y Fl(r)19 b Fm(\005\()p Fl(r)s Fm(\))j Fg(\001)g Fl(P)1321 4387 y Fk(dip)1421 4372 y Fm(\()p Fl(r)s Fm(\))27 b(=)g Fg(\000)p Fl(d)c Fg(\001)f Fl(E)6 b Fm(\()p Fl(R)q Fm(\))1502 b(\(22\))616 4569 y Fh(Z)715 4686 y Fl(d)766 4645 y Fj(3)805 4686 y Fl(r)20 b Fm(\002)945 4701 y Fk(\013;dip)1109 4686 y Fm(\()p Fl(r)s Fm(\))i Fg(^)h Fl(B)5 b Fm(\()p Fl(r)s Fm(\))27 b(=)1686 4619 y(1)p 1686 4663 49 4 v 1686 4755 a(2)1744 4686 y Fl(e)1789 4701 y Fk(\013)1839 4686 y Fm(\()p Fl(r)1921 4701 y Fk(\013)1992 4686 y Fg(\000)c Fl(R)q Fm(\))f Fg(^)h Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b(+)2676 4619 y(1)p 2676 4663 V 2676 4755 a(2)2777 4619 y Fl(m)2862 4634 y Fk(\013)p 2777 4663 135 4 v 2792 4755 a Fl(M)2939 4686 y(d)f Fg(^)i Fl(B)5 b Fm(\()p Fl(R)q Fm(\))276 b(\(23\))0 4935 y(Hence)34 b(the)f(dip)s(ole)e(appro)m (ximation)f(of)i Fl(H)1598 4899 y Fi(0)1590 4960 y Fk(mul)q(t)1745 4935 y Fm(,)g(denoted)i(b)m(y)f Fl(H)2397 4899 y Fi(0)2389 4960 y Fk(mul)q(t;dip)2660 4935 y Fm(,)f(is)g(giv)m(en)h(b)m(y)525 5230 y Fl(H)614 5189 y Fi(0)606 5255 y Fk(mul)q(t;dip)904 5230 y Fm(=)1079 5163 y(1)p 1017 5207 174 4 v 1017 5299 a(2)p Fl(m)1151 5314 y Fj(1)1217 5084 y Fh( )1283 5230 y Fl(p)22 b Fm(+)1462 5163 y Fl(m)1547 5178 y Fj(1)p 1462 5207 125 4 v 1472 5299 a Fl(M)1613 5230 y(P)36 b Fg(\000)1821 5163 y Fm(1)p 1821 5207 49 4 v 1821 5299 a(2)1897 5084 y Fh( )1963 5230 y Fl(e)2008 5245 y Fj(1)2069 5230 y Fg(\000)2169 5109 y Fh(\022)2240 5163 y Fl(m)2325 5178 y Fj(1)p 2240 5207 125 4 v 2250 5299 a Fl(M)2375 5109 y Fh(\023)2436 5132 y Fj(2)2492 5230 y Fl(e)2537 5245 y Fk(T)2592 5084 y Fh(!)2691 5230 y Fl(B)2765 5245 y Fj(0)2827 5230 y Fg(^)22 b Fl(x)525 5571 y Fg(\000)44 b Fl(e)691 5586 y Fk(T)756 5503 y Fl(m)841 5518 y Fj(1)p 756 5547 V 766 5639 a Fl(M)890 5571 y(A)p Fm(\()p Fl(R)q Fm(\))23 b Fg(\000)1246 5503 y Fl(m)1331 5518 y Fj(1)p 1246 5547 V 1256 5639 a Fl(M)1391 5503 y(e)1436 5518 y Fk(T)p 1391 5547 101 4 v 1417 5639 a Fm(2)1501 5571 y Fl(B)1575 5586 y Fj(0)1637 5571 y Fg(^)f Fl(R)i Fg(\000)1932 5503 y Fl(e)1977 5518 y Fj(1)p 1932 5547 85 4 v 1950 5639 a Fm(2)2069 5503 y Fl(m)2154 5518 y Fj(2)p 2069 5547 125 4 v 2079 5639 a Fl(M)2221 5571 y(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)g Fl(x)h Fg(\000)2763 5503 y Fl(m)2848 5518 y Fj(1)p 2749 5547 154 4 v 2749 5639 a Fm(2)p Fl(M)2912 5571 y(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)h Fl(d)3304 5449 y Fh(\023)3365 5472 y Fj(2)1865 5871 y Fm(7)p eop %%Page: 8 8 8 7 bop 525 339 a Fm(+)690 271 y(1)p 627 315 174 4 v 627 407 a(2)p Fl(m)761 422 y Fj(2)827 192 y Fh( )893 339 y Fg(\000)p Fl(p)23 b Fm(+)1150 271 y Fl(m)1235 286 y Fj(2)p 1150 315 125 4 v 1160 407 a Fl(M)1301 339 y(P)36 b Fm(+)1508 271 y(1)p 1508 315 49 4 v 1508 407 a(2)1583 192 y Fh( )1649 339 y Fl(e)1694 354 y Fj(2)1755 339 y Fg(\000)1855 217 y Fh(\022)1926 271 y Fl(m)2011 286 y Fj(2)p 1926 315 125 4 v 1936 407 a Fl(M)2061 217 y Fh(\023)2122 240 y Fj(2)2178 339 y Fl(e)2223 354 y Fk(T)2278 192 y Fh(!)2377 339 y Fl(B)2451 354 y Fj(0)2513 339 y Fg(^)22 b Fl(x)h Fg(\000)2789 271 y Fl(m)2874 286 y Fj(2)p 2789 315 V 2799 407 a Fl(M)2950 271 y(e)2995 286 y Fk(T)p 2950 315 101 4 v 2976 407 a Fm(2)3077 339 y Fl(B)3151 354 y Fj(0)3212 339 y Fg(^)g Fl(R)525 679 y Fg(\000)44 b Fl(e)691 694 y Fk(T)756 611 y Fl(m)841 626 y Fj(2)p 756 656 125 4 v 766 747 a Fl(M)890 679 y(A)p Fm(\()p Fl(R)q Fm(\))23 b(+)1245 611 y Fl(e)1290 626 y Fj(2)1329 611 y Fl(m)1414 626 y Fj(1)p 1245 656 210 4 v 1273 747 a Fm(2)p Fl(M)1481 679 y(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)g Fl(x)h Fg(\000)2023 611 y Fl(m)2108 626 y Fj(2)p 2009 656 154 4 v 2009 747 a Fm(2)p Fl(M)2172 679 y(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)h Fl(d)2564 558 y Fh(\023)2625 581 y Fj(2)525 897 y Fm(+)698 830 y Fl(e)743 845 y Fj(1)783 830 y Fl(e)828 845 y Fj(2)p 627 874 312 4 v 627 965 a Fm(4)p Fl(\031)e(")798 980 y Fj(0)837 965 y Fg(j)p Fl(r)s Fg(j)971 897 y(\000)h Fm(\()p Fl(M)1202 912 y Fj(1)1264 897 y Fm(+)g Fl(M)1456 912 y Fj(2)1496 897 y Fm(\))g Fg(\001)g Fl(B)5 b Fm(\()p Fl(R)q Fm(\))1770 b(\(24\))525 1239 y(+)627 1172 y(1)p 627 1216 49 4 v 627 1308 a(2)703 1122 y Fh(Z)802 1239 y Fl(d)853 1198 y Fj(3)892 1239 y Fl(r)972 1093 y Fh( )1048 1172 y Fm(\005\()p Fl(r)s Fm(\))1244 1136 y Fj(2)p 1048 1216 236 4 v 1123 1308 a Fl(")1169 1323 y Fj(0)1315 1239 y Fm(+)1423 1172 y Fl(B)5 b Fm(\()p Fl(r)s Fm(\))1625 1136 y Fj(2)p 1423 1216 242 4 v 1495 1308 a Fl(\026)1554 1323 y Fj(0)1674 1093 y Fh(!)1762 1239 y Fm(+)1860 1122 y Fh(Z)1960 1239 y Fl(d)2011 1198 y Fj(3)2050 1239 y Fl(r)2123 1172 y Fm(\005\()p Fl(r)s Fm(\))22 b Fg(\001)g Fl(P)14 b Fm(\()p Fl(r)s Fm(\))p 2123 1216 467 4 v 2314 1308 a Fl(")2360 1323 y Fj(0)2622 1239 y Fm(+)2772 1172 y(1)p 2730 1216 134 4 v 2730 1308 a(2)p Fl(")2825 1323 y Fj(0)2890 1122 y Fh(Z)2990 1239 y Fl(d)3041 1198 y Fj(3)3080 1239 y Fl(r)s(P)g Fm(\()p Fl(r)s Fm(\))3327 1198 y Fj(2)3365 1239 y 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y(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)h Fl(d)k Fm(=)2083 1914 y Fh( )2158 1993 y Fl(e)2203 2008 y Fj(2)p 2158 2037 85 4 v 2176 2129 a Fm(2)2275 2060 y Fg(\000)2385 1993 y Fm(1)p 2385 2037 49 4 v 2385 2129 a(2)2480 1993 y Fl(m)2565 1957 y Fj(2)2565 2018 y(2)p 2470 2037 145 4 v 2470 2129 a Fl(M)2574 2100 y Fj(2)2641 2060 y Fl(e)2686 2075 y Fk(T)2741 1914 y Fh(!)2840 2060 y Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)h Fl(x)370 b Fm(\(25\))0 2306 y(In)32 b(the)f(particular)f(case)i(where)h Fl(B)1296 2321 y Fj(0)1363 2306 y Fm(=)27 b(0)k(w)m(e)i(get)e(the)h(follo)m(wing) c(electric-dip)s(ole)h(appro)m(ximation)g(of)0 2427 y(the)k(m)m(ultip)s (olar)c(Hamiltonian)g(:)323 2682 y Fl(H)404 2697 y Fk(mul)q(t;dip)758 2682 y Fm(=)946 2614 y Fl(P)1023 2578 y Fj(2)p 927 2659 154 4 v 927 2750 a Fm(2)p Fl(M)1113 2682 y Fm(+)1244 2614 y Fl(p)1293 2578 y Fj(2)p 1221 2659 135 4 v 1221 2750 a Fm(2)p Fl(m)1387 2682 y Fg(\000)1614 2614 y Fl(e)1659 2578 y Fj(2)p 1497 2659 321 4 v 1497 2750 a Fm(4)p Fl(\031)20 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y(Hamiltonians,)d(w)m(e)i (ha)m(v)m(e)g(to)f(deal)f(with)h(the)g(b)s(eha)m(viour)g(of)g(the)g (electric)g(and)g(magnetic)f(dip)s(oles)0 4119 y(for)d(large)g Fg(j)p Fl(x)p Fg(j)p Fm(.)55 b(This)37 