Computational Physics

522 Computational methods for lattice field theories

One could ignore the doubling problem and live with the fact that the theory now
contains 2ddifferent species of fermions. However, it is also possible to lift up the
unwanted parts of the propagator by adding a term proportional to 1−cos(akμ)to
it, which forknear 0 does not affect the original propagator to lowest order, but
which lifts the parts forkμa=±πsuch that they are no longer picked up in the
continuum. The method is referred to as theWilson fermion method. The resulting
propagator is

Mk=m+

i

a

∑

μ

γμsin(akμ)+

r

a

∑

μ

[ 1 −cos(akμ)]. (15.145)

This form is very convenient because it requires only a minor adaptation of a pro-
gram with the original version of the propagator. In real space, and taking the lattice
constantaequal to 1, the Wilson propagator reads indspace-time dimensions:

Mnl=(m+ 4 r)δnl−

1

2

∑

μ

[(r+γμ)δl,n+μ+(r−γμ)δl,n−μ]. (15.146)

The disadvantage of this solution is that the extra terms destroy any chiral symmetry,
which is perhaps a bit too drastic.
In a more complicated method half of the unwanted states are removed by doub-
ling the period, so that the Brillouin zone of the lattice is cut-off atπ/( 2 a)instead
ofπ/a. This is done by putting different species of fermions on alternating sites
of the lattice. Although this removes the unwanted fermions, it introduces new fer-
mions which live on alternate sites of the lattice. The resulting method is called
thestaggered fermion method. The staggered fermion method respects the chiral
symmetry discussed above and is therefore a better option than the Wilson fermion
method. It is, however, more complicated than the Wilson fermion method and we
refrain from a discussion here, but refer to the original literature [ 47 , 48 ] and later
reviews [ 6 , 43 , 49 ].

Algorithms for dynamical fermions

If we want to include dynamical rather than quenched fermions into our lattice
field theory, we must generate configurations of anticommuting fermion fields. As
it is not clear how to do this directly and as this may cause negative probabilities,
various alternatives using results for Gaussian integrals over Grassmann variables
(seeSection 15.7.1) have been developed. We shall explain a few algorithms for
an action consisting of a bosonic part,SBoson, defined in terms of the boson field,
A(x), coupled to the fermion field,ψ,ψ ̄, via the fermion kernel,M(A):

S=SBoson(A)+

∫

ddxψ( ̄x)M(A)ψ(x). (15.147)

The QED Lagrangian in (15.115) has this form.