140
5. Quantum Field Theory I: The Free Scalar Field
We quantize these mode expressions by promoting φ → φ ˆ , π → π ˆ and assuming the commutator (5.117). Thus we write
∫ ∞
φ ˆ =
d
√
k [ˆ a(k)e
ikx−iωt + ˆ
a
† (k)e
−ikx+iωt ]
(5.129)
−∞ 2π 2ω
and similarly for π ˆ. The commutator (5.117) now determines the commutators
of the mode operators a ˆ and ˆ
a
† :
[ˆ a(k), a ˆ
† (k
′ )] = 2πδ(k − k
′ )
(5.130)
[ˆ a(k), a ˆ(k
′ )] = [ˆ a
† (k), a ˆ
† (k
′ )] = 0
as shown in problem 5.6. These are the desired continuum analogues of the
discrete oscillator commutation relations
[ˆ a r , a ˆ
†
s ] = δ rs
(5.131)
[ˆ a r , a ˆ s ] = [ˆ a
† , a ˆ
† ] = 0.
r s
The precise factor in front of the δ-function in (5.130) depends on the normalization choice made in the expansion of φ ˆ , (5.129). Problem 5.6 also shows
that the commutation relations (5.130) lead to (5.118) as expected.
The form of the ˆ
a, ˆ
a
† commutation relations (5.130) already suggests that
the ˆ
a(k) and ˆ
a
† (k) operators are precisely the single-quantum destruction and
creation operators for the continuum problem. To verify this interpretation
and find the eigenvalues of H ˆ , we now insert the expansion for φ ˆ and ˆ
π into
ˆ
H of (5.124). One finds the remarkable result (problem 5.7)
∫
(
)
∞
H ˆ =
dk 1 [ˆ a
† (k)ˆ a(k) + ˆ
a(k)ˆ a
† (k)]ω .
(5.132)
2π 2
−∞
Comparing this with the single-oscillator result
1
H ˆ = (ˆ a
† a ˆ + ˆ
aa ˆ
† )ω
(5.133)
2
shows that, as anticipated in section 5.1, each classical mode of the field can
be quantized, and behaves like a separate oscillator coordinate, with its own
frequency ω = k. The operator ˆ
a
† (k) creates, and ˆ
a(k) destroys, a quantum
of the k mode. The factor (2π)
−1 in H ˆ arises from our normalization choice.
We note that in the field operator φ ˆ of (5.129), those terms which destroy
quanta go with the factor e
−iωt , while those which create quanta go with
+iωt
e
. This choice is deliberate and is consistent with the ‘absorption’ and
‘emission’ factors e
±iωt of ordinary time-dependent perturbation theory in
quantum mechanics (cf equation (A.33) of appendix A).
What is the mass of these quanta? We know that their frequency ω is
related to their wavenumber k by (5.126), which – restoring ħ’s and c’s – can
be regarded as equivalent to ħω = ħck, or E = cp, where we use the Einstein
5. Quantum Field Theory I: The Free Scalar Field
We quantize these mode expressions by promoting φ → φ ˆ , π → π ˆ and assuming the commutator (5.117). Thus we write
∫ ∞
φ ˆ =
d
√
k [ˆ a(k)e
ikx−iωt + ˆ
a
† (k)e
−ikx+iωt ]
(5.129)
−∞ 2π 2ω
and similarly for π ˆ. The commutator (5.117) now determines the commutators
of the mode operators a ˆ and ˆ
a
† :
[ˆ a(k), a ˆ
† (k
′ )] = 2πδ(k − k
′ )
(5.130)
[ˆ a(k), a ˆ(k
′ )] = [ˆ a
† (k), a ˆ
† (k
′ )] = 0
as shown in problem 5.6. These are the desired continuum analogues of the
discrete oscillator commutation relations
[ˆ a r , a ˆ
†
s ] = δ rs
(5.131)
[ˆ a r , a ˆ s ] = [ˆ a
† , a ˆ
† ] = 0.
r s
The precise factor in front of the δ-function in (5.130) depends on the normalization choice made in the expansion of φ ˆ , (5.129). Problem 5.6 also shows
that the commutation relations (5.130) lead to (5.118) as expected.
The form of the ˆ
a, ˆ
a
† commutation relations (5.130) already suggests that
the ˆ
a(k) and ˆ
a
† (k) operators are precisely the single-quantum destruction and
creation operators for the continuum problem. To verify this interpretation
and find the eigenvalues of H ˆ , we now insert the expansion for φ ˆ and ˆ
π into
ˆ
H of (5.124). One finds the remarkable result (problem 5.7)
∫
(
)
∞
H ˆ =
dk 1 [ˆ a
† (k)ˆ a(k) + ˆ
a(k)ˆ a
† (k)]ω .
(5.132)
2π 2
−∞
Comparing this with the single-oscillator result
1
H ˆ = (ˆ a
† a ˆ + ˆ
aa ˆ
† )ω
(5.133)
2
shows that, as anticipated in section 5.1, each classical mode of the field can
be quantized, and behaves like a separate oscillator coordinate, with its own
frequency ω = k. The operator ˆ
a
† (k) creates, and ˆ
a(k) destroys, a quantum
of the k mode. The factor (2π)
−1 in H ˆ arises from our normalization choice.
We note that in the field operator φ ˆ of (5.129), those terms which destroy
quanta go with the factor e
−iωt , while those which create quanta go with
+iωt
e
. This choice is deliberate and is consistent with the ‘absorption’ and
‘emission’ factors e
±iωt of ordinary time-dependent perturbation theory in
quantum mechanics (cf equation (A.33) of appendix A).
What is the mass of these quanta? We know that their frequency ω is
related to their wavenumber k by (5.126), which – restoring ħ’s and c’s – can
be regarded as equivalent to ħω = ħck, or E = cp, where we use the Einstein