191
7.2. The Dirac field and the spin-statistics connection
to the story. Evaluation of [φ ˆ (x 1 ), φ ˆ† (x 2 )] reveals (problem 7.3) that, in the
region (x 1 − x 2 )
2 < 0, the commutator is the difference of two functions (not
field operators), one of which arises from the propagation of a particle from x 2
to x 1 , the other of which comes from the propagation of an antiparticle from
x 1 to x 2 (just as in figure 7.1). Both processes must exist for this difference
to be zero, and furthermore for cancellations between them to occur in the
space-like region the masses of the particle and antiparticle must be identical. In quantum field theory, therefore, ‘causality’ (in the sense of condition
(7.32) – cf (6.82)) requires that every particle has to have a corresponding
antiparticle, with the same mass and opposite quantum numbers. As we saw
in chapter 4, these requirements are guaranteed by the CPT theorem, which
is a consequence of very general principles of quantum field theory.
7.2 The Dirac field and the spin-statistics connection
I remember that when someone had tried to teach me about creation and
annihilation operators, that this operator creates an electron, I said ‘how
do you create an electron? It disagrees with the conservation of charge,’
and in that way I blocked my mind from learning a very practical scheme
of calculation.
—From the lecture delivered by Richard Feynman in Stockholm, Sweden,
on 11 December 1965, when he received the Nobel Prize in physics, which
he shared with Sin-itiro Tomonaga and Julian Schwinger. (Feynman 1966).
We now turn to the problem of setting up a quantum field which, in its
wave aspects, satisfies the Dirac equation (cf comment (5) in section 5.2.5),
and in its ‘particle’ aspects creates or annihilates fermions and antifermions.
Following the ‘Heisenberg–Lagrange–Hamilton’ approach of section 5.2.5, we
begin by writing down the Lagrangian which, via the corresponding Euler–
Lagrange equation, produces the Dirac equation as the ‘field equation’. The
answer (see problem 7.4) is
L D = iψ
† ψ ˙ + iψ
†
α · ∇ψ − mψ
† βψ.
(7.33)
The relativistic invariance of this is more evident in γ-matrix notation (problem 4.3):
¯
L D = ψ(iγ
μ ∂ μ − m)ψ.
(7.34)
We can now attempt to ‘quantize’ the field ψ by making a mode expansion
in terms of plane-wave solutions of the Dirac equation, in a fashion similar to
that for the complex scalar field in (7.16). We obtain (see problem 3.8 for the
definition of the spinors u and v, and the attendant normalization choice)
∫
d
3
k
∑
ψ ˆ =
√
[ˆ c s (k)u(k, s)e
−ik·x + d ˆ † (k)v(k, s)e
ik·x ],
(7.35)
s
(2π) 3 2ω s=1,2
7.2. The Dirac field and the spin-statistics connection
to the story. Evaluation of [φ ˆ (x 1 ), φ ˆ† (x 2 )] reveals (problem 7.3) that, in the
region (x 1 − x 2 )
2 < 0, the commutator is the difference of two functions (not
field operators), one of which arises from the propagation of a particle from x 2
to x 1 , the other of which comes from the propagation of an antiparticle from
x 1 to x 2 (just as in figure 7.1). Both processes must exist for this difference
to be zero, and furthermore for cancellations between them to occur in the
space-like region the masses of the particle and antiparticle must be identical. In quantum field theory, therefore, ‘causality’ (in the sense of condition
(7.32) – cf (6.82)) requires that every particle has to have a corresponding
antiparticle, with the same mass and opposite quantum numbers. As we saw
in chapter 4, these requirements are guaranteed by the CPT theorem, which
is a consequence of very general principles of quantum field theory.
7.2 The Dirac field and the spin-statistics connection
I remember that when someone had tried to teach me about creation and
annihilation operators, that this operator creates an electron, I said ‘how
do you create an electron? It disagrees with the conservation of charge,’
and in that way I blocked my mind from learning a very practical scheme
of calculation.
—From the lecture delivered by Richard Feynman in Stockholm, Sweden,
on 11 December 1965, when he received the Nobel Prize in physics, which
he shared with Sin-itiro Tomonaga and Julian Schwinger. (Feynman 1966).
We now turn to the problem of setting up a quantum field which, in its
wave aspects, satisfies the Dirac equation (cf comment (5) in section 5.2.5),
and in its ‘particle’ aspects creates or annihilates fermions and antifermions.
Following the ‘Heisenberg–Lagrange–Hamilton’ approach of section 5.2.5, we
begin by writing down the Lagrangian which, via the corresponding Euler–
Lagrange equation, produces the Dirac equation as the ‘field equation’. The
answer (see problem 7.4) is
L D = iψ
† ψ ˙ + iψ
†
α · ∇ψ − mψ
† βψ.
(7.33)
The relativistic invariance of this is more evident in γ-matrix notation (problem 4.3):
¯
L D = ψ(iγ
μ ∂ μ − m)ψ.
(7.34)
We can now attempt to ‘quantize’ the field ψ by making a mode expansion
in terms of plane-wave solutions of the Dirac equation, in a fashion similar to
that for the complex scalar field in (7.16). We obtain (see problem 3.8 for the
definition of the spinors u and v, and the attendant normalization choice)
∫
d
3
k
∑
ψ ˆ =
√
[ˆ c s (k)u(k, s)e
−ik·x + d ˆ † (k)v(k, s)e
ik·x ],
(7.35)
s
(2π) 3 2ω s=1,2