## Abstract

The overaraching goal of this paper is to elucidate the nature of superselection rules in a manner that is accessible to philosophers of science and that brings out the connections between superselection and some of the most fundamental interpretational issues in quantum physics. The formalism of von Neumann algebras is used to characterize three different senses of superselection rules (dubbed, weak, strong, and very strong) and to provide useful necessary and sufficient conditions for each sense. It is then shown how the Haag–Kastler algebraic approach to quantum physics holds the promise of a uniform and comprehensive account of the origin of superselection rules. Some of the challenges that must be met before this promise can be kept are discussed. The focus then turns to the role of superselection rules in solutions to the measurement problem and the emergence of classical properties. It is claimed that the role for “hard” superselection rules is limited, but “soft” (a.k.a. environmental) superselection rules or N. P. Landsman’s situational superselection rules may have a major role to play. Finally, an assessment is given of the recently revived attempts to deconstruct superselection rules.

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## Notes

Bohm concludes that “a sensible theory could be made for orbital angular momenta, if the angular momenta were either all integral, or half-integral, but not if both were present together” (p. 390). The Preface of Bohm’s book states that “Numerous discussions with students and faculty at Princeton University were helpful in clarifying the presentation” (p. v). Bohm was a colleague of Wightman and Wigner during the years 1947–1950 when he was an assistant professor at Princeton. It is interesting that Bohm’s argument is from single-valuedness and that Wightman dubbed the superselection rule at issue the univalence rule.

I am grateful to a Referee for bringing these quotations to my attention.

A *

*-algebra*is an algebra closed with respect to an involution \({\mathcal{A}}\ni A\,\mapsto\,A^{\ast}\in {\mathcal{A}}\) satisfying: \((A^{\ast})^{\ast}=A, (A+B)^{\ast}=A^{\ast}+B^{\ast }, (cA)^{\ast}=\bar{c}A^{\ast}\) and (*AB*)^{*}=*B*^{*}*A*^{*}for all \(A,B\in {\mathcal{A}}\) and all complex*c*(where the overbar denotes the complex conjugate). A*C*^{*}-*algebra*is a *-algebra equipped with a norm, satisfying \(\left\Vert A^{\ast}A\right\Vert =\left\Vert A\right\Vert ^{2}\) and \(\left\Vert AB\right\Vert \leq \left\Vert A\right\Vert \left\Vert B\right\Vert\) for all \(A,B\in {\mathcal{A}},\) and is complete in the topology induced by that norm.It will be assumed that the Hilbert space is separable. This assumption is used explicitly or implicitly in some of the key theorems used below.

A sequence of bounded operators

*O*_{1},*O*_{2}, ... converges in the weak topology to*O*just in case (ψ_{1},*O*_{ j }ψ_{2}) converges to (ψ_{1},*O*ψ_{2}) for all \(\psi_{1},\psi_{2}\in {\mathcal{H}}.\)The equivalence of these conditions is known as von Neumann’s double commutant theorem.

That is, if \({\mathfrak{X}}\subset {\mathfrak{B}}({\mathcal{H}})\) (the algebra of bounded linear operators on \({\mathcal{H}}\)), then the elements of \({\mathfrak{X}}^{\prime}\) consists of all of those elements of \({\mathfrak{B}}({\mathcal{H}})\) that commute with every element of \({\mathfrak{X}}.\)

Reducibility is taken as the criterion of the existence of superselection rules by Emch and Piron (1963), although they work in terms of a lattice of propositions rather than an algebra of observables.

Direct integral decompositions will be briefly discussed below.

That ω is positive means that for all \(A\in {\mathfrak{M}}, \omega (A^{\ast}A)=0\) implies that

*A*= 0.\(E\in {\mathfrak{M}}\) is a projector just in case it is self-adjoint and idempotent, i.e.

*E*^{2}=*E*. Two such projectors*E*and*F*are said to be orthogonal just in case they project onto orthogonal subspaces of \({\mathcal{H}},\) in which case*EF*=*FE*= 0.A factorial von Neumann algebra \({\mathfrak{R}}\) has a center \({\mathcal{Z}}({\mathfrak{R}}):={\mathfrak{R\cap R}}^{\prime}\) that consists of multiples of the identity. A factorial \({\mathfrak{R}}\) is of Type I iff it contains minimal projectors. For a non-factorial \({\mathfrak{R}}\) the definition of a Type I is a bit more complicated; namely, \({\mathfrak{R}}\) contains an abelian projector whose central carrier is the identity

*I*. That the projector \(E\in {\mathfrak{R}}\) is abelian means that \(E{\mathfrak{R}}E\) is an abelian algebra. The central carrier*C*_{ A }of*A*is the intersection of all projectors \(E\in{\mathcal{Z}} ({\mathfrak{R}})\) such that*EA*=*A*.The distinguishing feature of Type III algebras is that they contain properly infinite projectors and no finite projectors. For some of the implications for foundations issues, see Earman and Ruetsche (2007).

