Towards a geometrical understanding of the CPT theorem

Towards a Towards a geometrical geometrical

understanding of understanding of the CPT theoremthe CPT theorem

Hilary GreavesHilary Greaves1515thth UK and European Meeting on the UK and European Meeting on the

Foundations of PhysicsFoundations of PhysicsUniversity of Leeds, 30 March 2007University of Leeds, 30 March 2007

Outline of the talkOutline of the talk1. Spacetime theories

1. A puzzle about the CPT theorem

2. A classical ‘CPT’ theorem3. Towards a geometrical

understanding4. Summary so far; open questions





Spacetime theoriesSpacetime 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

M g

M D t h V

n

n

, , , ,

, , , , , , ,

1

1

(Trivial) general covariance(Trivial) general covariance• h:MM, manifold diffeomorphism

• Induces a map h:MKMK:

• General (diffeomorphism) covariance:

h D iff M

M M h M Mp D p D

( ),

, , , , , , 1 1

h M M h hp p, , , , * , , * 1 1

How to find nontrivial symmetriesHow to find nontrivial symmetries• Start from a generally covariant formulation of the theory

• Single out some subset Q of the objects as ‘special’

• For hDiff(M), define a map hQ:MKMK:

• CovarianceQ group: {hDiff(M):hQ is a symmetry}• Expect: covarianceQ group = invariance group of

Q

h M S S O O

M S S h O h O

Q n m

n m

, , , , , ,

, , , , * , , *

1 1

1 1

M p, , , 1

M S S O On m, , , , , ,1 1

Example: (special-relativistic) Example: (special-relativistic) electromagnetismelectromagnetism

• Fields: g (flat), F, J• Generally-covariant equations,

• Treat g as special

• The covariance{g} group is the Lorentz group

• Non-generally-covariant equations,

F J

F

abb

a

ab c

;

[ ; ]

4

0

F J

F

,

[ , ]

4

0

A puzzle about the CPT A puzzle about the CPT theoremtheorem

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

Q Invariance group of Q

g L (Lorentz group)

g, L+ (proper Lorentz group)

g,, L+ (restricted Lorentz group)

L+ L

-

L- L

+

L+

CPT theoremCPT 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?





A classical PT theorem A classical PT theorem (~Bell 1955)(~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”)





A ‘not nice’ theoryA ‘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.

S h L h : ( )( ) .

(Importance of the ‘innocuous (Importance of the ‘innocuous auxiliary constraints’)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-(A theory with PT-pseudo-objects)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?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 (A theory whose dynamics ‘involve existential quantification’)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 thefollowing condition holds:– There exists at least one vector field va such that

v a a 0.

Importance of the Lorentz Importance of the Lorentz groupgroup

• There is no Galilean PT theorem.

• There is a Galilean-invariant tensor field that defines a temporal orientation [and metric]: ta

t=t0

t=t1

t=t2

t=t3

t: time function

ta=dt (covector field)

(Counterexample to a (Counterexample to a Galilean PT-hypothesis)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:

t v haa ab

a b ; ;

v 0 2





SummarySummary• 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 puzzleA 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 prospectsFurther 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?

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Towards a geometrical understanding of the CPT theorem