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arXiv:1411.6048 · Journal of Mathematical Physics

Time dependent quantum generators for the Galilei group

G. Filippelli

Published Aug 1, 2011• 2 citations
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Abstract

In 1995, Doebner and Mann introduced an approach to the ray representations of the Galilei group in (1 + 1)-dimensions, giving rise to quantum generators with an explicit dependence on time. Recently (2004), Wawrzycki proposed a generalization of Bargmann's theory: in his paper, he introduce phase exponents that are explicitely dependent by 4-space point. In order to find applications of such generalization, we extend the approach of Doebner and Mann to higher dimensions: as a result, we determine the generators of the ray representation in (2 + 1) and (3 + 1) dimensions. The differences of the outcoming formal apparatus with respect to the smaller dimension case are established.

DOI
10.1063/1.3621518

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In 1995, Doebner and Mann introduced an approach to the ray representations of the Galilei group in (1 + 1)-dimensions, giving rise to quantum generators with an explicit dependence on time. Recently (2004), Wawrzycki proposed a generalization of Bargmann's theory: in his paper, he introduce phase exponents that are explicitely dependent by 4-space point. In order to find applications of such generalization, we extend the approach of Doebner and Mann to higher dimensions: as a result, we determine the generators of the ray representation in (2 + 1) and (3 + 1) dimensions. The differences of the outcoming formal apparatus with respect to the smaller dimension case are established.
The paper contains a complete theory, which is a generalization of the Bargmann’s theory of factors for ray representations. The theory is of primary importance in the investigations of covariance (in contradistinction to symmetry) of a quantum theory which possesses a nontrivial gauge freedom. In that case the group in question is not any symmetry group but it is a covariance group only – that case which has not been deeply investigated. It is shown on the paper that the factor of its representation depends on space and time when the system in question possesses a gauge freedom. In the nonrelativistic theories the factor depends on the time only. In connection with we explain two applications of this generalization: in the theory of a quantum particle in the gravitational field in the nonrelativistic limit and in the quantum electrodynamics. 1
The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
An elementary derivation of the system of phase factors defining the ray representations (those faithful to within a phase factor) of the Galilei group is presented. The proof employs the group elements themselves. In particular, the operation of conjugation (which corresponds to coordinate transformation) is used extensively to effect the desired result, i.e., in the notation of Levy-Leblond, U(b′,a′,v′,R′)U(b,a,v,R)=±exp [i(12m)(a′·R′v−v′·R′a+bv′·R′v)]×U(b′+b,a′+R′a+bv′,v′+R′v,R′R).
For a quantum mechanical system, modeled via unitary or antiunitary representations of a symmetry group G with Lie algebra g, the Hamiltonian H is of special interest. If H is an element of g or of the universal enveloping algebra U(g), a generic time dependence for any element in U(g) is given through the Heisenberg picture. As an example we consider a system with Gal(N) as symmetry group with H as one of the generators. For N≥3 one gets from ray representations the free Schrödinger equation. For N=1 a peculiarity occurs: the ‘‘free’’ equation has an interaction term, which results from the construction and parametrization of unitary ray representations. For N=2 there is another special feature: there exist ray representations of the universal covering group Gal(2)■, which induce no ray representations of Gal(2). Furthermore, for these representations it is not possible to construct a Schrödinger equation.