Skip to main content

Superselection Rules for Philosophers

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.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    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.

  2. 2.

    When it was realized that time reversal invariance might not be universally valid, a different proof using spatial rotations was given; see Hegerfeldt et al. (1968) and Sect. 12. Once again, all of the relevant considerations had been developed by Wigner in the 1930s.

  3. 3.

    See Aharonov and Susskind (1967a, 1967b), Rohlnick (1967), and Mirman (1969, 1970), and Lubkin (1970). A defense of the validity of the superselection rule for charge was given by Wick et al. (1970). The recent revival of the attacks on superselection rules will be discussed in Sect. 12.

  4. 4.

    See Doplicher et al. (1971, 1974). For an overview of the DHR program, see Halvorson (2007).

  5. 5.

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

  6. 6.

    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.

  7. 7.

    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.

  8. 8.

    A sequence of bounded operators O 1, O 2, ... converges in the weak topology to O just in case (ψ1, O j ψ2) converges to (ψ1, Oψ2) for all \(\psi_{1},\psi_{2}\in {\mathcal{H}}.\)

  9. 9.

    The equivalence of these conditions is known as von Neumann’s double commutant theorem.

  10. 10.

    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 \({\mathfrak{X}}.\)

  11. 11.

    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.

  12. 12.

    Direct integral decompositions will be briefly discussed below.

  13. 13.

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

  14. 14.

    \(E\in {\mathfrak{M}}\) 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.

  15. 15.

    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.

  16. 16.

    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).

  17. 17.

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

  18. 18.

    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 = α1ψ1 + α2ψ2 with |α1|2 + |α2|2 = 1, ||ψ1|| = ||ψ2|| = 1, and |(ψ1, ψ2)| = 0 such that the relative phase of α1 and α2 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.

  19. 19.

    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.”

  20. 20.

    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.

  21. 21.

    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 (ψ12) = (Uψ2, Uψ1). A unitary U is linear, i.e. U1ψ1 + α2ψ2) = α1 Uψ1 + α2 Uψ2, and (ψ12) = (Uψ1, Uψ2).

  22. 22.

    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).

  23. 23.

    For examples where “superselection rule” is taken to mean what is called here a very strong rule, see Bogolubov et al. (1975) and Wan (1980). Indeed, most texts simply assume that superselection rules are of the very strong variety.

  24. 24.

    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.

  25. 25.

    “Essentially unique” means unique up to measure zero.

  26. 26.

    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).

  27. 27.

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

  28. 28.

    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 × GU(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:GU(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.

  29. 29.

    These results come from a slight extension of Theorem 6.1 of Araki (1999).

  30. 30.

    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.

  31. 31.

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

  32. 32.

    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).

  33. 33.

    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.

  34. 34.

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

  35. 35.

    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.

  36. 36.

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

  37. 37.

    “[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).

  38. 38.

    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.

  39. 39.

    Strocchi and Wightman (1974) do not take Step 6.

  40. 40.

    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).

  41. 41.

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

  42. 42.

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

  43. 43.

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

  44. 44.

    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 superselection

  45. 45.

    See Sewell (2002, Sec. 2.3) and Landsman (2007, Sec. 6.4). Bub’s example does not fit well with the assumptions made here; in particular, the Hilbert space is non-separable and the commutant of the von Neumann algebra of observables is non-abelian.

  46. 46.

    See Robinson (1990, 1994) for more about the status the idealization of infinite systems as well as superselection solutions to the measurement problem.

  47. 47.

    At 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.

  48. 48.

    FAPP is John Bell’s acronym for “for all practical purposes.”

  49. 49.

    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.

  50. 50.

    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.

  51. 51.

    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).

  52. 52.

    I have no quarrel with the analysis of Bartlett et al. (2006, 2007) of the debate about the presence or absence of coherence in various experiments in quantum optics. But I am skeptical that this analysis extends to showing that all superselection can be undermined by introducing reference frames.

  53. 53.

    For a recent reappraisal of the Copenhagen interpretation of QM, see Landsman (2007, Sec. 3).

  54. 54.

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

References

  1. Aharonov, Y., & Susskind, L. (1967a). Charge superselection rule. Physical Review, 155, 1428–1431.

    Article  Google Scholar 

  2. Aharonov, Y., & Susskind, L. (1967b). Observability of the sign change of spinors under 2π rotations. Physical Review, 158, 1237–1238.

    Article  Google Scholar 

  3. Araki, H. (1999). Mathematical theory of quantum fields. Oxford: Oxford University Press.

    Google Scholar 

  4. Baez, J. C., Segal, I. E., & Zhou, Z. (1992). Introduction to algebraic and constructive quantum field theory. Princeton, NJ: Princeton University Press.

