108
4. Lorentz Transformations and Discrete Symmetries
odd amplitudes under T. An interesting example is provided by the amplitude
−id e ψ ¯ (x)σ
μν γ 5 ψ(x)F μν ,
(4.129)
where
σ
μν =
i (γ
μ γ
ν
− γ
ν γ
μ )
(4.130)
2
and where F μν is an external electric field with non-vanishing components
F 0i = E
i . In the representation (3.31),
(
)
σ i 0
σ
0i γ 5 = i
≡ iΣ i ,
(4.131)
0 σ i
and (4.129) reduces to
¯
d e ψ(x)Σψ(x) · E.
(4.132)
Problem 4.5 shows that the quantity (4.132) is odd under T, and it is easy
to check that it is also odd under P. A non-zero value of such a term would
correspond to an electric dipole moment for a spin-1/2 particle (compare the
¯
analogous quantity d m ψ(x)Σψ(x) · B for the magnetic dipole moment, which
is even under P and T). Experiment places very strong limits on possible
electric dipole moments (Nakamura et al. 2010) for the neutron, proton and
electron:
d n < 0.29 × 10
−25 e cm
(4.133)
d p < 0.54 × 10
−23 e cm
(4.134)
d e = (0.069 ± 0.074) × 10
−26 e cm
(4.135)
Although these numbers seem tiny, calculations of the d n in the Standard
Model produce a result some 6 or 7 orders of magnitude smaller than (4.133).
However, these experimental limits impose strong constraints on theories
which go beyond the Standard Model, and which may typically contain the
possibility of larger T and CP violating effects.
4.2.5 CPT
We denote the product CPT by θ, and the corresponding operator by θ ˆ . As
already mentioned, for any conventional quantum field theory, and certainly
for the Standard Model, the transformation θ is an invariance of the theory.
One immediate consequence of this invariance is the equality of particle and
antiparticle masses. This is easily demonstrated. Let |X, s z > be the state of
a particle X at rest with z-component of spin equal to s z . The mass of X is
given by the expectation value
M X =< X, s z |H ˆ |X, s z >,
(4.136)
where H ˆ is the total Hamiltonian. Clearly M X is real, and independent of
s z . Now the operator θ ˆ involves T ˆ , and therefore we must be careful to use
4. Lorentz Transformations and Discrete Symmetries
odd amplitudes under T. An interesting example is provided by the amplitude
−id e ψ ¯ (x)σ
μν γ 5 ψ(x)F μν ,
(4.129)
where
σ
μν =
i (γ
μ γ
ν
− γ
ν γ
μ )
(4.130)
2
and where F μν is an external electric field with non-vanishing components
F 0i = E
i . In the representation (3.31),
(
)
σ i 0
σ
0i γ 5 = i
≡ iΣ i ,
(4.131)
0 σ i
and (4.129) reduces to
¯
d e ψ(x)Σψ(x) · E.
(4.132)
Problem 4.5 shows that the quantity (4.132) is odd under T, and it is easy
to check that it is also odd under P. A non-zero value of such a term would
correspond to an electric dipole moment for a spin-1/2 particle (compare the
¯
analogous quantity d m ψ(x)Σψ(x) · B for the magnetic dipole moment, which
is even under P and T). Experiment places very strong limits on possible
electric dipole moments (Nakamura et al. 2010) for the neutron, proton and
electron:
d n < 0.29 × 10
−25 e cm
(4.133)
d p < 0.54 × 10
−23 e cm
(4.134)
d e = (0.069 ± 0.074) × 10
−26 e cm
(4.135)
Although these numbers seem tiny, calculations of the d n in the Standard
Model produce a result some 6 or 7 orders of magnitude smaller than (4.133).
However, these experimental limits impose strong constraints on theories
which go beyond the Standard Model, and which may typically contain the
possibility of larger T and CP violating effects.
4.2.5 CPT
We denote the product CPT by θ, and the corresponding operator by θ ˆ . As
already mentioned, for any conventional quantum field theory, and certainly
for the Standard Model, the transformation θ is an invariance of the theory.
One immediate consequence of this invariance is the equality of particle and
antiparticle masses. This is easily demonstrated. Let |X, s z > be the state of
a particle X at rest with z-component of spin equal to s z . The mass of X is
given by the expectation value
M X =< X, s z |H ˆ |X, s z >,
(4.136)
where H ˆ is the total Hamiltonian. Clearly M X is real, and independent of
s z . Now the operator θ ˆ involves T ˆ , and therefore we must be careful to use