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arXiv:quant-ph/0509177 · Journal of Physics A Mathematical and General

On superselection rules in Bohm–Bell theories

Samuel Colin, Thomas Durt, Roderich Tumulka

Published Nov 30, 2006• 17 citations
arXivHTMLPDF

Abstract

The meaning of superselection rules in Bohm–Bell theories (i.e., quantum theories with particle trajectories) is different from that in orthodox quantum theory. More precisely, there are two concepts of superselection rule, a weak and a strong one. Weak superselection rules exist both in orthodox quantum theory and in Bohm–Bell theories and represent the conventional understanding of superselection rules. We introduce the concept of strong superselection rule, which does not exist in orthodox quantum theory. It relies on the clear ontology of Bohm–Bell theories and is a sharper and, in the Bohm–Bell context, more fundamental notion. A strong superselection rule for the observable G asserts that one can replace every state vector by a suitable statistical mixture of eigenvectors of G without changing the particle trajectories or their probabilities. A weak superselection rule asserts that every state vector is empirically indistinguishable from a suitable statistical mixture of eigenvectors of G . We establish conditions on G for both kinds of superselection. For comparison, we also consider both kinds of superselection in theories of spontaneous wavefunction collapse.

DOI
10.1088/0305-4470/39/50/008

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The meaning of superselection rules in Bohm–Bell theories (i.e., quantum theories with particle trajectories) is different from that in orthodox quantum theory. More precisely, there are two concepts of superselection rule, a weak and a strong one. Weak superselection rules exist both in orthodox quantum theory and in Bohm–Bell theories and represent the conventional understanding of superselection rules. We introduce the concept of strong superselection rule, which does not exist in orthodox quantum theory. It relies on the clear ontology of Bohm–Bell theories and is a sharper and, in the Bohm–Bell context, more fundamental notion. A strong superselection rule for the observable G asserts that one can replace every state vector by a suitable statistical mixture of eigenvectors of G without changing the particle trajectories or their probabilities. A weak superselection rule asserts that every state vector is empirically indistinguishable from a suitable statistical mixture of eigenvectors of G . We establish conditions on G for both kinds of superselection. For comparison, we also consider both kinds of superselection in theories of spontaneous wavefunction collapse.
For orthodox quantum systems, pure states correspond one-to-one to all one-dimensional subspaces and observables correspond one-to-one to all selfadjoint operators in the state space. A relaxation of this one-to-one correspondence will bring about superselection rules. A general formulation of superselection rules is technically complicated. 1 We shall adopt an intuitive approach here and limit our discussion to some special cases to see how superselection rules can be incorporated in a Hilbert space structure of quantum theory.
Methods used to prove the existence of superselection rules are studied in detail. It is shown that the known methods of proving superselection rules are applicable only if the superselecting operator is an observable. In particular, place permutations are not observables, so the fact that place permutations commute with all observables does not lead to a superselection rule between vectors of different symmetry types. It is shown that if states not having a definite symmetry type can exist, then it is possible to have several different states which are eigenstates of the same observables with the same eigenvalues. In this case a maximal set of observables does not exist. Therefore a proof of the Symmetrization Postulate (SP) which assumes a maximal set of observables actually presupposes part of what is to be proved. A rigorous proof of the SP must depend only on the weaker assumption of a non-degenerate set of observables. The usual formulation of the Transition Probability Postulate (TPP) is used to construct a trivial proof of the assertion that physical states are represented by unique rays in Hilbert space; however the feature of the TPP which is essential to this proof is not essential to the TPP itself. The features of the TPP which are essential for computing transition probabilities are identified. These essential features of the TPP are then used to prove that states are represented by unique rays. The difficulties involved in rigorously defining the interchange operator Pij are discussed and a careful definition of Pij is given. The uniqueness of the ray is then used to show that states of systems containing several indistinguishable particles must be eigenstates of Pij and must satisfy the Symmetrization Postulate. Finally, the SP is used to prove the slightly stronger result that a given state must have the same symmetry type for each pair of the same species of particles.
We show that it is possible to explain the quantum measurement process within the framework of quantum mechanics without any additional postulates. The key concept of the theory is decoherence, which appears as an inherent characteristic of quantum mechanics and results from the uncertainty relation. In contrast to environment-induced decoherence, this decoherence exists prior to a measurement being made. To clarify our idea, we examine three elemental experiments: a Stern-Gerlach-like experiment, the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) experiment, and the double-slit experiment. By considering the first experiment, we explain how the uncertainty relation between position and momentum introduces decoherence prior to measurement. Consideration of the EPR-Bohm experiment leads us to conclude that the correlation of the EPR pair is not a consequence of what is known as the collapse of the wave function. Our theory of decoherence can also be applied to experiments with continuous eigenvalues, such as the double-slit experiment. Consideration of the double-slit experiment leads us to understand how pure quantum mechanics describes the fact that quanta behave as interfering particles.
In quantum theory, physically measurable quantities of a microscopic system are represented by self-adjoint operators. However, not all of the self-adjoint operators correspond to measurable quantities. The superselection rule is a criterion to distinguish measurable quantities. Any measurable quantity must obey the superselection rules. By contraposition, any quantity which does not obey the superselection rules cannot be measured. Although some of superselection rules were proved, the raizon d'être of the superselection rules has been still obscure. In this paper we deduce the superselection rules from an assumption on symmetry property of measurement process. We introduce the notion of covariant indicator, which is a macroscopic observable whose value indicates the value of a microscopic object observable. We prove that if an object system has a quantity that is conserved during the measurement process, other quantities that do not commute with the conserved quantity are non-measurable by the covariant indicator. Our derivation of superselection rules is compared with the uncertainty relation under the restriction by a conservation law. An implication of the color superselection rule for the color confinement is discussed. It is also argued that spontaneous symmetry breaking enables a measurement that the superselection rule prohibits.
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.
Superselection rules severely constrain the operations which can be implemented on a distributed quantum system. While the restriction to local operations and classical communication gives rise to entanglement as a nonlocal resource, particle number conservation additionally confines the possible operations and should give rise to a new resource. In Phys. Rev. Lett. 92, 087904 (2004) we showed that this resource can be quantified by a single additional number, the superselection-induced variance (SIV)(\mathrm{SIV})(SIV), without changing the concept of entanglement. In this paper, we give the results on pure states in greater detail; additionally, we provide a discussion of mixed-state nonlocality with superselection rules where we consider both formation and distillation. Finally, we demonstrate that SIV\mathrm{SIV}SIV is indeed a resource, i.e., that it captures how well a state can be used to overcome the restrictions imposed by the superselection rule.