    Google Scholar 

  5. Bargmann, V. (1964). Note on Wigner’s theorem on symmetry operations. Journal of Mathematical Physics, 5, 862–868.

    Article  Google Scholar 

  6. Bartlett, S. D., Rudolf, T., & Spekkens, R. W. (2006). Dialogue concerning two views on quantum coherence: Fascist and fictionalist. International Journal of Quantum Information, 4, 17–43.

    Article  Google Scholar 

  7. Bartlett, S. D., Rudolf, T., & Spekkens, R. W. (2007). Reference frames, superselection rules, and quantum information. Reviews of Modern Physics, 79, 555–609.

    Article  Google Scholar 

  8. Bartlett, S. D., & Wiseman, H. M. (2003). Entanglement constrained by superselection rules. Physical Review Letters, 91, 097903-1-4.

    Google Scholar 

  9. Beltrametti, E. G., & Cassinelli, G. (1981). The logic of quantum mechanics. Reading, MA: W. A. Benjamin.

    Google Scholar 

  10. Bogolubov, N. N., Logunov, A. A., & Todorov, I. T. (1975). Introduction to axiomatic quantum field theory. Reading, MA: W. A. Benjamin.

    Google Scholar 

  11. Bohm, D. (1951). Quantum theory. Englewood-Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  12. Bratelli, O., & Robinson, D. W. (1987). Operator algebras and quantum statistical mechanics 1 (2nd ed.). New York: Springer.

    Google Scholar 

  13. Bub, J. (1988). How to solve the measurement problem in quantum mechanics. Foundations of Physics, 18, 701–722.

    Article  Google Scholar 

  14. Bub, J. (1997). Interpreting the quantum world. Cambridge: Cambridge University Press.

    Google Scholar 

  15. Cisneros, R. P., Martínez-y-Romero, Núñez-Yépez, H. N., & Salas-Brito, A. L. (1998). Limitations on the superposition principle: superselection rules in non-relativistic quantum mechanics. European Journal of Physics, 19, 237–243.

  16. Colin, S., Durt, T., & Tumulka, R. (2006). On superselection rules in Bohm-Bell theories. Journal of Physics A, 39, 15403–15419.

    Article  Google Scholar 

  17. d’Espagnat, B. (1976). Conceptual foundations of quantum mechanics (2nd ed.). Reading, MA: W. A. Benjamin.

    Google Scholar 

  18. Divakaran, P. P. (1994). Symmetries and quantization: Structure of the state space. Reviews in Mathematical Physics, 6, 167–205.

    Article  Google Scholar 

  19. Doplicher, S., Haag, R., & Roberts, J. E. (1971). Local observables and particle statistics. I. Communications in Mathematical Physics, 23, 199–230.

    Article  Google Scholar 

  20. Doplicher, S., Haag, R., & Roberts, J. E. (1974). Local observables and particle statistics. I. Communications in Mathematical Physics, 35, 49–85.

    Article  Google Scholar 

  21. Earman, J., & Ruetsche, L. (2007). Probabilities in quantum field theory and quantum statistical mechanics. pre-print.

  22. Emch, G. (1972). Algebraic methods in statistical mechanics and quantum field theory. New York: Wiley.

    Google Scholar 

  23. Emch, G., & Piron, C. (1963). Symmetry in quantum theory. Journal of Mathematical Physics, 4, 469–473.

    Article  Google Scholar 

  24. Galindo, A., Morales, A., & Nuñez-Logos, R. (1962). Superselection principle and pure states of n identical particles. Journal of Mathematical Physics, 3, 324–328.

    Article  Google Scholar 

  25. Giulini, D. (1996). Galilei invariance in quantum mechanics. Annals of Physics, 249, 222–235.

    Article  Google Scholar 

  26. Giulini, D. (2003). Superselection rules and symmetries. In E. Joos et al. (Eds.), Decoherence and the appearance of a classical world in quantum theory (2nd ed., pp. 259–315). Berlin: Springer.

    Google Scholar 

  27. Giulini, D., Kiefer, C., & Zeh, D. H. (1995). Symmetries, superselection rules, and decoherence. Physics Letters A, 199, 291–298.

    Article  Google Scholar 

  28. Haag, R., & Kastler, J. (1964). An algebraic approach to quantum field theory. Journal of Mathematical Physics, 5, 848–861.

    Article  Google Scholar 

  29. Halvorson, H. (2007). Algebraic quantum field theory. In J. Butterfield & J. Earman (Eds.), Handbook of the philosophy of science. Philosophy of physics (pp.731–922). Amsterdam: Elsevier/North Holland.

