200
7. Quantum Field Theory III
classical mechanics (see for example Goldstein 1980) or classical field theory;
and they present major problems when it comes to canonical quantization.
It is actually at just this point that the ‘path-integral’ approach to quantization, alluded to briefly at the end of section 5.2.2, comes into its own. This
is basically because it does not involve non-commuting (or anticommuting)
operators and it is therefore to that extent closer to the classical case. This
means that the relatively straightforward procedures available for constrained
classical mechanics systems can – when suitably generalized! – be efficiently
brought to bear on the quantum problem. For an introduction to these ideas,
we refer to Swanson (1992).
However, we do not wish at this stage to take what would be a very long
detour, in setting up the path-integral quantization of QED. We shall continue
along the ‘canonical’ route. To see the kind of problems we encounter, let us
try and repeat for the A
ν field the ‘canonical’ procedure we introduced in
section 5.2.5. This was based, crucially, on obtaining from the Lagrangian the
momentum π conjugate to φ, and then imposing the commutation relation
(5.117) on the corresponding operators π ˆ and φ ˆ . But inspection of our Maxwell
Lagrangian (7.67) quickly reveals that
∂L A = 0
(7.89)
∂A ˙ 0
and hence there is no canonical momentum π
0 conjugate to A
0 . We appear
to be stymied before we can even start.
There is another problem as well. Following the procedure explained in
chapter 6, we expect that the Feynman propagator for the A ˆ μ field, namely
<0|T (A ˆ μ (x 1 )A ˆ ν (x 2 ))|0>, will surely appear, describing the propagation of a
photon between x 1 and x 2 . In the case of real scalar fields, problem 6.3
showed that the analogous quantity was actually a Green function for the
KG differential operator, (❗ + m
2 ). It turned out, in that case, that what
we really wanted was the Fourier transform of the Green function, which was
essentially (apart from the tricky ‘i∈ prescription’ and a trivial −i factor) the
inverse of the momentum–space operator corresponding to (❗ + m
2 ), namely
(−k
2 + m
2 )
−1 (see equation (6.98) and appendix G, and also (7.58)–(7.60) for
the Dirac case). Suppose, then, that we try to follow this route to obtaining
the propagator for the A ˆ ν field. For this it is sufficient to consider the classical
equations (7.68) with j em = 0, written in k space (problem 7.11(a)):
νμ + k
ν k
μ )A ˜ μ (k) ≡ M
νμ ˜
(−k
2 g
A μ (k) = 0
(7.90)
where A ˜ μ (k) is the Fourier transform of A μ (x). We therefore require the
inverse
νμ + k
ν k
μ )
−1
≡ (M
−1 )
νμ
(−k
2 g
.
(7.91)
Unfortunately it is easy to show that this inverse does not exist. From
Lorentz covariance, it has to transform as a second-rank tensor, and the only
7. Quantum Field Theory III
classical mechanics (see for example Goldstein 1980) or classical field theory;
and they present major problems when it comes to canonical quantization.
It is actually at just this point that the ‘path-integral’ approach to quantization, alluded to briefly at the end of section 5.2.2, comes into its own. This
is basically because it does not involve non-commuting (or anticommuting)
operators and it is therefore to that extent closer to the classical case. This
means that the relatively straightforward procedures available for constrained
classical mechanics systems can – when suitably generalized! – be efficiently
brought to bear on the quantum problem. For an introduction to these ideas,
we refer to Swanson (1992).
However, we do not wish at this stage to take what would be a very long
detour, in setting up the path-integral quantization of QED. We shall continue
along the ‘canonical’ route. To see the kind of problems we encounter, let us
try and repeat for the A
ν field the ‘canonical’ procedure we introduced in
section 5.2.5. This was based, crucially, on obtaining from the Lagrangian the
momentum π conjugate to φ, and then imposing the commutation relation
(5.117) on the corresponding operators π ˆ and φ ˆ . But inspection of our Maxwell
Lagrangian (7.67) quickly reveals that
∂L A = 0
(7.89)
∂A ˙ 0
and hence there is no canonical momentum π
0 conjugate to A
0 . We appear
to be stymied before we can even start.
There is another problem as well. Following the procedure explained in
chapter 6, we expect that the Feynman propagator for the A ˆ μ field, namely
<0|T (A ˆ μ (x 1 )A ˆ ν (x 2 ))|0>, will surely appear, describing the propagation of a
photon between x 1 and x 2 . In the case of real scalar fields, problem 6.3
showed that the analogous quantity was actually a Green function for the
KG differential operator, (❗ + m
2 ). It turned out, in that case, that what
we really wanted was the Fourier transform of the Green function, which was
essentially (apart from the tricky ‘i∈ prescription’ and a trivial −i factor) the
inverse of the momentum–space operator corresponding to (❗ + m
2 ), namely
(−k
2 + m
2 )
−1 (see equation (6.98) and appendix G, and also (7.58)–(7.60) for
the Dirac case). Suppose, then, that we try to follow this route to obtaining
the propagator for the A ˆ ν field. For this it is sufficient to consider the classical
equations (7.68) with j em = 0, written in k space (problem 7.11(a)):
νμ + k
ν k
μ )A ˜ μ (k) ≡ M
νμ ˜
(−k
2 g
A μ (k) = 0
(7.90)
where A ˜ μ (k) is the Fourier transform of A μ (x). We therefore require the
inverse
νμ + k
ν k
μ )
−1
≡ (M
−1 )
νμ
(−k
2 g
.
(7.91)
Unfortunately it is easy to show that this inverse does not exist. From
Lorentz covariance, it has to transform as a second-rank tensor, and the only