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arXiv:cond-mat/0102429

Charge superselection rule does not rule out pure states of subsystems to be coherent superpositions of states with different charges

Shimizu, Akira, Takayuki Miyadera

Published Feb 23, 2001• 1 citation

Abstract

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.

DOI: 10.48550/arxiv.cond-mat/0102429

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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.

We discuss the applicability of the programme of decoherence -- emergence of approximate classical behaviour through interaction with the environment -- to cases where it was suggested that the presence of symmetries would lead to exact superselection rules. For this discussion it is useful to make a distinction between pure symmetries and redundancies, which results from an investigation into the constraint equations of the corresponding theories. We discuss, in particular, superpositions of states with different charges, as well as with different masses, and suggest how the corresponding interference terms although they exist in principle, become inaccessible through decoherence.

A synthesis is attempted between the algebraic theory of superselection sectors (not necessarily of infinite systems), which is based on the representation theory of C* algebras (which we review), and recent ideas on the measurement problem, classical behavior in open quantum systems, and the peculiar nature of subsystems in quantum mechanics. The key point is a proper use of the notion of a state, in combination with an appropriate choice of algebras of observables for which the superposition “principle” may be invalid. In this way, a major part of the measurement problem becomes equivalent to the study of superselection sectors in models for a measurement situation, and can in principle be resolved unambiguously. Special emphasis is placed on the transition from pure to mixed states, and the (non)uniqueness of the decomposition of the latter into pure states.

It is shown that spontaneous symmetry breaking occurs in this scattering problem. As a result, the unitary, asymptotically free quantum dynamics of this process is unstable. It transforms into dynamics associated with the violation of

P

- and

T

-symmetries and the appearance of a superselection rule. Here, the non-unitary scattering state is a mixed vector state, which is a coherent superposition of transmission and reflection states related to different Hamiltonians. Superselection operators for the transmission and reflection subprocesses are defined.

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.