It is suggested that the fermion determinant for a vector-like gauge theory with strictly massless quarks can be represented on the lattice as $\det{{1+V}\over 2}$, where $V=X(X^\dagger X)^{-1/2}$ and $X$ is the Wilson-Dirac lattice operator with a negative mass term. There is no undesired doubling and no need for any fine tuning. Several other appealing features of the formula are pointed out.Comment: 7 pages, plain TeX; references correcte

Path integration over Euclidean chiral fermions is replaced by the quantum mechanics of an auxiliary system of non{interacting fermions. Our construction avoids the no{go theorem and faithfully maintains all the known important features of chiral fermions, including the violation of some perturbative conservation laws by gauge eld con gurations of non{trivial topology.

A practical implementation of the Overlap-Dirac operator 1+γ 5 ǫ(H) 2 is presented. The implementation exploits the sparseness of H and does not require full storage. A simple application to parity invariant three dimensional SU (2) gauge theory is carried out to establish that zero modes related to topology are exactly reproduced on the lattice.

An expression for the lattice e ective action induced by chiral fermions in any even dimensions in terms of an overlap of two states is shown to have promising properties in two dimensions: The correct abelian anomaly is reproduced and instantons are suppressed.

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We define smoothed Wilson loop operators on a four dimensional lattice and check numerically that they have a finite and nontrivial continuum limit. The continuum operators maintain their character as unitary matrices and undergo a phase transition at infinite N reflected by the eigenvalue distribution closing a gap in its spectrum when the defining smooth loop is dilated from a small size to a large one. If this large N phase transition belongs to a solvable universality class one might be able to calculate analytically the string tension in terms of the perturbative Λ-parameter. This would be achieved by matching instanton results for small loops to the relevant large-N -universal function which, in turn, would be matched for large loops to an effective string theory. Similarities between our findings and known analytical results in two dimensional space-time indicate that the phase transitions we found only affect the eigenvalue distribution, but the traces of finite powers of the Wilson loop operators stay smooth under scaling.

In a previous publication [hep-lat/9707022] I showed that the fermion determinant for strictly massless quarks can be written on the lattice as $\det D$, where $D$ is a certain finite square matrix explicitly constructed from the lattice gauge fields. Here I show that $D$ obeys the Ginsparg-Wilson relation $D\gamma_5 D = D\gamma_5 +\gamma_5 D$.Comment: 4 pages, plain Te