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Efforts during the late 1950s revealed the violation of P-symmetry by phenomena that involve the weak force, and there were well-known violations of C-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, which implied, by CPT invariance, violations of T-symmetry as well. The CPT theorem requires the preservation of CPT symmetry by all physical phenomena. It assumes 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, Gerhart Lüders and Wolfgang Pauli derived more explicit proofs, so this theorem is sometimes known as the Lüders-Pauli theorem. At about the same time, and independently, this 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.
Consider a Lorentz boost in a fixed direction z. This can be interpreted as a rotation of the time axis into the z axis, with an imaginary rotation parameter. If this rotation parameter were real, it would be possible for a 180° rotation to reverse the direction of time and of z. Reversing the direction of one axis is a reflection of space in any number of dimensions. If space has 3 dimensions, it is equivalent to reflecting all the coordinates, because an additional rotation of 180° in the x-y plane could be included.
This defines a CPT transformation if we adopt the Feynman-Stueckelberg interpretation of antiparticles as the corresponding particles traveling backwards in time. This interpretation requires a slight analytic continuation, which is well-defined only under the following assumptions:
- The theory is Lorentz invariant;
- The vacuum is Lorentz invariant;
- The energy is bounded below.
When the above hold, quantum theory can be extended to a Euclidean theory, defined by translating all the operators to imaginary time using the Hamiltonian. The commutation relations of the Hamiltonian, and the Lorentz generators, guarantee that Lorentz invariance implies rotational invariance, so that any state can be rotated by 180 degrees.
Since a sequence of two CPT reflections is equivalent to a 360-degree rotation, fermions change by a sign under two CPT reflections, while bosons do not. This fact 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 their positions reflected by an imaginary plane (corresponding to a parity inversion), all momenta reversed (corresponding to a time inversion) and with all matter replaced by antimatter (corresponding to a charge inversion)— would evolve under exactly our physical laws. The CPT transformation turns our universe into its "mirror image" and vice versa. 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.
In 2002 Oscar Greenberg proved that CPT violation implies the breaking of Lorentz symmetry. This implies that any study of CPT violation includes also Lorentz violation. The overwhelming majority of experimental searches for Lorentz violation have yielded negative results. A detailed tabulation of these results is given by Kostelecky and Russell.
See also 
- Poincaré symmetry and Quantum field theory
- Parity (physics), Charge conjugation and Time reversal symmetry
- CP violation and kaon
- Greenberg, O.W. (2002). "CPT Violation Implies Violation of Lorentz Invariance". Physical Review Letters 89 (23): 231602. arXiv:hep-ph/0201258. Bibcode:2002PhRvL..89w1602G. doi:10.1103/PhysRevLett.89.231602.
- Kostelecky, V.A.; Russell, N. (2011). "Data tables for Lorentz and CPT violation". Reviews of Modern Physics 83 (1): 11–31. arXiv:0801.0287. Bibcode:2011RvMP...83...11K. doi:10.1103/RevModPhys.83.11.
- Sozzi, M.S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.
- Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 0-471-60386-4.
- R. F. Streater and A. S. Wightman (1964). PCT, spin and statistics, and all that. Benjamin/Cummings. ISBN 0-691-07062-8.
- Background information on Lorentz and CPT violation by Alan Kostelecký at Theoretical Physics Indiana University
- Data Tables for Lorentz and CPT Violation at the arXiv
- The Pin Groups in Physics: C, P, and T at the arXiv
- Charge, Parity, and Time Reversal (CPT) Symmetry at LBL
- CPT Invariance Tests in Neutral Kaon Decay at LBL
- Space--Time Symmetry, CPT and Mirror Fermions at the arXiv
8-component theory for fermions in which T-parity can be a complex number with unit radius. The CPT invariance is not a theorem but a better to have property in these class of theories.