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the eigenvalues of the dirac operator are of great interest for the universality of important features of qcd and qed . on the one hand ,
the accumulation of small eigenvalues is , via the banks - casher formula  @xcite , related to the spontaneous breaking of chiral symmetry . on the other hand ,
the fluctuation properties of the eigenvalues in the bulk of the spectrum can be described by random matrix theory ( rmt ) , see ref .
for example , the nearest - neighbor spacing distribution @xmath0 , i.e. , the distribution of spacings @xmath1 between adjacent eigenvalues on the unfolded scale , agrees with the wigner surmise of rmt .
according to the bohigas - giannoni - schmit conjecture  @xcite , quantum systems whose classical counterparts are chaotic  @xcite have a nearest - neighbor spacing distribution given by rmt whereas systems whose classical counterparts are integrable obey a poisson distribution , @xmath2 .
therefore , the specific form of @xmath0 is often taken as a criterion for the presence or absence of `` quantum chaos '' .    in rmt
, one has to distinguish several universality classes which are determined by the symmetries of the system . for the case of the qcd dirac operator ,
this classification was done in ref .
@xcite . depending on the number of colors and the representation of the quarks ,
the dirac operator is described by one of the three chiral ensembles of rmt .
as far as the fluctuation properties in the bulk of the spectrum are concerned , the predictions of the chiral ensembles are identical to those of the ordinary ensembles  @xcite . in ref .
@xcite , the dirac matrix was studied for color - su(2 ) using both staggered and wilson fermions which correspond to the chiral symplectic ( chse ) and orthogonal ( choe ) ensemble , respectively .
here  @xcite , we study su(3 ) with staggered fermions which corresponds to the chiral unitary ensemble ( chue ) .
the rmt result for the nearest - neighbor spacing distribution can be expressed in terms of so - called prolate spheroidal functions , see ref .
@xcite . a very good approximation to @xmath0
is provided by the wigner surmise for the unitary ensemble , @xmath3    we generated gauge field configurations using the standard wilson plaquette action for su(3 ) with and without dynamical fermions in the kogut - susskind prescription .
we have worked on a @xmath4 lattice with various values of the inverse gauge coupling @xmath5 both in the confinement and deconfinement phase .
we typically produced 10 independent equilibrium configurations for each @xmath6 .
the dirac operator , @xmath7 , is anti - hermitian so that the eigenvalues @xmath8 of @xmath9 are real .
because of @xmath10 the nonzero @xmath8 occur in pairs of opposite sign .
all spectra were checked against the analytical sum rules @xmath11 and @xmath12 , where v is the lattice volume . to construct the nearest - neighbor spacing distribution from the eigenvalues ,
one first has to `` unfold '' the spectra  @xcite .
figure  [ fintemp ] compares @xmath0 of full qcd with @xmath13 flavors and quark mass @xmath14 to the rmt result . in the confinement
@xmath15 as well as in the deconfinement @xmath16 phase we observe agreement with rmt up to very high @xmath6 ( not shown ) . the observation that @xmath0 is not influenced by the presence of dynamical quarks is expected from the results of ref .
@xcite , which apply to the case of massless quarks .
our results , and those of ref .
@xcite , indicate that massive dynamical quarks do not affect @xmath0 either .
we have also investigated the staggered dirac spectrum of 4d u(1 ) gauge theory which corresponds to the chue of rmt . at @xmath17 u(1 )
gauge theory undergoes a phase transition between a confinement phase with mass gap and monopole excitations for @xmath18 and the coulomb phase which exhibits a massless photon for @xmath19  @xcite . as for su(2 ) and su(3 ) gauge groups , we expect the confined phase to be described by rmt , whereas free fermions yield the poisson distribution . the question arose whether the coulomb phase would be described by rmt or by the poisson distribution  @xcite .
the nearest - neighbor spacing distributions for an @xmath20 lattice at @xmath21 ( confined phase ) and at @xmath22 ( coulomb phase ) , averaged over 20 independent configurations , are depicted in fig .
both are well described by the chue of rmt .
no signs for a transition to poisson regularity are found .
the deconfinement phase transition does not seem to coincide with a transition in the spacing distribution .
for very large values of @xmath6 far into the deconfinement region , the eigenvalues start to approach the degenerate eigenvalues of the free theory , given by @xmath23 , where @xmath24 is the lattice constant , @xmath25 is the number of lattice sites in the @xmath26-direction , and @xmath27 . in this case
, the spacing distribution is neither wigner nor poisson .
it is possible to lift the degeneracies of the free eigenvalues using an asymmetric lattice where @xmath28 , @xmath29 , etc .
are relative primes and , for large lattices , the distribution is then poisson , @xmath2 , see fig .
we have continued our investigations with a study of the distribution of the small eigenvalues in the whole phase diagram .
the banks - casher formula @xcite relates the dirac eigenvalue density @xmath30 at @xmath31 to the chiral condensate , @xmath32 . the microscopic spectral density , @xmath33 should be given by the appropriate prediction of rmt  @xcite , which also generates the leutwyler - smilga sum rules  @xcite .
we present results in fig .
