Computational Physics

15.2 Quantum field theory 469

where we have assumed periodic boundary conditions in space and time (or van-
ishing fields at the integral boundaries, which are located at infinity) to neglect
integrated terms.
If, apart from a coupling to its neighbours, each particle had also been coupled
to an external harmonic potentialm^2 φ^2 /2, we would have obtained

S=−

∫

d^2 x

1

2

φ(x)(∂μ∂μ+m^2 )φ(x). (15.10)

Note that the Euler–Lagrange equations for the field are

(∂μ∂μ+m^2 )φ(x)=0; (15.11)

which is recognised as the Klein–Gordon equation, the straightforward covariant
generalisation of the Schrödinger equation.^1
Quantum field theory is often used as a theory for describing particles. The
derivation above started from a chain of particles, but these particles are merely
used to formulate our quantum field theory, and they should not be confused with
the physical particles which are described by the theory. The difference between
the two can be understood as follows. Condensed matter physicists treat wave-like
excitations of the chain (i.e. a one-dimensional ‘crystal’) as particles – they are
calledphonons. Note that the ‘real’ particles are the atoms of the crystal. In field
theory, the only ‘real’ particles are the excitations of the field.
In fact, we can imagine that a wave-like excitation pervades our sheet, for example
φ(t,x 1 )=eipx^1 −iωt (15.12)

(herex 1 denotes the spatial coordinate). This excitation carries a momentumpand
an energy
ω, and it is considered as a particle. We might have various waves as a
superposition running over the sheet: these correspond to as many particles.
Let us try to find the Hamiltonian corresponding to the field theory presented
above (the following analysis is taken up in some detail in Problems 15.2 and 15.3).
We do this by returning to the discretised version of the field theory. Let us first
consider the ordinary quantum description of a single particle of mass 1, moving
in one dimension in a potentialmx^2 /2. The Hamiltonian of this particle is given by

H=

p^2

2

+

m

2

x^2 (15.13)

with[p,x]=−i. In the example of the chain we have a large number of such
particles, but each particle can still be considered as moving in a harmonic potential,
and after some calculation we find for the Hamiltonian:

H=

∑

n

[ˆπn^2 +(φˆn−φˆn− 1 )^2 +m^2 φˆn^2 ], (15.14)

(^1) The Klein–Gordon equation leads to important problems in ordinary quantum mechanics, such as a
nonconserved probability density and an energy spectrum which is not bounded from below.