Computational Physics

15.7 Gauge field theories 511

have a more complicated structure. They include several kinds of particles, some of
which are fermions. Intermediate particles act as ‘messengers’ through which other
particles interact. It turns out that the action has a special kind of local symmetry,
the so-called ‘gauge symmetry’.
Global symmetries are very common in physics: rotational and translational sym-
metries play an important role in the solution of classical and quantum mechanical
problems. Such symmetries are associated with a transformation (rotation, trans-
lation) of the full space, which leaves the action invariant. Local symmetries are
operations which vary in space-time, and which leave the action invariant. You have
probably met such a local symmetry: electrodynamics is the standard example of a
system exhibiting a local gauge symmetry. The behaviour of electromagnetic fields
in vacuum is described by an action defined in terms of the four-vector potential
Aμ(x)(xis the space-time coordinate)[ 38 ]:

SEM=

1

4

∫

d^4 xFμνFμν≡

∫

d^4 xLEM(∂μAν) (15.107a)

with
Fμν=∂μAν−∂νAμ. (15.107b)

LEM =^14 FμνFμν is the electromagnetic Lagrangian. The gauge symmetry of
electrodynamics is a symmetry with respect to a particular class of space-time-
dependent shifts of the four-vector potentialAμ(x):

Aμ(x)→Aμ(x)+∂μχ(x), (15.108)

whereχ(x)is some scalar function. It is easy to check that the action(15.107a)is
indeed invariant under the gauge transformation(15.108). If sourcesjμare present
[j=(ρ,j)whereρis the charge density, andjthe current density], the action reads

SEM=

1

4

∫

d^4 x(FμνFμν+jμAμ). (15.109)

The Maxwell equations are found as the Euler–Lagrange equations for this action.
The action is gauge-invariant if the current is conserved, according to

∂μjμ(x)=0. (15.110)
A quantum theory for the electromagnetic field (in the absence of sources) is
constructed proceeding in the standard way, by using the action(15.107a)in the
path integral. If we fix the gauge, for example by setting∂μAμ=0 (Lorentz gauge),
the transition probability for going from an initial field configurationAiattitoAfat
tffor imaginary times (we use Euclidean metric throughout this section) is given by

〈Af;tf|Ai;ti〉=

∫

[DAμ]exp

[

−

1

∫tf

ti

dtLEM(∂μAν)

]

(15.111)