Computational Physics

15.2 Quantum field theory 467

describe what quantum field theory is and present several examples. In the follow-
ing section, the procedure of numerical quantum field theory will be described in
the context of renormalisation theory. Then we shall describe several algorithms
for simulating field theory, and in particular methods for reducing critical slow-
ing down, a major problem in numerical field theory computations. Finally, we
shall consider some applications in quantum electrodynamics (QED) and quantum
chromodynamics (QCD).

15.2 Quantum field theory

To understand quantum field theory, it is essential to be accustomed to the path-
integral formalism (Section 12.4), so let us recall this concept briefly.
Consider a single particle in one dimension. The particle can move along the
x-axis and its trajectory can be visualised in( 1 + 1 )-dimensional space-time. Fixing
initial and final positions and time to(ti,xi),(tf,xf)respectively, there is (in general)
one particular curve in the(t,x)-plane, theclassical trajectory, for which the action
Sis stationary. The action is given by

S(xi,xf;ti,tf)=

∫tf

ti

dtL(x,x ̇,t) (15.1)

whereL(x,x ̇,t)is the Lagrangian. In quantum mechanics, nonstationary paths are
allowed too, and the probability of going from an initial positionxito a final position
xfis given by
∫
[Dx(t)]e−iS/
=〈xf|e−i(ti−tf)H/
|xi〉, (15.2)

whereHis the Hamiltonian of the system. The integral

∫

[Dx(t)]is over all possible
paths with fixed initial and final valuesxiandxfrespectively. If we send Planck’s
constant
to zero, the significant contributions to the path integral will be more
and more concentrated near the stationary paths, and the stationary path with the
lowest action is the only one that survives when
=0.
Now consider a field. The simplest example of a field is a one-dimensional
string, which we shall consider as a chain of particles with massm, connected
by springs such that in equilibrium the chain is equidistant with spacinga. The
particles can move along the chain, and the displacement of particlenwith respect
to the equilibrium position is calledφn. Fixed, free or periodic boundary conditions
can be imposed. The chain is described by the action

S=

1

2

∫tf

ti

{

∑

n

1

2

mφ ̇^2 n(t)−A

[

φn+ 1 (t)−φn(t)

a

] 2 }

dt. (15.3)