Quantum Mechanics for Mathematicians

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))