212 From Classical Mechanics to Quantum Field Theory. A Tutorial
The vacuum state becomes trivial
Ψ 0 =1,which now is normalizable with respect to the Gaussian measure Eq. (3.44), i.e
(Ψ 0 ,Ψ 0 )=1.
The new renormalized Hamiltonian is
Hˆren=−∫
d^3 x(
δ
δφ(x)− 2
√
−∇^2 +m^2 φ(x))
δ
δφ(x),
and the excited states are just field polynomials.
3.5.1 Fockspace
The space of physical states is generated by polynomials of field operators, e.g.
F(f 1 ,f 2 ,···,fn)=φ(f 1 )φ(f 2 )···φ(fn). (3.45)The simplest state is a degree zero polynomial: the vacuum state. The degree one
monomialsφ(f) correspond to one-particle states, wherefis the quantum wave
packet state of the particle. The functional
Ff(φ)=φ(f)associated to one-particle states is a linear map in the space of quantum field
φ∈S′(R^3 ). In mathematical terms, one-particle state constitute the dual space
of the configuration space of classical fields^7.
Higher order monomials correspond to linear combinations of quantum states
with different number of particles. To pick up only states with a defined number
of particles, one has to proceed as in the harmonic oscillator case where the eigen-
states of the Hamiltonian are given by Hermite polynomials which involve suitable
combinations of monomials.
For such a reason to identify physical states in terms of particles, it is convenient
to introduce a coherent state basis. This can be achieved in terms of creation and
annihilation operators,
a(f)=φ(√
−∇^2 +m^2 f)+iˆπ(f),a(f)†=φ(√
−∇^2 +m^2 f)−iˆπ(f)†.It is easy to show that
[a(f),a(g)] = [a(f)†,a(g)†]=0,
[
a(f),a(g)†]
=2(f,√
−∇^2 +m^2 g).(^7) This explains why in the case of gauge fields, where the configuration space of classical gauge
fields modulo gauge transformations is a curved manifold, the particle interpretation of quantum
states is so difficult.