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Journal of Mathematical Physics

Ray representations of N(≤2)+1-dimensional Galilean group

Heinz Dietrich Doebner, Hans-Jürgen Mann

Published Jul 1, 1995• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.531026

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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.
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.
We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.
The aim of the present chapter is to obtain concrete expressions for observables like position, linear momentum, orbital and spin angular momenta, energy etc., through an investigation of the projective unitary antiunitary (p.u.a.) representations of the groups under which the description of the quantum mechanical system is assumed to be covariant. (See sections 1.4 and 1.5). Special emphasis is laid on the Galilean group and the Poincare group. (See sections 4.2–4.5). The basic feature of these groups is that they are semidirect products of a vector group and a Lie group which acts on the vector group as nonsingular linear transformations.
Physical states in quantum mechanics are rays in a Hilbert space. Projective representations of a relativity group transform between the quantum physical states that are in the admissible class. The physical observables of position, time, energy and momentum are the Hermitian representation of the generators of the algebra of the Weyl–Heisenberg group. We show that there is a consistency condition that requires the relativity group to be a subgroup of the group of automorphisms of the Weyl–Heisenberg algebra. This, together with the requirement of the invariance of classical time, results in the inhomogeneous Hamilton group. The Hamilton group is the relativity group for noninertial frames in classical Hamilton's mechanics. The projective representation of a group is equivalent to unitary representations of the central extension of the group. The central extension of the inhomogeneous Hamilton group and its corresponding Casimir invariants are computed. One of the Casimir invariants is a generalized spin that is invariant for noninertial states. It is the familiar inertial Galilean spin with additional terms that may be compared to noninertial experimental results.
The representation theory of the Galilean group extended by a one-dimensional group of dilations (which leave invariant the free Schrodinger equation) is studied. The existence of position and time operators is investigated from the standpoint of the imprimitivity theorem, for quantum particles whose states carry a projective representation of the extended group. Position and time operators are shown to exist for some two-component direct sums with non-trivial multipliers and for vector representations with zero helicity.
Universal quantum integrals of motion are introduced, and their relation with the universal quantum invariants is established. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, which are preserved in time independently of the concrete form of the coefficients of the Schrödinger equation, provided the Hamiltonian is either a generic quadratic form of the coordinate and momenta operators, or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics in terms of the Wigner function, the relations between the quantum invariants and the classical universal integral invariants by Poincaré and Cartan are elucidated. Examples of the `universal invariant solutions' of the Schrödinger equation, i.e. self-consistent eigenstates of the universal integrals of motion, are given. Applications to the physics of optical and particle beams are discussed.