Towards a Geometrical Understanding of the CPT Theorem

The CPT theorem of quantum field theory states that any relativistic (Lorentz-invariant) quantum field theory must also be invariant under CPT, the composition of charge conjugation, parity reversal and time reversal. This paper sketches a puzzle that seems to arise when one puts the existence of this sort of theorem alongside a standard way of thinking about symmetries, according to which spacetime symmetries (at any rate) are associated with features of the spacetime structure. The puzzle is, roughly, that the existence of a CPT theorem seems to show that it is not possible for a well-formulated theory that does not make use of a preferred frame or foliation to make use of a temporal orientation. Since a manifold with only a Lorentzian metric can be temporally orientable—capable of admitting a temporal orientation—this seems to be an odd sort of necessary connection between distinct existences. The paper then suggests a solution to the puzzle: it is suggested that the CPT theorem arises because temporal orientation is unlike other pieces of spacetime structure, in that one cannot represent it by a tensor field. To avoid irrelevant technical details, the discussion is carried out in the setting of classical field theory, using a little-known classical analog of the CPT theorem. – 1. Introduction; – 2. The Connection between Dynamical Symmetries and Spacetime Structure; – 3. A Puzzle about the CPT Theorem; – 4. A Classical PT Theorem : Bell's theorem; Auxiliary constraints; – 5. Resolution of the Puzzle; – 6. Galilean-Invariant Field Theories : Temporal orientation in Galilean spacetime; Counterexample to the Galilean PT hypothesis; – 7. Conclusions. M.-M. V.