CPT Invariance

Many of the profound ideas in nature manifest themselves as symmetries. A symmetry in a physical experiment suggests that something is conserved, or remains constant, during the experiment. So conservation laws and symmetries are strongly linked.

Three of the symmetries which usually, but not always, hold are those of charge conjugation (C), parity (P), and time reversal (T):

• Charge conjugation(C): reversing the electric charge and all the internal quantum numbers.
  电荷共轭(C)：反转电荷和所有内部量子数。
• Parity (P): space inversion; reversal of the space coordinates, but not the time.
  宇称（P）：空间反演；反转变量空间坐标，但不反转变量时间。
• Time reversal (T): replacing t by -t. This reverses time derivatives like momentum and angular momentum.
  时间反演（T）：将时间t替换为-t。这会反转像动量和角动量这样的时间导数。

Examples in nature can be cited for the violation of each of these symmetries individually. It was thought for a time that CP (parity transformation plus charge conjugation) would always leave a system invariant, but the notable example of the neutral kaons has shown a slight violation of CP symmetry.

We are left with the combination of all three, CPT, a profound symmetry consistent with all known experimental observations.

On the theoretical side, CPT invariance has received a great deal of attention. Georg Ludens, Wolfgang Pauli and Julian Schwinger independently showed that invariance under Lorentz transformations implies CPT invariance. CPT invariance itself has implications which are at the heart of our understanding of nature and which do not easily arise from other types of considerations.

• Integer spin particles obey Bose-Einstein statistics and half-integer spin particles obey Fermi-Dirac statistics. Operators with integer spins must be quantized using commutation relations, while anticommutation relations must be used for operators with half integer spin.
  整数自旋粒子服从玻色-爱因斯坦统计，半整数自旋粒子服从费米-狄拉克统计。整数自旋算符必须用对易关系进行量化，而半整数自旋算符则必须用反对易关系进行量化。
• Particles and antiparticles have identical masses and lifetimes. This arises from CPT invariance of physical theories.
  粒子和反粒子具有相同的质量和寿命。这源于物理理论的CPT不变性。
• All the internal quantum numbers of antiparticles are opposite to those of the particles.
  所有反粒子的内部量子数都与粒子的相反。

Charge Conjugation

Associated with the conservation laws which govern the behavior of physical particles, charge conjugation (C), parity (P) and time reversal (T) combine to constitute a fundamental symmetry called CPT invariance.

Classically, charge conjugation may seem like a simple idea: just replace positive charges by negative charges and vice versa. Since electric and magnetic fields have their origins in charges, you also must reverse these fields.

In quantum mechanical systems, charge conjugation has some further implications. It also involves reversing all the internal quantum numbers like those for lepton number, baryon number and strangeness. It does not affect mass, energy, momentum or spin.

Thinking of charge conjugation as an operator, C, then electromagnetic processes are invariant under the C operation since Maxwell's equations are invariant under C. This restricts some kinds of particle processes. Das and Ferbel proceed by defining a charge parity of η_C(γ) = -1 for a photon since the C operation reverses the electric field. This constrains the electromagnetic decay of a neutral particle like the π⁰. The decay of the π⁰ is:

This implies that the charge parity or behavior under charge conjugation for a π⁰ is:

Charge conjugation symmetry would imply that the π⁰ will not decay by

which we already know because it can't conserve momentum, but the decay

can conserve momentum. This decay cannot happen because it would violate charge conjugation symmetry.

While the strong and electromagnetic interactions obey charge conjugation symmetry, the weak interaction does not. As an example, neutrinos are found to have intrinsic parities: neutrinos have left-handed parity and antineutrinos right-handed. Since charge conjugation would leave the spatial coordinates untouched, then if you operated on a neutrino with the C operator, you would produce a left-handed antineutrino. But there is no experimental evidence for such a particle; all antineutrinos appear to be right-handed. The combination of the parity operation P and the charge conjugation operation C on a neutrino do produce a right-handed antineutrino, in accordance with observation. So it appears that while beta decay does not obey parity or charge conjugation symmetry separately, it is invariant under the combination CP.

Time Reversal

Associated with the conservation laws which govern the behavior of physical particles, charge conjugation (C), parity (P) and time reversal combine to constitute a fundamental symmetry called CPT invariance.

In simple classical terms, time reversal just means replacing t by -t, inverting the direction of the flow of time. Reversing time also reverses the time derivatives of spatial quantities, so it reverses momentum and angular momentum.Newton's second law is quadratic in time and is invariant under time reversal. It's invariance under time reversal holds for either gravitational or electromagnetic forces.

Very sensitive experimental tests have been done to put upper bounds on any violation of time-reversal symmetry. One experiment described by Das and Ferbel is the search for a dipole moment for the neutron. Even though the neutron is neutral, it is viewed as made up of charged quarks and therefore could conceivably have a dipole moment. Experimental evidence is consistent with zero dipole moment, so time reversal symmetry seems to hold in this case.

CP Invariance

Associated with the conservation laws which govern the behavior of physical particles, charge conjugation (C), parity (P) and time reversal (T) combine to constitute a fundamental symmetry called CPT invariance.

The strong and electromagnetic interactions leave systems invariant under any of the three operations applied alone, but the weak interaction does not. The beta decay of cobalt-60 established the violation of parity in 1957, and led to our understanding that the weak interaction violates both charge conjugation and parity invariance. However, the weak interaction does appear to leave systems invariant under the combination CP. Examination of the case of the neutrino is instructive at this point. The parity operation on a neutrino would leave its spin in the same direction while reversing space coordinates.

Neither of these things is observed to happen in nature; neutrinos are always left-handed, anti-neutrinos always right-handed. But if you add the charge conjugation operation, the result of the combined operation gives you back the original particle.

CP invariance was thought to be a general conservation principle until the details of the neutral kaon decay process were examined by Cronin and Fitch. After intense study over many years, the consensus is that CP is violated by a small amount. In 2001 CP violation was confirmed in B-meson decay. It is thought possible by some investigators that in CP violation is to be found the reason for the vast excess of matter over antimatter in the universe.