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CPT Invariance, the Spin-Statistics Connection, and the Ontology of Relativistic Quantum Field Theories

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Abstract

CPT invariance and the spin-statistics connection are typically taken to be essential properties in relativistic quantum field theories (RQFTs), insofar as the CPT and Spin-Statistics theorems entail that any state of a physical system characterized by an RQFT must possess these properties. Moreover, in the physics literature, they are typically taken to be properties of particles. But there is a Received View among philosophers that RQFTs cannot fundamentally be about particles. This essay considers what proofs of the CPT and Spin-Statistics theorems suggest about the ontology of RQFTs, and the extent to which this is compatible with the Received View. I will argue that such proofs suggest the Received View’s approach to ontology is flawed.

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Notes

  1. Readers who might object to viewing the spin-statistics connection as a property may view it instead as a principle, law, or disposition in the following discussion. The issue at stake is not so much how to characterize it, but rather whether it is essential, and what it is predicated of.

  2. Typical statements to this effect are found in Peskin and Schroeder (1995): "This conclusion is part of a more general result, first derived by Pauli… particles of integer spin obey Bose-Einstein statistics, while particles of half-odd-integer spin obey Fermi-Dirac statistics “(pp. 57–58).” “At the same time that we discuss P and T, it will be convenient to discuss a third (non-spacetime) discrete operation: charge conjugation, denoted by C. Under this operation, particles and antiparticles are interchanged” (p. 64). See, also, Weinberg (1995, pp. 191, 238), Sterman (1993), Jost (1965, pp. 100, 106).

  3. Suppose |Φ〉 is a multi-particle state, and let |Φ′〉 be a multi-particle state obtained from |Φ〉 by permuting its single-particle substates. |Φ〉 is symmetric just when |Φ′〉 = |Φ〉. |Φ〉 is anti-symmetric just when |Φ′〉 = −|Φ〉. |Φ〉 is permutation invariant just when, for any linear operator A representing an observable quantity, the expectation value of A is the same for |Φ〉 and |Φ′〉: 〈Φ|A|Φ〉 = 〈Φ′|A|Φ′〉.

  4. Haag (1996, p. 97) takes the first route. Streater and Wightman (2000, p. 147) suggest the second route: "A natural way to arrive at Bose-Einstein statistics is to describe the system in question by a field which commutes for space-like separations, while the analogous way for Fermi-Dirac statistics is to use a field which anti-commutes for space-like separations." The "natural way", evidently, would be to demonstrate that Local Commutativity entails that Fock space creation/annihilation operators corresponding to the fields (when they exist) satisfy (1).

  5. Here ϕ(x) is intended to represent a generic quantum field of arbitrary spin and |Ω〉 is the corresponding unique vacuum state. In a more precise formulation, the field would be defined as an operator-valued distribution, and the corresponding Wightman function as a tempered distribution. Expositions of the CPT and Spin-Statistics theorems in the axiomatic approach are given in Araki (1999), Haag (1996), and Streater and Wightman (2000).

  6. More precisely, the spectrum of the momentum operator associated with \( L_{ + }^{ \uparrow } \) is confined to the forward lightcone.

  7. As Wightman (1999, p. 744) points out, this is a remarkable fact: "…although only invariance under \( L_{ + }^{ \uparrow } \) was assumed for the n-point vacuum expectation values, the associated analytic function is invariant under space–time inversion.”

  8. One must also assume that the fields have finitely many components to avoid counterexamples of infinite fields that do not possess the spin-statistics connection or CPT invariance (Streater 1967; Oksak and Todorov 1968).

  9. Massimi and Redhead (2003) compare Weinberg's approach to the Spin-Statistics theorem with the standard textbook approach reviewed in Sect. 2.3 below.

  10. The latter commutativity condition guarantees that time-ordered products of \( {\mathfrak{H}}_{\text{int}} (x) \) are restricted Lorentz invariant.

  11. The existence of a conserved charge entails that \( {\mathfrak{H}}_{\text{int}} ( x ) \) must commute with the charge operator Q. This entails that \( {\mathfrak{H}}_{\text{int}}( x ) \) must be formed out of fields ψ m that have simple commutation relations with Q. To accomplish this, it suffices to construct ψ m as a sum \( \psi_{m} = \psi_{m}^{ + }( x ) + \psi_{m}^{ + c\dag }( x) \), where \( \psi_{m}^{ + }( x) \) and \( \psi_{m}^{ + c}( x) \) are linear combinations of creation/annihilation operators a, a c for particle states with the same mass and spin, but opposite charge; i.e., ψ m is a sum of fields associated with particles and their antiparticles.

