196
7. Quantum Field Theory III
As in (7.31), the i∈ takes care of the particle/antiparticle, emission/absorption
business. Formula (7.60) is the fermion analogue of ‘rule (ii)’ in (6.103).
The reader should note carefully one very important difference between
(7.60) and (7.31), which is that (7.60) is a 4×4 matrix. What we are re¯ ˆ
ally saying (cf (6.98)) is that the Fourier transform of <0|T (ψ ˆ α (x 1 )ψ β (x 2 ))|0>,
where α and β run over the four components of the Dirac field, is equal to the
(α, β) matrix element of the matrix i(k / − m + i∈)
−1 :
∫
¯ ˆ
d
4 (x 1 − x 2 ) e
ik·(x1−x2)
<0|T (ψ ˆ α (x 1 )ψ β (x 2 ))|0> = i(k / − m + i∈)
−1
(7.61)
αβ .
The form (7.61) can be made to look more like (7.31) by making use of the
result (problem 7.7)
(k / − m)(k / + m) = (k
2
− m
2 )
(7.62)
(where the 4×4 unit matrix is understood on the right-hand side) so as to
write (7.61) as
i(k / + m) .
(7.63)
k 2 − m 2 + i∈
As in the scalar case, (7.61) can be directly verified by inserting the field
expansion (7.35) into the left-hand side, and following steps analogous to those
in equations (6.92)–(6.98). In following this through one will meet the expres∑
∑
sions
u(k, s)¯ u(k, s) and
v(k, s)¯ v(k, s), which are also 4 × 4 matrices.
s
s
Problem 7.8 shows that these quantities are given by
∑
∑
u α (k, s)¯ u β (k, s) = (k / + m) αβ
v α (k, s)¯ v β (k, s) = (k / − m) αβ . (7.64)
s
s
With these results, and remembering the minus sign in (7.57), one can check
(7.63) (problem 7.9).
One might now worry that the adoption of anticommutation relations for
Dirac fields might spoil ‘causality’, in the sense of the discussion after (7.32).
¯
One finds, indeed, that the fields ψ ˆ and ψ ˆ anticommute at space-like separation, but this is enough to preserve causality for physical observables, which
will involve an even number of fermionic fields.
We now turn to the problem of quantizing the Maxwell (electromagnetic)
field.
7.3 The Maxwell field A
μ (x)
7.3.1 The classical field case
Following the now familiar procedure, our first task is to find the classical field
Lagrangian which, via the corresponding Euler–Lagrangian equations, yields
7. Quantum Field Theory III
As in (7.31), the i∈ takes care of the particle/antiparticle, emission/absorption
business. Formula (7.60) is the fermion analogue of ‘rule (ii)’ in (6.103).
The reader should note carefully one very important difference between
(7.60) and (7.31), which is that (7.60) is a 4×4 matrix. What we are re¯ ˆ
ally saying (cf (6.98)) is that the Fourier transform of <0|T (ψ ˆ α (x 1 )ψ β (x 2 ))|0>,
where α and β run over the four components of the Dirac field, is equal to the
(α, β) matrix element of the matrix i(k / − m + i∈)
−1 :
∫
¯ ˆ
d
4 (x 1 − x 2 ) e
ik·(x1−x2)
<0|T (ψ ˆ α (x 1 )ψ β (x 2 ))|0> = i(k / − m + i∈)
−1
(7.61)
αβ .
The form (7.61) can be made to look more like (7.31) by making use of the
result (problem 7.7)
(k / − m)(k / + m) = (k
2
− m
2 )
(7.62)
(where the 4×4 unit matrix is understood on the right-hand side) so as to
write (7.61) as
i(k / + m) .
(7.63)
k 2 − m 2 + i∈
As in the scalar case, (7.61) can be directly verified by inserting the field
expansion (7.35) into the left-hand side, and following steps analogous to those
in equations (6.92)–(6.98). In following this through one will meet the expres∑
∑
sions
u(k, s)¯ u(k, s) and
v(k, s)¯ v(k, s), which are also 4 × 4 matrices.
s
s
Problem 7.8 shows that these quantities are given by
∑
∑
u α (k, s)¯ u β (k, s) = (k / + m) αβ
v α (k, s)¯ v β (k, s) = (k / − m) αβ . (7.64)
s
s
With these results, and remembering the minus sign in (7.57), one can check
(7.63) (problem 7.9).
One might now worry that the adoption of anticommutation relations for
Dirac fields might spoil ‘causality’, in the sense of the discussion after (7.32).
¯
One finds, indeed, that the fields ψ ˆ and ψ ˆ anticommute at space-like separation, but this is enough to preserve causality for physical observables, which
will involve an even number of fermionic fields.
We now turn to the problem of quantizing the Maxwell (electromagnetic)
field.
7.3 The Maxwell field A
μ (x)
7.3.1 The classical field case
Following the now familiar procedure, our first task is to find the classical field
Lagrangian which, via the corresponding Euler–Lagrangian equations, yields