b(means)f(that)h(the)g(m)m (ultip)s(olar)c(Hamiltonians)g(are)k(more)f(con)m(v)m(enien)m(t)i(for)e (the)0 4239 y(description)28 b(of)h(phenomena)g(asso)s(ciated)f(with)h (con\014ned)h(systems)g(of)e(c)m(harges)i(with)f(resp)s(ect)h(to)e(the) 0 4359 y(in)m(ternal)j(v)-5 b(ariables.)0 4513 y(Let)26 b(us)g(no)m(w)g(consider)g(the)g(three)h(b)s(o)s(dy)e(system.)43 b(W)-8 b(e)26 b(restrict)f(ourselv)m(es)i(to)e(the)i(case)f(of)f(the)h (helium)0 4633 y(atom,)35 b(sa)m(y)h Fl(H)540 4597 y Fj(4)532 4658 y Fk(e)579 4633 y Fm(,)f(for)g(simplicit)m(y)-8 b(.)48 b(Here)36 b Fl(m)1593 4648 y Fj(1)1668 4633 y Fm(is)e(the)i(mass)f(of)f(the)i(electron)f(and)g Fl(m)3113 4648 y Fj(2)3188 4633 y Fm(is)f(the)i(mass)f(of)0 4754 y(the)e(n)m(ucleus.)0 4907 y(W)-8 b(e)33 b(no)m(w)g(in)m(tro)s(duce)f 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Fm(Then)35 b(it)e(is)h(straigh)m(tforw)m(ard)f(to)h(get)g(the)h(m)m (ultip)s(olar)30 b(Hamiltonian)h(for)i Fl(H)2877 4818 y Fj(4)2869 4879 y Fk(e)2950 4854 y Fm(without)h(an)g(exterior)0 4974 y(electromagnetic)d(\014eld,)i(in)e(the)i(dip)s(ole)e(appro)m (ximation,)g(using)h(\(2.23\))g(:)63 5241 y Fl(H)144 5258 y Fk(H)207 5239 y Fe(4)202 5274 y Ff(e)241 5258 y Fk(;dip)444 5241 y Fm(=)675 5174 y(1)p 613 5218 174 4 v 613 5309 a(2)p Fl(m)747 5324 y Fj(1)813 5120 y Fh(\022)884 5174 y Fl(m)969 5189 y Fj(1)p 884 5218 125 4 v 894 5309 a Fl(M)1019 5241 y(P)j Fg(\000)1227 5174 y Fm(1)p 1227 5218 49 4 v 1227 5309 a(2)1286 5241 y Fl(p)1335 5200 y Fk(a)1398 5241 y Fg(\000)23 b Fl(p)1547 5256 y Fk(a)1611 5241 y Fg(\000)1722 5174 y Fl(e)p 1720 5218 V 1720 5309 a Fm(2)1796 5120 y Fh(\022)1867 5174 y Fl(m)1952 5189 y Fj(2)p 1867 5218 125 4 v 1877 5309 a Fl(M)2002 5241 y(r)2049 5200 y Fk(a)2112 5241 y Fm(+)2220 5174 y(1)p 2220 5218 49 4 v 2220 5309 a(2)2279 5241 y Fl(r)2323 5256 y Fk(a)2364 5120 y Fh(\023)2448 5241 y Fg(^)f Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(\000)2898 5174 y Fl(em)3028 5189 y Fj(1)p 2898 5218 170 4 v 2931 5309 a Fl(M)3078 5241 y(r)3125 5200 y Fk(a)3188 5241 y Fg(^)h Fl(B)5 b Fm(\()p Fl(R)q Fm(\))3507 5120 y Fh(\023)3568 5136 y Fj(2)603 5572 y Fm(+)768 5505 y(1)p 705 5549 174 4 v 705 5640 a(2)p Fl(m)839 5655 y Fj(1)906 5451 y Fh(\022)977 5505 y Fl(m)1062 5520 y Fj(1)p 977 5549 125 4 v 987 5640 a Fl(M)1111 5572 y(P)36 b Fg(\000)1319 5505 y Fm(1)p 1319 5549 49 4 v 1319 5640 a(2)1378 5572 y Fl(p)1427 5531 y Fk(a)1491 5572 y Fm(+)22 b Fl(p)1638 5587 y Fk(a)1702 5572 y Fg(\000)1813 5505 y Fl(e)p 1811 5549 V 1811 5640 a Fm(2)1887 5451 y Fh(\022)1958 5505 y Fl(m)2043 5520 y Fj(2)p 1958 5549 125 4 v 1968 5640 a Fl(M)2092 5572 y(r)2139 5531 y Fk(a)2203 5572 y Fg(\000)2312 5505 y Fm(1)p 2312 5549 49 4 v 2312 5640 a(2)2371 5572 y Fl(r)2415 5587 y Fk(a)2457 5451 y Fh(\023)2540 5572 y Fg(^)h Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(\000)2990 5505 y Fl(em)3120 5520 y Fj(1)p 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4130 a Fl(\026)1888 4145 y Fj(0)2009 3916 y Fh(!)2097 4062 y Fm(+)2247 3994 y(1)p 2205 4039 134 4 v 2205 4130 a(2)p Fl(")2300 4145 y Fj(0)2365 3945 y Fh(Z)2465 4062 y Fl(dr)2563 4021 y Fj(3)2602 4062 y Fl(P)2679 4021 y Fj(2)2717 4062 y Fm(\()p Fl(r)s Fm(\))22 b(+)g(2)p Fl(er)3101 4021 y Fk(a)3165 4062 y Fg(\001)f Fl(E)6 b Fm(\()p Fl(R)q Fm(\))0 4422 y Fd(2.2)135 b(The)45 b(Breit-F)-11 b(ermi)45 b(m)l(ultip)t(olar)h(Hamiltonian)0 4640 y Fm(Let)33 b Fl(h)231 4655 y Fk(B)s(F)379 4640 y Fm(b)s(e)f(the)h(Breit-F)-8 b(ermi)30 b(Hamiltonian)f(for)j(t)m(w)m (o)h(fermions)f(:)616 4834 y Fl(h)672 4849 y Fk(B)s(F)871 4834 y Fm(=)84 b(\()p Fl(c\013)1173 4849 y Fj(1)1234 4834 y Fg(\001)22 b Fl(p)1333 4849 y Fj(1)1395 4834 y Fm(+)g Fl(\014)1548 4849 y Fj(1)1587 4834 y Fl(m)1672 4849 y Fj(1)1712 4834 y Fl(c)1754 4793 y Fj(2)1793 4834 y Fm(\))g Fg(\012)h Fm(1)-33 b(1)2018 4849 y Fj(2)2079 4834 y Fm(+)22 b(1)-33 b(1)2242 4849 y Fj(1)2303 4834 y Fg(\012)22 b Fm(\()p Fl(c\013)2544 4849 y Fj(2)2606 4834 y Fg(\001)g Fl(p)2705 4849 y Fj(2)2766 4834 y Fm(+)g Fl(\014)2919 4849 y Fj(2)2959 4834 y Fl(mc)3086 4793 y Fj(2)3126 4834 y Fm(\))1226 5069 y(+)1383 5001 y Fl(e)1428 5016 y Fj(1)1467 5001 y Fl(e)1512 5016 y Fj(2)p 1312 5045 312 4 v 1312 5137 a Fm(4)p Fl(\031)e(")1482 5152 y Fj(0)1521 5137 y Fg(j)p Fl(r)s Fg(j)1650 5069 y Fm(1)-33 b(1)1715 5084 y Fj(1)1776 5069 y Fg(\012)22 b Fm(1)-33 b(1)1940 5084 y Fj(2)2001 5069 y Fg(\000)2182 5001 y Fl(e)2227 5016 y Fj(1)2266 5001 y Fl(e)2311 5016 y Fj(2)p 2111 5045 V 2111 5137 a Fm(8)p Fl(\031)20 b(")2281 5152 y Fj(0)2320 5137 y Fg(j)p Fl(r)s Fg(j)2432 5069 y Fl(\013)2494 5084 y Fj(1)2556 5069 y Fg(\012)i Fl(\013)2717 5084 y Fj(2)3606 5069 y Fm(\(33\))870 5372 y Fg(\000)1112 5305 y Fl(e)1157 5320 y Fj(1)1196 5305 y Fl(e)1241 5320 y Fj(2)p 1041 5349 V 1041 5440 a Fm(8)p Fl(\031)e(")1211 5455 y Fj(0)1250 5440 y Fg(j)p Fl(r)s Fg(j)1362 5372 y Fm(\(\()p Fl(\013)1500 5387 y Fj(1)1562 5372 y Fg(\001)26 b Fm(^)-54 b Fl(r)s Fm(\))22 b Fg(\012)h Fm(\()p Fl(\013)1918 5387 y Fj(2)1979 5372 y Fg(\001)j Fm(^)-53 b Fl(r)s Fm(\)\))0 5622 y(\(see)34 b([B.S]\))1841 5871 y(11)p eop %%Page: 12 12 12 11 bop 0 299 a Fm(where)34 b Fl(r)c Fm(=)e Fl(r)504 314 y Fj(1)565 299 y Fg(\000)23 b Fl(r)709 314 y Fj(2)781 299 y Fm(and)37 b(^)-54 b Fl(r)31 b Fm(=)1186 232 y Fl(r)p 1158 276 103 4 v 1158 368 a Fg(j)p Fl(r)s Fg(j)1270 299 y Fm(.)0 509 y(Here)d Fl(\013)287 524 y Fj(1)354 509 y Fm(and)g Fl(\014)594 524 y Fj(1)661 509 y Fm(\(resp.)43 b Fl(\013)1004 524 y Fj(2)1071 509 y Fm(and)28 b Fl(\014)1311 524 y Fj(2)1350 509 y Fm(\))g(are)f(the)h(4)p Fg(\002)g Fm(4)f(Dirac)g(matrices)f(for)h(particle)g(1)g(\(resp.)43 b(particle)0 629 y(2\).)g(1)-33 b(1)222 644 y Fj(1)293 629 y Fm(and)33 b(1)-33 b(1)548 644 y Fj(2)619 629 y Fm(are)33 b(4)p Fg(\002)g Fm(4)f(unit)g(matrices.)0 783 y Fl(h)56 798 y Fk(B)s(F)204 783 y Fm(is)g(a)g(self)g(adjoin)m(t)g(op)s (erator)g(in)g Fl(L)1458 747 y Fj(2)1498 783 y Fm(\()p Fl(I)-25 b(R)1629 747 y Fj(3)1668 783 y Fm(;)17 b Fl(I)-59 b(C)1773 747 y Fj(4)1812 783 y Fm(\))22 b Fg(\012)h Fl(L)2038 747 y Fj(2)2078 783 y Fm(\()p Fl(I)-25 b(R)2209 747 y Fj(3)2248 783 y Fm(;)17 b Fl(I)-59 b(C)2353 747 y Fj(4)2392 783 y Fm(\).)0 936 y Fl(\013)62 951 y Fj(1)124 936 y Fg(\012)22 b Fm(1)-33 b(1)288 951 y Fj(2)327 936 y Fm(,)33 b(1)-33 b(1)452 951 y Fj(1)513 