CPT Symmetry

Prerequisites: Parity, Lorentz Group. Full path (10 pages)

Best explanations

Concrete Why is CPT Fundamental? by O. W. Greenberg

Intuitive
Concrete
Abstract
Why is it interesting?
Research

Intuitive

CPT symmetry means 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.

A nice introduction is this video by Veritasium.

Concrete

We use $\hbar = c = 1$ and the metric signature $(+,-,-,-)$ .

The combined transformation $\Theta = CPT$ maps a field at the point $x$ to a field at $-x = (-t,-\vec x)$ . For a Dirac field it acts as $\Theta\, \psi(x)\, \Theta^{-1} = \eta\, \gamma^5\, \psi^*(-x)$ with a phase $\eta$ , in a representation where $\gamma^5$ is real (here $\psi^*$ means $(\psi^\dagger)^T$ ), and it is antiunitary, so complex numbers in front of operators are conjugated. The easiest way to see what CPT does to a Lagrangian is to look at the Dirac bilinears. Under CPT, each bilinear at $x$ goes into the same bilinear at $-x$ , times a sign:

scalar $\bar\psi\psi$ and pseudoscalar $i\bar\psi\gamma^5\psi$ : $+1$
vector $\bar\psi\gamma^\mu\psi$ and axial vector $\bar\psi\gamma^\mu\gamma^5\psi$ : $-1$
tensor $\bar\psi\sigma^{\mu\nu}\psi$ : $+1$
derivative $\partial_\mu$ : $-1$

The rule is simple: an object with $n$ Lorentz indices picks up $(-1)^n$ . The same holds for scalar, vector and tensor boson fields with the standard phase choices. A term in the Lagrangian is a Lorentz scalar, so all indices are contracted and the total number of indices is even. Hence every Hermitian, Lorentz invariant term transforms as $\mathcal L(x) \to \mathcal L(-x)$ , and the action $\int d^4x\, \mathcal L$ is invariant. Hermiticity matters here: a complex coupling $g$ is turned into $g^*$ by the antiunitary $\Theta$ , and this is compensated because the term comes together with its Hermitian conjugate. The CPT theorem is the general version of this argument.

Consequences. $\Theta$ commutes with the Hamiltonian and maps a one-particle state $|a; \vec p, s\rangle$ to the antiparticle state $|\bar a; \vec p, -s\rangle$ up to a phase (P and T each reverse $\vec p$ , so the momentum is unchanged, while the spin component is reversed by T). Therefore:

particle and antiparticle have the same mass,
they have the same total decay width, hence the same lifetime,
their electric charges and magnetic moments are equal in size and opposite in sign.

Partial decay rates into a specific final state and its conjugate can differ if CP is violated, but the sums over all final states must agree. These relations are the basis of experimental tests, for example comparisons of the electron and positron magnetic moments and of the masses of the neutral kaon and its antiparticle.

See WHY IS CPT FUNDAMENTAL? by O.W. Greenberg

Abstract

The symmetry. In a relativistic quantum theory the symmetries are realized on the Hilbert space by unitary or antiunitary operators (Wigner's theorem). Transformations that reverse the direction of time and preserve positivity of energy must be antiunitary. CPT is the antiunitary operator $\Theta$ with $\Theta\, U(a,\Lambda)\, \Theta^{-1} = U(-a, \Lambda)$ for every element $(a,\Lambda)$ of the identity component of the Poincaré group (or its covering): $\Theta$ commutes with all Lorentz transformations and reverses all translations. Equivalently, it implements the total inversion $x \to -x$ , which lies in the full Lorentz group $O(1,3)$ but not in its identity component.

Why it is special. The total inversion $-\mathbb 1$ is not in the identity component of the real Lorentz group, but it is in the identity component of the complexified group $SO(4,\mathbb C)$ , since it is a rotation by $\pi$ in two orthogonal complex planes. Analyticity of vacuum correlation functions, which follows from positivity of energy, lets one reach $-\mathbb 1$ by a continuous path through complex Lorentz transformations. This is why CPT, but none of C, P, T separately, follows from Lorentz invariance, locality and the spectrum condition. In Euclidean signature the same transformation is an ordinary rotation by $\pi$ in two orthogonal planes. The Wightman-axiomatic statement and proof are on the CPT theorem page.

S-matrix. Because $\Theta$ maps in-states to out-states, $\Theta S \Theta^{-1} = S^\dagger$ , and antiunitarity gives $\langle f|S|i\rangle = \langle \Theta i|S|\Theta f\rangle ,$ so the amplitude for $i \to f$ equals the amplitude for the CPT-conjugate process with initial and final states exchanged. Combined with unitarity of $S$ , this yields the equality of total widths of particles and antiparticles.

The standard reference is "PCT, spin and statistics, and all that" by R. F. Streater and A. S. Wightman.

Why is it interesting?

According to the famous CPT theorem the symmetry CPT holds for all physical phenomena under quite reasonable assumptions.

This makes CPT symmetry one of the most important symmetries of modern physics and especially of the standard model of particle physics.

Research

For the experimental status of CPT symmetry see the Data Tables for Lorentz and CPT Violation by Alan Kostelecky, Neil Russell