  12. This assumes there is a corresponding Hamiltonian density.

  13. Ω is cyclic for \( \Re \) just when \( \{ A\Upomega \, :A \in \Re \} \) is dense in \( \mathcal{H}_{0} \). Ω is separating for \( \Re \) just when AΩ = 0 and \( A \in \Re \) entails A = 0.

  14. A wedge region in Minkowski spacetime M is any Poincaré transformation of the region \( \{ x \in M:x_{1} >| {x_{0} }|\} \), where (x 1, x 2, x 3, x 0) is an inertial coordinate system. That a modular operator exists is entailed by the Tomita-Takesaki theorem (see, e.g., Halvorson and Müger 2006, p. 738; Haag 1996, p. 217). The latter demonstrates that a von Neumann algebra \( \Re \) of bounded linear operators on a Hilbert space \( \mathcal{H} \) with a cyclic and separating vacuum vector Ω possesses a modular operator Δ and a modular conjugate operator J such that \( J\Upomega = \Upomega = \Updelta \Upomega , \, \Updelta^{it} \Re \Updelta^{ - it} = \Re \), and \( J\Re J = \Re^{\prime } \), where \( \Re^{\prime } \) is the commutant of \( \Re \) (i.e., the set of bounded linear operators that commute with all elements of \( \Re \)).

  15. In general, a representation of \( \Re \) consists of a pair \( ({\mathcal{H}},\pi) \) where \( \mathcal{H} \) is a Hilbert space and π is a map that takes elements of \( \Re \) to bounded linear operators on \( \mathcal{H} \). A state ω on \( \Re \) is a linear map that takes elements of \( \Re \) to complex numbers. The GNS theorem entails that any state can be associated with a unique representation (Araki 1999, p. 34; Halvorson and Müger 2006, p. 734).

  16. More precisely, Θ = J W R W , where J W is the modular conjugate operator of \( \Re \) restricted to the wedge W, and R W implements rotations that leave W invariant.

  17. Doplicher et al. (1974) derived the spin-statistics connection for irreducible, restricted Poincaré-invariant DHR representations with finite statistics, positive masses, and finitely many components, under the assumptions of Microcausality, Haag Duality, and Property B (for definitions of the latter, see Araki 1999, pp. 163–64, or Halvorson and Müger 2006, p. 784). Guido and Longo recover this result in the following way: They first demonstrate that Microcausality, Weak Additivity, and MC entail Essential Duality, which is a weaker form of Haag Duality that still allows Doplicher, Haag and Robert's analysis to go through. They further demonstrate that Microcausality, Weak Additivity, and MC entail the existence of a unique unitary representation of the restricted Poincaré group that acts on \( \Re \) and satisfies the Spectrum Condition. This has two consequences. First, the uniqueness of this representation rules out counterexamples to the spin-statistics theorem of fields with infinitely many components (Guido and Longo 1995, p. 519). Second, Microcausality, the Spectrum Condition, and Weak Additivity entail Property B (Halvorson and Müger 2006, p. 748).

  18. For defenses of positions associated with (A) and (B), see Wallace (2011) and Fraser (2011), respectively.

  19. Pragmatists may respond to the consequences of Haag's theorem by adopting what Wallace (2011) refers to as "conventional", or "cutoff" QFT. See Sect. 5.1 below for a discussion.

  20. In the axiomatic approach, the relation between restricted Lorentz invariance and CPT invariance is tighter in an interacting theory, appropriately construed, than a free theory. In the LSZ formalism, time-ordered Wightman functions (or "τ-functions") are used to calculate the elements of the S-matrix of an interacting theory. Greenberg (2002) demonstrates that violation of CPT invariance of any Wightman function entails that the corresponding τ-function is not restricted Lorentz invariant. Thus, "[i]f CPT invariance is violated in an interacting quantum field theory, then that theory also violates Lorentz invariance" (pp. 231602-1, 231602-2), where Greenberg takes Lorentz invariance as the condition that both Wightman and τ-functions be restricted Lorentz invariant.

