210
7. Quantum Field Theory III
The application of the formalism of chapter 6 is not completely straightforward in this scalar case. The problem is that L ˆ int of (7.139) involves derivatives of the fields and, in particular, their time derivatives. Hence the canonical momenta will be changed from their non-interacting forms. This, in turn,
implies that the additional (interaction) term in the Hamiltonian is not just
−L ˆ int , as in the Dirac case, but is given by (problem 7.15)
ˆ ′
− ˆ
2 (A ˆ 0 )
2 φ ˆ † ˆ
H S = L int − q
φ.
(7.142)
The problem here is that the Hamiltonian and −L ˆ int differ by a term which is
non-covariant (only A ˆ 0 appears).This seems to threaten the whole approach
of chapter 6. Fortunately, another subtlety rescues the situation. There is
a second source of non-covariance arising from the time-ordering of terms
involving time derivatives, which will occur when (7.142) is used in the Dyson
series (6.42). In particular, one can show (problem 7.16) that
ˆ
ˆ
<0|T (∂ 1μ φ(x 1 )∂ 2ν φ
† (x 2 ))|0>
= ∂ 1μ ∂ 2ν <0|T (φ ˆ (x 1 )φ ˆ † (x 2 ))|0> − ig μ0 g ν0 δ
4 (x 1 − x 2 ) (7.143)
which also exhibits a non-covariant piece. A careful analysis (Itzykson and
Zuber 1980, section 6.1.4) shows that the two covariant effects exactly com′
pensate, so that in the Dyson series we may use H ˆ = −L ˆ int after all. The
S
Feynman rules for charged scalar electrodynamics are given in appendix L.
7.5 P, C and T in quantum field theory
We end this chapter by completing the discussion of the discrete symmetries
which we began in section 4.2, extending it from the single particle (wavefunction) theory to quantum fields. We begin with the parity transformation.
7.5.1 Parity
The algebraic manipulations of section 4.2.1 apply equally well to the equations of motion for the quantum field, and we can take over the results by
replacing a transformed wavefunction such as ψ P (x, t) by the corresponding
transformed field ψ ˆ P (x, t) = P ˆ ψ ˆ (x, t)P ˆ −1 where P ˆ is a unitary quantum field
operator (which we shall not need to calculate explicitly). Thus we have
ˆ
ˆ
φ P (x, t) = φ(−x, t)
(7.144)
ˆ
β ˆ
ψ P (x, t) =
ψ(−x, t),
(7.145)
for the KG and Dirac fields, and
A ˆ P (x, t) = −A ˆ (−x, t),
A ˆ 0
P (x, t) = A ˆ0 (−x, t)
(7.146)
7. Quantum Field Theory III
The application of the formalism of chapter 6 is not completely straightforward in this scalar case. The problem is that L ˆ int of (7.139) involves derivatives of the fields and, in particular, their time derivatives. Hence the canonical momenta will be changed from their non-interacting forms. This, in turn,
implies that the additional (interaction) term in the Hamiltonian is not just
−L ˆ int , as in the Dirac case, but is given by (problem 7.15)
ˆ ′
− ˆ
2 (A ˆ 0 )
2 φ ˆ † ˆ
H S = L int − q
φ.
(7.142)
The problem here is that the Hamiltonian and −L ˆ int differ by a term which is
non-covariant (only A ˆ 0 appears).This seems to threaten the whole approach
of chapter 6. Fortunately, another subtlety rescues the situation. There is
a second source of non-covariance arising from the time-ordering of terms
involving time derivatives, which will occur when (7.142) is used in the Dyson
series (6.42). In particular, one can show (problem 7.16) that
ˆ
ˆ
<0|T (∂ 1μ φ(x 1 )∂ 2ν φ
† (x 2 ))|0>
= ∂ 1μ ∂ 2ν <0|T (φ ˆ (x 1 )φ ˆ † (x 2 ))|0> − ig μ0 g ν0 δ
4 (x 1 − x 2 ) (7.143)
which also exhibits a non-covariant piece. A careful analysis (Itzykson and
Zuber 1980, section 6.1.4) shows that the two covariant effects exactly com′
pensate, so that in the Dyson series we may use H ˆ = −L ˆ int after all. The
S
Feynman rules for charged scalar electrodynamics are given in appendix L.
7.5 P, C and T in quantum field theory
We end this chapter by completing the discussion of the discrete symmetries
which we began in section 4.2, extending it from the single particle (wavefunction) theory to quantum fields. We begin with the parity transformation.
7.5.1 Parity
The algebraic manipulations of section 4.2.1 apply equally well to the equations of motion for the quantum field, and we can take over the results by
replacing a transformed wavefunction such as ψ P (x, t) by the corresponding
transformed field ψ ˆ P (x, t) = P ˆ ψ ˆ (x, t)P ˆ −1 where P ˆ is a unitary quantum field
operator (which we shall not need to calculate explicitly). Thus we have
ˆ
ˆ
φ P (x, t) = φ(−x, t)
(7.144)
ˆ
β ˆ
ψ P (x, t) =
ψ(−x, t),
(7.145)
for the KG and Dirac fields, and
A ˆ P (x, t) = −A ˆ (−x, t),
A ˆ 0
P (x, t) = A ˆ0 (−x, t)
(7.146)