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Physical Review

Charge Superselection Rule

Yakir Aharonov, Leonard Susskind

Published Mar 25, 1967• 242 citations
Publisher

Abstract

Questions concerning superselection rules are considered. Two experiments are discussed. In the first, coherent superpositions of different angular-momentum states are constructed. In the second, coherent superpositions of states with different charge are constructed in complete analogy with the angular-momentum case. We suggest that, contrary to a widespread belief, interference may be possible between states with different charges.

DOI
10.1103/physrev.155.1428

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Aharonov and Susskind denied the concept of the superselection rules, and suggested the experimental feasibility of measuring interference between states with different charges. Their reasoning is shown to be incorrect.

Progress of Theoretical Physics•Jun 1, 1972•1 citation
Questions concerning superselection rules are considered. Two experiments are discussed. In the first, coherent superpositions of different angular-momentum states are constructed. In the second, coherent superpositions of states with different charge are constructed in complete analogy with the angular-momentum case. We suggest that, contrary to a widespread belief, interference may be possible between states with different charges.
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.
The experimental meaning of coherent superposition of state vectors is analyzed. It is shown that the involvement of the measuring instrument is of basic importance and that it is erroneous to study superposition without it. From this analysis it is found that there are no such things as superselection rules. The meaning of the sign change of spinors under $2\ensuremath{\pi}$ rotation is considered, and it is found to have experimental consequences. The nonmeasurability of the relative parity of states with different zzz components of angular momentum is shown. An appendix discusses the application of some of the considerations of the paper to decaying states and elementary particles.
The customary notion of superselection rule stipulates the vanishing of those matrix elements of the density matrix that connect states separated by the superselection rule, i.e., the relative phases of which are, according to the superselection rule, meaningless. It is shown that, if any two states for which an additive conserved quantity has different values are separated for all physical systems by a superselection rule at one time, the separation will persist for all times. It is concluded that the states with different electric (or baryonic) charges are so separated. The reason for the difference between electric charge and other additive conserved quantities, such as momentum, is brought out.
We consider a huge quantum system that is subject to the charge superselection rule, which requires that any pure state must be an eigenstate of the total charge. We regard some parts of the system as "subsystems," and the rest as an environment E. We assume that one does not measure anything of E, i.e., one is only interested in observables of the subsystems. We show that there exist states with the following properties: (i) Its reduced density operator is completely equivalent to a vector state of the subsystems for any gauge-invariant observable of the subsystems. (ii) The vector state is a simple product of vector states of individual subsystems, each of which is not an eigenstate of the charge of each subsystem. Furthermore, one can associate to each subsystem its vector state, which is a pure state, and observables which are not necessarily gauge invariant in each subsystem. These results justify taking (a) superpositions of states with different charges, and (b) non-gauge-invariant operators, such as the order parameter of the breaking of the gauge symmetry, as observables, for subsystems.