G.-C. Wick, A. S. Wightman, and E. P. Wigner, 'Superselection Rule for Charge' (Phys. Rev. D 1, 3267, 1970), prove a permanence theorem: if states with different eigenvalues of an additive conserved quantity are separated by a superselection rule at one time, the separation persists under composition of independent systems, evolution under a quantity-conserving Hamiltonian, and separation into subsystems; it concludes that electric and baryonic charges obey a superselection rule, given the empirical assumption that only charge-diagonal ('permissible') states are naturally given. The open problem the paper itself leaves — stated in its Conclusion and its remark on the Wigner–Yanase skew information — is the deeper reason for the asymmetry between charge and other additive conserved quantities: nature supplies coherent superpositions of different momenta whose relative phases are interpretable as position, but no naturally given superpositions of different charges and no known interpretation of such phases, and the Wigner–Yanase skew information (a measure of the information a state contains about a conserved observable) is 'well available' for momentum but vanishes for charge in all naturally given states. Later work produced framework-relative partial answers — the Gauss-law derivation of the charge superselection rule in local quantum field theory (Strocchi–Wightman 1974), decoherence-based programs for dynamically induced superselection rules (Zurek 1982, 2003; Giulini 2000), and relational/reference-frame reformulations (Aharonov–Susskind 1967; Bartlett–Rudolph–Spekkens 2007) — but no accepted derivation of the initial charge-diagonality of naturally given states from more primitive principles, no demonstration that it can be violated, and no consensus account of the charge–momentum asymmetry; the question is actively contested in current quantum-reference-frame work on the accessibility of total charge.

Background

In quantum mechanics, a superselection rule is the statement that certain pairs of states can never be connected by any physical measurement: the density-matrix elements between them (equivalently, the relative phase of a formal superposition) are in principle unobservable. The notion was introduced by Wick, Wightman, and Wigner in 1952, who proposed, among others, a superselection rule for electric charge. An additive conserved quantity is a quantity like electric charge, baryon number, or a component of momentum whose value for a composite system is the sum of its values for the parts and which is conserved in time. A density matrix (the general mixed-state description of a quantum system) is called permissible, in the terminology of the 1970 source paper, when it commutes with the charge operator, so that it is block-diagonal in the charge eigenvalue, Rqn;q′n′=δqq′ ρnn′q,R_{qn;q'n'}=\delta_{qq'}\,\rho^{q}_{nn'}, where qq labels the charge and nn labels the degeneracy within each charge sector.

By 1970 the charge superselection rule was contested. Aharonov and Susskind (1967) and Mirman (1969) argued that coherent superpositions of different charges can be produced: a subsystem interacting with a second subsystem prepared in a charge-coherent state can be driven into a charge superposition. The reply by Wick, Wightman, and Wigner, 'Superselection Rule for Charge' (Phys. Rev. D 1, 3267 (1970)), proves a permanence theorem: writing the subsystem charge operators on the composite Hilbert space as Q1=Q1⊗1Q^1=Q_1\otimes 1 and Q2=1⊗Q2Q^2=1\otimes Q_2, a permissible density matrix remains permissible under the three elementary processes into which any set of events can be decomposed — union of statistically independent systems (direct product of density matrices), time evolution under a charge-conserving Hamiltonian ([H,Q]=0[H,Q]=0), and separation into subsystems (partial trace, with separation performed last). Hence a superselection separation that holds at one time persists for all times, and no ordinary physical process can manufacture charge coherence from charge-diagonal ingredients; the Aharonov–Susskind scheme works only if some subsystem is already charge-coherent to begin with.

The demonstration rests on a critical empirical assumption that the paper does not derive: that, to begin with, only permissible states are 'naturally given' — nature provides no states that are superpositions (rather than mixtures) of different charges. The paper's Conclusion contrasts this with momentum: naturally given states do include superpositions of different momenta (every localized state is one, and all macroscopic systems are), and their relative phases have a classical interpretation — they give position in space. For charge, no such interpretation is known. The paper adds an information-theoretic version of the same asymmetry using the skew information of Wigner and Yanase (1963), a measure of the information a state contains about a conserved observable, defined through the commutator [ρ,A][\sqrt{\rho},A] and vanishing exactly when the state commutes with that observable: skew information is 'well available' with respect to momentum and has a classical interpretation, but not with respect to electric or baryonic charge; and if it is zero initially it remains zero throughout. The paper closes by admitting: 'We do not know the deeper reason for this difference, but it seems to be present.'

Three later research programs grew around this gap. In local relativistic quantum field theory, Strocchi and Wightman (1974) proved from Gauss's law (the relation, at the operator level, between the charge in a region and the electric flux through its boundary) plus locality and causality that the charge commutes with all quasilocal observables (observables attached to bounded spacetime regions) — a symmetry-based derivation of the charge superselection rule within that framework, reflecting that electric charge generates a local gauge symmetry while momentum generates a global spacetime symmetry. In the decoherence program, Zurek (1982, 2003) showed that environments can dynamically induce 'effective' or 'soft' superselection rules — einselection, Zurek's term for this environment-induced selection of robust, effectively classical states; Giulini (2000) asked whether the 'hard' charge rule might also have a dynamical origin. In the relational program descending from Aharonov and Susskind, Bartlett, Rudolph, and Spekkens (2007) recast superselection rules as the lack of a shared reference frame — a physical system, such as a phase standard, relative to which phases are defined, whose quantum version is a quantum reference frame — on which view charge coherence is a matter of available resources rather than fundamental impossibility. What no program has delivered is what the source asks for: a derivation of the charge-diagonality of naturally given states from more primitive principles — or a demonstration that it can be violated — together with an account of why momentum behaves differently.

Problem Statement

Explain, from principles more primitive than the empirical fact itself, the asymmetry stated by Wick, Wightman, and Wigner (1970) between additive conserved quantities: naturally given quantum states — the initial states and resources that nature supplies without artificial preparation, in the empirical sense of the source paper, whose membership the source leaves without a formal criterion — include coherent superpositions of different momenta (and angular momenta), whose relative phases have a classical interpretation as position, but no naturally given states are coherent superpositions of different electric or baryonic charges, and no interpretation is known for the relative phases between different charge eigenstates. The single scientific objective — the origin and foundational status of the charge superselection rule — comprises three linked facets: (i) identify the deeper reason, whether dynamical, symmetry-based, or informational, for the validity of the charge superselection rule for naturally given states, for its possible violation by resources outside that class, and for the contrast with momentum; (ii) determine, in particular, whether the critical initial-state assumption of the Wick–Wightman–Wigner demonstration — that all physically given density matrices are permissible, i.e., commute with the charge operator and are therefore block-diagonal in the charge eigenvalue — can be derived from more primitive principles, or whether and how it could be violated by fundamental physics or by novel experimental resources; and (iii) state and, where possible, derive the corresponding information-theoretic statement, that the Wigner–Yanase skew information — the information a state carries about a conserved observable, defined through the commutator [ρ,A][\sqrt{\rho},A] and vanishing exactly when the state commutes with that observable — vanishes with respect to charge for all naturally given states and, being zero initially, remains zero under all allowed processes, whereas with respect to momentum it is well available and classically interpretable. The objective is to supply a criterion for the naturally-given class and show that it yields the charge–momentum asymmetry; a re-derivation of the permanence of an initially assumed charge-diagonality is not the target, and a derivation confined to a single framework is relative to that framework.

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