Here the two different integrandsH(x) are related (as in the non-relativistic
case) by integration by parts, so these just differ by boundary terms that are
assumed to vanish.
In terms of theA(p),A(p), the Hamiltonian will be
h=
∫
R^3
ωpA(p)A(p)d^3 p
The equations of motion are
d
dt
A={A,h}=
∫
R^3
ωp′A(p′){A(p),A(p′)}d^3 p′=iωpA
d
dt
A={A,h}=−iωpA
with solutions
A(p,t) =eiωptA(p,0), A(p,t) =e−iωptA(p,0)
Digression.Taking as starting point the Lagrangian formalism, the action for
the Klein-Gordon theory is
S=
∫
M^4
Ld^4 x
where
L=
1
2
((
∂
∂t
φ
) 2
−(∇φ)^2 −m^2 φ^2
)
This action is a functional of fields on Minkowski spaceM^4 and is Poincar ́e
invariant. The Euler-Lagrange equations give as equation of motion the Klein-
Gordon equation 43.2. One recovers the Hamiltonian formalism by seeing that
the canonical momentum forφis
π=
∂L
∂φ ̇
=φ ̇
and the Hamiltonian density is
H=πφ ̇−L=
1
2
(π^2 + (∇φ)^2 +m^2 φ^2 )
43.4 Quantization of the Klein-Gordon theory
Given the description we have found in momentum space of real solutions of
the Klein-Gordon equation and the choice of complex structureJrdescribed
in the last section, we can proceed to construct a quantum field theory by the
Bargmann-Fock method in a manner similar to the non-relativistic quantum
field theory case. Quantization takes
A(α(p))∈M+Jr=H 1 →a†(α(p))