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

New gauge transformation for a generalized quantum representation

A. D. Jannussis, V. Papatheou, P. Filippakis

Published Aug 1, 1980• 2 citations
Publisher
DOI
10.1007/bf02776200

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A new method for finding the dynamical invariant for a general time-dependent harmonic oscillator is proposed by making use of two linearly independent solutions to the classical equation of motion. It is shown that the dynamical invariant for different gauged Hamiltonians are connected by time-dependent gauge transformations. Therefore, the representation whose bases are the instantaneous eigenstates of the invariant operator is a good representation for the quantum nonautonomous system. In this representation, the wavefunction of the system is gauge covariant and thus any observable physical effect is naturally independent of the choice of gauged Hamiltonians. The exact even and odd coherent states for a time-dependent harmonic oscillator are constructed in terms of these gauge covariant bases. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.
Here, the ordinary gauge transformations in configuration space are generalized to the phase space. The corresponding gauge functions are separable in terms of those of the configuration and the momentum spaces. Furthermore, we consider the gauge transformations in phase space for which their gauge functions are not, in general, separable. It is shown that, the assumption of the general global gauge symmetry in phase space immediately yields the quantization of the electric charge. We obtain this result without assuming the concept of the magnetic monopole that Dirac did in his approach to the problem of electric charge quantization and which is not detected yet.
The equivalence of representations of symmetry groups operating upon wavefunctions in configuration space is studied with regard to the (intuitive) notion of physical equivalence. A refinement of the usual projective equivalence relation is introduced, called gauge equivalence, for which the allowed unitary equivalence transformations are gauge transformations. For a Euclidean as well as for a Newton-Hooke symmetry group the gauge equivalence classes of unitary multiplier representations are determined. These examples support the assertion that equivalence from a physical viewpoint corresponds better to this new gauge equivalence concept than to the usual notion of projective equivalence.