142
5. Quantum Field Theory I: The Free Scalar Field
one quantum of momentum k 1 and another of momentum k 2 may be written
(cf (5.81))
|k 1 , k 2 > ∝ a ˆ
† (k 1 )ˆ a
† (k 2 )|0>.
(5.138)
A general state will contain an arbitrary number of quanta.
Once again, and this time more formally, we have completed the programme outlined in section 5.1, ending up with the ‘quantization’ of a classical
field φ(x, t), as exemplified in the basic expression (5.129), together with the
interpretation of the operators a ˆ(k) and ˆ
a
† (k) as destruction and creation operators for mode quanta. We have, at least implicitly, still retained up to this
point the ‘mechanical model’ of some material object oscillating – some kind
of infinitely extended ‘jelly’. We now throw away the mechanical props and
embrace the unadorned quantum field theory! We do not ask what is waving,
we simply postulate a field – such as φ – and quantize it. Its quanta of excitation are what we call particles – for example, photons in the electromagnetic
case.
We end this long section with some further remarks about the formalism,
and the physical interpretation of our quantum field φ ˆ .
Comment (1)
The alert reader, who has studied appendix I, may be worried about the
following (possible) consistency problem. The fields φ ˆ and ˆ
π are Heisenberg
picture operators, and obey the equations of motion
φ ˆ ˙ (x, t) = −i[φ ˆ (x, t), H ˆ ]
(5.139)
˙
ˆ
π ˆ(x, t) = −i[ˆ π(x, t), H]
(5.140)
where H ˆ is given by (5.132). It is a good exercise to check (problem 5.8(a))
˙
ˆ
that (5.139) yields just the expected relation φ(x, t) = ˆ
π(x, t) (cf (5.122)).
Thus (5.140) becomes
¨ ˆ
ˆ
φ(x, t) = −i[ˆ π(x, t), H].
(5.141)
However, we have assumed in our work here that φ ˆ obeyed the wave equation
(cf.(5.121))
∂
2
¨ ˆ
ˆ
φ =
φ(x, t)
(5.142)
∂x 2
as a consequence of the quantized version of the Euler–Lagrange equation (5.96).
Thus the right-hand sides of (5.141) and (5.142) need to be the same, for consistency – and they are: see problem 5.8(b). Thus – at least in this case –
the Heisenberg operator equations of motion are consistent with the Euler–
Lagrange equations.
Comment (2)
Following on from this, we may note that this formalism encompasses both
the wave and the particle aspects of matter and radiation. The former is evi
5. Quantum Field Theory I: The Free Scalar Field
one quantum of momentum k 1 and another of momentum k 2 may be written
(cf (5.81))
|k 1 , k 2 > ∝ a ˆ
† (k 1 )ˆ a
† (k 2 )|0>.
(5.138)
A general state will contain an arbitrary number of quanta.
Once again, and this time more formally, we have completed the programme outlined in section 5.1, ending up with the ‘quantization’ of a classical
field φ(x, t), as exemplified in the basic expression (5.129), together with the
interpretation of the operators a ˆ(k) and ˆ
a
† (k) as destruction and creation operators for mode quanta. We have, at least implicitly, still retained up to this
point the ‘mechanical model’ of some material object oscillating – some kind
of infinitely extended ‘jelly’. We now throw away the mechanical props and
embrace the unadorned quantum field theory! We do not ask what is waving,
we simply postulate a field – such as φ – and quantize it. Its quanta of excitation are what we call particles – for example, photons in the electromagnetic
case.
We end this long section with some further remarks about the formalism,
and the physical interpretation of our quantum field φ ˆ .
Comment (1)
The alert reader, who has studied appendix I, may be worried about the
following (possible) consistency problem. The fields φ ˆ and ˆ
π are Heisenberg
picture operators, and obey the equations of motion
φ ˆ ˙ (x, t) = −i[φ ˆ (x, t), H ˆ ]
(5.139)
˙
ˆ
π ˆ(x, t) = −i[ˆ π(x, t), H]
(5.140)
where H ˆ is given by (5.132). It is a good exercise to check (problem 5.8(a))
˙
ˆ
that (5.139) yields just the expected relation φ(x, t) = ˆ
π(x, t) (cf (5.122)).
Thus (5.140) becomes
¨ ˆ
ˆ
φ(x, t) = −i[ˆ π(x, t), H].
(5.141)
However, we have assumed in our work here that φ ˆ obeyed the wave equation
(cf.(5.121))
∂
2
¨ ˆ
ˆ
φ =
φ(x, t)
(5.142)
∂x 2
as a consequence of the quantized version of the Euler–Lagrange equation (5.96).
Thus the right-hand sides of (5.141) and (5.142) need to be the same, for consistency – and they are: see problem 5.8(b). Thus – at least in this case –
the Heisenberg operator equations of motion are consistent with the Euler–
Lagrange equations.
Comment (2)
Following on from this, we may note that this formalism encompasses both
the wave and the particle aspects of matter and radiation. The former is evi