Towards a Geometrical Understanding of the CPT Theorem
The CPT theorem states that any relativistic QFT must also be invariant under CPT, the composition of charge conjugation, parity reversal and time reversal. This paper sketches a puzzle that arises when one puts the existence of this theorem alongside a standard way of thinking about symmetries, according to which spacetime symmetries are associated with features of the spacetime structure. The puzzle is 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, this is an odd sort of necessary connection between distinct existences. The paper then suggests a solution to the puzzle: the CPT theorem arises because temporal orientation is unlike other pieces of spacetime structure, in that one cannot represent it by a tensor field.