Towards a geometrical understanding of the CPT theorem Hilary Greaves 15 th UK and European Meeting on the Foundations of Physics University of Leeds, 30 March 2007

Outline of the talk 1.Spacetime theories 1.A puzzle about the CPT theorem 2.A classical ‘CPT’ theorem 3.Towards a geometrical understanding 4.Summary so far; open questions

Outline of the talk 1.Spacetime theories 1.A puzzle about the CPT theorem 2.A classical ‘CPT’ theorem 3.Towards a geometrical understanding 4.Summary so far; open questions

Spacetime theories Spacetime theory T: intended models of the form Coordinate-independent formalism Realism about spacetime structure M K: set of kinematically allowed structures M D  M K : set of dynamically allowed structures Symmetry of T: a map M K  M K leaving M D invariant

(Trivial) general covariance h:M  M, manifold diffeomorphism Induces a map h:M K  M K : General (diffeomorphism) covariance:

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 h Q :M K  M K : Covariance Q group: {hDiff(M):h Q is a symmetry} Expect: covariance Q group = invariance group of Q

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,

A puzzle about the CPT theorem Some geometrical objects that a spacetime theory might(?) invoke: –g, metric (flat, Lorentzian) –, total orientation –, temporal orientation QInvariance group of Q gL (Lorentz group) g, L + (proper Lorentz group) g,, L  + (restricted Lorentz group) L+L+ L-L- L-L- L+L+ L+L+

CPT theorem CPT theorem: –If T is L  + -covariant Q, then T is also CPT-covariant Q. PT theorem: –If T is L  + -covariant Q, then T is actually L + -covariant Q. –I.e. “a nice theory cannot use a temporal orientation.” Why not?

Outline of the talk 1.Spacetime theories 1.A puzzle about the CPT theorem 2.A classical ‘CPT’ theorem 3.Towards a geometrical understanding 4.Summary so far; open questions

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: 1.The ‘ordinary’ fields are tensors (of arbitrary rank). 2.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”)

Outline of the talk 1.Spacetime theories 1.A puzzle about the CPT theorem 2.A classical ‘CPT’ theorem 3.Towards a geometrical understanding 4.Summary so far; open questions

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.

(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.

(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.

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.

(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 v a  such that

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]: t a t=t 0 t=t 1 t=t 2 t=t 3 t: time function t a =dt (covector field)

(Counterexample to a Galilean PT-hypothesis) Spacetime structure (‘special fields’): –D, affine connection (flat) –t a, temporal metric+orientation –h ab, spatial metric ‘Ordinary’ fields: –, a scalar field –v a, a vector field Generally covariant equation: Non-generally-covariant equation:

Outline of the talk 1.Spacetime theories 1.A puzzle about the CPT theorem 2.A classical ‘CPT’ theorem 3.Towards a geometrical understanding 4.Summary so far; open questions

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.

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??

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?