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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

A group-theoretical representation of the generalized Heisenberg relations

ELIZABETH A. RAUSCHER

Published Dec 1, 1972• 3 citations
Publisher
DOI
10.1007/bf02777992

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A set of generalized Heisenberg relations and a possible new form of quantization

ELIZABETH A. RAUSCHER

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento•Aug 1, 1972•5 citations
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.
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.
We define general lowering and raising operators of the eigenstates for one-dimensional quantum mechanical potential problems leading to discrete energy spectra and investigate their associative algebra. The Hamilton operator is quadratic in these lowering and raising operators and corresponding representations of operators for action and angle are found. The normally ordered representation of general operators using combinatorial elements such as partitions is derived. The introduction of generalized coherent states is discussed. Linear laws for the spacing of the energy eigenvalues lead to the Heisenberg–Weyl group and general quadratic laws of level spacing to unitary irreducible representations of the Lie group SU (1, 1) that is considered in detail together with a limiting transition from this group to the Heisenberg–Weyl group. The relation of the approach to quantum deformations is discussed. In two appendices, the classical and quantum mechanical treatment of the squared tangent potential is presented as a special case of a system with quadratic level spacing.