936 y Fg(\012)22 b Fl(\013)674 951 y Fj(2)714 936 y Fm(,)32 b Fl(\013)835 951 y Fj(1)897 936 y Fg(\012)23 b Fl(\013)1059 951 y Fj(2)1098 936 y Fm(,)33 b Fl(\013)1220 951 y Fj(1)1281 936 y Fg(\012)23 b Fl(\013)1443 951 y Fj(2)1515 936 y Fm(and)33 b Fl(\013)1767 951 y Fj(1)1828 936 y Fg(\001)26 b Fm(^)-53 b Fl(r)25 b Fg(\012)d Fl(\013)2108 951 y Fj(2)2170 936 y Fg(\001)k Fm(^)-53 b Fl(r)35 b Fm(are)e(matrices)e(in)h Fl(I)-59 b(C)3027 900 y Fj(4)3089 936 y Fg(\012)23 b Fl(I)-59 b(C)3250 900 y Fj(4)3290 936 y Fm(.)0 1090 y(The)34 b(Breit-F)-8 b(ermi)29 b(Hamiltonian)g(in)j(Q.E.D.,)h(denoted)g(b)m(y)h Fl(H)2344 1105 y Fk(B)s(F)2458 1090 y Fm(,)f(is)f(giv)m(en)h(b)m(y)330 1258 y Fl(H)411 1273 y Fk(B)s(F)609 1258 y Fm(=)83 b(\()p Fl(c\013)910 1273 y Fj(1)972 1258 y Fg(\001)21 b Fm(\()p Fl(p)1108 1273 y Fj(1)1170 1258 y Fg(\000)i Fl(e)1315 1273 y Fj(1)1354 1258 y Fl(A)1427 1273 y Fk(ext)1529 1258 y Fm(\()p Fl(r)1611 1273 y Fj(1)1651 1258 y Fm(\))f Fg(\000)g Fl(e)1855 1273 y Fj(1)1895 1258 y Fl(A)p Fm(\()p Fl(r)2050 1273 y Fj(1)2090 1258 y Fm(\)\))g(+)g Fl(m)2371 1273 y Fj(1)2410 1258 y Fl(c)2452 1217 y Fj(2)2492 1258 y Fl(\014)2547 1273 y Fj(1)2586 1258 y Fm(\))g Fg(\012)h Fm(1)-33 b(1)2811 1273 y Fj(2)609 1470 y Fm(+)83 b(1)-33 b(1)833 1485 y Fj(1)894 1470 y Fg(\012)23 b Fm(\()p Fl(c\013)1136 1485 y Fj(2)1197 1470 y Fg(\001)f Fm(\()p Fl(p)1334 1485 y Fj(2)1395 1470 y Fg(\000)h Fl(e)1540 1485 y Fj(2)1580 1470 y Fl(A)1653 1485 y Fk(ext)1755 1470 y Fm(\()p Fl(r)1837 1485 y Fj(2)1876 1470 y Fm(\))f Fg(\000)h Fl(e)2081 1485 y Fj(2)2121 1470 y Fl(A)p Fm(\()p Fl(r)2276 1485 y Fj(2)2315 1470 y Fm(\)\))f(+)g Fl(m)2596 1485 y Fj(2)2636 1470 y Fl(c)2678 1429 y Fj(2)2717 1470 y Fl(\014)2772 1485 y Fj(2)2812 1470 y Fm(\))609 1705 y(+)849 1637 y Fl(e)894 1652 y Fj(1)934 1637 y Fl(e)979 1652 y Fj(2)p 778 1682 312 4 v 778 1773 a Fm(4)p Fl(\031)e(")948 1788 y Fj(0)987 1773 y Fg(j)p Fl(r)s Fg(j)1099 1705 y Fm(1)-33 b(1)1164 1720 y Fj(1)1225 1705 y Fg(\012)23 b Fm(1)-33 b(1)1390 1720 y Fj(2)1451 1705 y Fg(\000)1632 1637 y Fl(e)1677 1652 y Fj(1)1716 1637 y Fl(e)1761 1652 y Fj(2)p 1560 1682 V 1560 1773 a Fm(8)p Fl(\031)21 b(")1731 1788 y Fj(0)1770 1773 y Fg(j)p Fl(r)s Fg(j)1898 1705 y Fl(\013)1960 1720 y Fj(1)2022 1705 y Fg(\012)i Fl(\013)2184 1720 y Fj(2)2245 1705 y Fg(\000)2418 1637 y Fl(e)2463 1652 y Fj(1)2502 1637 y Fl(e)2547 1652 y Fj(2)p 2355 1682 295 4 v 2355 1773 a Fm(8)p Fl(\031)t(")2509 1788 y Fj(0)2548 1773 y Fg(j)p Fl(r)s Fg(j)2660 1705 y Fm(\(\()p Fl(\013)2798 1720 y Fj(1)2859 1705 y Fg(\001)j Fm(^)-53 b Fl(r)s Fm(\))22 b Fg(\012)g Fm(\()p Fl(\013)3215 1720 y Fj(2)3277 1705 y Fg(\001)k Fm(^)-53 b Fl(r)r Fm(\)\))609 2047 y(+)778 1980 y(1)p 778 2024 49 4 v 778 2115 a(2)853 1930 y Fh(Z)953 2047 y Fl(d)1004 2006 y Fj(3)1043 2047 y Fl(r)1106 1901 y Fh( )1182 1980 y Fm(\005\()p Fl(r)s Fm(\))1378 1944 y Fj(2)p 1182 2024 236 4 v 1257 2115 a Fl(")1303 2130 y Fj(0)1449 2047 y Fm(+)1557 1980 y Fl(B)5 b Fm(\()p Fl(r)s Fm(\))1759 1944 y Fj(2)p 1557 2024 242 4 v 1629 2115 a Fl(\026)1688 2130 y Fj(0)1808 1901 y Fh(!)1891 2047 y Fl(;)212 b(r)2174 2062 y Fj(1)2213 2047 y Fl(;)17 b(r)2301 2062 y Fj(2)2368 2047 y Fg(2)28 b Fl(I)-25 b(R)2555 2006 y Fj(3)3606 2047 y Fm(\(34\))0 2278 y(By)34 b(applying)e(the)i (unitary)g(transform)e Fl(U)45 b Fm(\(see)34 b(\(2.7\)\))f(to)g Fl(H)2296 2293 y Fk(B)s(F)2445 2278 y Fm(w)m(e)i(get)e(the)h(follo)m (wing)d(m)m(ultip)s(olar)0 2398 y(Hamiltonian)e(denoted)k(b)m(y)h Fl(H)1146 2413 y Fk(B)s(F)r(;mul)q(t)1456 2398 y Fm(:)0 2566 y Fl(H)81 2581 y Fk(B)s(F)r(;mul)q(t)442 2566 y Fm(=)84 b Fl(U)678 2525 y Fi(\000)p Fj(1)773 2566 y Fl(H)854 2581 y Fk(B)s(F)969 2566 y Fl(U)442 2818 y Fm(=)602 2697 y Fh(\022)663 2818 y Fl(c\013)767 2833 y Fj(1)829 2818 y Fg(\001)21 b Fm(\()p Fl(p)965 2833 y Fj(1)1027 2818 y Fg(\000)h Fl(e)1171 2833 y Fj(1)1211 2818 y Fl(A)1284 2833 y Fk(ext)1386 2818 y Fm(\()p Fl(r)1468 2833 y Fj(1)1508 2818 y Fm(\))g Fg(\000)g Fl(e)1712 2833 y Fk(T)1778 2751 y Fl(m)1863 2766 y Fj(1)p 1778 2795 125 4 v 1788 2887 a Fl(M)1912 2818 y(A)p Fm(\()p Fl(R)q Fm(\))h(+)2257 2701 y Fh(Z)2356 2818 y Fl(d)2407 2777 y Fj(3)2446 2818 y Fl(r)s Fm(\002)2569 2833 y Fj(1)2608 2818 y Fm(\()p Fl(r)s Fm(\))f Fg(^)h Fl(B)5 b Fm(\()p Fl(r)s Fm(\)\))21 b(+)h Fl(m)3286 2833 y Fj(1)3326 2818 y Fl(c)3368 2777 y Fj(2)3407 2818 y Fl(\014)3462 2833 y Fj(1)3502 2697 y Fh(\023)3585 2818 y Fg(\012)h Fm(1)-33 b(1)3750 2833 y Fj(2)442 3117 y Fm(+)84 b(1)-33 b(1)667 3132 y Fj(1)728 3117 y Fg(\012)827 2996 y Fh(\022)889 3117 y Fl(c\013)993 3132 y Fj(2)1054 3117 y Fg(\001)22 b Fm(\()p Fl(p)1191 3132 y Fj(2)1252 3117 y Fg(\000)h Fl(e)1397 3132 y Fj(2)1437 3117 y Fl(A)1510 3132 y Fk(ext)1612 3117 y Fm(\()p Fl(r)1694 3132 y Fj(1)1733 3117 y Fm(\))f Fg(\000)h Fl(e)1938 3132 y Fk(T)2003 3050 y Fl(m)2088 3065 y Fj(2)p 2003 3094 V 2013 3186 a Fl(M)2138 3117 y(A)p Fm(\()p Fl(R)q Fm(\))f(+)2482 3000 y Fh(Z)2582 3117 y Fl(d)2633 3076 y Fj(3)2672 3117 y Fl(r)s Fm(\002)2795 3132 y Fj(2)2834 3117 y Fm(\()p Fl(r)s Fm(\))f Fg(^)i Fl(B)5 b Fm(\()p Fl(r)s Fm(\)\))22 b(+)g Fl(m)3512 3132 y Fj(2)3552 3117 y Fl(c)3594 3076 y Fj(2)3633 3117 y Fl(\014)3688 3132 y Fj(2)3728 2996 y Fh(\023)441 3438 y Fg(\000)717 3371 y Fl(e)762 3334 y Fj(2)p 612 3415 295 4 v 612 3506 a Fm(4)p Fl(\031)t(")766 3521 y Fj(0)805 3506 y Fg(j)p Fl(r)s Fg(j)917 3438 y Fm(1)-33 b(1)982 3453 y Fj(1)1042 3438 y Fg(\012)23 b Fm(1)-33 b(1)1207 3453 y Fj(2)1268 3438 y Fm(+)1481 3371 y Fl(e)1526 3334 y Fj(2)p 1376 3415 V 1376 3506 a Fm(8)p Fl(\031)t(")1530 3521 y Fj(0)1569 3506 y Fg(j)p Fl(r)s Fg(j)1681 3438 y Fl(\013)1743 3453 y Fj(1)1804 3438 y Fg(\012)23 b Fl(\013)1966 3453 y Fj(2)2028 3438 y Fm(+)2241 3371 y Fl(e)2286 3334 y Fj(2)p 2136 3415 V 2136 3506 a Fm(8)p Fl(\031)t(")2290 3521 y Fj(0)2328 3506 y Fg(j)p Fl(r)s Fg(j)2440 3438 y Fm(\(\()p Fl(\013)2578 3453 y Fj(1)2640 3438 y Fg(\001)j Fm(^)-53 b Fl(r)r Fm(\))22 b Fg(\012)h Fm(\()p Fl(\013)2996 3453 y Fj(2)3058 3438 y Fg(\001)j Fm(^)-54 b Fl(r)s Fm(\)\))402 b(\(35\))442 3780 y(+)612 3713 y(1)p 612 3757 49 4 v 612 3849 a(2)687 3663 y Fh(Z)787 3780 y Fl(d)838 3739 y Fj(3)877 3780 y Fl(r)940 3634 y Fh( )1016 3713 y Fm(\005\()p Fl(r)s Fm(\))1212 3677 y Fj(2)p 1016 3757 236 4 v 1091 3849 a Fl(")1137 3864 y Fj(0)1283 3780 y Fm(+)1391 