Computational Physics

532 Computational methods for lattice field theories

masses that can currently be included are still too high too predict the instability of
theρ-meson, for example.
At the time of writing, many interesting results on lattice QCD have been obtained
and much is still to be expected. A very important breakthrough is the formulation of
improved staggered fermion (ISF) actions, which approximate the continuum action
to higher order in the latice constant than the straightforward lattice formulations
discussed so far [57–59]. This makes it possible to obtain results for heavy quark,
and even for lighter ones, important properties have been or are calculated [59],
such as decay constants for excited hadron states.
An interesting state of matter is thequark–gluon plasma, which is the QCD
analog of the Kosterlitz-Thouless phase transition: the hadrons can be viewed as
bound pairs or triplets of quarks, but for high densities and high temperatures, the
‘dielectric’ system may ‘melt’ into a ‘conducting’, dense system of quarks and
gluons. This seems to have been observed very recently after some ambiguous
indications. It turns out that this state of matter resembles a liquid. Lattice gauge
theorists try to match these results in their large-scale QCD calculations. For a
recentreview,seeRef. [60].
To conclude, we describe how to update gauge fields in a simulation. In a Met-
ropolis approach we want to change the matricesUμ(n)and then accept or reject
these changes. A way to do this is to fill a list with ‘random SU(3)’ matrices, which
are concentrated near the unit matrix. We multiply our link matrixUμ(n)by a
matrix taken randomly from the list. For this step to be reversible, the list must
contain the inverse of each of its elements. The list must be biased towards the unit
matrix because otherwise the changes in the matrices become too important and
the acceptance rate becomes too small. Creutz has developed a clever heat bath
algorithm for SU(2) [6, 61]. Cabibbo and Marinari have devised an SU(3) variant
of this method in which the heat bath is successively applied to SU(2) subgroups
of SU(3) [62].

Exercises

15.1 Consider the Gaussian integral

I 1 =

∫∞

−∞

dx 1 ...dxNe−xAx

wherex=(x 1 ,...,xN)is a real vector andAis a Hermitian and positiveN×N

matrix (positive means that all the eigenvaluesλiofAare positive).

(a) By diagonalisingA, show that the integral is equal to

I 1 =

√

( 2 π)N

∏N

i= 1 λi

=

√

( 2 π)N

det(A)

.