Quantum Mechanics for Mathematicians

and integrate the analog ofμLoverR^3 to get an appropriate moment map for
the field theory case. This gives a quadratic functional on the fields that will have
the desired Poisson bracket with the fields for each value ofx. We will denote
the result byQ, since it is an observable that will have a physical interpretation
as electric charge when this theory is coupled to the electromagnetic field (see
chapter 45):

Q=

∫

R^3

(Π 2 (x)Φ 1 (x)−Π 1 (x)Φ 2 (x))d^3 x

One can use the field Poisson bracket relations

{Φj(x),Πk(x′)}=δjkδ(x−x′)

to check that
{
Q,

(

Φ 1 (x)

Φ 2 (x)

)}

=

(

−Φ 2 (x)

Φ 1 (x)

)

,

{

Q,

(

Π 1 (x)

Π 2 (x)

)}

=

(

−Π 2 (x)

Π 1 (x)

)

Quantization of the classical field theory gives a unitary representationUof
SO(2) on the multi-particle state space, with

U′(L) =−iQ̂=−i

∫

R^3

(Π̂ 2 (x)Φ̂ 1 (x)−Π̂ 1 (x)Φ̂ 2 (x))d^3 x

The operator

U(θ) =e−iθ

Q̂

will act by conjugation on the fields:

U(θ)

(

Φ̂ 1 (x)

Φ̂ 2 (x)

)

U(θ)−^1 =

(

cosθ sinθ

−sinθ cosθ

)(

Φ̂ 1 (x)

Φ̂ 2 (x)

)

U(θ)

(

Π̂ 1 (x)

Π̂ 2 (x)

)

U(θ)−^1 =

(

cosθ sinθ

−sinθ cosθ

)(̂

Π 1 (x)

Π̂ 2 (x)

)

It will also give a representation ofSO(2) on states, with the state space de-
composing into sectors each labeled by the integer eigenvalue of the operatorQ̂
(which will be called the “charge” of the state).
Using the definitions of̂Φ andΠ (43.19 and 43.20),̂ Q̂can be computed in
terms of annihilation and creation operators, with the result

Q̂=i

∫

R^3

(a† 2 (p)a 1 (p)−a† 1 (p)a 2 (p))d^3 p (44.1)

One expects that since the time evolution action on the classical field space
commutes with theSO(2) action, the operatorQ̂should commute with the
Hamiltonian operatorĤ. This can readily be checked by computing [H,̂Q̂]
using

Ĥ=

∫

R^3

ωp(a† 1 (p)a 1 (p) +a† 2 (p)a 2 (p))d^3 p