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Physical Review Letters

Inadequacy of the Usual Galilean Transformation in Quantum Mechanics

Daniel M. Greenberger

Published Aug 21, 2001• 48 citations
Publisher

Abstract

We show that the superselection rule in the Galilean transformation, forbidding the superposition of states of different mass, in inconsistent with the nonrelativistic limit of the Lorentz transformation. We also point out that the extra Galilean phase is merely the residue of the "twin-paradox" effect, which does not vanish nonrelativistically. In general, there are phase effects due to proper time differences, and effects due to mass-energy equivalence, that do not vanish nonrelativistically but that are not handled adequately by the Galilean transformation.

DOI
10.1103/physrevlett.87.100405

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We show that the superselection rule in the Galilean transformation, forbidding the superposition of states of different mass, in inconsistent with the nonrelativistic limit of the Lorentz transformation. We also point out that the extra Galilean phase is merely the residue of the "twin-paradox" effect, which does not vanish nonrelativistically. In general, there are phase effects due to proper time differences, and effects due to mass-energy equivalence, that do not vanish nonrelativistically but that are not handled adequately by the Galilean transformation.
Abstract It is shown that the generally accepted statement that one cannot superpose states of different mass in non-relativistic quantum mechanics is inconsistent. It is pointed out that the extra phase induced in a moving system, which was previously thought to be unphysical, is merely the non-relativistic residue of the "twin-paradox" effect. In general, there are phase effects due to proper time differences between moving frames that do not vanish non-relativistically. There are also effects due to the equivalence of mass and energy in this limit. The remedy is to include both proper time and rest energy non-relativis-tically. This means generalizing the meaning of proper time beyond its classical meaning, and introduc­ ing the mass as its conjugate momentum. The result is an uncertainty principle between proper time and mass that is very general, and an integral role for both concepts as operators in non-relativistic physics.
The covariance of the Schr\"odinger equation under Galilei boosts and the compatibility of nonrelativistic quantum mechanics with Einstein's equivalence principle have been constrained for so long to the existence of a superselection rule which would prevent a quantum particle from being found in superposition states of different masses. In an effort to avoid this expedient, and thus allow nonrelativistic quantum mechanics to account for unstable particles, recent works have suggested that the usual Galilean transformations are inconsistent with the nonrelativistic limit implied by the Lorentz transformation. Here we approach the issue in a fundamentally different way. Using a formalism of unitary transformations and employing quantum reference frames rather than immaterial coordinate systems, we show that the Schr\"odinger equation, although form variant, is fully compatible with the aforementioned principles of relativity.
The Schroedinger equation is up-to-a-phase invariant under the Galilei group. This phase leads to the Bargmann's superselection rule, which forbids the existence of the superposition of states with different masses and implies that unstable particles cannot be described consistently in non-relativistic quantum mechanics. In this paper we claim that Bargmann's rule neglects physical effects and that a proper description of non-relativistic quantum mechanics requires to take into account this phase through the Extended Galilei group and the definition of its action on spacetime coordinates.
The Galilean transformation carries one inertial frame into another with a different velocity. The extended Galilean transformation carries an inertial frame into a rigid frame with uniform spatial, but arbitrary translational acceleration x′=x+ξ (t). Besides being useful in discussing the equivalence principle, this transformation provides a physical interpretation for the theorem in nonrelativistic quantum mechanics, that we cannot coherently combine states of different mass. This restriction comes about because we cannot distinguish between coordinate and proper time in nonrelativistic physics.
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
The correspondence of the previously given relativistic quantum mechanics of spinless or spin 1/2 particle with the quantum nonrelativistic and the classical relativistic mechanics is proved. In the Galilean quantum case the appearance of the familiar transformation phase factor is explained and the origin of the mass superselection rule indicated. In the relativistic classical case the equation for trajectories and the covariant equation of motion for spin are obtained. The conditions are found under which the usual theory of relativistic wave equations is of relevance (in some particular reference frame).