Journal Article

Towards a Geometrical Understanding of the CPT Theorem

Hilary Greaves

in The British Journal for the Philosophy of Science

Published on behalf of British Society for the Philosophy of Science

Volume 61, issue 1, pages 27-50
Published in print March 2010 | ISSN: 0007-0882
Published online February 2009 | e-ISSN: 1464-3537 | DOI:
Towards a Geometrical Understanding of the CPT Theorem

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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.


The Connection between Dynamical Symmetries and Spacetime Structure

A Puzzle about the CPT Theorem

A Classical PT Theorem 4.1

Bell's theorem


Auxiliary constraints

Resolution of the Puzzle

Galilean-Invariant Field Theories 6.1

Temporal orientation in Galilean spacetime


Counterexample to the Galilean PT hypothesis


Journal Article.  9907 words.  Illustrated.

Subjects: Philosophy of Science ; Science and Mathematics

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