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Annals of Physics

Gauge transformations and quantum mechanics I. Gauge invariant interpretation of quantum mechanics

Kuo‐Ho Yang

Published Sep 1, 1976• 177 citations
Publisher
DOI
10.1016/0003-4916(76)90275-x

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Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations

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In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Nov 1, 1996•134 citations
The conventional interpretation of the quantum mechanical probability amplitudes and probabilities is examined in detail with respect to gauge transformations and the Foldy-Wouthuysen transformations. It is shown that the conventional interpretation is gauge dependent and, when applied to the relativistic (Dirac) and the non-relativistic quantum mechanics, cannot be consistent with itself through the Foldy-Wouthuysen transformations. It is also shown that these two difficulties are not present in a gauge invariant formulation that explicitly incorporates Poynting's theorem and the conservation law of energy into the definition of probability amplitudes.
In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.