  21. Taken as a subset \( \Re \subset {\mathfrak{B}}({\mathcal{H}}_{0}) \) of the concrete algebra of bounded linear operators on the vacuum Hilbert space \( {\mathcal{H}}_{0},\Re \) is Poincaré covariant just when there is a unitary representation \( U_{0} :P_{+}^{\uparrow} \to {\mathfrak{B}} ({\mathcal{H}}_{0}) \) such that \( U_{0}(g )\Re ({\mathcal{O}})U_{0} (g)^{*} = \Re (g{\mathcal{O}}),g \in P_{+}^{\uparrow} \). A DHR representation \( ({\mathcal{H}},\pi) \) of \( \Re \) is unitarily equivalent to a localized morphism \( \rho :\Re \to {\mathfrak{B}}({\mathcal{H}}_{0}) \) defined by \( \rho ( A) = V\pi( A )V^{*} \), for unitary map \( V:{\mathcal{H}} \to {\mathcal{H}}_{0} \). A DHR representation is then said to be Poincaré covariant just when there is a unitary representation \( U_{\rho} :\tilde{P}_{+}^{\uparrow} \to {\mathfrak{B}}({\mathcal{H}}_{0}) \) of the universal covering \( \tilde{P}^{ \uparrow }_{ + } \)of the restricted Poincaré group such that \( U_{\rho }( h )\rho( A)U_{\rho }( h)^{*} = \rho (U_{0} (\sigma( h ))AU_{0} (\sigma ( h))^{*}) \), where \( h \in \tilde{P}^{ \uparrow }_{ + } \), and \( \sigma :\tilde{P}_{ + }^{ \uparrow } \to P_{ + }^{ \uparrow } \), is the covering map.

  22. If \( \Re \) is Poincaré covariant, so are its DHR representations, but the converse is not true: Guido and Longo (1992, p. 534) show that every DHR representation with finite statistics is Poincaré covariant with positive energy, provided \( \Re \) has a certain regularity property.

  23. Additivity (as opposed to Weak Additivity) is the requirement \( \Re = \bigcup_{i} \Re ({\mathcal{O}}_{i}) \).

  24. This result, combined with Guido and Longo's analysis, suggests another version of an algebraic CPT theorem; namely, for a von Neumann algebra of local observables with a cyclic vacuum representation, the conjunction of Poincaré Covariance, Weak Additivity, Wedge Duality, and the Reality Condition entails CPT invariance (Borchers 2000, p. 32).

  25. This is typically interpreted as a thermalized multi-particle state, although Earman (2011) and Arageorgis et al. (2003) argue that this unjustified.

  26. Greenberg (2006) demonstrates that restricted Lorentz invariance of τ-functions (see footnote 20) evaluated at Jost points entails LC. Thus, "…if we take Lorentz covariance of time-ordered products as the condition of Lorentz covariance of the field theory, then… local commutativity is not an independent assumption of the theory" (p. 087701-1).

  27. Greenberg (1998, p. 145) also associates the "Spin-Statistics" theorem with Weinberg's approach: "[Fierz and Pauli] used locality of observables as the crucial condition for integer-spin particles and positivity of the energy as the crucial condition for the odd half-integer case. Weinberg showed that one can use the locality of observables for both cases if one requires positive-frequency modes to be associated with annihilation operators and negative-frequency modes to be associated with creation operators." However, for Weinberg, "locality of observables" (i.e., commutativity at spacelike separated distances) is only imposed on the interacting Hamiltonian density \( {\mathfrak{H}}_{\text{int}}( x) \) and only to formally secure Lorentz invariance of the S-matrix (see Sect. 2.2). For Weinberg, "causality" as applied to observable quantities other than the S-matrix is explicitly renounced: "The point of view taken here is that [LC] is needed for the Lorentz invariance of the S-matrix, without any ancillary assumptions about measurability or causality" (Weinberg 1995, p. 198).

  28. This follows the intuitions of Halvorson and Clifton (2002, pp. 17–18). This aspect of the Received View should thus be made distinct from concepts of localized particles that require the existence of position operators and/or localized states.

  29. Streater and Wightman (2000, p. 139). This, in turn, is a consequence of the Reeh-Schlieder theorem and Local Commutativity.

  30. Note that Baker and Halvorson are not expressly concerned with Thesis (I), even restricted to the concept of antimatter. Rather, they are only concerned with divorcing the concept of antimatter from the concept of particle: "[T]here may be a fundamental matter–antimatter distinction to be drawn in QFT. Whether there is does not depend on whether particles play any part in the theory's fundamental ontology" (p. 94).

