The QCD Dirac Operator Spectrum and Finite-Volume Scaling

The succesful analytical computations of the so-called microscopic QCD Dirac operator spectrum from random matrix theory [1–3] have raised an important question. Why can random matrix theory, which seems to be very far from the quantum field theory of QCD, apparently be used to compute spectral properties of the Dirac operator? Within the last year this issue has been much clarified, with important consequences also for lattice gauge theory. The results obtained in random matrix theory are now seen to be exact predictions of QCD itself, in an appropriate finite-volume scaling regime. This is defined by [4] 1/ΛQCD << L << 1/mπ, where L = V 1/4 is the linear extent of the four-volume V . In this region one is probing finite-volume effects of QCD to the extreme. Only static modes of the lowest hadronic excitations contribute to the euclidean partition function of QCD, and if chiral symmetry is spontaneously broken according to SU(Nf)× SU(Nf) → SU(Nf), it reads [4]