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\topmatter
\title
Quantum dynamics of transition processes in the interacting systems
\endtitle
\author
Denis V. Juriev
\endauthor
\address Thalassa Aitheria, Ltd.\newline
Res Center for Math Phys and Informatics,\newline
Miklukho-Maklaya 20-180 Moscow 117437 Russia
\endaddress
\email denis\@juriev.msk.ru
\endemail
\abstract The dynamical inverse problem of representation theory (the Wigner
problem) [1] is solved for a certain class of interactions of Hamiltonian
systems. The constructed quantum dynamics describes transition processes and
stationary regimes on an equal footing. Among other numerous applications the
results may be useful for the description of the simultaneous quantum dynamics
of the open and closed string fields in the unified string field theory with
their noncanonical coupling.
\endabstract
\endtopmatter
\document

\

The aim of this short note is to describe the quantum picture for
transition processes in a certain class of pairs of interacting systems.

\proclaim{Theorem 1} Let $(V_1,V_2)$ be an arbitrary isotopic pair [2],
$\Der(V_1,V_2)$ be the Lie algebra of its derivations, and $L$ be an arbitrary
polynomial mapping from $V_1\oplus V_2$ into $\Der(V_1,V_2)$ (a nonlinear
sectional operator -- cf.[3]), then
the equations
$$\left\{\aligned
\dot X_t=& L(X_t,A_t)^{(1)}X_t\\
\dot A_t=& L(X_t,A_t)^{(2)}A_t
\endaligned\right.$$
where $X_t\in V_1$, $A_t\in V_2$,
$(L(X_t,A_t)^{(1)},L(X_t,A_t)^{(2)})\in\Der(V_1,V_2)$, preserves the
isocommutation relations.
\endproclaim

\proclaim{Corollary 1} Let $(V_1,V_2)$ be an arbitrary isotopic pair, $T$ be
its representation in the space $U$ [2], i.e. a homomorphism of the isotopic
pair $(V_1,V_2)$ into the operator isotopic pair $(\End(U),\End(U))$. Let
$L$ be an arbitrary polynomial mapping from $\End(U)\oplus\End(U)$ into
$\Der(\End(U),\End(U))$, then the equations
$$\left\{\aligned
\dot T_t(A)=& L(T_t(X),T_t(A))^{(1)}T_t(A)\\
\dot T_t(X)=& L(T_t(X),T_t(A))^{(2)}T_t(X)
\endaligned\right.$$
where $X\in V_1$, $A\in V_2$, define a family $T_t$ ($T_0=T$) of
representations of the isotopic pair $(V_1,V_2)$ in the space $U$. It means
that the operators $\hat X_t=T_t(X)$ and $\hat A_t=T_t(A)$ obey the initial
isocommutation relations.
\endproclaim

\remark{Remark 1} An operator isotopic pair in Corollary 1 may be changed to
any isotopic pair $(W_1,W_2)$. The representation $T$ should be changed to
a homomorphism $\pi$ from $(V_1,V_2)$ to $(W_1,W_2)$. The most natural
candidates for $(W_1,W_2)$ are geometric isotopic pairs or asymmetric
isotopic pairs $(S^2(U),\Lambda^2(U^*))$.
\endremark

\proclaim{Theorem 2} An arbitrary derivation of the operator isotopic pair
$(\End(U),\End(U))$ has the form
$$\aligned
LX=& [H,X]+S\circ A\\
LA=& [H,A]-S\circ X
\endaligned
$$
where $H,S\in\End(H)$, $[X,Y]=XY-YX$, $X\circ Y=\frac12(XY+YX)$. So the
Lie algebra $\Der(\End(U),\End(U))$ is isomorphic to $\gla(n)\oplus\gla(n)$
($n=\dim U$).
\endproclaim