For a general discussion of the superposition principle in the algebraic formulation of QM, see Horuzhy (1975).

The habit, developed from working in ordinary QM, of identifying pure states with vector states is not easily broken. Witness the exposition of one of the discoverers of superselection rules: A superselection rule “means that there exist pure states [sic] described by a wave function ψ = α

_{1}ψ_{1}+ α_{2}ψ_{2}with |α_{1}|^{2}+ |α_{2}|^{2}= 1, ||ψ_{1}|| = ||ψ_{2}|| = 1, and |(ψ_{1}, ψ_{2})| = 0 such that the relative phase of α_{1}and α_{2}is unobservable” (Wightman and Glance 1989, p. 202). I have changed the notation to correct an obvious misprint. Wightman and Glance go on to say correctly that ψ may also be defined by a density matrix that describes a mixed state.One of the few clear statements in the philosophical literature of the limitation imposed by superselection rules on the superposition principle comes from van Fraassen (1991, p. 186): “[T]the principle of superposition is curtailed: what looks mathematically like a superposition of pure states may actually represent a mixed state.”

See Hamhalter (2003, Ch. 5) for a proof. A von Neumann algebra is of Type I

_{ n }if the unit element can be written as the sum of*n*abelian projectors.For a rigorous proof, see Bargmann (1964). An anti-unitary

*U*is anti-linear transformation, i.e. for all \(\psi_{1},\psi_{2}\in {\mathcal{H}}\) and all \(\alpha_{1},\alpha_{2}\in {\mathbb{C}}, U(\alpha_{1}\psi_{1} +\alpha_{2}\psi_{2})=\alpha_{1}^{\ast}U\psi_{1}+\alpha_{2}^{\ast}U\psi_{2},\) such that (ψ_{1},ψ_{2}) = (*U*ψ_{2},*U*ψ_{1}). A unitary*U*is linear, i.e.*U*(α_{1}ψ_{1}+ α_{2}ψ_{2}) = α_{1}*U*ψ_{1}+ α_{2}*U*ψ_{2}, and (ψ_{1},ψ_{2}) = (*U*ψ_{1},*U*ψ_{2}).Sometimes a non-trivial center is taken as the definition of superselection rules; see, for example, Piron (1976). Speaking of the lattice of propositions Piron says: the system is classical if the center is whole lattice. “In the pure quantum case [ordinary QM] the center contains only 0 and

*I*. In physics there exist a large number of intermediate cases where the center is strictly smaller than the whole lattice but contains nontrivial propositions. We shall then say that the system possesses superselection rules” (p. 29).There is a direct and easy way to see this. If \({\mathfrak{M}}^{\prime}\) is abelian then the identity

*I*is an abelian projector whose central carrier is*I*and, thus, \({\mathfrak{M}}^{\prime}\) is Type I. But for any von Neumann algebra \({\mathfrak{M}}, {\mathfrak{M}}\) and \({\mathfrak{M}}^{\prime}\) are the same type. The proof given below leads eventually to the same conclusion while showing some other useful things along the way.“Essentially unique” means unique up to measure zero.

If it is demanded that the gauge group have an action on the Hilbert space, then commutativity of superselection rules fails and parastatistics are precluded; see Galindo et al. (1962).

Or in more sophisticated language, by the dictum that the state space is the projective Hilbert space.

An accessible introduction can be found in Giulini (2003). What makes the analysis complicated is that for many of the symmetry groups encountered in physics, the faithful unitary representations are projective (a.k.a. ray representations or representations up to a phase factor). In a projective unitary representation of a group

*G*,*U*(*g*_{1})*U*(*g*_{2}) = γ(*g*_{1},*g*_{2})*U*(*g*_{1},*g*_{2}),*g*_{1},*g*_{2}∈*G*, where the*U*(*g*) are unitary operators and γ :*G*×*G*→*U*(1) (the complex numbers of unit modulus). The multipliers γ must satisfy the cocycle condition γ(*g*_{1},*g*_{2})γ (*g*_{1}*g*_{2},*g*_{3}) = γ(*g*_{1},*g*_{2}*g*_{3})γ (*g*_{2},*g*_{3}). The multipliers γ and γ ′ are said to be similar just in case there is an*f*:*G*→*U*(1) such that*f*(*e*) = 1 (*e*being the identity element of*G*) and γ ′(*g*_{1},*g*_{2}) = γ(*g*_{1},*g*_{2})*f*(*g*_{1}*g*_{2})*f*(*g*_{1})^{−1}*f*(*g*_{2})^{−1}for all*g*_{1},*g*_{2}∈*G*. The classification of non-similar multipliers is one of the keys to the superselection rule associated with*G*.These results come from a slight extension of Theorem 6.1 of Araki (1999).

