Towards a geometrical understanding of the CPT theorem

1. 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

2. 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

3. 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

4. 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 MKleavingMD invariant

5. (Trivial) general covariance • h:MM, manifold diffeomorphism • Induces a map h:MKMK: • General (diffeomorphism) covariance:

6. 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 hQ:MKMK: • CovarianceQ group: {hDiff(M):hQ is a symmetry} • Expect: covarianceQ group = invariance group of Q

7. Example: (special-relativistic) electromagnetism • Fields: g (flat), F, J • Generally-covariant equations, • Treat g as special • The covariance{g} group is the Lorentz group • Non-generally-covariant equations,

8. A puzzle about the CPT theorem L+ L- • Some geometrical objects that a spacetime theory might(?) invoke: • g, metric (flat, Lorentzian) • , total orientation • , temporal orientation L+ L- L+

9. 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?

10. 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

11. 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”)

12. 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

13. 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.

14. (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.

15. (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.

16. 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.

17. (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

18. 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 t=t3 t=t2 t=t1 t=t0 t: time function ta=dt (covector field)

19. (Counterexample to a Galilean PT-hypothesis) • Spacetime structure (‘special fields’): • D, affine connection (flat) • ta, temporal metric+orientation • hab, spatial metric • ‘Ordinary’ fields: • , a scalar field • va, a vector field • Generally covariant equation: • Non-generally-covariant equation:

20. 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

21. 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.

22. 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??

23. 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?