Abstract
An influential claim in the physics literature states that the violation of CPT invariance in an interacting RQFT entails the violation of Lorentz invariance. This claim is surprising since standard proofs of the CPT theorem require more assumptions than Lorentz invariance, and are restricted to non-interacting, or at best, unrealistic interacting theories. This essay analyzes this claim in the context of the debate between pragmatist approaches to RQFTs, which trade mathematical rigor for the ability to derive predictions from realistic interacting theories, and purist approaches, which trade the ability to formulate realistic interacting theories for mathematical rigor.
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Notes
- 1.
- 2.
A Jost point (x 1, …, x n ) is a convex set of points that are spacelike separated from each other. In other words, the difference variables \( {\xi}_i\equiv {x_i}_{-1} - {x}_i \) satisfy \( {\left(\sum {\lambda}_j{\xi}_j\right)}^2 < 0,\ \mathrm{f}\mathrm{o}\mathrm{r}{\lambda}_{\mathrm{j}}\ \ge\ 0,\sum {\lambda}_{\mathrm{j}} > 0 \) (Streater and Wightman 1964, p. 71).
- 3.
This is reflected in Weinberg’s (1995, p. 198) view of the axiomatic assumption of local commutativity (i.e., fields at spacelike separated points commute): “The point taken here is that [local commutativity of fields] is needed for the Lorentz invariance of the S-matrix, without any ancillary assumptions about measurability or causality.”
- 4.
In other words, a model of the Wightman axioms need not take the form of a Fock space representation of the canonical (anti-) commutation relations. Note that the basic objects of the Wightman approach are tempered distributions (i.e., Wightman functions), but Wightman’s (1956) reconstruction theorem indicates that these can be interpreted as vacuum expectation values of unordered products of fields.
- 5.
Lévy-Leblond (1967, p. 165) explains the failure of the CPT theorem in GQFTs as due to the fact that GQFTs do not satisfy local commutativity: “This situation [i.e., the GQFT case] is to be contrasted with the relativistic case where the requirements of local commutativity on a free field… impose both the existence of a TCP [i.e., CPT] operation… and the spin-statistics relation, as has been shown in a very illuminating way, for this free-field case, by Weinberg…”
- 6.
The Hamiltonian is related to the Hamiltonian density by \( H(t) = \int {d}^3\mathbf{x}\mathcal{H}\left(\mathbf{x},t\right) \).
- 7.
ϕ I (x) is defined by \( {\phi}_I\left(\mathbf{x},t\right)\equiv {e}^{i{H_0}\left(t-t_0\right)}\phi \left(\mathbf{x},{t}_0\right){e}^{-i{H_0}\left(t-t_0\right)} \), where ϕ(x, t 0) is a non-interacting field at time t 0.
- 8.
As Weinberg (1995, p. 441) states, “…the renormalization of masses and fields has nothing directly to do with the presence of infinities, and would be necessary even in a theory in which all momentum space integrals were convergent.”
- 9.
An important exception to this is QCD, which is not weakly coupled.
- 10.
Causal perturbation theory consists of both a regularization scheme to address UV divergences in power series expansions, and an axiomatic scheme underwriting such expansions. These schemes can be separated; in particular, the regularization scheme can be adopted by pragmatists independently of the axiomatic scheme (Helling 2012; Falk et al. 2010). Conversely, the axiomatic scheme can be adopted by purists to extend purist axiomatic systems to include perturbative techniques (Brunetti and Fredenhagen 2000).
- 11.
The proof of this claim rests on the fact that it is always possible to choose two Lorentz transformations that time-order a Jost point (x 1, …, x n ) in opposite ways.
- 12.
The first entailment is based on the fact that the time-ordering of two points is RLI unless the points are spacelike separated. Thus if a field is RLI, then so are time-ordered products of it, except when it is evaluated at spacelike separated points. But if the field commutes when it is evaluated at spacelike separated points, then time-ordering will not violate RLI even at such points. This also holds for sums of products of fields, and hence for ℋ int (x). The second entailment follows since if time-ordered products of ℋ int (x) are RLl, then so is the S-matrix in the form (1), since all other quantities in (1) are manifestly RLI.
- 13.
Chaichian et al. (2011, p. 178) provide examples of non-local interaction Hamiltonian densities that are restricted Lorentz invariant and violate CPT invariance (thanks to a referee for pointing this out).
- 14.
Here is another concern about the feasibility of Premise I in pragmatist approaches. If an interacting RQFT is in the business of calculating S-matrix elements, then τ-functions play an important role, as the discussion of the LSZ and Gell-Mann/Low formulas indicated, and this seems to make Premise I initially plausible. However, if there are other methods for calculating S-matrix elements that do not rely on τ-functions, and, moreover, if there are other testable predictions of RQFTs that can be derived without the use of τ-functions, then Premise I will again loose traction with pragmatists.
- 15.
For the purist, this problem manifested itself in the fact that currently there are no examples of realistic interacting τ-functions in the form of well-defined analytic functions. For the pragmatist who allows τ-functions to take the form of divergent power series expansions obtained via the Gell-Mann/Low formula, the problem is that such expressions do not satisfy the Spectrum Condition (in the sense that the fields that occur in them do not satisfy the Spectrum Condition).
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Bain, J. (2015). Pragmatists and Purists on CPT Invariance in Relativistic Quantum Field Theories. In: Mäki, U., Votsis, I., Ruphy, S., Schurz, G. (eds) Recent Developments in the Philosophy of Science: EPSA13 Helsinki. European Studies in Philosophy of Science, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-23015-3_17
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