An equivalent characterization of disjoint representations can be given using this concept; namely, two representations are disjoint just in case they do not share any unitarily equivalent subrepresentations.

The Weyl CCR entail the familiar CCR but not vice versa.

On the advantages of the

*C*^{*}-algebra approach is that it is supposed to be representation independent. It is somewhat embarrassing, therefore, that typical constructions of the*C*^{*}-algebra version of the concrete Weyl CCR algebra use Hilbert space representations. However, the embarrassment is overcome by showing that the*C*^{*}-algebra is independent of the representation used in the construction in the sense that unitarily equivalent representations lead to the same*C*^{*}-algebra. For a discussion of these matters, see Baez et al. (1992, Ch. 5).The Stone-von Neumann theorem breaks down for an infinite number of degrees of freedom—as encountered in QFT and quantum statistical mechanics—and unitarily inequivalent representations of the Weyl CCR exist in abundance.

For an overview of superselection rules in non-relativistic QM, see Cisneros et al. (1998).

It is natural to wonder whether the univalence superselection rule survives when the rotation and translation groups are imbedded in an even larger group—for example, the Galilean or the Poincaré group. The answer is yes; see below.

Here the notion is disjointness is extended to states by saying that two states are disjoint just in case they determine disjoint GNS representations.

“[W]e will simply use the phrase physically admissible to indicate a state selected according to some appropriate criteria. The important point is that, once the definition of physically admissible has been fixed, the superselection sectors of a Haag–Kastler [local quantum field theory] are given by the unitarily inequivalent physically admissible representations of the quasilocal algebra” (Strocchi and Wightman 1974, p. 2198).

Note that if π and \(\hat{\pi}\) belong to the same unitary equivalence class, then \(\pi ({\mathcal{A}})^{\prime \prime }\) and \(\hat{\pi}({\mathcal{A}})^{\prime \prime }\) are *-isomorphic. Thus, the algebras of observables \(\oplus_{j}\pi_{j}({\mathcal{A}})^{\prime \prime }\) and \(\oplus _{j}\hat{\pi}_{j}({\mathcal{A}})^{\prime \prime }\) are the same.

Strocchi and Wightman (1974) do

*not*take Step 6.This argument has been labeled heuristic because to bring it into the formalism described above would require the introduction of a mass operator for non-relativistic QM. For how this can be done and the implications for the status of a superselection rule for mass, see Giulini (1996).

Some remarks on the problem with extending the formalism to include continuous superselection rules are given in Sect. 13.

The importance of the uniqueness is emphasized by Landsman (1995).

For sake of simplicity the object observable is assumed to have a pure discrete and non-degenerate spectrum.

There is also another worry. The superselection sectors for the measurement apparatus are not just superselection sectors for pointer position but for all the other quantities that are subject to (very strong) superselection rules. Thus, as the state vector moves from a sector corresponding to the null pointer position

*p*_{0}to a sector corresponding to, say, pointer position*p*_{15}, the values of all of the other superselected quantities must change, which is contrary to everything that is believed about superselectionAt least not if the

*C*^{*}-algebra is the total algebra of observables. But if the relevant algebra is some subalgebra of the total algebra, the situation changes; see below.FAPP is John Bell’s acronym for “for all practical purposes.”

Or somewhat more liberally, a self-adjoint

*A*can be deemed to have a definite value if*A*is affiliated with the center of the algebra of observables.Or at least this is the general consensus in the philosophy of science community. But in the physics literature one occassionally sees falsifiability used as the touchstone of the physically meaningful.

From the discussion in Sect. 8 it follows that it is more accurate to say that the univalence superselection rule results from invariance under the Euclidean group, the product of spatial translations and spatial rotations

*SO*(3); for if*SO*(3) were the relevant symmetry group, the superselection rules would be more fine grained. The univalence superselection rule survives under relativization in the sense that it remains a valid superselection rule when the relevant symmetry group is the Poincaré group; see Divakaran (1994, Sec. 5.2).For a recent reappraisal of the Copenhagen interpretation of QM, see Landsman (2007, Sec. 3).

For comments about continuous superselection rules, see Piron (1969).

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## Acknowledgments

I am grateful to Jeremy Butterfield, Klaas Landsman, and Laura Ruetsche for a number of helpful comments; needless to say, however, this does not imply that they share the opinions expressed herein. Thanks are also due to two anonymous referees for correcting several errors and for suggestions that led to material improvements.

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Earman, J. Superselection Rules for Philosophers.
*Erkenn* **69**, 377–414 (2008). https://doi.org/10.1007/s10670-008-9124-z

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### Keywords

- Algebraic Approach
- Canonical Commutation Relation
- Superselection Rule
- Superselection Sector
- Spatial Translation