Towards a geometrical understanding of the CPT theorem

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Towards a geometrical understanding of the CPT
theorem

• Hilary Greaves
• 15th UK and European Meeting on the Foundations
  of Physics
• University of Leeds, 30 March 2007

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Outline of the talk

• Spacetime theories
• A puzzle about the CPT theorem
• A classical CPT theorem
• Towards a geometrical understanding
• Summary so far open questions

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Outline of the talk

• Spacetime theories
• A puzzle about the CPT theorem
• A classical CPT theorem
• Towards a geometrical understanding
• Summary so far open questions

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Spacetime theories

• Spacetime theory T intended models of the form
• Coordinate-independent formalism
• Realism about spacetime structure
• MK set of kinematically allowed structures
• MD? MK set of dynamically allowed structures
• Symmetry of T a map MK ? MK leaving MD invariant

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(Trivial) general covariance

• hM?M, manifold diffeomorphism
• Induces a map hMK?MK
• General (diffeomorphism) covariance

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How to find nontrivial symmetries

• Start from a generally covariant formulation of
  the theory
• Single out some subset Q of the objects as
  special
• For h?Diff(M), define a map hQMK?MK
• CovarianceQ group h?Diff(M)hQ is a symmetry
• Expect covarianceQ group invariance group of Q

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Example (special-relativistic) electromagnetism

• Fields g (flat), F, J
• Generally-covariant equations,
• Treat g as special
• The covarianceg group is the Lorentz group
• Non-generally-covariant equations,

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A puzzle about the CPT theorem

• Some geometrical objects that a spacetime theory
  might(?) invoke
• g, metric (flat, Lorentzian)
• ?, total orientation
• ?, temporal orientation

L?
L?-
L
L?-
L?
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CPT theorem

• CPT theorem
• If T is L? -covariantQ, then T is also
  CPT-covariantQ.
• PT theorem
• If T is L? -covariantQ, then T is actually
  L-covariantQ.
• I.e. a nice theory cannot use a temporal
  orientation.
• Why not?

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Outline of the talk

• Spacetime theories
• A puzzle about the CPT theorem
• A classical CPT theorem
• Towards a geometrical understanding
• Summary so far open questions

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A classical PT theorem (Bell 1955)

• Let T be a spacetime theory according to which
  there are n ordinary fields ?i.
• Suppose that the following two conditions hold
• The ordinary fields are tensors (of arbitrary
  rank).
• In some fixed coordinate system, the dynamical
  equations for the ?i take the form F(j)0,
  where each F(j) is a functional that is
  polynomial in the components of the ?i and their
  coordinate derivatives.
• Then, if the set S of solutions to the dynamical
  equations is invariant under L?, S is actually
  invariant under all of L.
• ( rational and integral)

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Outline of the talk

• Spacetime theories
• A puzzle about the CPT theorem
• A classical CPT theorem
• Towards a geometrical understanding
• Summary so far open questions

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A not nice theory

• Let ? be some particular scalar field, with no
  interesting symmetries.
• Let S be given by
• Then, S is L?-invariant (by construction), but
  is not invariant under PT.

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(Importance of the innocuous auxiliary
constraints)

• The theorem will only go through for theories
• whose objects transform as tensor fields
• and
• whose dynamics are given by PDEs in the usual
  fashion.

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(A theory with PT-pseudo-objects)

• A simple pseudo-object counterexample
• Let ? be a PT-pseudo-scalar field.
• Dynamics ?1.
• A (slightly) more realistic one
• Let ? be a PT-pseudoscalar, ? a scalar.
• Dynamics ?(? - ?)0.

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A geometrical explanation?

• Observation there is no tensor field that
• defines a temporal orientation, and also
• is L? -invariant.
• If there were, we could use it to violate the
  PT-theorem.
• Idea If there exists a set Q of tensor fields
  whose invariance group is X, then it is possible
  to write down a nice theory whose covariance
  group is X.

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(A theory whose dynamics involve existential
quantification)

• Take the temporal orientation ? to be the set of
  all nowhere vanishing, future-directed timelike
  vector fields.
• Let there be (besides the temporal orientation,
  total orientation and metric) a scalar field ?.
• Say that ? is dynamically allowed iff
  the following condition holds
• There exists at least one vector field va?? such
  that

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Importance of the Lorentz group

• There is no Galilean PT theorem.
• There is a Galilean-invariant tensor field that
  defines a temporal orientation and metric ta

tt3
tt2
tt1
tt0
t time function tadt (covector field)
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(Counterexample to a Galilean PT-hypothesis)

• Spacetime structure (special fields)
• D, affine connection (flat)
• ta, temporal metricorientation
• hab, spatial metric
• Ordinary fields
• ?, a scalar field
• va, a vector field
• Generally covariant equation
• Non-generally-covariant equation

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Outline of the talk

• Spacetime theories
• A puzzle about the CPT theorem
• A classical CPT theorem
• Towards a geometrical understanding
• Summary so far open questions

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Summary

• From the spacetime point of view, a PT theorem is
  prima facie puzzling it seems to assert that it
  is not possible for a nice theory to use a
  temporal orientation, over and above a Lorentzian
  metric and total orientation.
• The solution to this puzzle lies in the
  observation that there is no Lorentz-invariant
  tensor-field way of representing temporal
  orientation.
• This is a peculiarity of the Lorentz-temporal
  combination. The analogous phenomenon does not
  occur
• For total orientation, or
• In the Galilean case.

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A residual puzzle

• My explanation of the PT-theorem concerned the
  nonexistence of a tensor-field temporal
  orientation.
• The proof of the PT theorem is based on the fact
  that the identity and total-reflection components
  of L are connected in the complex Lorentz group.
• What is the connection??

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Further prospects

• Can one prove a generalized PT theorem?
• Can one prove a coordinate-independent PT
  theorem?
• What, exactly, is the relationship between the
  classical PT theorem and the quantum CPT
  theorem?