    Google Scholar 

  30. Hamhalter, J. (2003). Quantum measure theory. Dordecht: Kluwer Academic.

    Google Scholar 

  31. Hartle, J. B., & Taylor, J. R. (1969). Quantum mechanics of paraparticles. Physical Review, 178, 2043–2051.

    Article  Google Scholar 

  32. Hegerfeldt, G. C., & Kraus, K. (1968). Critical remark on the observability of the sign change of spinors under 2π rotations. Physical Review, 170, 1185–1186.

    Article  Google Scholar 

  33. Hegerfeldt, G. C., Kraus, K., & Wigner, E. P. (1968). Proof of the Fermion superselection rule without the assumption of time-reversal invariance. Journal of Mathematical Physics, 9, 2029–2031.

    Article  Google Scholar 

  34. Henneaux, M., & Teitelboim, C. (1992). Quantization of gauge theories. Princeton: Princeton University Press.

    Google Scholar 

  35. Hepp, K. (1972). Quantum theory of measurement and macroscopic observables. Helvetica Physica Acta, 45, 237–248.

    Google Scholar 

  36. Horuzhy, S. S. (1975). Superposition principle in algebraic quantum field theory. Theoretical and Mathematical Physics, 23, 413–421.

    Article  Google Scholar 

  37. Hughes, R. I. G. (1989). The structure and interpretation of quantum mechanics. Cambridge, MA: Harvard University Press.

    Google Scholar 

  38. Jauch, J. M. (1960). Systems of observables in quantum mechanics. Helvetica Physica Acta, 33, 711–726.

    Google Scholar 

  39. Jauch, J. M., & Misra, B. (1961). Supersymmetries and essential observables. Helvetica Physica Acta, 34, 699–709.

    Google Scholar 

  40. Kadison, R. V., & Ringrose, J. R. (1991). Fundamentals of the theory of operator algebras, Vols. I and II. Providence, RI: American Mathematical Society.

    Google Scholar 

  41. Kaempffer, F. A. (1965). Concepts in quantum mechanics. New York: Academic Press.

    Google Scholar 

  42. Kastler, D. (Ed.) (1990). The algebraic theory of superselection sectors: Introduction and recent results. In Proceedings of the Convegno Internationale Algebraic Theory of Superselection Sectors and Field Theory. Singapore: World Scientific.

  43. Klein, A. G., & Opat, I. (1976). Observation of 2π rotations by Fresnel diffraction of neutrons. Physical Review Letters, 37, 238–240.

    Article  Google Scholar 

  44. Landsman, N. P. (1991). Algebraic theory of superselection sectors and the measurement problem in quantum mechanics. International Journal of Theoretical Physics A, 30, 5349–5371.

    Google Scholar 

  45. Landsman, N. P. (1995). Observation and superselection in quantum mechanics. Studies in History and Philosophy of Modern Physics, 26, 45–73.

    Article  Google Scholar 

  46. Landsman, N. P. (2007). Between classical and quantum. In J. Butterfield & J. Earman (Eds.), Handbook of the philosophy of science. Philosophy of physics (pp. 417–553). Amsterdam: Elsevier/North-Holland.

    Google Scholar 

  47. Lubkin, E. (1970). On violation of the superselection rules. Annals of Physics, 56, 69–80.

    Article  Google Scholar 

  48. Mackey, G. W. (1998). The relationship between classical mechanics and quantum mechanics. In L. A. Coburn & M. A. Rieffel (Eds.), Perspectives on quantization (pp. 91–109). Providence, RI: American Mathematical Society.

    Google Scholar 

  49. Messiah, A. (1962). Quantum mechanics (Vol. 2). New York: Wiley.

    Google Scholar 

  50. Mirman, R. (1969). Coherent superposition of charge states. Physical Review, 186, 1380–1383.

    Article  Google Scholar 

  51. Mirman, R. (1970). Analysis of the experimental meaning of coherent superposition and the nonexistence of superselection rules. Physical Review D, 1, 3349–3363.

    Article  Google Scholar 

  52. Mirman, R. (1979). Nonexistence of superselection rules: Definition of the term frame of reference. Foundations of Physics, 9, 283–299.

    Article  Google Scholar 

  53. Orear, J., Rosenfeld, A. H., & Schulter, R. A. (Eds.) (1951). Nuclear physics and the physics of fundamental particles; Proceedings. Chicago: Institute for Nuclear Studies.

