# CPT Theorem

## CPT theorem

A fundamental ingredient in quantum field theories, which dictates that all interactions in nature, all the force laws, are unchanged (invariant) on being subjected to the combined operations of particle-antiparticle interchange (socalled charge conjugation, *C*), reflection of the coordinate system through the origin (parity, *P*), and reversal of time, *T*. In other words, the *CPT* operator commutes with the hamiltonian. The operations may be performed in any order; *TCP*, *TPC*, and so forth, are entirely equivalent. If an interaction is not invariant under any one of the operations, its effect must be compensated by the other two, either singly or combined, in order to satisfy the requirements of the theorem. *See* Quantum field theory

The *CPT* theorem appears implicitly in work by J. Schwinger in 1951 to prove the connection between spin and statistics. Subsequently, G. Lüders and W. Pauli derived more explicit proofs, and it is sometimes known as the Lüders-Pauli theorem. The proof is based on little more than the validity of special relativity and local interactions of the fields. The theorem is intrinsic in the structure of all the successful field theories. *See* Quantum statistics, Relativity, Spin (quantum mechanics)

*CPT* assumed paramount importance in 1957, with the discovery that the weak interactions were not invariant under the parity operation. Almost immediately afterward, it was found that the failure of *P* was attended by a compensating failure of *C* invariance. Initially, it appeared that *CP* invariance was preserved and, with the application of the *CPT* theorem, invariance under time reversal. Then, in 1964 an unmistakable violation of *CP* was discovered in the system of neutral *K* mesons. *See* Parity (quantum mechanics)

One question immediately posed by the failure of parity and charge conjugation invariance is why, as one example, the π^{+} and π^{-} mesons, which decay through the weak interactions, have the same lifetime and the same mass. It turns out that the equality of particle-antiparticle masses and lifetimes is a consequence of *CPT* invariance and not *C* invariance alone. *See* Elementary particle, Meson, Symmetry laws (physics)

## CPT Theorem

a theorem of quantum field theory stating that the equations of the theory are invariant under the *CPT* transformation—that is, they do not change form if the three transformations of charge conjugation (*C*), space inversion (*P*), and time reversal (*T*) are performed simultaneously. In charge conjugation, particles are replaced by the corresponding antiparticles. Space inversion, or reflection, involves the replacement of the space coordinates *r* by – *r*. Time reversal means the replacement of the time coordinate *t* by *—t*.

The *CPT* theorem was formulated and proved by the German physicist G. Lüders between 1952 and 1954 and by the Swiss physicist W. Pauli in 1955. The theorem follows from the fundamental principles of quantum field theory. If some process occurs in nature, the *CPT* theorem implies that there can also occur in nature, with the same probability, the process obtained by replacing the particles in the original process with the corresponding antiparticles, by changing the signs of the projections of the particles’ spins, and by interchanging the initial and final states of the original process.

It follows, in particular, from the *CPT* theorem that (1) a particle and the corresponding antiparticle have equal masses and equal lifetimes, (2) the electric charges of a particle and the corresponding antiparticle differ only in sign, as do the magnetic moments, (3) the interaction of a particle and of the antiparticle with a gravitational field is identical (that is, there is no “antigravitation”), and (4) in those cases where the interaction of particles in the final state is negligible, the energy spectra and angular distributions of the decay products are the same for particles and anti-particles, and the projections of the spins are of opposite sign.

Not a single instance of violation of the *CPT* theorem has been observed. The equality of the masses of a particle and the corresponding antiparticle has been tested for the kaons K^{0} and K̅^{0} with an accuracy of approximately 10^{–15}, which exceeds by ten orders of magnitude the best accuracy achieved for the masses of other particles: ~10^{–5} for the electron (e^{–}) and positron (e^{+}), ~10^{–4} for the muons µ^{–} and µ^{+}, and ~10^{–3} for the kaons K^{–} and K^{+}. The equality of the lifetimes of particles and antiparticles has been verified with an accuracy not exceeding 10^{–3}. The equality of the magnetic moments has been tested with an accuracy of ~10^{–6} for µ^{–} and µ^{+} and ~10^{–5} for e^{–} and e^{+}. The accuracy of the comparison of spectra and polarization in the decays of particles and antiparticles apparently does not exceed 10^{–2}.

Until 1956 the laws of nature were believed to be symmetric under each of the *C*, *P*, and *T* transformations performed separately. The discovery in 1956 of violations of *P* and *C* invariance and the discovery in 1964 of a violation of *CP* invariance (*see*COMBINED INVERSION) had little effect on the theoretical apparatus of physics, which was able to make allowance for these discoveries in a natural manner, without violating the fundamental principles of the theory. In contrast to the violation of *P, C,* and *CP* invariance, the detection of a violation of *CPT* invariance would require changes in the foundations of quantum field theory. The violation of the *CPT* theorem would “break” the connection between particles and antiparticles. Within the framework of traditional quantum field theory, the bases of the *CPT* theorem—such as relativistic invariance, the localization of interactions, and the relation between spin and statistics—are such that it cannot yet be seen how even one of them could be sacrificed without radically altering the entire theory. The same statement can be made with regard to axiomatic quantum field theory. The experimental search for effects reflecting a violation of *CPT* invariance is thus all the more interesting.

### REFERENCES

Lapidus, L. I. “Sledstviia*CPT*-invariantnosti i eksperiment.”

*Uspekhi fizicheskikh nauk,*1968, vol. 95, issue 4.

Fainberg, V. la. “Teoreticheskie osnovy

*CPT*-invariantnosti.”

*Uspekhi fizicheskikh nauk,*1968, vol. 95, issue 3.

L. B. OKUN’

## CPT theorem

[¦sē¦pē¦tē ′thir·əm]*C,*space inversion

*P,*and time reversal

*T*.