?Classical? equations of motion in quantum mechanics with gauge fields

Belov, V. V.; Kondrat'eva, M. F.

doi:10.1007/bf01018700

Cited by 12 publications

(8 citation statements)

References 15 publications

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“…The non-Abelian case also requires appreciable complication of the usual WKB-method in the analysis of Eq. (5.3) that is beyond the scope of this work (see, for example, [28,31,71]). A solution of equation (5.3) in the semiclassical limit is defined as a series in powers of…”

Section: Mapping Into Lagrangian With a Local Supersymmetrymentioning

confidence: 99%

Mapping between the classical and pseudoclassical models of a relativistic spinning particle in external bosonic and fermionic fields. I

Markov

Markova

2015

Nuclear Physics A

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Abstract. The problem on mapping between two Lagrangian descriptions (using a commuting c-number spinor ψ α or anticommuting pseudovector ξ µ and pseudoscalar ξ 5 variables) of the spin degrees of freedom of a color spinning massive particle interacting with background non-Abelian gauge field, is considered. A general analysis of the mapping between a pair of Majorana spinors (ψ α , θ α ) (θ α is some auxiliary anticommuting spinor) and a real anticommuting tensor aggregate (S, V µ , * T µν , A µ , P ), is presented. A complete system of bilinear relations between the tensor quantities, is obtained. The analysis we have given is used for the above problem of the equivalence of two different ways of describing the spin degrees of freedom of the relativistic particle. The mapping of the kinetic term (i /2)(θθ)(ψψ −ψψ), the term (1/e)(θθ)ẋ µ (ψγ µ ψ) that provides a couple of the spinning variable ψ and the particle velocitẏ x µ , and the interaction term (θθ)Q a F a µν (ψσ µν ψ) with an external non-Abelian gauge field, are considered in detail. In the former case a corresponding system of bilinear identities including both the tensor variables and their derivatives (Ṡ,V µ , * Ṫ µν ,Ȧ µ ,Ṗ ), is defined. A detailed analysis of the local bosonic symmetry of the Lagrangian with the commuting spinor ψ α , is carried out. A connection of this symmetry with the local SUSY transformation of the Lagrangian containing anticommuting pseudovector and pseudoscalar variables, is considered. The approach of obtaining a supersymmetric Lagrangian in terms of the even ψ α and odd θ α spinors, is offered. *

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Section: Mapping Into Lagrangian With a Local Supersymmetrymentioning

confidence: 99%

Mapping between the classical and pseudoclassical models of a relativistic spinning particle in external bosonic and fermionic fields. I

Markov

Markova

2015

Nuclear Physics A

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“…For the scalar SchrSdinger equation, a finite 2-system was first obtained in [7] in another way; for N = 1, a finite 2-system given on a separate symplectic leaf was obtained independently in [14]. The generalization of a 2-system was obtained in [8] for the vector Pauli equation, and in [9] for the SchrSdinger and Dirac equations with the gauge group SU(2).…”

Section: =0mentioning

confidence: 99%

Hamiltonian systems of equations for quantum means

Belov¹,

Kondrat'eva²

1994

Math Notes

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“…For operators with symbols (4.10), averaged over such states, the following estimate is valid [16, 10, where eqkr are the structural constants of a Lie algebra isomorphic to su (2). Therefore, the matrix Q(z) defines a Poisson structure on the dual space g* of the Lie algebra g isomorphic to the direct sum of the Lie algebras sp(2N, R) and su (2) , the classical equations as the semiclassical limit of the equations of motion for quantum averages were derived in [16][17][18][19] where it was also shown that for Hamiltonians of the form (4.8) and (4.9) the systems of equations for quantum averages lead to the classical Frenkel [22] and Wong [23] equations, respectively. In contrast to [18], in the present paper we obtain the closed Hamiltonian system (4.10)-(4.13) for the set 7/(z) of quantum averages of position and momentum operators, their second moments, and spin, whereas in [18] the equation for second moments is considered as a relation between the parameters of a closed Hamiltonian system for the quantum averages of the position, momentum and spin operators.…”

Section: Nmentioning

confidence: 99%

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

Belov

Kondrat'eva

1995

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ABSTRACT. An infinite system of ordinary differential equations for ~,/~, and for averages of a set of operators is derived for quantum-mechanical problems with a (K x K) matrix Hamiltonian 74(~,/~), z E R N 9 The set of operators is chosen to be basis in the space MatKC @ UOSYN) , where U(WN) is the universal enveloping algebra of the Heisenberg-Weyl algebra FVN, generated by the time-dependent operators ], ~ -e(t) 9 I, and /~ -/~(t) 9 ], where ] is the identity operator and ~ and /~ are the averages of the position and momentum operators. The system in question can be written in Hamiltonian form; the corresponding Poisson bracket is degenerate and is equal to the sum of the standard bracket on R ~N with respect to the variables (~',/~) and the generalized Dirac bracket with respect to the other variables. The possibility of obtaining finite-dimensional approximations to the infinite-dimensional system in the semiclassical limit h ---, 0 is investigated.

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?Classical? equations of motion in quantum mechanics with gauge fields

Cited by 12 publications

References 15 publications

Mapping between the classical and pseudoclassical models of a relativistic spinning particle in external bosonic and fermionic fields. I

Mapping between the classical and pseudoclassical models of a relativistic spinning particle in external bosonic and fermionic fields. I

Hamiltonian systems of equations for quantum means

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

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