# CPT symmetry

CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously.

## History

Efforts in the late 1950s revealed the violation of P-symmetry by phenomena that involve the weak force, and there are well known violations of C-symmetry and T-symmetry as well. For a short time, the CP-symmetry was believed to be preserved by all physical phenomena, but that was later found to be false too. On the other hand, there is a theorem that derives the preservation of CPT symmetry for all of physical phenomena assuming the correctness of quantum laws and Lorentz invariance. Specifically, the CPT theorem states that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.
The CPT theorem appeared for the first time, implicitly, in the work of Julian Schwinger in 1951 to prove the connection between spin and statistics. In 1954 Gerhard Lüders and Wolfgang Pauli derived more explicit proofs so that the theorem is sometimes known as the Lüders-Pauli theorem. At about the same time and independently the theorem was also proved by John Stewart Bell. These proofs are based on the validity of Lorentz invariance and the Principle of locality in the interaction of quantum fields. Subsequently Res Jost gave a more general proof in the framework of axiomatic quantum field theory.

## Derivation

Consider a Lorentz boost in a fixed direction z. This can be interpreted as a rotation of the time axis into the z axis, but the rotation parameter is imaginary. If the rotation parameter were real, it would be possible to do a 180 degree rotation, and the rotation would reverse the direction of time and the direction of z. Reversing the direction of one axis is a reflection of space in any number of dimensions, and in three space dimensions it is equivalent to reflecting all the coordinates, because you can throw in an additional rotation of 180 degrees in the x-y plane for good measure.

This defines a CPT transformation when the antiparticles are interpreted as particles travelling backwards in time. This interpretation requires a slight analytic continuation, which is only well-defined under the following assumptions:

1. the theory is Lorentz invariant
2. the vacuum is Lorentz invariant
3. the energy is bounded below.

When this is true, the quantum theory can be extended to a Euclidean theory, which is defined by translating all the operators to imaginary time using the Hamiltonian. The commutation relations of the Hamiltonian and Lorentz generators guarantee that the Lorentz invariance implies rotation invariance, and any state can be rotated by 180.

Since two CPT-reflections in a sequence are equivalent to a 360 degree rotation, fermions change by a sign under two CPT reflections, while bosons do not. This can be used to prove the spin-statistics theorem.

## Consequences and Implications

A consequence of this derivation is that a violation of CPT automatically indicates a Lorentz violation.

The implication of CPT symmetry is that a mirror-image of our universe — with all objects having momenta and positions reflected by an imaginary plane (corresponding to a parity inversion), with all matter replaced by antimatter (corresponding to a charge inversion)— would evolve exactly like our universe. At any moment of corresponding times, the two universes would be identical, and the CPT transformation would simply turn one into the other. CPT symmetry is recognized to be a fundamental property of physical laws.

In order to preserve this symmetry, every violation of the combined symmetry of two of its components (such as CP) must have a corresponding violation in the third component (such as T); in fact, mathematically, these are the same thing. Thus violations in T symmetry are often referred to as CP violations.

The CPT theorem can be generalized to take into account pin groups.