213
7.5. P, C and T in quantum field theory
corresponds physically to fermion and antifermion, spin up and down, but
the Majorana fermion is the same as its antiparticle. The free field expansion
corresponding to (7.35) for a Majorana field is
∫
d
3
k
∑
†
ψ ˆ M (x) =
√
[ˆ c λ (k)u(k, λ)e
−ik·x + ˆ
c (k)v(k, λ)e
ik·x ]. (7.158)
λ
(2π) 3 2ω λ=1,2
¯ ˆ
The Lagrangian for a free Majorana field may be taken to be ψ M (i∂ / −
m)ψ ˆ M , which the reader can rewrite in terms of ˆ
χ. For example, the mass
term is
¯
−mψ ˆ
M ψ ˆ M = −mχ ˆ
T iσ 2 χ ˆ + Hermitian conjugate.
(7.159)
We note that this expression will vanish unless the components ˆ
χ 1 and ˆ
χ 2
anticommute with each other.
7.5.3 Time reversal
In section 4.2.4 we found that the time reversal transformation for the single
particle theories was not represented by a unitary operator, but rather by the
product of a unitary operator and the complex conjugation operator. We can
see that the same must be true in quantum field theory by considering the
equation of motion (6.18) for a scalar field (for simplicity), in the interaction
picture:
∂φ ˆ (x, t) = i[ H ˆ 0 , φ ˆ (x, t)].
(7.160)
∂t
Suppose the field φ ˆ T in the time reversed frame were related to φ ˆ by a unitary quantum field operator U ˆ T so that (suppressing the spatial argument)
†
†
U ˆ T φ ˆ (t)U ˆ = φ ˆ T (t
′ ). Then applying U ˆ T . . . U ˆ T to equation (7.160) we would
T
obtain
∂φ ˆ T (t
′ )
†
= i[U ˆ T H ˆ 0 U ˆ T , φ ˆ T (t
′ )]
(7.161)
∂t
or equivalently
∂φ ˆ T (t
′ )
†
= −i[U ˆ T H ˆ 0 U ˆ T , φ ˆ T (t
′ )].
(7.162)
∂t ′
To restore (7.162) to the form (7.160) – i.e. for covariance to hold – would
require that U ˆ T transforms H ˆ 0 to −H ˆ 0 . But this is unacceptable on physical
grounds, because the eigenvalues of H ˆ 0 must be positive relative to the vacuum, both before and after the transformation. We must therefore write the
transformation as
ˆ
ˆ
T = U T K
(7.163)
where, as in section 4.2.4, K takes the complex conjugate of ordinary numbers
and functions (i.e. it replaces i by -i). The operator U ˆ T depends on the field
involved, but we shall not need to exhibit it explicitly.
7.5. P, C and T in quantum field theory
corresponds physically to fermion and antifermion, spin up and down, but
the Majorana fermion is the same as its antiparticle. The free field expansion
corresponding to (7.35) for a Majorana field is
∫
d
3
k
∑
†
ψ ˆ M (x) =
√
[ˆ c λ (k)u(k, λ)e
−ik·x + ˆ
c (k)v(k, λ)e
ik·x ]. (7.158)
λ
(2π) 3 2ω λ=1,2
¯ ˆ
The Lagrangian for a free Majorana field may be taken to be ψ M (i∂ / −
m)ψ ˆ M , which the reader can rewrite in terms of ˆ
χ. For example, the mass
term is
¯
−mψ ˆ
M ψ ˆ M = −mχ ˆ
T iσ 2 χ ˆ + Hermitian conjugate.
(7.159)
We note that this expression will vanish unless the components ˆ
χ 1 and ˆ
χ 2
anticommute with each other.
7.5.3 Time reversal
In section 4.2.4 we found that the time reversal transformation for the single
particle theories was not represented by a unitary operator, but rather by the
product of a unitary operator and the complex conjugation operator. We can
see that the same must be true in quantum field theory by considering the
equation of motion (6.18) for a scalar field (for simplicity), in the interaction
picture:
∂φ ˆ (x, t) = i[ H ˆ 0 , φ ˆ (x, t)].
(7.160)
∂t
Suppose the field φ ˆ T in the time reversed frame were related to φ ˆ by a unitary quantum field operator U ˆ T so that (suppressing the spatial argument)
†
†
U ˆ T φ ˆ (t)U ˆ = φ ˆ T (t
′ ). Then applying U ˆ T . . . U ˆ T to equation (7.160) we would
T
obtain
∂φ ˆ T (t
′ )
†
= i[U ˆ T H ˆ 0 U ˆ T , φ ˆ T (t
′ )]
(7.161)
∂t
or equivalently
∂φ ˆ T (t
′ )
†
= −i[U ˆ T H ˆ 0 U ˆ T , φ ˆ T (t
′ )].
(7.162)
∂t ′
To restore (7.162) to the form (7.160) – i.e. for covariance to hold – would
require that U ˆ T transforms H ˆ 0 to −H ˆ 0 . But this is unacceptable on physical
grounds, because the eigenvalues of H ˆ 0 must be positive relative to the vacuum, both before and after the transformation. We must therefore write the
transformation as
ˆ
ˆ
T = U T K
(7.163)
where, as in section 4.2.4, K takes the complex conjugate of ordinary numbers
and functions (i.e. it replaces i by -i). The operator U ˆ T depends on the field
involved, but we shall not need to exhibit it explicitly.