CPT symmetry

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Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T that is observed to be an exact symmetry of nature at the fundamental level.[1] The CPT theorem says that CPT symmetry holds for all physical phenomena, or more precisely, 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.[2] In 1954, Gerhart Lüders and Wolfgang Pauli derived more explicit proofs,[3][4] 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.[5] These proofs are based on the principle 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.

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.

Derivation of the CPT theorem[edit]

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 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[edit]

A consequence of this derivation is that a violation of CPT automatically indicates a Lorentz violation.[citation needed]

The implication of CPT symmetry is that a "mirror-image" of our universe — with all objects having their positions reflected by an arbitrary 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 published an apparent proof that CPT violation implies the breaking of Lorentz symmetry.[6] If correct, this would imply that any study of CPT violation also includes Lorentz violation. However, Chaichian et al later disputed the validity of Greenberg's result.[7] Greenberg replied that the model used in their paper meant that their "proposed objection was not relevant to my result".[8]

The overwhelming majority of experimental searches for Lorentz violation have yielded negative results. A detailed tabulation of these results was given in 2011 by Kostelecky and Russell.[9]

See also[edit]


  1. ^ Kostelecký, V. A. (1998). "The Status of CPT". arXiv:hep-ph/9810365Freely accessible [hep-ph]. 
  2. ^ Schwinger, Julian (1951). The Theory of Quantized Fields I. Physical Review. 82. p. 914. doi:10.1103/PhysRev.82.914. 
  3. ^ Lüders, G. (1954). "On the Equivalence of Invariance under Time Reversal and under Particle-Antiparticle Conjugation for Relativistic Field Theories". Kongelige Danske Videnskabernes Selskab, Matematisk-Fysiske Meddelelser. 28 (5): 1–17. 
  4. ^ Pauli, W.; Rosenfelf, L.; Weisskopf, V., eds. (1955). Niels Bohr and the Development of Physics. McGraw-Hill. LCCN 56040984. 
  5. ^ Bell, J. S. (1954). (Thesis). Birmingham University.  Missing or empty |title= (help)
  6. ^ Greenberg, O. W. (2002). "CPT Violation Implies Violation of Lorentz Invariance". Physical Review Letters. 89 (23): 231602. Bibcode:2002PhRvL..89w1602G. PMID 12484997. arXiv:hep-ph/0201258Freely accessible. doi:10.1103/PhysRevLett.89.231602. 
  7. ^ Chaichian, M.; Dolgov, A. D.; Novikov, V. A.; Tureanu, A. (2011). "CPT Violation Does Not Lead to Violation of Lorentz Invariance and Vice Versa". Physics Letters B. 699 (3): 177–180. Bibcode:2011PhLB..699..177C. arXiv:1103.0168Freely accessible. doi:10.1016/j.physletb.2011.03.026. 
  8. ^ Greenberg, O. W. (4 May 2011). "Remarks on a challenge to the relation between CPT and Lorentz violation". Bibcode:2011arXiv1105.0927G. arXiv:1105.0927Freely accessible. The objection [arXiv:1103.0168] to my theorem [arXiv:hep-ph/0201258] that violation of CPT symmetry implies violation of Lorentz covariance is based on a nonlocal model in which time-ordered products are not well defined. I used covariance of time-ordered products as the condition for Lorentz covariance; therefore the proposed objection is not relevant to my result. 
  9. ^ Kostelecký, V. A.; Russell, N. (2011). "Data tables for Lorentz and CPT violation". Reviews of Modern Physics. 83 (1): 11–31. Bibcode:2011RvMP...83...11K. arXiv:0801.0287Freely accessible. doi:10.1103/RevModPhys.83.11. 


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