\proclaim{Corollary 2} Let $(V_1,V_2)$ be an arbitrary isotopic pair, $T$ be
its representation in the space $U$, and $\hat H$, $\hat S$ be polynomial
expressions of operators $\hat X$, $\hat A$, then the dynamics
$$\left\{\aligned
\tfrac{\partial}{\partial t}\hat X_t=& [\hat H(\hat X_t,\hat A_t),\hat X_t]
+\hat S(\hat X_t,\hat A_t)\circ\hat X_t\\
\tfrac{\partial}{\partial t}\hat A_t=& [\hat H(\hat X_t,\hat A_t),\hat A_t]
-\hat S(\hat X_t,\hat A_t)\circ\hat A_t
\endaligned\right.$$
({\it the coupled Manakov dynamics}, cf.[4]) preserves the initial
isocommutation relations between operators $\hat X$ and $\hat A$.
\endproclaim

\remark{Remark 2} The coupled Manakov dynamics may be written in the
alternative form
$$\left\{\aligned
\tfrac{\partial}{\partial t}\hat X_t=& \hat M(\hat X_t,\hat A_t)\hat X_t
-\hat X_t\hat N(\hat X_t,\hat A_t)\\
\tfrac{\partial}{\partial t}\hat A_t=& \hat N(\hat X_t,\hat A_t)\hat A_t
-\hat A_t\hat M(\hat X_t,\hat A_t)
\endaligned\right.$$
where $\hat M=\hat H+\hat S$ and $\hat N=\hat H-\hat S$.
\endremark

\remark{Remark 3} The equations of Corollary 2 are the quantum counterparts of
classical dynamical equations for two systems with potential Hamiltonian,
magnetic--type (gyroscopic) and Rayleigh--Lienart type nonHamiltonian
interactions. The classical equations describe transition processes as well
as stationary regimes in pairs of the interacting systems with
auto--oscillations. It is remarkable that both cases are described on an
equal footing in the quantum picture.
\endremark

\remark{Remark 4} Quantum dynamical equations of the paper [5] is a particular
case of the equations of Corollary 2.
\endremark

\remark{Remark 5} All results allow a supergeneralization (cf.[2]).
\endremark

\remark{Remark 6} The results may be useful for the description of the
simultaneous quantum dynamics of closed and open string fields in the unified
string field theory with their {\it noncanonical coupling}. It seems that the
algebraic structure of the quantum string field theory [6] supports such
description.
\endremark

\

\

\Refs
\roster
\item"[1]" Wigner E., Do the equations of motion determine the quantum
mechanical commutation relations? Phys. Rev. 77 (1950), 711-712;\newline
Juriev D., Dynamical inverse problem of representation theory and the
noncommutative geometry. Report RCMPI-95/03 (1995) [e-print version (SISSA
Electronic Archive on Funct. Anal.): {\it funct-an/9507001\/} (1995)].
\item"[2]" Juriev D., Topics in isotopic pairs and their representations.
Theor. Math. Phys. 105(1) (1995), 18-28 [e-print version (Texas Univ.
Electronic Archive on Math. Phys.): {\it mp\_arc/94-267\/} (1994)]; Topics
in isotopic pairs and their representations. II. A general supercase. Report
RCMPI-95/06 (1995) [e-print version (Duke Univ. Electronic Archive on Quant.
Alg.): {\it q-alg/9511012\/} (1995)].
\item"[3]" Trofimov V.V., Fomenko A.T., Algebra and geometry of integrable
Hamiltonian dynamical systems. Moscow, 1995.
\item"[4]" Manakov S.V., The inverse scattering problem method and
two--dimensional evolution equations. Russian Math. Surveys 31(5) (1976),
245-246.
\item"[5]" Juriev D., Classical and quantum dynamics of noncanonically
coupled oscillators and Lie superalgebras. Russian J. Math. Phys., to appear
[e-print version (SISSA Electronic Archive on Funct. Anal.): {\it
funct-an/9409003\/} (1994)].
\item"[6]" Juriev D., Infinite dimensional geometry and quantum field theory
of strings. II. Infinite dimensional noncommutative geometry of a
self--interacting string field. Russian J. Math. Phys. 4 (1996), to appear
[e-print version (LANL Electronic Archive on Theor. High Energy Phys.): {\it
hep-th/9403148\/} (1994)].
\endroster
\endRefs
\enddocument