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arXiv:2012.02324 · Physical Review A

Galilean covariance of quantum-classical hybrid systems of the Sudarshan type

A. D. Bermúdez Manjarres, Nathaly Marín–Medina

Published Oct 22, 2020• 5 citations
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Abstract

We revisit quantum-classical hybrid systems of the Sudarshan type under the light of Galilean covariance. We show that these kinds of hybrids cannot be given as a unitary representation of the Galilei group and at the same time conserve the total linear momentum unless the interaction term only depends on the relative canonical velocities.

DOI
10.1103/physreva.102.042221

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We revisit quantum-classical hybrid systems of the Sudarshan type under the light of Galilean covariance. We show that these kinds of hybrids cannot be given as a unitary representation of the Galilei group and at the same time conserve the total linear momentum unless the interaction term only depends on the relative canonical velocities.
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.
Galilei transformations are formulated in a 5-dimensional form and on such basis the non-relativistic quantum mechanics is reconstructed. Our approach has the following advantages over others'. (1) The Schrödinger wavefunctions are taken to be vector (not projective) representations of the Galilei group, so that (2) the problem here is considerably simplified. (3) The formulation proceeds in a way quite parallel to that of relativistic quantum mechanics, and what is most important, (4) Galilei covariance is manifest throughout.
It is known that the Schroedinger equation is not covariant under Galilei boosts, unless the phase of its solutions are shifted simultaneously. It is argued that the phase shift is not a coordinate transformation, because it depends on the mass of the Schroedinger particle. The phase shift also cannot be derived from low speed Lorentz boost. It is proposed to extend the Galilei boost with two terms of order v^2/c^2 to avoid these issues and to guarantee covariance of the Schroedinger kinetic energy and momentum. The extensions imply that proper time and relativity of simultaneity are essential features of Schroedinger quantum mechanics.
Textbook treatments of the Galilean covariance of the time-dependent Schrödinger equation for a spinless particle seem invariably to cover the case of a free particle or one in the presence of a scalar potential. The principal objective of this paper is to examine the situation in the case of arbitrary forces, including the velocity-dependent variety resulting from a vector potential. To this end, we revisit the 1964 theorem of Jauch which purports to determine the most general form of the Hamiltonian consistent with “Galilean-invariance,” and argue that the proof is less than compelling. We then show systematically that the Schrödinger equation in the case of a Jauch-type Hamiltonian is Galilean covariant, so long as the vector and scalar potentials transform in a certain way. These transformations, which to our knowledge have appeared very rarely in the literature on quantum mechanics, correspond in the case of electrodynamical forces to the “magnetic” nonrelativistic limit of Maxwell’s equations in the sense of Le Bellac and Lévy-Leblond (1973). Finally, this Galilean covariant theory sheds light on Feynman’s “proof” of Maxwell’s equations, as reported by Dyson in 1990.