Computational Physics

15.7 Gauge field theories 513

Grassmann variables are anticommuting numbers – Grassmann numbersaandb
have the properties:
ab+ba=0. (15.117)

In particular, takinga=b, we see thata^2 =0. We do not go into details concerning
Grassmann algebra [ 4 , 5 , 39 ] but mention only the result of a Gaussian integration
over Grassman variables. For a Gaussian integral over a vectorψψψwe have the
following result for the components ofψψψbeing ordinary commuting, or Grassmann
anticommuting numbers:

∫

dψ 1 ...dψNexp(−ψψψTAψψψ)=







√

( 2 π)N

det(A)

commuting;

√

det(A) anticommuting.

(15.118)

The matrixAis symmetric. In quantum field theories such as QED, we need a
Gaussian integral over complex commuting and noncommuting variables, with the
result:
∫
dψ 1 dψ 1 ∗...dψNdψN∗exp(−ψψψ†Aψψψ)=

{

( 2 π)N/det(A) commuting;
det(A) anticommuting
(15.119)
for a Hermitian matrixA. Fortunately the Lagrangian depends only quadratically
on the fermionic fields, so only Gaussian integrals over Grassmann variables occur
in the path integral.

15.7.2 Electromagnetism on a lattice: quenched compact QED

Physical quantities involving interactions between photons and electrons, such as
scattering amplitudes, masses and effective interactions, can be derived from the
QED Lagrangian in a perturbative analysis. This leads to divergences similar to
those mentioned in connection with scalar fields, and these divergences should be
renormalised properly by choosing values for the bare coupling constanteand
massmoccurring in the Lagrangian such that physical mass and coupling constant
become finite; more precisely, they become equal to the experimental electron mass
and the charge which occurs in the large-distance Coulomb law in three spatial
dimensions:

V(r)=

e^2

4 π 0 r

(15.120)

(for short distances, this formula is no longer valid as a result of quantum
corrections).
Instead of following the perturbative route, we consider the discretisation of
electrodynamics on a lattice (the Euclidean metric is most convenient for lattice