3713 y Fl(B)5 b Fm(\()p Fl(r)s Fm(\))1593 3677 y Fj(2)p 1391 3757 242 4 v 1463 3849 a Fl(\026)1522 3864 y Fj(0)1642 3634 y Fh(!)1730 3780 y Fm(+)1856 3713 y(1)p 1838 3757 86 4 v 1838 3849 a Fl(")1884 3864 y Fj(0)1950 3663 y Fh(Z)2049 3780 y Fl(d)2100 3739 y Fj(3)2139 3780 y Fl(r)20 b Fm(\005\()p Fl(r)s Fm(\))h Fg(\001)h Fl(P)14 b Fm(\()p Fl(r)s Fm(\))21 b(+)2842 3713 y(1)p 2799 3757 134 4 v 2799 3849 a(2)p Fl(")2894 3864 y Fj(0)2960 3663 y Fh(Z)3059 3780 y 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Fl(\026)51 b(>)h Fm(0,)e Fl(\021)55 b(>)d Fm(0,)e Fl(\026)c Fm(and)h Fl(\021)j Fm(are)d(dimensionless)f(constan)m(ts.)87 b(This)47 b(scaling)e(induces)j(a)e(unitary)0 1150 y(transformation)30 b(that)j(w)m(e)g(no)m(w)h(compute.)0 1303 y(The)g(dilation)29 b Fl(k)i Fg(!)c Fl(\021)t(k)36 b Fm(induces)d(an)f(unitary)h(op)s (erator)e(in)h Fl(L)2305 1267 y Fj(2)2345 1303 y Fm(\()p Fl(I)-25 b(R)2476 1267 y Fj(3)2515 1303 y Fm(\))33 b(denoted)g(b)m(y)h (\000)3151 1318 y Fk(\021)3225 1303 y Fm(:)971 1483 y(\(\000)1070 1498 y Fk(\021)1112 1483 y Fl(f)11 b Fm(\))17 b(\()p Fl(k)s Fm(\))27 b(=)g Fl(\021)1538 1442 y Fj(3)p Fk(=)p Fj(2)1648 1483 y Fl(f)11 b Fm(\()p Fl(\021)t(k)s Fm(\))p Fl(;)211 b(\021)32 b(>)27 b Fm(0)p Fl(;)114 b(k)31 b Fg(2)d Fl(I)-25 b(R)2769 1442 y Fj(3)3606 1483 y Fm(\(37\))0 1663 y(where)34 b Fl(f)k Fg(2)28 b Fl(L)528 1627 y Fj(2)568 1663 y Fm(\()p Fl(I)-25 b(R)699 1627 y Fj(3)738 1663 y Fm(\).)0 1817 y(W)-8 b(e)33 b(ha)m(v)m(e)1357 1975 y(\(\000)1456 1934 y Fi(\000)p Fj(1)1456 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b(the)g(n)m(um)m(b)s(er)g(of)f (photons)h(op)s(erator.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)1255 4393 y Fl(N)38 b Fm(=)1508 4310 y Fh(X)1474 4495 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)1677 4276 y Fh(Z)1777 4393 y Fl(d)1828 4352 y Fj(3)1867 4393 y Fl(k)20 b(a)1989 4408 y Fk(\025)2034 4393 y Fl(a)2085 4352 y Fi(\003)2125 4393 y Fm(\()p Fl(k)s Fm(\))d Fl(a)2323 4408 y Fk(\025)2368 4393 y Fm(\()p Fl(k)s Fm(\))p Fl(:)0 4653 y(N)43 b Fm(is)32 b(ob)m(viously)h(in)m(v)-5 b(arian)m(t)31 b(b)m(y)i(dilation.)0 4807 y(Therefore)863 4987 y(\000)924 4945 y Fi(\000)p Fj(1)924 5011 y Fk(\021)1034 4987 y Fl(N)28 b Fm(\000)1201 5002 y Fk(\021)1325 4987 y Fm(=)1518 4904 y Fh(X)1484 5088 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)1687 4869 y Fh(Z)1787 4987 y Fl(d)1838 4945 y Fj(3)1877 4987 y Fl(k)20 b Fm(\000)2009 4945 y Fi(\000)p Fj(1)2009 5011 y Fk(\021)2103 4987 y Fl(a)2154 4945 y Fi(\003)2154 5011 y Fk(\025)2199 4987 y Fm(\()p Fl(k)s Fm(\)\000)2390 5002 y Fk(\021)2432 4987 y Fm(\000)2493 4945 y Fi(\000)p Fj(1)2493 5011 y 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474 y Fk(\021)1874 459 y Fm(=)83 b Fl(\021)2085 418 y Fj(3)p Fk(=)p Fj(2)2195 459 y Fl(a)2246 474 y Fk(\025)2291 459 y Fm(\()p Fl(\021)t(k)s Fm(\))1307 621 y(\000)1368 580 y Fi(\000)p Fj(1)1368 646 y Fk(\021)1462 621 y Fl(a)1513 580 y Fi(\003)1513 646 y Fk(\025)1558 621 y Fm(\()p Fl(k)s Fm(\)\000)1749 636 y Fk(\021)1874 621 y Fm(=)g Fl(\021)2085 580 y Fj(3)p Fk(=)p Fj(2)2195 621 y Fl(a)2246 580 y Fi(\003)2246 646 y Fk(\025)2291 621 y Fm(\()p Fl(\021)t(k)s Fm(\))0 792 y(b)s(ecause)34 b Fl(a)412 755 y Fi(\003)412 816 y Fk(\025)457 792 y Fm(\()p Fl(k)s Fm(\))f(is)f(the)h(adjoin)m(t)f(of)g Fl(a)1379 807 y Fk(\025)1424 792 y Fm(\()p Fl(k)s Fm(\).)0 945 y(Note)44 b(that)g(the)g(righ)m(t)f(mathematical)e(de\014nition)i (of)g(the)i(energy)g(op)s(erator)e(for)g(the)i(photons)f(is)0 1066 y(giv)m(en)33 b(b)m(y)1352 1172 y(1)p 1352 1216 49 4 v 1352 1308 a(2)1427 1122 y Fh(Z)1527 1240 y Fl(d)1578 1198 y Fj(3)1617 1240 y Fl(r)e Fm(:)1756 1172 y(\005\()p Fl(r)s Fm(\))1952 1136 y Fj(2)p 1756 1216 236 4 v 1831 1308 a Fl(")1877 1323 y Fj(0)2024 1240 y Fm(+)2131 1172 y Fl(B)5 b Fm(\()p Fl(r)s Fm(\))2333 1136 y Fj(2)p 2131 1216 242 4 v 2203 1308 a Fl(\026)2262 1323 y Fj(0)2410 1240 y Fm(:)0 1443 y(where)34 b(:)e(:)44 b(is)32 b(the)h(Wic)m(k)g (normal)e(order)h(\(see)i([S]\).)0 1597 y(Let)f Fl(H)256 1612 y Fk(ph)368 1597 y Fm(denote)h(the)f(energy)g(op)s(erator)f(for)g (the)h(photons.)44 b(W)-8 b(e)33 b(then)g(ha)m(v)m(e)h(:)565 1829 y Fl(H)646 1844 y Fk(ph)754 1829 y Fm(=)867 1762 y(1)p 867 1806 49 4 v 867 1898 a(2)942 1712 y Fh(Z)1042 1829 y Fl(d)1093 1788 y Fj(3)1132 1829 y Fl(r)c Fm(:)1271 1762 y(\005\()p Fl(r)s Fm(\))1467 1726 y Fj(2)p 1271 1806 236 4 v 1346 1898 a Fl(")1392 1913 y Fj(0)1538 1829 y Fm(+)1646 1762 y Fl(B)5 b Fm(\()p Fl(r)s Fm(\))1848 1726 y Fj(2)p 1646 1806 242 4 v 1718 1898 a Fl(\026)1777 1913 y Fj(0)1925 1829 y Fm(:=)2089 1746 y Fh(X)2056 1931 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)2259 1829 y Fl(c)p Fm(\026)-50 b Fl(h)2372 1712 y Fh(Z)2472 1829 y Fl(d)2523 1788 y Fj(3)2562 1829 y Fl(r)19 b Fg(j)p Fl(k)s Fg(j)p Fl(a)2786 1788 y Fi(\003)2786 1854 y Fk(\025)2831 1829 y Fm(\()p Fl(k)s Fm(\))p Fl(a)3012 1844 y Fk(\025)3058 1829 y Fm(\()p Fl(k)s Fm(\))p Fl(:)0 2080 y Fm(Th)m(us)622 2250 y(\000)683 2209 y Fi(\000)p Fj(1)683 2275 y Fk(\021)777 2250 y Fl(H)858 2265 y Fk(ph)955 2250 y Fm(\000)1016 2265 y Fk(\021)1141 2250 y Fm(=)1333 2167 y Fh(X)1300 2352 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)1503 2250 y Fl(c)p Fm(\026)-50 b Fl(h)1616 2133 y Fh(Z)1716 2250 y Fl(d)1767 2209 y Fj(3)1806 2250 y Fl(k)s Fg(j)p Fl(k)s Fg(j)p Fm(\000)2031 2209 y Fi(\000)p Fj(1)2031 2275 y Fk(\021)2125 2250 y Fl(a)2176 2209 y Fi(\003)2176 2275 y Fk(\025)2221 2250 y Fm(\()p Fl(k)s Fm(\)\000)2412 2265 y Fk(\021)2454 2250 y Fm(\000)2515 2209 y Fi(\000)p Fj(1)2515 2275 y Fk(\021)2610 2250 y Fl(a)2661 2265 y Fk(\025)2706 2250 y Fm(\()p Fl(k)s Fm(\)\000)2897 2265 y Fk(\021)1141 2590 y Fm(=)1333 2507 y Fh(X)1300 2691 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)1503 2590 y Fl(c)p Fm(\026)g Fl(h)1616 2473 y Fh(Z)1716 2590 y Fl(d)1767 2549 y Fj(3)1806 2590 y Fl(k)s Fg(j)p Fl(k)s Fg(j)p Fl(\021)2022 2549 y Fj(3)2061 2590 y Fl(a)2112 2549 y Fi(\003)2112 2614 y Fk(\025)2157 2590 y Fm(\()p Fl(\021)t(k)s Fm(\))p Fl(a)2390 2605 y Fk(\025)2435 2590 y Fm(\()p Fl(\021)t(k)s Fm(\))989 b(\(40\))1141 2944 y(=)1311 2876 y(1)p 1310 2921 52 4 v 1310 3012 a Fl(\021)1421 2861 y Fh(X)1388 3045 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)1591 2944 y Fl(c)p Fm(\026)-50 b Fl(h)1705 2827 