137
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
and the Hamiltonian is
∞
∑ 2
r
H ρ =
p +
e ρω
2 A r
2 .
(5.111)
r
eρ 4
r=1
We may cast (5.111) into nicer form by the change of variables
√
√
P r = 2/e p r , Q r = e/2 A r ,
(5.112)
in terms of which
∞
∑ P
2
r
H ρ =
+
1 ρω r
2 Q
2
r
(5.113)
2ρ
2
r=1
just as in (5.56), with N → ∞.
5.2.5 Heisenberg–Lagrange–Hamilton quantum field
mechanics
Finally, we are ready to quantize classical field formalism, and arrive at a
quantum field mechanics – at least for the scalar field φ(x, t). If we were
dealing with the case in which φ(x, t) represented the displacement of a onedimensional stretched string, quantization would be straightforward. We
would take the classical Hamiltonian (5.113) and promote the mode coordinates Q r and their conjugate momenta P r to operators satisfying commutation
relations of the form (5.85). The rest of the analysis would be exactly as in
equations (5.86) to (5.89), except that the number of modes N is infinite. But
in the case of the general scalar field, we do not want to impose the boundary
conditions φ(0, t) = φ(e, t) = 0, which led to the mode expansion (5.34). It is
then not so clear how to proceed.
Fortunately, the Lagrange-Hamilton field formalism does indicate the way
forward, which is one good reason for developing it in the first place. (Another
is that it is very well suited to the analysis of symmetries, a crucial aspect
of gauge theories – see chapter 7.) In the previous section we introduced the
‘coordinate-like’ field φ(x, t) and (via the Lagrangian) the ‘momentum-like’
field π(x, t). To pass to the quantized version of the field theory, we mimic
the procedure followed in the discrete case and promote both the quantities φ
and π to operators φ ˆ and ˆ
π, in the Heisenberg picture. As usual, the distinctive
feature of quantum theory is the non-commutativity of certain basic quantities
in the theory – for example, the fundamental commutator (ħ = 1)
[ˆ q r (t), p ˆ s (t)] = iδ rs
(5.114)
of the discrete case. Thus we expect that the operators φ ˆ and ˆ
π will obey
some commutation relation which is a continuum generalization of (5.114).
The commutator will be of the form [φ ˆ (x, t), π ˆ(y, t)], since – recalling figure 5.5 – the discrete index r or s becomes the continuous variable x or y; we
5.2. The quantum field: (ii) Lagrange–Hamilton formulation
and the Hamiltonian is
∞
∑ 2
r
H ρ =
p +
e ρω
2 A r
2 .
(5.111)
r
eρ 4
r=1
We may cast (5.111) into nicer form by the change of variables
√
√
P r = 2/e p r , Q r = e/2 A r ,
(5.112)
in terms of which
∞
∑ P
2
r
H ρ =
+
1 ρω r
2 Q
2
r
(5.113)
2ρ
2
r=1
just as in (5.56), with N → ∞.
5.2.5 Heisenberg–Lagrange–Hamilton quantum field
mechanics
Finally, we are ready to quantize classical field formalism, and arrive at a
quantum field mechanics – at least for the scalar field φ(x, t). If we were
dealing with the case in which φ(x, t) represented the displacement of a onedimensional stretched string, quantization would be straightforward. We
would take the classical Hamiltonian (5.113) and promote the mode coordinates Q r and their conjugate momenta P r to operators satisfying commutation
relations of the form (5.85). The rest of the analysis would be exactly as in
equations (5.86) to (5.89), except that the number of modes N is infinite. But
in the case of the general scalar field, we do not want to impose the boundary
conditions φ(0, t) = φ(e, t) = 0, which led to the mode expansion (5.34). It is
then not so clear how to proceed.
Fortunately, the Lagrange-Hamilton field formalism does indicate the way
forward, which is one good reason for developing it in the first place. (Another
is that it is very well suited to the analysis of symmetries, a crucial aspect
of gauge theories – see chapter 7.) In the previous section we introduced the
‘coordinate-like’ field φ(x, t) and (via the Lagrangian) the ‘momentum-like’
field π(x, t). To pass to the quantized version of the field theory, we mimic
the procedure followed in the discrete case and promote both the quantities φ
and π to operators φ ˆ and ˆ
π, in the Heisenberg picture. As usual, the distinctive
feature of quantum theory is the non-commutativity of certain basic quantities
in the theory – for example, the fundamental commutator (ħ = 1)
[ˆ q r (t), p ˆ s (t)] = iδ rs
(5.114)
of the discrete case. Thus we expect that the operators φ ˆ and ˆ
π will obey
some commutation relation which is a continuum generalization of (5.114).
The commutator will be of the form [φ ˆ (x, t), π ˆ(y, t)], since – recalling figure 5.5 – the discrete index r or s becomes the continuous variable x or y; we