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arXiv:quant-ph/9809059 · European Journal of Physics

Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics

C. Cisneros, R. P. Martı́nez-y-Romero, H. N. Núñez‐Yépez, A. L. Salas‐Brito

Published May 1, 1998• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0143-0807/19/3/005

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This is a comment to the paper, Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics by C Cisneros et al 1998 Eur. J. Phys. 19 237. doi:10.1088/0143-0807/19/3/005. The proof that the authors construct for the limitation on the superposition of state vectors corresponding to different sectors of the Hilbert space, partitioned by a superoperator has a flaw as outlined below.

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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.
Some physicists believe that superselection rules should be implemented to get rid of inconsistencies when a theory is framed in terms of a new mathematical formulation, whilst others think that this new formulation should be modified instead of implementing those rules, at the expense of introducing additional mathematical structure. The outcome, however, is that we are still uncertain whether these rules should be implemented and how they should be interpreted and assessed from the philosophical point of view. Based on a detailed examination of the group-theoretic reformulation of (relativistic and non-relativistic) quantum mechanics that prompts physicists to impose superselection rules, I shall argue that the implementation of these rules involves serious heuristic and epistemological concerns. Considering this argument, I shall conclude that there are suitable philosophical reasons to claim that the implementation of superselection rules should be rejected and that there are certain circumstances when the formulation of a theory should be modified.
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.
A superselection rule advanced in the course of a quantum-mechanical treatment of some phenomenon is an assertion to the effect that the superposition principle of quantum mechanics is to be restricted in the application at hand. Superselection accounts of measurement all have in common a decision to represent the indicator states of detectors by eigenspaces of superselection operators named in a superselection rule, on the grounds that the states in question are states of a so-called classical quantity and therefore not subject to quantum interference effects. By this strategy superselectionists of measurement expect to dispense with use of projection postulates in treatments of measurement. I shall argue that superselection accounts of measurement are self-contradictory, and that treatments of infinite systems, if they can avoid the contradiction, are not true superselection accounts.
The formal structure of quantum mechanics allows systems to exist in a superposition of different states. A curious feature is that while some superpositions, such as position or momentum, are routinely observed, others such as mass or charge are not. This anomaly is at least partly due to the fact that it is much easier to carry out interference experiments for certain observables than others. Here we focus particularly on the superposition of two different number states for photons or bosonic atoms. This can equally well be thought of as a superposition of different masses in the case of atoms. We present different interferometry schemes that should, at least in principle, allow us to observe interference between these states. This would provide evidence of a superposition that we would normally consider to be prevented by superselection rules.