y Fh(Z)1804 2944 y Fl(d)1855 2903 y Fj(3)1894 2944 y Fm(\()p Fl(\021)t(k)s Fm(\))p Fg(j)p Fl(\021)t(k)s Fg(j)p Fl(a)2289 2903 y Fi(\003)2289 2968 y Fk(\025)2334 2944 y Fm(\()p Fl(\021)t(k)s Fm(\))p Fl(a)2567 2959 y Fk(\025)2612 2944 y Fm(\()p Fl(\021)t(k)s Fm(\))27 b(=)2936 2876 y(1)p 2935 2921 V 2935 3012 a Fl(\021)2996 2944 y(H)3077 2959 y Fk(ph)0 3193 y Fm(Moreo)m(v)m(er)442 3425 y(\000)503 3383 y Fi(\000)p Fj(1)503 3449 y Fk(\021)597 3425 y Fl(A)p Fm(\()p Fl(r)s Fm(\))17 b(\000)871 3440 y Fk(\021)995 3425 y Fm(=)1187 3341 y Fh(X)1154 3526 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)1357 3307 y Fh(Z)1457 3425 y Fl(d)1508 3383 y Fj(3)1547 3425 y Fl(k)1617 3278 y Fh( )1941 3357 y Fm(\026)-50 b Fl(h)p 1693 3401 552 4 v 1693 3493 a Fm(2)p Fl(")1788 3508 y Fj(0)1827 3493 y Fm(\(2)p Fl(\031)t Fm(\))2011 3464 y Fj(3)2050 3493 y Fl(!)t Fm(\()p Fl(k)s Fm(\))2254 3278 y Fh(!)2320 3301 y Fj(1)p Fk(=)p Fj(2)2447 3425 y Fm([)p Fl(\021)2526 3383 y Fj(3)p Fk(=)p Fj(2)2635 3425 y Fm(\()p Fl(a)2724 3440 y Fk(\025)2770 3425 y Fm(\()p Fl(\021)t(k)s Fm(\))p Fl(")2998 3440 y Fk(\025)3042 3425 y Fm(\()p Fl(k)s Fm(\))p Fl(e)3217 3383 y Fk(ik)r Fi(\001)p Fk(r)1154 3734 y Fm(+)33 b Fl(a)1314 3693 y Fi(\003)1314 3759 y Fk(\025)1359 3734 y Fm(\()p Fl(\021)t(k)s Fm(\))17 b Fl(")1604 3749 y Fk(\025)1648 3734 y Fm(\()p Fl(k)s Fm(\))g Fl(e)1840 3693 y Fi(\000)p Fk(ik)r Fi(\001)p Fk(r)2015 3734 y Fm(])0 3905 y(Note)33 b(that)f Fl(")493 3920 y Fk(\025)538 3905 y Fm(\()p Fl(\021)t(k)s Fm(\))27 b(=)h Fl(")897 3920 y Fk(\025)942 3905 y Fm(\()p Fl(k)s Fm(\),)k Fl(\025)c Fm(=)g(1)p Fl(;)17 b Fm(2.)42 b(W)-8 b(e)33 b(then)g(get)443 4161 y(\000)504 4120 y Fi(\000)p Fj(1)504 4186 y Fk(\021)598 4161 y Fl(A)p Fm(\()p Fl(r)s Fm(\))17 b(\000)872 4176 y Fk(\021)996 4161 y Fm(=)1167 4094 y(1)p 1165 4138 52 4 v 1165 4229 a Fl(\021)1276 4078 y Fh(X)1243 4263 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)1446 4044 y Fh(Z)1546 4161 y Fl(d)1597 4120 y Fj(3)1636 4161 y Fl(k)1707 4015 y Fh( )2056 4094 y Fm(\026)-50 b Fl(h)p 1782 4138 603 4 v 1782 4229 a Fm(2)p Fl(")1877 4244 y Fj(0)1916 4229 y Fm(\(2)p Fl(\031)t Fm(\))2100 4201 y Fj(3)2139 4229 y Fl(!)t Fm(\()p Fl(\021)t(k)s Fm(\))2395 4015 y Fh(!)2461 4038 y Fj(1)p Fk(=)p Fj(2)2587 4161 y Fm([\()p Fl(a)2703 4176 y Fk(\025)2749 4161 y Fm(\()p Fl(\021)t(k)s Fm(\))p Fl(")2977 4176 y Fk(\025)3022 4161 y Fm(\()p Fl(k)s Fm(\))p Fl(e)3197 4115 y Fk(ik)r Fi(\001)3291 4087 y Ff(r)p 3289 4099 34 4 v 3289 4141 a(\021)1155 4478 y Fm(+)32 b Fl(a)1314 4437 y Fi(\003)1314 4503 y Fk(\025)1360 4478 y Fm(\()p Fl(\021)t(k)s Fm(\))17 b Fl(")1605 4493 y Fk(\025)1649 4478 y Fm(\()p Fl(k)s Fm(\))g Fl(e)1841 4432 y Fi(\000)p Fk(ik)r Fi(\001)1990 4405 y Ff(r)p 1988 4417 V 1988 4458 a(\021)2036 4478 y Fm(])996 4752 y(=)1167 4685 y(1)p 1165 4729 52 4 v 1165 4821 a Fl(\021)1243 4752 y(A)1333 4606 y Fh( )1411 4685 y Fl(r)p 1409 4729 V 1409 4821 a(\021)1470 4606 y Fh(!)0 4980 y Fm(In)33 b(the)g(same)f(w)m(a)m(y)i(w)m(e)g(get)1301 5207 y(\000)1362 5166 y Fi(\000)p Fj(1)1362 5232 y Fk(\021)1456 5207 y Fl(B)5 b Fm(\()p Fl(r)s Fm(\))17 b(\000)1736 5222 y Fk(\021)1861 5207 y Fm(=)2051 5140 y(1)p 2029 5184 92 4 v 2029 5275 a Fl(\021)2081 5247 y Fj(2)2163 5207 y Fl(B)2259 5061 y Fh( )2337 5140 y Fl(r)p 2334 5184 52 4 v 2334 5275 a(\021)2396 5061 y Fh(!)3606 5381 y Fm(\(41\))1302 5556 y(\000)1363 5515 y Fi(\000)p Fj(1)1363 5580 y Fk(\021)1457 5556 y Fl(E)6 b Fm(\()p Fl(r)s Fm(\))17 b(\000)1736 5571 y Fk(\021)1861 5556 y Fm(=)2051 5488 y(1)p 2029 5532 92 4 v 2029 5624 a Fl(\021)2081 5595 y Fj(2)2163 5556 y Fl(E)2258 5410 y Fh( )2336 5488 y Fl(r)p 2333 5532 52 4 v 2333 5624 a(\021)2395 5410 y Fh(!)1841 5871 y Fm(14)p eop %%Page: 15 15 15 14 bop 0 289 a Fm(W)-8 b(e)33 b(also)f(ha)m(v)m(e,)i(for)e Fl(r)e Fg(2)e Fl(I)-25 b(R)1026 253 y Fj(3)1065 289 y Fm(,)1253 406 y Fh( )1318 552 y Fm(\000)1379 511 y Fi(\000)p Fj(1)1379 577 y Fk(\026)1524 485 y Fl(@)p 1484 529 138 4 v 1484 621 a(@)5 b(r)1584 636 y Fk(j)1648 552 y Fm(\000)1709 567 y Fk(\026)1755 406 y Fh(!)1854 552 y Fl(')p Fm(\()p Fl(r)s Fm(\))27 b(=)h Fl(\026)2265 485 y(@)5 b(')p 2257 529 V 2257 621 a(@)g(r)2357 636 y Fk(j)2404 552 y Fm(\()p Fl(r)s Fm(\))0 816 y(Therefore)1526 936 y(\000)1587 895 y Fi(\000)p Fj(1)1587 961 y Fk(\026)1681 936 y Fg(r)1764 951 y Fk(r)1819 936 y Fm(\000)1880 951 y Fk(\026)1954 936 y Fm(=)27 b Fl(\026)17 b Fg(r)2216 951 y Fk(r)0 1105 y Fm(F)-8 b(uthermore,)32 b(for)g(an)m(y)h(function)f Fl(V)22 b Fm(\()p Fl(r)s Fm(\),)32 b(w)m(e)i(ha)m(v)m(e)1431 1374 y(\000)1492 1333 y Fi(\000)p Fj(1)1492 1399 y Fk(\026)1587 1374 y Fl(V)21 b Fm(\()p Fl(r)s Fm(\))c(\000)1866 1389 y Fk(\026)1939 1374 y Fm(=)28 b Fl(V)2138 1228 y Fh( )2220 1307 y Fl(r)p 2214 1351 59 4 v 2214 1442 a(\026)2283 1228 y Fh(!)3606 1374 y Fm(\(42\))0 1643 y(Let)22 b(us)g(no)m(w)h (consider)f(the)g(transformed)f(m)m(ultip)s(olar)d(hamiltonian)g(under) 23 b(the)f(scaling)f(\()p Fl(r)3343 1658 y Fj(1)3382 1643 y Fl(;)c(r)3470 1658 y Fj(2)3509 1643 y Fl(;)g(k)s Fm(\))27 b Fg(\000)-16 b(!)0 1680 y Fh(\020)60 1737 y Fk(r)92 1746 y Fe(1)p 60 1753 67 4 v 72 1810 a Fk(\026)136 1776 y Fl(;)190 1737 y Fk(r)222 1746 y Fe(2)p 190 1753 V 202 1810 a Fk(\026)266 1776 y Fl(;)17 b(\021)t(k)416 1680 y Fh(\021)465 1776 y Fm(,)33 b Fl(\026)27 b(>)h Fm(0,)k Fl(\021)f(>)d Fm(0.)43 b(Again)32 b Fl(\026)g Fm(and)h Fl(\021)j Fm(are)c(dimensionless)g(constan)m(ts.)0 1929 y(F)-8 b(or)34 b(simplicit)m(y)e(let)i(us)h(consider)g(the)g (electric)e(dip)s(ole)g(appro)m(ximation)g(of)h(the)h(m)m(ultip)s(olar) c(Hamil-)0 2050 y(tonian)g(for)i(b)s(oth)f Fl(e)728 2065 y Fk(T)811 2050 y Fm(=)27 b(0,)33 b Fl(B)1097 2065 y Fj(0)1164 2050 y Fm(=)27 b(0)33 b(and)f Fl(M)1632 2065 y Fj(1)1700 2050 y Fm(=)27 b Fl(M)1897 2065 y Fj(2)1965 2050 y Fm(=)g(0.