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The British Journal for the Philosophy of Science Advance Access published online on February 19, 2009
The British Journal for the Philosophy of Science, doi:10.1093/bjps/axp004
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© The Author (2009). Published by Oxford University Press. For Permissions, please email: journals.permissions@oxfordjournals.org

Towards a Geometrical Understanding of the CPT Theorem

Hilary Greaves

Merton College, Oxford OX1 4JD

hilary.greaves{at}philosophy.ox.ac.uk


  Abstract

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
    4.1 Bell's theorem
    4.2 Auxiliary constraints

  5. Resolution of the Puzzle
  6. Galilean-Invariant Field Theories
    6.1 Temporal orientation in Galilean spacetime
    6.2 Counterexample to the Galilean PT hypothesis

  7. Conclusions


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