Quantum Mechanics for Mathematicians

To make physical sense of the quanta in the relativistic theory, assigning
all non-vacuum states a positive energy, we take such quanta as having two
physically equivalent descriptions:

• A positive energy particle moving forward in time with momentump.


• A positive energy antiparticle moving backwards in time with momentum
  −p.

The operatora†(p) adds such quanta to a state, the operatora(p) destroys
them. Note that for a theory of quantized real-valued Klein-Gordon fields, the
field Φ has components in bothM+Jr andM−Jr so its quantization will both
create and destroy quanta.
Just as in the non-relativistic case (see equation 37.1) quantum field opera-
tors can be defined using the momentum space decomposition and annihilation
and creation operators:

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

Φ(̂x) =^1

(2π)^3 /^2

∫

R^3

(a(p)eip·x+a†(p)e−ip·x)

d^3 p

√

2 ωp

(43.19)

Π(̂x) =^1

(2π)^3 /^2

∫

R^3

(−iωp)(a(p)eip·x−a†(p)e−ip·x)

d^3 p

√

2 ωp

(43.20)

By essentially the same computation as for Poisson brackets, the commuta-
tion relations are

[Φ(̂x),Π(̂x′)] =iδ^3 (x−x′), [Φ(̂x),Φ(̂x′)] = [Π(̂x),Π(̂x′)] = 0 (43.21)

These can be interpreted as the distributional form of the relations of a unitary
representation of a Heisenberg Lie algebra onM⊕R, whereMis the space of
solutions of the Klein-Gordon equation.
The Hamiltonian operator will be quadratic in the field operators and can
be chosen to be

Ĥ=

∫

R^3

1

2

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

This operator is normal ordered, and a computation (see for instance chapter 5
of [16]) shows that in terms of momentum space operators this is the expected

Ĥ=

∫

R^3

ωpa†(p)a(p)d^3 p (43.22)

The dynamical equations of the quantum field theory are now

∂

∂t

Φ = [̂ Φ̂,−iĤ] =Π̂