CPT symmetry
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CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity and time simultaneously.
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[edit] 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.
[edit] Derivation
For a handwaving argument, take a Lorentz boost in a fixed direction, let's call it z. If we complexify the Lorentz group, an imaginary boost with a boost parameter of iπ will result in t going to -t and z going to -z. If we later perform an addition rotation by π in the xy-plane, we get a combination of P and CT. The combination CT appears here instead of T because we are dealing with a unitary transformation, not an antiunitary one. Assuming that the operation of taking a complex boost is valid as a symmetry, we still get a state which is described by the same laws. This gives us the CPT theorem.
[edit] 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), and reversed in time — 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.
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C, P and T Symmetries |
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| C-symmetry | P-symmetry | T-symmetry | L-symmetry | | |
| CP-symmetry | CPT symmetry | |
| pin group | |
[edit] See also
- Poincaré symmetry and Quantum field theory
- Parity (physics), Charge conjugation and Time reversal symmetry
- CP violation and Kaon
[edit] References
- 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.
[edit] 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 propert in these class of theories.
