Quantum Mechanics for Mathematicians

Digression. Quantum field theory textbooks often contain a discussion of a
non-positive Hermitian inner product on the space of Klein-Gordon solutions,
given by

〈φ 1 ,φ 2 〉=i

∫

R^3

(

φ 1 (t,x)

∂

∂t

φ 2 (t,x)−

(

∂

∂t

φ 2 (t,x)

)

φ 1 (t,x)

)

d^3 x

which can be shown to be independent oft. This is defined onM⊗C, the com-
plexified Klein-Gordon solutions and is zero on the real-valued solutions, so does
not provide an inner product on those. It does not use the relativistic complex
structure. If we start with a theory of complex Klein-Gordon fields, the function
〈φ,φ〉will be the moment map for theU(1)action by phase transformations on
the fields. It will carry an interpretation as charge, and give after quantization
the charge operator. This will be discussed in section 44.1.2. To compare the
formula for the charge to be found there (equation 44.4) with the formula above,
use the equation of motion

Π =

∂

∂t

Φ

43.3 Hamiltonian and dynamics of the Klein-Gordon theory

The Klein-Gordon equation forφ(t,x) in Hamiltonian form is the following pair
of first-order equations

∂

∂t

φ=π,

∂

∂t

π= (∆−m^2 )φ

which together imply
∂^2
∂t^2

φ= (∆−m^2 )φ

To get these as equations of motion, we need to find a Hamiltonian functionh
such that

∂

∂t

φ={φ,h}=π

∂

∂t

π={π,h}= (∆−m^2 )φ

One can show that two choices of Hamiltonian function with this property are

h=

∫

R^3

H(x)d^3 x

where

H=

1

2

(π^2 −φ∆φ+m^2 φ^2 ) or H=

1

2

(π^2 + (∇φ)^2 +m^2 φ^2 )