  31. A wavefunctional state Ψ[χ] is a probability distribution over classical field configurations χ(x) (see, e.g., Wallace 2006, pp. 40–41). A field operator \( \hat{\phi }(x) \) (distinguished here with a hat) acts on Ψ[χ] and produces the field configuration χ(x).

  32. The literature on this is vast. A partial survey is given in Duck and Sudarshan (1997).

  33. One attempt at the former is Bain (2000). A view that might be identified with the latter is Ruetsche's (2011, p. 147) "coalescence" approach to the interpretation of RQFTs, which declares an RQFT appropriately interpreted only after an application is specified. One might thus attempt to argue that in applications of RQFTs in which CPT invariance and the spin-statistics connection are important, particle interpretations (of either the sort sanctioned by the Received View or otherwise) are available.

References

  • Arageorgis, A., Earman, J., & Ruetsche, L. (2003). Fulling non-uniqueness and the Unruh effect: A primer on some aspects of quantum field theory. Philosophy of Science, 70, 164–202.

    Article  Google Scholar 

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

    Google Scholar 

  • Bain, J. (2000). Against particle/field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who’s Afraid of Haag’s Theorem?). Erkenntnis, 53, 375–406.

    Article  Google Scholar 

  • Bain, J. (2011). Quantum field theories in classical spacetimes and particles. Studies in History and Philosophy of Modern Physics, 42, 98–106.

    Article  Google Scholar 

  • Baker, D. (2009). Against field interpretations of quantum field theory. British Journal of Philosophy of Science, 60, 585–609.

    Google Scholar 

  • Baker, D., & Halvorson, H. (2010). Antimatter. British Journal of Philosophy of Science, 61, 93–121.

    Article  Google Scholar 

  • Bell, J. (1955). Time reversal in field theory. Proceedings of the Royal Society of London A, 231, 479–495.

    Article  Google Scholar 

  • Bisognano, J., & Wichmann, E. (1976). On the duality condition for quantum fields. Journal of Mathematical Physics, 17, 303–321.

    Article  Google Scholar 

  • Borchers, H.-J. (2000). Modular groups in quantum field theory. In P. Breitenlohner & D. Maison (Eds.), Quantum field theory (Lecture notes in physics) (pp. 26–46). Berlin: Springer.

    Chapter  Google Scholar 

  • Brunetti, R., Guido, D., & Longo, R. (1993). Modular structure and duality in conformal quantum field theory. Communications in Mathematical Physics, 156, 201–219.

    Article  Google Scholar 

  • Burgoyne, N. (1958). On the connection of spin and statistics. Nuovo Cimento, 8, 607–609.

    Article  Google Scholar 

  • Clifton, R., & Halvorson, H. (2001). Are Rindler quanta real? Inequivalent particle concepts in quantum field theory. British Journal for the Philosophy of Science, 52, 417–470.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Duck, I., & Sudarshan, E. (1997). Pauli and the spin-statistics theorem. Singapore: World Scientific.

    Google Scholar 

  • Earman, J. (2011). ‘The Unruh effect for philosophers. Studies in History and Philosophy of Modern Physics, 42, 81–97.

    Article  Google Scholar 

  • Earman, J., & Fraser, D. (2006). Haag’s theorem and its implications for the foundations of quantum field theory. Erkenntnis, 64, 305–344.

    Article  Google Scholar 

  • Fierz, M. (1939). Über die relativistische Theorie kräftefeier Teilchen mit beliebigem Spin. Helvetica Physica Acta, 12, 3–37.

    Article  Google Scholar 

  • Fraser, D. (2008). The fate of “Particles” in quantum field theories with interactions. Studies in History and Philosophy of Modern Physics, 39, 841–859.

    Article  Google Scholar 

  • Fraser, D. (2011). How to take particle physics seriously: A further defence of axiomatic quantum field theory. Studies in History and Philosophy of Modern Physics, 42, 126–135.

    Article  Google Scholar 

  • Georgi, H. (1993). Effective field theory. Annual Review of Nuclear and Particle Science, 43, 209–252.

    Article  Google Scholar 

  • Greaves, H. (2010). Towards a geometrical understanding of the CPT theorem. British Journal of Philosophy of Science, 61, 27–50.

    Article  Google Scholar 

  • Greenberg, O. (1998). Spin-statistics, spin-locality, and TCP: Three distinct theorems. Physics Letters B, 416, 144–149.