[ fig4 ] for su(3 ) theory and the staggered dirac operator on a @xmath34 lattice from 5000 configurations for @xmath35 and 3000 configurations for @xmath36 and @xmath37 around the critical temperature @xmath38 . in the confinement phase , both the microscopic spectral density @xmath39 and
the distribution @xmath40 of the smallest eigenvalue agree with the rmt predictions of the chue for topological charge @xmath41 @xcite .
our analog presentation for u(1 ) theory is from 10000 configurations on a @xmath34 lattice around the critical coupling . in the left plot of fig .
[ fig5 ] a comparison with rmt for the microscopic spectral density @xmath42 yields again quite satisfactory agreement in the confinement .
the analytical rmt result for the ( quenched ) chue and @xmath43 is given by @xcite @xmath44/2 $ ] , where @xmath45 denotes the bessel function .
the chiral condensate @xmath46 can be obtained by extrapolating the histogram for @xmath30 to @xmath31 and using the banks - casher relation  @xcite .
the right plot in fig .
[ fig5 ] exhibits the distribution @xmath47 of the smallest eigenvalue , being in the chirally broken phase in accordance with the prediction of the ( quenched ) chue of rmt for topological charge @xmath43 , @xmath48 .
the quasi - zero modes which are responsible for the chiral condensate @xmath49 build up when we cross from the deconfinement into the confined phase
. figures  [ fig4 ] and  [ fig5 ] demonstrate that both @xmath30 and @xmath50 plotted with varying @xmath6 on identical scales , respectively , can serve as an indicator for the phase transition .    in fig .
[ fig6 ] we turn to a discussion of the spectrum in the quark - gluon plasma and coulomb phase . from rmt a functional form of @xmath51
is expected at the onset of the eigenvalue density  @xcite .
a fit to the data in the regime up to @xmath52 yields @xmath53 for the non - abelian and @xmath54 for the abelian theory , in agreement with recent studies  @xcite .
this suggests that both theories correspond to universality class @xmath55 . for this class
a microscopic level density involving the airy function can be deduced from rmt  @xcite .
a rescaling of our data from the @xmath34 lattice to this functional form is not satisfactory for both theories  @xcite
. the reason might be ( i ) that the lowest eigenvalue is still influenced by quasi - zero modes from the confinement and ( ii ) that already the first eigenvalues lie above an analogue of the thouless energy . with increasing lattice size these effects should decrease .
further , we checked that the high - end of the spectrum behaves similar to the low - end and thus deviates from the results of the ordinary ue of rmt  @xcite .
again the question arises whether larger lattices possess spectral edges closer to the microscopic semi - circle universality .
the aim of this contribution was to work out two different types of universalities inherent in quantum field theories with a covariant derivative and their interpretation in terms of rmt .
the first type concerns the bulk of the spectrum of the dirac operator . the nearest - neighbor spacing distribution @xmath0 agrees with the rmt result in both the confinement and the deconfinement phase of pure gauge theory and of full qcd , except for extremely large values of @xmath6 where the eigenvalues are known analytically .
the nearest - neighbor spacing distribution of 4d u(1 ) quenched lattice gauge theory is described by the chue of rmt in both the confinement and the coulomb phase . even in the deconfinement phase ,
gauge fields retain a considerable degree of randomness , which apparently gives rise to quantum chaos in these theories .    the second type of universality concerns the low - lying spectra of the dirac operators of both qcd and qed . in all cases considered
, one finds that in the phase in which chiral symmetry is spontaneously broken the distribution @xmath47 and the microscopic spectral density @xmath39 are described by chiral rmt .
when chiral symmetry is restored one has to rely on ordinary rmt with the related space - time symmetries , but one finds for our lattice size only universal behavior of the macroscopic density @xmath30 .
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we investigate the eigenvalue spectrum of the staggered dirac matrix in su(3 ) gauge theory and in full qcd as well as in quenched u(1 ) theory . as a measure of the fluctuation properties of the eigenvalues , 
 we consider the nearest - neighbor spacing distribution . 
 we find that in all regions of their phase diagrams , compact lattice gauge theories have bulk spectral correlations given by random matrix theory , which is an indication for quantum chaos . in the confinement phase , 
 the low - lying dirac spectrum of these quantum field theories is well described by random matrix theory , exhibiting universal behavior . 
 related results for gauge theories with minimal coupling are now discussed also in the chirally symmetric phase .    
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