Computational Physics

15.7 Gauge field theories 521

15.7.4 Including dynamical fermions

In real problems studied by particle physicists, fermions are to be included into
the lattice action, and not in a quenched fashion as done in the previous section.
In this section we focus on dynamical fermions, which cause two problems. First, a
straightforward discretisation of the fermion action leads to 2dspecies of uncoupled
fermions to be included into the problem (indspace-time dimensions) instead of the
desired number of fermion species (‘flavours’). Second, we have not yet discussed
the problem of including the fermion character in a path integral simulation. We
first consider the ‘fermion doubling problem’ and then sketch how simulations can
actually be carried out for theories including fermions.

Fermions on a lattice

When calculating the path integral for free fermions for which the Lagrangian is
quadratic in the fermion fields, the following Gaussian Grassmann integral must be
evaluated: ∫

[DψDψ ̄]e−ψ ̄Mψ, (15.142)

where the kernelMis given as

M=m+γμ∂μ. (15.143)

The expression in the exponent is shorthand for an integral over the space-time
coordinates. Discretising the theory on the lattice and Fourier-transforming the
fields andMwe find that the latter becomes diagonal:

Mk,k′=

(

m+

i

a

∑

μ

γμsin(kμa)

)

δ(k+k′). (15.144)

The lattice version ofMis therefore the Fourier transform of this function.
There is a problem with this propagator. The continuum limit singles out only
the minima of the sine as a result of the factor 1/ain front of it. These are found
not only neark =0 but also nearka=(±π,0,0,0)(in four dimensions) etc.,
because of the sine function having zeroes at 0 andπ. This causes the occurrence
of two different species of fermions per dimension, adding up to 16 species for
four-dimensional space-time. It turns out that this degeneracy can be lifted only at
the expense of breaking the so-called ‘chiral symmetry’ for massless fermions[46].
Chiral symmetry is a particular symmetry which is present in the Dirac equation
(and action) for massless particles. Suppose chiral symmetry is present in the lattice
version of the action. This symmetry forbids a mass term to be present, and the
renormalised theory should therefore havem=0. A lattice action which violates
chiral symmetry might generate massive fermions under renormalisation.