194
7. Quantum Field Theory III
One may well wonder why things have to be this way – ‘bosons commute,
fermions anticommute’. To gain further insight, we turn again to a consideration of symmetries and the question of particle and antiparticle – this time
for the Dirac field, rather than the Dirac wavefunction discussed in chapter 4.
The Dirac field ψ ˆ is a complex field, as is reflected in the two distinct mode
operators in the expansion (7.35); as in the complex scalar field case, there
is only one mass parameter and we expect the quanta to be interpretable as
particle and antiparticle. The symmetry operator which distinguishes them is
found by analogy with the complex scalar field case. We note that L ˆ D ( the
quantized version of (7.34)) is invariant under the global U(1) transformation
ψ ˆ → ψ ˆ ′ = e
−iα ψ ˆ
(7.48)
which is
ψ ˆ → ψ ˆ ′ = ψ ˆ − i∈ψ ˆ
(7.49)
in infinitesimal form. The corresponding (Noether) symmetry current can be
calculated as
μ
ˆ
¯ ˆ
N = ψγ
μ ψ ˆ
(7.50)
ψ
and the associated symmetry operator is
∫
ˆ
ψ ˆ † ψ ˆ d
3
N ψ =
x.
(7.51)
ˆ
N ψ is clearly a number operator for the fermion case. As for the complex
scalar field, invariance under a global U(1) phase transformation is associated
with a number conservation law.
Inserting the plane-wave expansion (7.35), we obtain, after some effort
(problem 7.6),
∫ d
3
k ∑ †
N ˆ ψ =
[ˆ c (k)ˆ c s (k) + d ˆ s (k)d ˆ † (k)].
(7.52)
s
s
(2π) 3
s=1,2
Similarly the Dirac Hamiltonian may be shown to have the form (problem 7.6)
∫ d
3
k ∑ †
H ˆ D =
[ˆ c (k)ˆ c s (k) − d ˆ s (k)d ˆ † (k)]ω.
(7.53)
s
s
(2π) 3
s=1,2
It is important to state that in obtaining (7.52) and (7.53), we have not assumed either commutation or anticommutation relations for the mode opera†
tors ˆ
c, ˆ
c , d ˆ and d ˆ† , only properties of the Dirac spinors; in particular, neither
(7.52) nor (7.53) has been normally ordered. Suppose now that we assume
commutation relations, so as to rewrite the last terms in (7.52) and (7.53) in
normally ordered form as d ˆ† (k)d ˆ s (k). We see that H ˆ D will then contain the
s
difference of two number operators for ‘c’ and ‘d’ particles, and is therefore
not positive-definite as we require for a sensible theory. Moreover, we suspect
7. Quantum Field Theory III
One may well wonder why things have to be this way – ‘bosons commute,
fermions anticommute’. To gain further insight, we turn again to a consideration of symmetries and the question of particle and antiparticle – this time
for the Dirac field, rather than the Dirac wavefunction discussed in chapter 4.
The Dirac field ψ ˆ is a complex field, as is reflected in the two distinct mode
operators in the expansion (7.35); as in the complex scalar field case, there
is only one mass parameter and we expect the quanta to be interpretable as
particle and antiparticle. The symmetry operator which distinguishes them is
found by analogy with the complex scalar field case. We note that L ˆ D ( the
quantized version of (7.34)) is invariant under the global U(1) transformation
ψ ˆ → ψ ˆ ′ = e
−iα ψ ˆ
(7.48)
which is
ψ ˆ → ψ ˆ ′ = ψ ˆ − i∈ψ ˆ
(7.49)
in infinitesimal form. The corresponding (Noether) symmetry current can be
calculated as
μ
ˆ
¯ ˆ
N = ψγ
μ ψ ˆ
(7.50)
ψ
and the associated symmetry operator is
∫
ˆ
ψ ˆ † ψ ˆ d
3
N ψ =
x.
(7.51)
ˆ
N ψ is clearly a number operator for the fermion case. As for the complex
scalar field, invariance under a global U(1) phase transformation is associated
with a number conservation law.
Inserting the plane-wave expansion (7.35), we obtain, after some effort
(problem 7.6),
∫ d
3
k ∑ †
N ˆ ψ =
[ˆ c (k)ˆ c s (k) + d ˆ s (k)d ˆ † (k)].
(7.52)
s
s
(2π) 3
s=1,2
Similarly the Dirac Hamiltonian may be shown to have the form (problem 7.6)
∫ d
3
k ∑ †
H ˆ D =
[ˆ c (k)ˆ c s (k) − d ˆ s (k)d ˆ † (k)]ω.
(7.53)
s
s
(2π) 3
s=1,2
It is important to state that in obtaining (7.52) and (7.53), we have not assumed either commutation or anticommutation relations for the mode opera†
tors ˆ
c, ˆ
c , d ˆ and d ˆ† , only properties of the Dirac spinors; in particular, neither
(7.52) nor (7.53) has been normally ordered. Suppose now that we assume
commutation relations, so as to rewrite the last terms in (7.52) and (7.53) in
normally ordered form as d ˆ† (k)d ˆ s (k). We see that H ˆ D will then contain the
s
difference of two number operators for ‘c’ and ‘d’ particles, and is therefore
not positive-definite as we require for a sensible theory. Moreover, we suspect