## Quantisation of Fields

As position is no longer a quantum observable but free particles do not seem to be in contradiction with relativistic invariance, we can try to introduce such a free particle as a quantum observable. This observable is hence described by a plane wave

ϕk(x, t) = e−i(ko t−x·k)/h¯ , (2.1) which satisfies the Klein-Gordon equation

−h¯2 ∂ 2ϕ(x, t)

∂t2 = −h¯2c2 ∂

2ϕ(x, t) ∂x2 + m

2c4ϕ(x, t), (2.2)

where k0 = √

c2k2 + m2c4 is the energy of the free particle. By superposition of these plane waves, we can make a superposition of free particles, which is therefore described by a field

ϕ(x, t) = (2πh¯)− 32 ∫

d3k ϕ˜(k, t)eik·x/h¯ . (2.3)

It satisfies the Klein-Gordon equation if the Fourier components ϕ˜(k, t) satisfy the harmonic equation

−h¯2 ∂ 2ϕ˜(k, t) ∂t2

= (c2k2 + m2c4)ϕ˜(k, t) ≡ k2o (k)ϕ˜(k, t). (2.4)

Its solutions split in positive and negative frequency components

ϕ˜(k, t) = ϕ˜+(k)e−iko t/h¯ + ϕ˜−(k)eiko t/h¯ . (2.5) The wave function, or rather the wave functional (ϕ), describes the distribution over the various free particle states. The basic dynamical variables are ϕ˜(k). These play the role the coordinates used to play in ordinary quantum mechanics and will require quantisation. As they satisfy a simple harmonic equation in time, it is natural to quantise them as harmonic oscillators.