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 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 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, 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:

1. The theory is Lorentz invariant;
2. The vacuum is Lorentz invariant;
3. 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°.

Since a sequence of two CPT reflections is equivalent to a 360° 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.

See also

• Poincaré symmetry and Quantum field theory
• Parity (physics), Charge conjugation and Time reversal symmetry
• CP violation and Kaon

References

• 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 statistics and all that. Benjamin/Cummings. ISBN 0-691-07062-8. 

External links

• http://www.arxiv.org/abs/math-ph/0012006
• http://www.lbl.gov/abc/wallchart/chapters/05/2.html
• Particle data group on CPT
• 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.