Quantum Mechanics for Mathematicians

Using these creation and annihilation operators, we can define position space
field operators analogous to the ones given by equations 43.19 and 43.20 in the
real scalar field case. NowΦ(̂x) will not be self-adjoint, but its adjoint will be a
fieldΦ̂†(x) which will act on states by increasing the charge by 1, with one term
that creates particles and another that annihilates antiparticles. We define

Definition(Complex scalar quantum field).The complex scalar quantum field
operators are the operator-valued distributions defined by

Φ(̂x) =^1

(2π)^3 /^2

∫

R^3

(a(p)eip·x+b†(p)e−ip·x)

d^3 p

√

2 ωp

Φ̂†(x) =^1

(2π)^3 /^2

∫

R^3

(b(p)eip·x+a†(p)e−ip·x)

d^3 p

√

2 ωp

Π(̂x) =^1

(2π)^3 /^2

∫

R^3

(−iωp)(a(p)eip·x−b†(p)e−ip·x)

d^3 p

√

2 ωp

Π̂†(x) =^1

(2π)^3 /^2

∫

R^3

(−iωp)(b(p)eip·x−a†(p)e−ip·x)

d^3 p

√

2 ωp

These satisfy the commutation relations

[Φ(̂x),Π̂†(x′)] = [Π(̂x),Π̂†(x′)] = [Φ(̂x),Π̂†(x′)] = [Φ̂†(x),Π(̂x′)] = 0

[Φ(̂x),Π(̂x′)] = [Φ̂†(x),Π̂†(x′)] =iδ^3 (x−x′) (44.2)

In terms of these field operators, the Hamiltonian operator will be

Ĥ=

∫

R^3

:(Π̂†(x)Π(̂x) + (∇Φ̂†(x))(∇Φ(̂x)) +m^2 ̂Φ†(x)Φ(̂x)):d^3 x

and the charge operator will be

Q̂=−i

∫

R^3

:(Π(̂x)Φ(̂x)−Π̂†(x)Φ̂†(x)):d^3 x

TakingL=ias a basis element foru(1), one gets a unitary representationUof
U(1) using
U′(L) =−iQ̂

and
U(θ) =e−iθ
Q̂

Uacts by conjugation on the fields:

U(θ)Φ̂U(θ)−^1 =e−iθΦ̂, U(θ)Φ̂†U(θ)−^1 =eiθ̂Φ†

U(θ)Π̂U(θ)−^1 =eiθΠ̂, U(θ)Π̂†U(θ)−^1 =e−iθΠ̂†