  54. Piron, C. (1969). Les régles de superséletion continues. Helvetica Physica Acta, 42, 330–338.

    Google Scholar 

  55. Piron, C. (1976). Foundations of quantum physics. Reading, MA: W. A. Benjamin.

    Google Scholar 

  56. Reed, M., & Simon, B. (1975). Methods of mathematical physics (Vol. 2). New York: Academic Press.

    Google Scholar 

  57. Reed, M., & Simon, B. (1980). Methods of mathematical physics (Vol. 1). New York: Academic Press (revised edition).

    Google Scholar 

  58. Roberts, J. E., & Roepstorff, G. (1969). Some basic concepts of algebraic quantum theory. Communications in Mathematical Physics, 11, 321–338.

    Article  Google Scholar 

  59. Robinson, D. (1990). The infinite apparatus in the quantum theory of measurement. In A. Fine, M. Forbes & L. Wessels (Eds.), PSA 1990 (Vol. 1, pp. 251–261). East Lansing, MI: Philosophy of Science Association.

    Google Scholar 

  60. Robinson, D. (1994). Can superselection rules solve the measurement problem? British Journal for the Philosophy of Science, 45, 79–93.

    Article  Google Scholar 

  61. Rolnick, W. B. (1967). Does charge obey a superselection rule? Physical Review Letters, 19, 717–718.

    Google Scholar 

  62. Roman, P. (1965). Advanced quantum theory. Reading, MA: Addison-Wesley.

    Google Scholar 

  63. Sewell, G. L. (2002). Quantum mechanics and its emergent macrophysics. Princeton, NJ: Princeton University Press.

    Google Scholar 

  64. Streater, R. F., & Wightman, A. S. (1964). PCT, spin and statistics, and all that. New York: W. A. Benjamin.

    Google Scholar 

  65. Strocchi, F., & Wightman, A. S. (1974). Proof of the charge superselection rule in local relativistic quantum field theory. Journal of Mathematical Physics, 15, 2198–2224.

    Article  Google Scholar 

  66. Thalos, M. (1998). The trouble with superselection accounts of measurement. Philosophy of Science, 65, 518–544.

    Article  Google Scholar 

  67. van Fraassen, B. C. (1991). Quantum mechanics: An empiricist approach. Oxford: Clarendon Press.

    Google Scholar 

  68. Vermaas, P. E. (1999). A philosopher’s understanding of quantum mechanics: Possibilities and impossibilities of a modal interpretation. Cambridge: Cambridge University Press.

    Google Scholar 

  69. Verstraete, F., & Cirac, J. I. (2003). Quantum nonlocality in the presence of superselection rules and data hiding protocols. Physical Review Letters, 91, 010404–14.

    Article  Google Scholar 

  70. von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer.

    Google Scholar 

  71. Wald, R. M. (1994). Quantum field theory in curved spacetime and black hole thermodynamics. Chicago: University of Chicago Press.

    Google Scholar 

  72. Wan, K.-K. (1980). Superselection rules, quantum measurement, and Schrödinger’s cat. Canadian Journal of Physics, 58, 976–982.

    Google Scholar 

  73. Weingard, R., & Smith, G. (1982). Spin and space. Synthese, 50, 313–331.

    Article  Google Scholar 

  74. Werner, S. A., Colella, R., Overhauser, A. W., & Eagen, C. F. (1975). Observation of the phase shift of a neutron due to precession in a magnetic field. Physical Review Letters, 35, 1050–1055.

    Article  Google Scholar 

  75. Wick, G. C., Wightman, A. S., & Wigner, E. P. (1952). The intrinsic parity of elementary particles. Physical Review, 88, 101–105.

    Article  Google Scholar 

  76. Wick, G. C., Wightman, A. S., & Wigner, E. P. (1970). Superselection rule for charge. Physical Review D, 1, 3267–3269.

    Article  Google Scholar 

  77. Wightman, A. S. (1959). Relativistic invariance and quantum mechanics. Nuovo Cimento, 14(supplemento), 81–94.

    Google Scholar 

  78. Wightman, A. S. (1995). Superselection rules; Old and new. Nuovo Cimento B, 110(N. 5-6), 751–769.

    Article  Google Scholar 

  79. Wightman, A. S., & Glance, N. (1989). Superselection rules in molecules. Nuclear Physics B (Proc. Suppl.), 6, 202–206.

    Article  Google Scholar 

  80. Wigner, E. P. (1931). Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Braunsweig: Friedrich Vieweg.

    Google Scholar 

  81. Wigner, E. P. (1973). Epistemological perspective on quantum theory. In C. A. Hooker (Ed.), Contemporary research in the foundations and philosophy of quantum theory (pp. 369–385). Dordrecht: D. Reidel.

    Google Scholar 

  82. Zurek, W. H. (1982). Environment induced superselection rules. Physical Review D, 26, 1862–1880.

    Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to John Earman.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Keywords

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