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(:)444 2310 y Fl(H)525 2325 y Fk(mul)q(t;dip)878 2310 y Fm(=)1066 2243 y Fl(P)1143 2207 y Fj(2)p 1047 2287 154 4 v 1047 2379 a Fm(2)p Fl(M)1233 2310 y Fm(+)1383 2243 y(1)p 1341 2287 135 4 v 1341 2379 a(2)p Fl(m)1501 2310 y(p)1550 2269 y Fj(2)1612 2310 y Fg(\000)1835 2243 y Fl(e)1880 2207 y Fj(2)p 1721 2287 312 4 v 1721 2379 a Fm(4)p Fl(\031)21 b(")1892 2394 y Fj(0)1931 2379 y Fg(j)p Fl(r)s Fg(j)2065 2310 y Fm(+)2196 2227 y Fh(X)2163 2412 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)2366 2310 y Fl(c)p Fm(\026)-50 b Fl(h)2479 2193 y Fh(Z)2579 2310 y Fl(d)2630 2269 y Fj(3)2669 2310 y Fl(k)20 b Fg(j)p Fl(k)s Fg(j)d Fl(a)2918 2269 y Fi(\003)2918 2335 y Fk(\025)2962 2310 y Fm(\()p Fl(k)s Fm(\))g Fl(a)3160 2325 y Fk(\025)3206 2310 y Fm(\()p Fl(k)s Fm(\))1037 2607 y Fg(\000)g Fl(d)22 b Fg(\001)g Fl(E)6 b Fm(\()p Fl(R)q Fm(\))1037 2863 y Fg(\000)1193 2796 y Fm(1)p 1141 2840 154 4 v 1141 2932 a(2)p Fl(M)1305 2863 y Fm(\()p Fl(P)35 b Fg(\001)22 b Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)h Fl(d)f Fm(+)g Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)g Fl(d)g Fg(\001)g Fl(P)14 b Fm(\))1037 3187 y Fg(\000)1184 3119 y Fm(1)p 1141 3163 135 4 v 1141 3255 a(4)p Fl(m)1302 3065 y Fh(\022)1363 3187 y Fm(1)22 b Fg(\000)1543 3119 y Fm(4)p Fl(m)p 1543 3163 V 1558 3255 a(M)1687 3065 y Fh(\023)1749 3081 y Fj(1)p Fk(=)p Fj(2)1875 3187 y Fm(\()p Fl(p)g Fg(\001)g Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)h Fl(d)f Fm(+)g Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)g Fl(d)g Fg(\001)g Fl(p)p Fm(\))1037 3493 y(+)1182 3425 y(1)p 1139 3469 V 1139 3561 a(8)p Fl(m)1300 3493 y Fm(\()p Fl(B)5 b Fm(\()p Fl(R)q Fm(\))22 b Fg(^)h Fl(d)p Fm(\))1768 3451 y Fj(2)3606 3493 y Fm(\(43\))0 3755 y(The)48 b(scaling)f(\()p Fl(r)633 3770 y Fj(1)672 3755 y Fl(;)17 b(r)760 3770 y Fj(2)799 3755 y Fm(\))53 b Fg(\000)-16 b(!)1104 3659 y Fh(\020)1163 3716 y Fk(r)1195 3725 y Fe(1)p 1163 3732 67 4 v 1175 3790 a Fk(\026)1240 3755 y Fl(;)1293 3716 y Fk(r)1325 3725 y Fe(2)p 1293 3732 V 1305 3790 a Fk(\026)1370 3659 y Fh(\021)1419 3755 y Fm(,)51 b Fl(r)1541 3770 y Fj(1)1581 3755 y Fl(;)17 b(r)1669 3770 y Fj(2)1761 3755 y Fg(2)53 b Fl(I)-25 b(R)1973 3719 y Fj(3)2013 3755 y Fm(,)51 b(induces)d(the)g(scaling)e(\()p Fl(r)m(;)17 b(R)q Fm(\))53 b Fg(\000)-16 b(!)3474 3659 y Fh(\020)3538 3716 y Fk(r)p 3533 3732 43 4 v 3533 3790 a(\026)3586 3755 y Fl(;)3639 3716 y Fk(R)p 3639 3732 54 4 v 3645 3790 a(\026)3703 3659 y Fh(\021)3752 3755 y Fm(,)0 3892 y Fl(r)m(;)17 b(R)29 b Fg(2)f Fl(I)-25 b(R)375 3856 y Fj(3)414 3892 y Fm(.)0 4046 y(Therefore)614 4253 y(\(\000)713 4268 y Fk(\026)782 4253 y Fg(\012)22 b Fm(\000)942 4268 y Fk(\026)1011 4253 y Fg(\012)h Fm(\000)1172 4268 y Fj(1)p Fk(=\021)1284 4253 y Fm(\))1322 4212 y Fi(\000)p Fj(1)1433 4253 y Fl(H)1514 4268 y Fk(mul)q(t;dip)1784 4253 y Fm(\(\000)1883 4268 y Fk(\026)1952 4253 y Fg(\012)f Fm(\000)2112 4268 y Fk(\026)2181 4253 y Fg(\012)h Fm(\000)2342 4268 y Fj(1)p Fk(=\021)2454 4253 y Fm(\))455 4531 y(=)83 b Fl(\026)673 4490 y Fj(2)741 4463 y Fl(P)818 4427 y Fj(2)p 722 4508 154 4 v 722 4599 a Fm(2)p Fl(M)908 4531 y Fm(+)22 b Fl(\026)1065 4490 y Fj(2)1137 4463 y Fl(p)1186 4427 y Fj(2)p 1114 4508 135 4 v 1114 4599 a Fm(2)p Fl(m)1280 4531 y Fg(\000)1474 4463 y Fl(e)1519 4427 y Fj(2)1558 4463 y Fl(\026)p 1390 4508 312 4 v 1390 4599 a Fm(4)p Fl(\031)e(")1560 4614 y Fj(0)1599 4599 y Fg(j)p Fl(r)s Fg(j)1733 4531 y Fm(+)i Fl(\021)1933 4448 y Fh(X)1900 4632 y Fk(\025)p Fj(=1)p Fk(;)p Fj(2)2102 4531 y Fl(c)p Fm(\026)-50 b Fl(h)2216 4414 y Fh(Z)2316 4531 y Fl(d)2367 4490 y Fj(3)2406 4531 y Fl(k)20 b Fg(j)p Fl(k)s Fg(j)d Fl(a)2655 4490 y Fi(\003)2655 4555 y Fk(\025)2699 4531 y Fm(\()p Fl(k)s Fm(\))g Fl(a)p Fm(\()p Fl(k)s Fm(\))614 4903 y Fg(\000)718 4835 y Fl(\021)770 4799 y Fj(2)p 718 4879 92 4 v 734 4971 a Fl(\026)836 4903 y(E)930 4756 y Fh( )1010 4835 y Fl(\021)p 1006 4879 59 4 v 1006 4971 a(\026)1075 4903 y(R)1150 4756 y Fh(!)1238 4903 y Fg(\001)22 b Fl(d)f Fg(\000)1522 4835 y Fm(1)p 1470 4879 154 4 v 1470 4971 a(2)p Fl(M)1650 4903 y(\021)1702 4861 y Fj(2)1758 4756 y Fh( )1824 4903 y Fl(P)35 b Fg(\001)22 b Fl(B)2068 4756 y Fh( )2147 4835 y Fl(\021)p 2143 4879 59 4 v 2143 4971 a(\026)2212 4903 y(R)2287 4756 y Fh(!)2375 4903 y Fg(^)h Fl(d)e Fm(+)h Fl(B)2730 4756 y Fh( )2809 4835 y Fl(\021)p 2806 4879 V 2806 4971 a(\026)2874 4903 y(R)2949 4756 y Fh(!)3037 4903 y Fg(^)h Fl(d)f Fg(\001)g Fl(P)3326 4756 y Fh(!)614 5191 y Fg(\000)761 5123 y Fm(1)p 718 5167 135 4 v 718 5259 a(4)p Fl(m)879 5069 y Fh(\022)940 5191 y Fm(1)g Fg(\000)1120 5123 y Fm(4)p Fl(m)p 1120 5167 V 1135 5259 a(M)1264 5069 y Fh(\023)1326 5085 y Fj(1)p Fk(=)p Fj(2)1452 5191 y Fl(\021)1504 5149 y Fj(2)1560 5044 y Fh( )1626 5191 y Fl(p)g Fg(\001)g Fl(B)1842 5044 y Fh( )1922 5123 y Fl(\021)p 1918 5167 59 4 v 1918 5259 a(\026)1987 5191 y(R)2062 5044 y Fh(!)2150 5191 y Fg(^)g Fl(d)g Fm(+)g Fl(B)2505 5044 y Fh( )2584 5123 y Fl(\021)p 2580 5167 V 2580 5259 a(\026)2649 5191 y(R)2724 5044 y Fh(!)2812 5191 y Fg(^)h Fl(d)e Fg(\001)h Fl(P)3100 5044 y Fh(!)614 5556 y Fm(+)759 5488 y(1)p 717 5532 135 4 v 717 5624 a(8)p Fl(m)891 5488 y(\021)943 5452 y Fj(4)p 887 5532 99 4 v 887 5624 a Fl(\026)946 5595 y Fj(2)1028 5410 y Fh( )1094 5556 y Fl(B)1190 5410 y Fh( )1269 5488 y Fl(\021)p 1266 5532 59 4 v 1266 5624 a(\026)1334 5410 y Fh(!)1422 5556 y Fg(^)h Fl(d)1562 5410 y Fh(!)1627 5433 y Fj(2)3606 5556 y Fm(\(44\))1841 5871 y(15)p eop %%Page: 16 16 16 15 bop 0 289 a Fm(W)-8 b(e)43 b(no)m(w)f(supp)s(ose)i(that)e Fl(e)g Fm(is)g(the)g(c)m(harge)h(of)f(the)g(electron.)73 b(Let)42 b Fl(\013)h Fm(denote)g(the)f(\014ne)h(structure)0 463 y(constan)m(t)514 395 y Fl(e)559 359 y Fj(2)p 403 439 307 4 v 403 531 a Fm(4)p Fl(\031)21 b(")574 546 y Fj(0)613 531 y Fm(\026)-50 b Fl(hc)720 463 y Fm(.)0 656 y(Setting)1615 776 y Fl(\026)1674 735 y Fj(2)1740 776 y Fm(=)28 b Fl(\013)17 b(\026)28 b Fm(=)f Fl(\021)0 941 y Fm(w)m(e)34 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b(w)m(e)f(ha)m(v)m(e)h (omitted)d(the)i(spins)g(terms.)0 3663 y(Setting)f(as)h(b)s(efore)f(:) 44 b Fl(\026)878 3626 y Fj(2)945 3663 y Fm(=)27 b(2)p Fl(\026\013)h Fm(=)g Fl(\021)t Fm(,)k(w)m(e)i(get)e(:)44 b Fl(\026)27 b Fm(=)g(2)p Fl(\013)33 b Fm(and)g Fl(\021)e Fm(=)d(4)p Fl(\013)2656 3626 y Fj(2)2695 3663 y Fm(.)0 3816 y(Hence)34 b(w)m(e)g(obtain)d(:)303 3973 y(1)p 252 4017 151 4 v 252 4109 a(4)p Fl(\013)364 4080 y Fj(2)429 4040 y Fm(\(\000)528 4055 y Fk(\026)597 4040 y Fg(\012)23 b Fm(\000)758 4055 y Fk(\026)826 4040 y Fg(\012)g Fm(\000)987 4055 y Fk(\026)1056 4040 y Fg(\012)f Fm(\000)1216 4056 y Fj(1)p Fk(=\021)1329 4040 y Fm(\))1367 3999 y Fi(\000)p Fj(1)1461 4040 y Fl(H)1542 4057 y Fk(H)1605 4038 y Fe(4)1600 4074 y Ff(e)1639 4057 