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arXiv:1106.3757 · AIP conference proceedings

Unstable particles in non-relativistic quantum mechanics?

H. Hernandez-Coronado, Luis Arturo Ureña-López, Hugo Aurelio Morales-Técotl, Román Linares-Romero, Elí Santos-Rodríguez, Sendic Estrada-Jiménez

Published Jan 1, 2011• 0 citations
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Abstract

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.

DOI
10.1063/1.3647538

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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.
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.
We reinvestigate Bargmann's superselection rule for the overall mass of n particles in ordinary quantum mechanics with Galilei invariant interaction potential. We point out that in order for mass to define a superselection rule it should be considered as a dynamical variable. We present a minimal extension of the original dynamics in which mass it treated as dynamical variable. Here the classical symmetry group turns out to be given by an \\reals-extension of the Galilei group which formerly appeared only at the quantum level. There is now no obstruction to implement an action of the classical symmetry group on Hilbert space. We include some critical comments of a general nature on formal derivations of superselection rules without dynamical context.
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.
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.
The superposition principle is a very basic ingredient of quantum theory. What may come as a surprise to many students, and even to many practitioners of the quantum craft, is tha superposition has limitations imposed by certain requirements of the theory. The discussion of such limitations arising from the so-called superselection rules is the main purpose of this paper. Some of their principal consequences are also discussed. The univalence, mass and particle number superselection rules of non-relativistic quantum mechanics are also derived using rather simple methods.
Mass-superselection rule (MSR) states that in the non-relativistic quantum theory superpositions of states with different masses are unphysical. While MSR features even in textbooks, its validity, physical content and consequences remain debated. Its original derivation is known to be inconsistent, while a consistent approach does not yield a superselection rule. Yet, superpositions of masses do not seem to be present in Newtonian physics. Here we offer a resolution of the MSR puzzle. Crucial for the result is the understanding of two issues: how the notion of a mass parameter arises from the fundamentally dynamical relativistic notion of mass-energy, and what is the correct Newtonian limit of relativistic dynamics of composite particles. The result explains the physical content of the MSR without invoking any non-standard physics and clarifies the relation between MSR and a formalism describing relativistic composite particles, developed for studying relativistic effects in table top experiments.