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Journal of Physics A Mathematical and General

Central extensions and realizations of one-dimensional Galilean systems and quantization

Miguel A. Martín, M. A. del Olmo

Published Feb 7, 1996• 6 citations
Publisher

Abstract

The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei group, (1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, (1 + 1). As an application we construct the Stratonovich - Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.

DOI
10.1088/0305-4470/29/3/019

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The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei group, (1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, (1 + 1). As an application we construct the Stratonovich - Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.
Abstract A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in the Koopman–von Neumann formulation of classical mechanics. This classical representation is characterized by the vanishing of the central charge of the Galilei algebra. This is in contrast to the quantum case where the mass plays the role of the central charge. Here we show, by direct construction, that classical mechanics also allows for a projective representation of the Galilei group where the mass is the central charge of the algebra. We extend the result to certain kind of quantum–classical hybrid systems.
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl [Formula: see text]-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
Mathjax On | Off Abstract In the preceding chapter, we showed how one-particle quantum states can be obtained from unitary irreducible representations of the Galilei group, the transformation group of Newton relativity. Since the Galilei group defines transformations amongst inertial reference frames, this description of one-particle quantum states is limited to inertial observers. In particular, observers who are accelerating are excluded in this description. This motivates the question of whether it is possible to provide a description of one-particle quantum states that holds for non-inertial observers by expanding the notion of Newton relativity to include non-inertial reference frames.
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.