y Fk(;dip)1759 4040 y Fm(\(\000)1858 4055 y Fk(\026)1927 4040 y Fg(\012)g Fm(\000)2087 4055 y Fk(\026)2156 4040 y Fg(\012)h Fm(\000)2317 4055 y Fk(\026)2385 4040 y Fg(\012)g Fm(\000)2546 4056 y Fj(1)p Fk(=\021)2658 4040 y Fm(\))83 4367 y(=)271 4300 y 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Fh( )551 5181 y Fl(M)c(m)740 5196 y Fj(2)803 5181 y Fg(\000)23 b Fm(4)p Fl(m)1037 5145 y Fj(2)1037 5206 y(1)p 551 5225 525 4 v 637 5317 a Fl(M)10 b(m)826 5332 y Fj(1)866 5317 y Fl(m)951 5332 y Fj(2)1086 5102 y Fh(!)1169 5248 y Fg(f)o Fl(p)1267 5207 y Fk(a)1331 5248 y Fg(\001)22 b Fm(\()p Fl(r)1466 5207 y Fk(a)1529 5248 y Fg(^)h Fl(B)5 b Fm(\(2)p Fl(\013)q(R)q Fm(\)\))21 b(+)h(\()p Fl(r)2202 5207 y Fk(a)2266 5248 y Fg(^)g Fl(B)5 b Fm(\(2)p Fl(\013)q(R)q Fm(\)\))22 b Fg(\001)g Fl(p)2855 5207 y Fk(a)2896 5248 y Fg(g)242 5572 y Fm(+)435 5505 y Fl(e)p 371 5549 174 4 v 371 5640 a Fm(4)p Fl(m)505 5655 y Fj(1)571 5572 y Fg(f)p Fl(p)670 5587 y Fk(a)734 5572 y Fg(\001)f Fm(\()p Fl(r)865 5587 y Fk(a)929 5572 y Fg(^)i Fl(B)5 b Fm(\(2)p Fl(\013)q(R)q Fm(\)\))21 b(+)h(\()p Fl(r)1599 5587 y Fk(a)1663 5572 y Fg(^)g Fl(B)5 b Fm(\(2)p Fl(\013)q(R)q Fm(\)\))22 b Fg(\001)g Fl(p)2252 5587 y Fk(a)2293 5572 y Fg(g)2343 5451 y Fh(\025)3790 5572 y Fm(\(48\))1841 5871 y(17)p eop %%Page: 18 18 18 17 bop 242 339 a Fm(+)17 b Fl(\013)398 297 y Fj(4)491 271 y Fl(e)536 235 y Fj(2)p 446 315 174 4 v 446 407 a Fm(4)p Fl(m)580 422 y Fj(1)647 192 y Fh(")705 271 y Fl(m)790 286 y Fj(2)830 271 y Fl(M)934 235 y Fj(2)996 271 y Fm(+)22 b(8)p Fl(m)1228 235 y Fj(3)1228 296 y(1)p 705 315 563 4 v 852 407 a Fl(m)937 422 y Fj(2)977 407 y Fl(M)1081 378 y Fj(2)1278 339 y Fm(\()p Fl(r)1363 297 y Fk(a)1426 339 y Fg(^)h Fl(B)5 b Fm(\(2)p Fl(\013)q(R)q Fm(\)\))1895 297 y Fj(2)1956 339 y Fm(+)22 b(\()p Fl(r)2136 354 y Fk(a)2199 339 y Fg(^)h Fl(B)5 b Fm(\(2)p Fl(\013)q(R)q Fm(\)\))2668 297 y Fj(2)2707 192 y Fh(#)0 587 y Fm(Th)m(us)34 b(the)f(commen)m(ts)g(in)e(the)i(three)h(b)s(o)s(dy)e(case)i(are)e(the) h(same)g(as)g(in)e(the)i(t)m(w)m(o)h(b)s(o)s(dy)e(case.)0 740 y(In)25 b(the)g(relativistic)d(case,)28 b(one)d(easily)e(sho)m(ws)k (that)d(the)h(electric-dip)s(ole)d(and)j(magnetic-dip)s(ole)d(terms)0 861 y(are)k(of)f(order)g(1)g(with)h(resp)s(ect)g(to)f(the)h(\014ne)h (structure)f(constan)m(t,)i(in)d(con)m(trast)h(to)f(the)h(non)g (relativistic)0 981 y(case.)0 1343 y Fn(App)t(endix)0 1595 y Fm(In)41 b(this)g(app)s(endix)g(w)m(e)h(giv)m(e)f(a)f(detailed)g (pro)s(of)g(of)h(the)g(expression)h(of)e(the)i(magnetization)c (\014elds)0 1716 y(giv)m(en)33 b(b)m(y)g(\(2.9\).)43 b(W)-8 b(e)33 b(ha)m(v)m(e)0 1982 y(\026)-50 b Fl(h)137 1914 y(@)p 65 1958 201 4 v 65 2050 a(@)5 b(r)165 2065 y Fk(\014)s(;j)275 1982 y Fl(S)89 b Fm(=)664 1914 y Fl(@)p 593 1958 V 593 2050 a(@)5 b(r)693 2065 y Fk(\014)s(;j)910 1874 y(N)881 1899 y Fh(X)873 2080 y Fk(\013)p Fj(=1)829 2136 y Fk(i)p Fj(=1)p Fk(;)p Fj(2)p Fk(;)p Fj(3)1080 1982 y Fl(e)1125 1997 y Fk(\013)1174 1982 y Fm(\()p Fl(r)1256 1997 y Fk(\013;i)1372 1982 y Fg(\000)22 b Fl(R)1545 1997 y Fk(i)1574 1982 y Fm(\))1629 1865 y Fh(Z)1728 1982 y Fl(d)1779 1941 y Fj(3)1818 1982 y Fl(r)1881 1865 y Fh(Z)1964 1891 y Fj(1)1927 2053 y(0)2020 1982 y Fl(d\025)17 b(\016)t Fm(\()p Fl(r)24 b Fg(\000)f Fl(R)g Fg(\000)g Fl(\025)p Fm(\()p Fl(r)2734 1997 y Fk(\013)2805 1982 y Fg(\000)g Fl(R)q Fm(\)\))p Fl(A)3129 1997 y Fk(i)3157 1982 y Fm(\()p Fl(r)s Fm(\))424 2419 y(=)673 2311 y Fk(N)645 2336 y Fh(X)637 2517 y Fk(\013)p Fj(=1)593 2573 y Fk(i)p Fj(=1)p Fk(;)p Fj(2)p Fk(;)p Fj(3)843 2419 y Fl(e)888 2434 y Fk(\013)938 2419 y Fm(\()p Fl(r)1020 2434 y Fk(\013;i)1135 2419 y Fg(\000)g Fl(R)1309 2434 y Fk(i)1337 2419 y Fm(\))1392 2302 y Fh(Z)1491 2419 y Fl(d)1542 2378 y Fj(3)1581 2419 y Fl(r)1645 2302 y Fh(Z)1728 2328 y Fj(1)1691 2490 y(0)1784 2419 y Fl(d\025)1909 2273 y Fh( )2055 2351 y Fl(@)p 1984 2395 V 1984 2487 a(@)5 b(r)2084 2502 y Fk(\014)s(;j)2194 2419 y Fl(\016)t Fm(\()p Fl(r)24 b Fg(\000)f Fl(R)g Fg(\000)g Fl(\025)p Fm(\()p Fl(r)2783 2434 y Fk(\013)2854 2419 y Fg(\000)g Fl(R)q Fm(\)\))3105 2273 y Fh(!)3187 2419 y Fl(A)3260 2434 y Fk(i)3288 2419 y Fm(\()p Fl(r)s Fm(\))583 2828 y(+)17 b Fl(e)721 2843 y Fk(\014)784 2711 y Fh(Z)884 2828 y Fl(d)935 2787 y Fj(3)974 2828 y Fl(r)1037 2711 y Fh(Z)1120 2737 y Fj(1)1083 2899 y(0)1176 2828 y Fl(d\025\016)t Fm(\()p Fl(r)25 b Fg(\000)d Fl(R)h Fg(\000)g Fl(\025)p Fm(\()p Fl(r)1873 2843 y Fk(\014)1942 2828 y Fg(\000)g Fl(R)q Fm(\)\))p Fl(A)2266 2843 y Fk(j)2302 2828 y Fm(\()p Fl(r)s Fm(\))583 3161 y Fg(\000)687 3093 y Fl(m)772 3108 y Fk(\014)p 687 3137 133 4 v 701 3229 a Fl(M)899 3053 y Fk(N)870 3078 y Fh(X)862 3259 y Fk(\013)p Fj(=1)1015 3161 y Fl(e)1060 3176 y Fk(\013)1126 3043 y Fh(Z)1225 3161 y Fl(d)1276 3119 y Fj(3)1316 3161 y Fl(r)1379 3043 y Fh(Z)1462 3070 y Fj(1)1425 3232 y(0)1518 3161 y Fl(d\025)17 b(\016)t Fm(\()p Fl(r)24 b Fg(\000)e Fl(R)i Fg(\000)e Fl(\025)p Fm(\()p Fl(r)2231 3176 y Fk(\013)2303 3161 y Fg(\000)g Fl(R)q Fm(\)\))p Fl(A)2626 3176 y Fk(j)2663 3161 y Fm(\()p Fl(r)s Fm(\))780 b(\()p Fl(A:)p Fm(1\))0 3416 y(and)365 3549 y Fk(N)337 3574 y Fh(X)329 3755 y Fk(\013)p Fj(=1)285 3811 y Fk(i)p Fj(=1)p Fk(;)p Fj(2)p Fk(;)p Fj(3)535 3657 y Fl(e)580 3672 y Fk(\013)630 3657 y Fm(\()p Fl(r)712 3672 y Fk(\013;i)827 3657 y Fg(\000)23 b Fl(R)1001 3672 y 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Fk(\013)2554 1699 y Fg(\000)g Fl(R)q Fm(\)\))p Fl(A)2878 1714 y Fk(j)2914 1699 y Fm(\()p Fl(r)s Fm(\))280 2055 y(=)83 b Fg(\000)545 1972 y Fh(X)533 2156 y Fk(\013)p Fi(6)p Fj(=)p Fk(\014)693 2055 y Fl(e)738 2070 y Fk(\013)804 1938 y Fh(Z)904 2055 y Fl(d)955 2014 y Fj(3)994 2055 y Fl(r)1057 1938 y Fh(Z)1140 1964 y Fj(1)1104 2126 y(0)1196 2055 y Fl(d\025)1314 1987 y(m)1399 2002 y Fk(\014)p 1314 2031 133 4 v 1328 2123 a Fl(M)1456 2055 y Fm(\()p Fl(\025)22 b Fg(\000)h Fm(1\)\(\()p Fl(r)1880 2070 y Fk(\013)1951 2055 y Fg(\000)g Fl(R)q Fm(\))f Fg(^)h Fl(B)5 b Fm(\()p Fl(r)s Fm(\)\))2515 2070 y Fk(j)2567 2055 y Fl(\016)t Fm(\()p Fl(r)25 b Fg(\000)e Fl(R)g Fg(\000)f Fl(\025)p Fm(\()p Fl(r)3156 2070 y Fk(\013)3228 2055 y Fg(\000)h Fl(R)q Fm(\)\))439 2411 y(+)525 2343 y Fl(m)610 2358 y Fk(\014)p 525 2387 V 539 2479 a Fl(M)696 2328 y Fh(X)684 2512 y Fk(\013)p Fi(6)p Fj(=)p Fk(\014)844 2411 y Fl(e)889 2426 y Fk(\013)955 2293 