512 Computational methods for lattice field theories
where the path integral is over all vector potential fields that are compatible with
the Lorentz gauge and with the initial and final vector potential fields at timesti
andtfrespectively. If we do not fix the gauge, this integral diverges badly, whereas
for a particular choice of gauge, the integral converges.
Just as in the case of scalar fields, the excitations of the vector potential field
are considered as particles. These particles are massless: they are the well-known
photons. The electromagnetic field theory is exactly solvable: the photons do
not interact, so we have a situation similar to the free field theory. The theory
becomes more interesting when electrons and positrons are coupled to the field.
These particles are described by vector fieldsψ(x)withD= 2 [d/^2 ]components
ford-dimensional space-time ([x]denotes the integer part ofx), soD =4in
four-dimensional space-time (d=4). The first two components of the four-vector
correspond to the spin-up and spin-down states of the fermion (e.g. the electron) and
the third and fourth components to the spin-up and -down components of the anti-
fermion (positron). The Euler–Lagrange equation for a fermion system interacting
with an electromagnetic field is the famousDirac equation:
[γμ(∂μ−ieAμ)+m]ψ(x)=0. (15.112)The objectsγμ are HermitianD×Dmatrices obeying the anti-commutation
relations:
[γμ,γν]+=γμγν+γνγμ= 2 δμν (15.113)
(in Minkowski metric,δμνis to be replaced bygμν). The Dirac equation is invariant
under the gauge transformation (15.108) if it is accompanied by the following
transformation of theψ:
ψ(x)→eieχ(x)ψ(x). (15.114)
The action from which the Dirac equation can be derived as the Euler–Lagrange
equation is the famous quantum electrodynamics (QED) action:
SQED=∫
d^4 x[
−ψ( ̄x)(γμ∂μ+m)ψ(x)+ieAμ(x)ψ( ̄ x)γμψ(x)−1
4
Fμν(x)Fμν(x)]
. (15.115)
Here,ψ(x)andψ( ̄x)are independent fields. The Dirac equation corresponds to the
Euler–Lagrange equation of this action with
ψ( ̄ x)=ψ†(x)γ^0. (15.116)The QED action itself does not show the fermionic character of theψ-field, which
should, however, not disappear in the Lagrangian formulation. The point is that
theψfield is not an ordinaryc-number field, but a so-calledGrassmann field.