theorems:cpt

# CPT Theorem

## Intuitive

The CPT theorem tells us that a mirror-image of our universe where we reverse all momenta (corresponding to the reversal of time) and with all matter replaced by anti-matter would evolve under exactly the same physical laws.

## Concrete

The CPT theorem states that the product of charge conjugation, parity, and time rever- sal transformations is under quite general assumptions a valid symmetry.

The assumptions are:

• We are dealing with a quantum field theory.
• It is based on a Hermitian, local, normal-ordered Lagrangian.
• The Lagrangian is invariant under Lorentz transformations.
• The canonical commutation or anti-commutation rules hold for the fields.

Formulated differently, the CPT theorem says that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must be CPT symmetric.

From these assumptions it follows that the Lagrangian is also invariant under the produce of C, P, and T , taken in any order. Take note that $C$, $P$, $T$ or any other product of them can be violated, which CPT is intact. This means concretely that we can always choose the phases which appear in C, P, T transformations such that the product of those operators is a symmetry of our theory.

In this sense, the combination CPT is more fundamental than the three component transformations.

To test the CPT symmetry we compare the masses, lifetimes, electric charges and anomalous magnetic moments of particles with their corresponding antiparticles. Another possibility are detailed analysis of the behaviour of neutral flavoured meson systems.

• For a nice discussion see Chapter 5 in “Discrete Symmetries and CP Violation: From Experiment to Theory” by Marco Sozzi

A story has it that in the early sixties, Feynman was asked to give an evening talk to physics students at Caltech, explaining the basic idea of the CPT theorem: the celebrated result in quantum field theory that states that any relativistic (i.e. Lorentz-invariant) quantum field theory must be invariant under CPT, the composition of charge conjugation, parity reversal and time reversal. Feynman agreed to commit to doing this, commenting that if one cannot explain something to second year Caltech undergraduates then one does not understand it. The story goes that Feynman spent a month or two trying to plan the talk, and then, in despair, cancelled the commitment.Towards a geometrical understanding of the CPT theorem by Hilary Greaves

## Abstract

• The standard reference is "PCT, spin and statistics, and all that" by R. F. Streater and A. S. Wightman.
• For a nice discussion of the symmetry groups involved see The Pin Groups in Physics: C, P, and T by M. Berg, C. DeWitt-Morette, S. Gwo, E. Kramer

## Why is it interesting?

The CPT theorem tells us that CPT symmetry holds for all physical phenomena.

## FAQ

Why, exactly, is the CPT theorem considered so holy?

Because the CPT theorem is an almost-consequence of Lorentz invariance.

If you have a Lorentz invariant theory, then you can change the coordinates of space time with the following matrix and leave the theory the same

 cosh(y) 0 0 sinh(y) 0 1 0 0 0 0 1 0 sinh(y) 0 0 cosh(y)

and this is true for any y.

An interesting property of quantum mechanics is that the amplitudes you calculate are analytic functions of the variables in the problem, this is an obvious fact in perturbation theory, where the amplitudes are just rational functions of the momenta coming in and going out, but its more general than that. The amplitudes are analytic in a wide range of circumstances.

And this means that the theory is invariant for any complex value of y, by the principle of analytic continuation. In particular, for y=−1=i and if you stick in y=i, you get that the matrix above does a PT transformation. So that a theory of a bunch of scalar particles is PT invariant.

A more general result, if you allow charged particles, is that a general theory is CPT invariant; the argument is essentially the same.Ron Maimon 