y Fh(Z)1055 2411 y Fl(d)1106 2370 y Fj(3)1145 2411 y Fl(r)1208 2293 y Fh(Z)1291 2320 y Fj(1)1254 2482 y(0)1347 2411 y Fl(d\025)32 b(\016)t Fm(\()p Fl(r)25 b Fg(\000)e Fl(R)g Fg(\000)g Fl(\025)p Fm(\()p Fl(r)2077 2426 y Fk(\013)2148 2411 y Fg(\000)g Fl(R)q Fm(\)\))p Fl(A)2472 2426 y Fk(j)2508 2411 y Fm(\()p Fl(r)s Fm(\))439 2807 y Fg(\000)526 2740 y Fl(m)611 2755 y Fk(\014)p 526 2784 V 540 2876 a Fl(M)686 2636 y Fh(0)686 2786 y(@)770 2724 y(X)758 2909 y Fk(\013)p Fi(6)p Fj(=)p Fk(\014)918 2807 y Fl(e)963 2822 y Fk(\013)1013 2636 y Fh(1)1013 2786 y(A)1119 2807 y Fl(A)1192 2822 y Fk(j)1228 2807 y Fm(\()p Fl(R)q Fm(\))p Fl(:)1951 b Fm(\()p Fl(A:)p Fm(7\))0 3114 y(W)-8 b(e)33 b(\014nally)e(get)i(from)e (\(A.1\),)i(\(A.6\))f(and)h(\(A.7\))307 3384 y(\026)-50 b Fl(h)443 3317 y(@)p 372 3361 201 4 v 372 3452 a(@)5 b(r)472 3467 y Fk(\014)s(;j)582 3384 y Fl(S)33 b Fm(=)28 b Fl(e)824 3399 y Fk(\014)888 3267 y Fh(Z)987 3384 y Fl(d)1038 3343 y Fj(3)1078 3384 y Fl(r)1141 3267 y Fh(Z)1224 3293 y Fj(1)1187 3455 y(0)1280 3384 y Fl(d\025)k(\025)p Fm(\(\()p Fl(r)1597 3399 y Fk(\014)1666 3384 y Fg(\000)23 b Fl(R)q Fm(\))f Fg(^)g Fl(B)5 b Fm(\()p Fl(r)s Fm(\)\))2229 3399 y Fk(j)2282 3384 y Fl(\016)t Fm(\()p Fl(r)25 b Fg(\000)d Fl(R)i Fg(\000)e Fl(\025)p Fm(\()p Fl(r)2871 3399 y Fk(\014)2940 3384 y Fg(\000)h Fl(R)q Fm(\)\))307 3741 y Fg(\000)437 3633 y Fk(N)409 3658 y Fh(X)401 3839 y Fk(\013)p Fj(=1)553 3741 y Fl(e)598 3756 y Fk(\013)664 3624 y Fh(Z)764 3741 y Fl(d)815 3700 y Fj(3)854 3741 y Fl(r)917 3624 y Fh(Z)1000 3650 y Fj(1)963 3813 y(0)1056 3741 y Fl(d\025)1206 3674 y(m)1291 3689 y Fk(\014)p 1206 3718 133 4 v 1220 3810 a Fl(M)1349 3741 y Fm(\()p Fl(\025)f Fg(\000)g Fm(1\))17 b(\(\()p Fl(r)1789 3756 y Fk(\013)1860 3741 y Fg(\000)23 b Fl(R)q Fm(\))f Fg(^)h Fl(B)5 b Fm(\()p Fl(r)s Fm(\)\))2424 3756 y Fk(j)2476 3741 y Fl(\016)t Fm(\()p Fl(r)25 b Fg(\000)e Fl(R)g Fg(\000)f Fl(\025)p Fm(\()p Fl(r)3065 3756 y Fk(\014)3135 3741 y Fg(\000)g Fl(R)q Fm(\)\))307 3991 y(+)32 b Fl(e)460 4006 y Fk(\014)524 3991 y Fl(A)597 4006 y Fk(j)634 3991 y Fm(\()p Fl(r)716 4006 y Fk(\014)763 3991 y Fm(\))22 b Fg(\000)h Fl(e)968 4006 y Fk(T)1033 3924 y Fl(m)1118 3939 y Fk(\014)p 1033 3968 V 1047 4060 a Fl(M)1175 3991 y(A)1248 4006 y Fk(j)1285 3991 y Fm(\()p Fl(R)q Fm(\))p Fl(:)1951 b Fm(\()p Fl(A:)p Fm(8\))0 4236 y(Setting)368 4525 y(\002)444 4540 y Fk(\014)491 4525 y Fm(\()p Fl(r)s Fm(\))27 b(=)781 4417 y Fk(N)753 4442 y Fh(X)745 4623 y Fk(\013)p Fj(=1)897 4525 y Fl(e)942 4540 y Fk(\013)1008 4407 y Fh(Z)1091 4434 y Fj(1)1054 4596 y(0)1147 4525 y Fl(d\025)p Fm(\()p Fl(\025)17 b(\016)1410 4540 y Fk(\014)s(\013)1524 4525 y Fg(\000)1634 4457 y Fl(m)1719 4472 y Fk(\014)p 1634 4501 V 1648 4593 a Fl(M)1776 4525 y Fm(\()p Fl(\025)22 b Fg(\000)h Fm(1\)\))17 b(\()p Fl(r)2217 4540 y Fk(\013)2288 4525 y Fg(\000)22 b Fl(R)q Fm(\))p Fl(\016)t Fm(\()p Fl(r)j Fg(\000)e Fl(R)g Fg(\000)f Fl(\025)p Fm(\()p Fl(r)3089 4540 y Fk(\013)3161 4525 y Fg(\000)g Fl(R)q Fm(\)\))0 4808 y(w)m(e)34 b(then)f(get)711 4953 y(\026)-50 b Fl(h)17 b Fg(r)866 4968 y Fk(r)898 4980 y Ff(\014)960 4953 y Fl(S)33 b Fm(=)28 b Fl(e)1202 4968 y Fk(\014)1266 4953 y Fl(A)p Fm(\()p Fl(r)1421 4968 y Fk(\014)1468 4953 y Fm(\))22 b Fg(\000)h Fl(e)1673 4968 y Fk(T)1754 4886 y Fl(m)1839 4901 y Fk(\014)p 1754 4930 V 1768 5021 a Fl(M)1913 4953 y(A)p Fm(\()p Fl(R)q Fm(\))g(+)2258 4836 y Fh(Z)2357 4953 y Fl(d)2408 4912 y Fj(3)2447 4953 y Fl(r)d Fm(\002)2587 4968 y Fk(\014)2634 4953 y Fm(\()p Fl(r)s Fm(\))h Fg(^)i Fl(B)5 b Fm(\()p Fl(r)s Fm(\))0 5158 y(Th)m(us)34 b(\(2.9\))e(is)g(pro)m(v)m(ed.)0 5430 y Fb(Akno)m(wledgmen)m(ts)38 b Fm(:)57 b(One)40 b(of)f(us)h(\(J.-C.)f (G.\))g(w)m(ould)g(lik)m(e)g(to)g(thank)h(V)-8 b(olk)m(er)39 b(Bac)m(h)h(for)f(helpful)0 5550 y(discussions.)1841 5871 y(20)p eop %%Page: 21 21 21 20 bop 0 289 a Fn(References)49 541 y Fm([B.B.L])37 b Fb(C.)42 b(Baxter,)h(M.)g(Babik)m(er)f(and)h(R.)f(London)p Fm(.)154 b(Ph)m(ys.)58 b(Rev.)f(A.,)38 b(47)p 3188 554 98 4 v -1 w(,)g(n)3404 505 y Fi(\016)3444 541 y Fm(2,)f(1278,)244 661 y(\(1993\).)49 865 y([B.F.S])32 b Fb(V.)37 b(Bac)m(h,)h(J.)f(F)-9 b(r\177)-56 b(ohlic)m(h)35 b(and)k(I.M.)e(Segal)p Fm(.)98 b(Adv.)44 b(Math.,)33 b(137)p 2810 878 147 4 v -1 w(,)g(p.299-395,)d (\(1998\).)49 1068 y([B.S])g Fb(H.A.)k(Bethe)g(and)h(E.)f(Salp)s(eter)p Fm(.)98 b Fa(Quantum)32 b(Me)-5 b(chanics)32 b(of)g(one)f(-)i(and)e (two)i(-)f(ele)-5 b(ctr)g(on)244 1189 y(atoms)p Fm(,)32 b(Plen)m(um)g(Rosetta)h(Edition,)e(\(1977\).)49 1392 y([C.T,D.R,G])i Fb(C.)38 b(Cohen-T)-9 b(annoudji,)40 b(J.)f(Dup)s(on)m(t-Ro)s(c)f(et)g(G.)g(Gryn)m(b)s(erg)p Fm(.)98 b Fa(Photons)35 b(et)244 1512 y(A)n(tomes.)43 b(Intr)-5 b(o)g(duction)31 b(\022)-50 b(a)31 b(l')n(\023)-47 b(ele)-5 b(ctr)g(o)g(dynamque)29 b(quantique)p Fm(,)h(\(1987\),)f(In)m (ter-Editions,)g(CNRS.)244 1633 y(F)-8 b(rance.)49 1836 y([Cr.Th])35 b Fb(D.P)-9 b(.)38 b(Craig)h(and)h(T.)f(Thirunamac)m (handram)p Fm(.)98 b Fa(Mole)-5 b(cular)36 b(Quantum)h(Ele)-5 b(ctr)g(o)g(dy-)244 1956 y(namics)p Fm(,)31 b(Academic)h(Press,)j (London,)d(\(1984\).)49 2160 y([F])g Fb(A.)37 b(F)-9 b(elhi)p Fm(.)140 b(Ph.)44 b(D.,)32 b(to)h(b)s(e)f(published.)49 2363 y([F.F.G])46 b Fb(C.)54 b(F)-9 b(e\013erman,)60 b(J.)54 b(F)-9 b(r\177)-56 b(ohlic)m(h)52 b(and)k(G.M.)e(Graf)p Fm(.)99 b(Com.)87 b(Math.)g(Ph)m(ys.,)53 b(190)p 3606 2376 V -1 w(,)244 2484 y(p.309-330,)31 b(\(1997\).)49 2687 y([G.H])h Fb(H.)37 b(Grotc)m(h)g(and)i(R.A.)d(Hegstrom)p Fm(.)140 b(Ph)m(ys.)46 b(Rev.)e(A.4)p 2594 2700 49 4 v(,)32 b(59,)g(\(1971\).)49 2890 y([L.B.B.L])d Fb(V.E.)j(Lem)m(b)s (essis,)i(M.)f(Babik)m(er,)h(C.)f(Baxter)f(and)i(R.London)p Fm(.)183 b(Ph)m(ys.)44 b(Rev.)244 3011 y(A.,)33 b(48)p 404 3024 98 4 v -1 w(,)g(n)615 2975 y Fi(\016)655 3011 y Fm(2,)f(1594,)g(\(1993\).)49 3214 y([S])i Fb(S.S.)39 b(Sc)m(h)m(w)m(eb)s(er)p Fm(.)98 b Fa(A)n(n)36 b(Intr)-5 b(o)g(duction)35 b(to)h(R)-5 b(elativistic)35 b(Quantum)h(Field)f(The) -5 b(ory)p Fm(.)46 b(Ro)m(w,)34 b(P)m(e-)244 3335 y(terson)f(and)g (Compan)m(y)-8 b(,)33 b(\(1961\).)49 3538 y([W])g Fb(M.)38 b(Wilk)m(ens)p Fm(.)139 b(Ph)m(ys.)45 b(Rev.)g(A.,)32 b(48)p 1645 3551 V(,)h(n)1857 3502 y Fi(\016)1896 3538 y Fm(6,)g(570,)f(\(1994\).)1841 5871 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0106050727560--