2Free field theory
III Quantum Field Theory
2.2 The quantum field
We are now going to use canonical quantization to promote our classical fields
to quantum fields. We will first deal with the case of a real scalar field.
Definition
(Real scalar quantum field)
.
A (real, scalar) quantum field is an
operator-valued function of space
φ
, with conjugate momentum
π
, satisfying the
commutation relations
[φ(x), φ(y)] = 0 = [π(x), π(y)]
and
[φ(x), π(y)] = iδ
3
(x − y).
In case where we have many fields labelled by
a ∈ I
, the commutation relations
are
[φ
a
(x), φ
b
(y)] = 0 = [π
a
(x), π
b
(y)]
and
[φ
a
(x), π
b
(y)] = iδ
3
(x − y)δ
b
a
.
The evolution of states is again given by Schr¨odinger equation.
Definition (Schr¨odinger equation). The Schr¨odinger equation says
i
d
dt
|ψi = H |ψi.
However, we will, as before, usually not care and just look for eigenvalues of
H.
As in the case of the harmonic oscillator, our plan is to rewrite the field in
terms of creation and annihilation operators. Note that in quantum mechanics, it
is always possible to write the position and momentum in terms of some creation
and annihilation operators for any system. It’s just that if the system is not
a simple harmonic oscillator, these operators do not necessarily have the nice
properties we want them to have. So we are just going to express everything in
creation and annihilation operators nevertheless, and see what happens.