97
4.2. Discrete transformations: P, C and T
It is also interesting to look at the behaviour of the spinors φ and χ in the
representation (3.40), where they satisfy the equations (4.14) and (4.15). Under parity p → −p, so we can immediately see that φ P = χ and χ P = φ. Thus
the 2-component spinors φ and χ are (in this representation) interchanged under parity.
The analysis leading to (4.68) may be extended to the case of the Dirac
equation (3.102) for a particle of charge q in the field A
μ . As already noted,
A is a polar vector, transforming under like x or ∇; the scalar potential A
0 is
invariant under parity. The combination (−i∇ − qA) therefore changes sign
under parity, and the manipulations following (4.65) proceed as before.
We may introduce a corresponding parity operator P ˆ , which is unitary
and acts on wavefunctions so as to change ψ into ψ P ; then
P ˆ ψ(x, t) = βψ(−x, t) = βP ˆ 0 ψ(x, t),
(4.71)
so that
ˆ βP ˆ 0 .
P =
(4.72)
Applying P ˆ twice, we find
P ˆ 2 ψ(x, t) = ψ(x, t)
(4.73)
which implies that the eigenvalues of P ˆ are ±1.
For example, the positive energy rest-frame spinors ((3.73) with p = 0))
are eigenstates of P ˆ with eigenvalue +1, and the negative energy rest-frame
spinors are eigenstates of P ˆ with eigenvalue −1. Such rest-frame eigenvalues
of P ˆ are called intrinsic parities. The correspondence between negative energy
solutions and antiparticles, discussed in the preceding section, then suggests
that a fermion and its antiparticle have opposite intrinsic parity (note that
the parity eigenvalue is multiplicative). We shall be able to derive this result
after quantization of the Dirac field, in chapter 7.
As usual in quantum mechanics, we may consider the action of P ˆ on operators as well as wavefunctions. In particular, the parity transform of a Dirac
Hamiltonian H ˆ (x) will be
†
P ˆ H ˆ (x)P ˆ † = βP ˆ 0 H ˆ (x)P ˆ β.
(4.74)
0
If the Hamiltonian is invariant under parity, the right hand side of (4.74) will
equal H ˆ and the operator P ˆ will commute with H ˆ ; the eigenvalue of P ˆ will
then be conserved. The reader may easily check that the Hamiltonian for the
†
charged particle in a field A
μ is parity invariant, using P ˆ 0 AP ˆ = −A.
0
With the rule (4.68) in hand, we can examine how various bilinear covariants, such as ¯
ψγ
μ ψ, transform under parity. For example,
ψψ or ¯
¯
′
′
ψ P (x , t)ψ P (x , t) = ψ
† (x, t)βββψ(x, t) = ψ ¯ (x, t)ψ(x, t),
(4.75)
showing that ¯
ψψ is a scalar. Similarly, for a 4-vector
¯
v
μ (x, t) = (v
0 (x, t), v(x, t)) = ψ(x, t)γ
μ ψ(x, t),
(4.76)
4.2. Discrete transformations: P, C and T
It is also interesting to look at the behaviour of the spinors φ and χ in the
representation (3.40), where they satisfy the equations (4.14) and (4.15). Under parity p → −p, so we can immediately see that φ P = χ and χ P = φ. Thus
the 2-component spinors φ and χ are (in this representation) interchanged under parity.
The analysis leading to (4.68) may be extended to the case of the Dirac
equation (3.102) for a particle of charge q in the field A
μ . As already noted,
A is a polar vector, transforming under like x or ∇; the scalar potential A
0 is
invariant under parity. The combination (−i∇ − qA) therefore changes sign
under parity, and the manipulations following (4.65) proceed as before.
We may introduce a corresponding parity operator P ˆ , which is unitary
and acts on wavefunctions so as to change ψ into ψ P ; then
P ˆ ψ(x, t) = βψ(−x, t) = βP ˆ 0 ψ(x, t),
(4.71)
so that
ˆ βP ˆ 0 .
P =
(4.72)
Applying P ˆ twice, we find
P ˆ 2 ψ(x, t) = ψ(x, t)
(4.73)
which implies that the eigenvalues of P ˆ are ±1.
For example, the positive energy rest-frame spinors ((3.73) with p = 0))
are eigenstates of P ˆ with eigenvalue +1, and the negative energy rest-frame
spinors are eigenstates of P ˆ with eigenvalue −1. Such rest-frame eigenvalues
of P ˆ are called intrinsic parities. The correspondence between negative energy
solutions and antiparticles, discussed in the preceding section, then suggests
that a fermion and its antiparticle have opposite intrinsic parity (note that
the parity eigenvalue is multiplicative). We shall be able to derive this result
after quantization of the Dirac field, in chapter 7.
As usual in quantum mechanics, we may consider the action of P ˆ on operators as well as wavefunctions. In particular, the parity transform of a Dirac
Hamiltonian H ˆ (x) will be
†
P ˆ H ˆ (x)P ˆ † = βP ˆ 0 H ˆ (x)P ˆ β.
(4.74)
0
If the Hamiltonian is invariant under parity, the right hand side of (4.74) will
equal H ˆ and the operator P ˆ will commute with H ˆ ; the eigenvalue of P ˆ will
then be conserved. The reader may easily check that the Hamiltonian for the
†
charged particle in a field A
μ is parity invariant, using P ˆ 0 AP ˆ = −A.
0
With the rule (4.68) in hand, we can examine how various bilinear covariants, such as ¯
ψγ
μ ψ, transform under parity. For example,
ψψ or ¯
¯
′
′
ψ P (x , t)ψ P (x , t) = ψ
† (x, t)βββψ(x, t) = ψ ¯ (x, t)ψ(x, t),
(4.75)
showing that ¯
ψψ is a scalar. Similarly, for a 4-vector
¯
v
μ (x, t) = (v
0 (x, t), v(x, t)) = ψ(x, t)γ
μ ψ(x, t),
(4.76)