CPT Theorem

Prerequisites: Lorentz Group, Canonical Quantum Field Theory. Full path (22 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

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 reversal 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 product of C, P, and T , taken in any order. Take note that $C$, $P$, $T$ or any other product of them can be violated, while 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 sharpest form of the theorem is in axiomatic quantum field theory, in the framework of the Wightman axioms: fields are operator-valued tempered distributions on a Hilbert space with a unitary representation of the (covering of the) Poincaré group, a unique invariant vacuum $|0\rangle$, the spectrum condition (energy-momentum in the closed forward cone), and local commutativity, with the normal connection between spin and statistics. No Lagrangian is needed.

Statement (Jost). For a Hermitian scalar field the CPT theorem says $$\langle 0|\phi(x_1)\cdots\phi(x_n)|0\rangle = \langle 0|\phi(-x_n)\cdots\phi(-x_1)|0\rangle$$ for all $n$, which is equivalent to the existence of an antiunitary operator $\Theta$ with $\Theta|0\rangle = |0\rangle$ and $\Theta\,\phi(x)\,\Theta^{-1} = \phi(-x)$. The order of the fields is reversed because $\Theta$ is antiunitary. For fields in a general finite-dimensional representation of $SL(2,\mathbb{C})$, $\Theta$ maps a field to the Hermitian conjugate of the field at $-x$, times a phase that depends on the representation. Jost showed that CPT invariance holds if and only if weak local commutativity holds: the vacuum expectation value of the fields at totally spacelike configurations equals that with the order reversed. In particular it follows from ordinary local commutativity, so every Wightman theory is CPT invariant.

Idea of the proof. By the spectrum condition, the Wightman function $W(x_1,\dots,x_n)$ is the boundary value of a function $\mathcal W(\zeta_1,\dots,\zeta_{n-1})$ of the complex difference variables, holomorphic in the forward tube. Lorentz covariance and the Bargmann-Hall-Wightman theorem extend it to the extended tube, and the extension is invariant under the complexified proper Lorentz group $L_+(\mathbb{C})$. In four dimensions the total inversion $x \mapsto -x$ lies in the identity component of $L_+(\mathbb{C})$: it is the product of a boost by imaginary rapidity $i\pi$ in the $(t,z)$ plane, which reverses $t$ and $z$, and a rotation by $\pi$ in the $(x,y)$ plane. Hence $\mathcal W(\zeta) = \mathcal W(-\zeta)$ on the extended tube. The extended tube contains real points, the Jost points, which are configurations whose difference vectors and all their positive combinations are spacelike. There local commutativity lets us reverse the order of the fields, and the identity theorem for holomorphic functions extends the resulting equality to all of the domain and then to its boundary values.

Generalisations. The argument uses that $-1$ is in the identity component of the complexified Lorentz group, which holds in even spacetime dimension. In any dimension one obtains instead a symmetry that reverses time and one spatial direction, usually called CRT. In Euclidean signature this symmetry is the continuation of a rotation by $\pi$ in a plane containing Euclidean time, and its antiunitarity is tied to reflection positivity. In algebraic quantum field theory, the Bisognano-Wichmann theorem identifies the modular conjugation of the von Neumann algebra of a wedge region with the CRT operator for the reflection $(t,x^1) \mapsto (-t,-x^1)$. This gives an operator-algebraic origin of the CPT operator. The group-theoretic question of how $C$, $P$ and $T$ act on spinors, including the choice between the Pin groups, is discussed in the Pin groups reference below.

• 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

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