    Article  Google Scholar 

  • Greenberg, O. (2002). CPT violation implies violation of Lorentz invariance. Physical Review Letters, 89, 231602/1–231602/4.

    Article  Google Scholar 

  • Greenberg, O. (2006). Covariance of time-ordered products implies local commutativity of fields. Physical Review D, 73, 087701/1–087701/3.

    Google Scholar 

  • Guido, D., & Longo, R. (1992). Relativistic invariance and charge conjugation in quantum field theory. Communications in Mathematical Physics, 148, 521–551.

    Article  Google Scholar 

  • Guido, D., & Longo, R. (1995). An algebraic spin and statistics theorem. Communications in Mathematical Physics, 172, 517–533.

    Article  Google Scholar 

  • Haag, R. (1996). Local quantum physics (2nd ed.). Berlin: Springer.

    Book  Google Scholar 

  • Halvorson, H., & Clifton, R. (2002). No place for particles in relativistic quantum theories? Philosophy of Science, 69, 1–28.

    Article  Google Scholar 

  • Halvorson, H., & Müger, M. (2006). Algebraic quantum field theory. In J. Butterfield & J. Earman (Eds.), Philosophy of physics (pp. 731–922). Amsterdam: Elsevier Press.

    Google Scholar 

  • Jost, R. (1957). Eine Bemerkung zum CTP theorem. Helvetica Physica Acta, 30, 409–416.

    Google Scholar 

  • Jost, R. (1965). The general theory of quantized fields. Providence: American Mathematical Society.

    Google Scholar 

  • Kaku, M. (1993). Quantum field theory. Oxford: Oxford University Press.

    Google Scholar 

  • Lüders, G., & Zumino, B. (1958). Connection between spin and statistics. Physical Review, 110, 1450–1453.

    Article  Google Scholar 

  • Massimi, M., & Redhead, M. (2003). Weinberg’s proof of the spin-statistics theorem. Studies in History and Philosophy of Modern Physics, 34, 621–650.

    Article  Google Scholar 

  • Morgan, J. (2004). Spin and statistics in classical mechanics. American Journal of Physics, 72, 1408–1417.

    Google Scholar 

  • Oksak, A. I., & Todorov, I. T. (1968). Invalidity of TCP-theorem for infinite-component fields. Communications in Mathematical Physics, 11, 125–130.

    Article  Google Scholar 

  • Pauli, W. (1940). The connection between spin and statistics. Physical Review, 58, 716–722.

    Article  Google Scholar 

  • Peskin, M., & Schroeder, D. (1995). An Introduction to quantum field theory. Reading, MA: Addison-Wesley.

    Google Scholar 

  • Ruetsche, L. (2011). Interpreting quantum theories: The art of the possible. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Sterman, G. (1993). An introduction to quantum field theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Streater, R. (1967). Local fields with the wrong connection between spin and statistics. Communications in Mathematical Physics, 5, 88–96.

    Article  Google Scholar 

  • Streater, R., & Wightman, A. (2000). PCT, spin and statistics, and all that. Princeton: Princeton University Press.

    Google Scholar 

  • Wallace, D. (2006). In defense of naiveté: The conceptual status of lagrangian quantum field theory. Synthese, 151, 33–80.

    Article  Google Scholar 

  • Wallace, D. (2009). QFT, antimatter, and symmetry. Studies in History and Philosophy of Modern Physics, 40, 209–222.

    Article  Google Scholar 

  • Wallace, D. (2011). Taking particle physics seriously: A critique of the algebraic approach to quantum field theory. Studies in History and Philosophy of Modern Physics, 42, 116–125.

    Article  Google Scholar 

  • Weinberg, S. (1964). Feynman rules for any spin. Physical Review, 133, B1318–B1332.

    Article  Google Scholar 

  • Weinberg, S. (1995). The quantum theory of fields (Vol. 1). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Wightman, A. (1999). Review of I. Duck and E. C. G. Sudarshan Pauli and the spin-statistics theorem. American Journal of Physics, 67, 742–746.

    Article  Google Scholar 

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    Jonathan Bain

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Bain, J. CPT Invariance, the Spin-Statistics Connection, and the Ontology of Relativistic Quantum Field Theories. Erkenn 78, 797–821 (2013). https://doi.org/10.1007/s10670-011-9324-9

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