Chiral fermion operators on the lattice

We only require generalized chiral symmetry and $\gamma_5$-hermiticity, which leads to a large class of Dirac operators describing massless fermions on the lattice, and use this framework to give an overview of developments in this field. Spectral representations turn out to be a powerful tool for obtaining detailed properties of the operators and a general construction of them. A basic unitary operator is seen to play a central r\^ole in this context. We discuss a number of special cases of the operators and elaborate on various aspects of index relations. We also show that our weaker conditions lead still properly to Weyl fermions and to chiral gauge theories.Comment: 25 pages, invited review article for Int. J. Mod. Phys.

Dirac operator normality and chiral fermions

Normality of the Dirac operator is shown to be necessary for chiral properties. From the global chiral Ward identity, which in the continuum limit gives the index theorem, a sum rule results which constrains the spectrum. The Ginsparg-Wilson relation is to be restricted to its simple form and is a member of a set of spectral constraints. A family of alternative chiral transformations is introduced. The one of L\"uscher is a special case which transports only the anomaly term to the measure. An alternative transformation would also be needed to correct Fujikawa's path-integral approach. From a general function of the hermitean Wilson-Dirac operator the one of Neuberger follows.Comment: 12 pages, LaTeX, cjp.sty included, talk at CHIRAL '99, Taipei, Sep. 13-18, 199

Form and index of Ginsparg-Wilson fermions

We clarify the questions rised by a recent example of a lattice Dirac operator found by Chiu. We show that this operator belongs to a class based on the Cayley transformation and that this class on the finite lattice generally does not admit a nonvanishing index, while in the continuum limit, due to operator properties in Hilbert space, this defect is no longer there. Analogous observations are made for the chiral anomaly. We also elaborate on various aspects of the underlying sum rule for the index.Comment: 10 pages; v2: equation corrected, conclusions unchange

General chiral gauge theories

Only requiring that Dirac operators decribing massless fermions on the lattice decompose into Weyl operators we arrive at a large class of them. After deriving general relations from spectral representations we study correlation functions of Weyl fermions for any value of the index, stressing the related conditions for basis transformations and getting the precise behaviors under gauge and CP transformations. Using the detailed structure of the chiral projections we also obtain a form of the correlation functions with a determinant in the general case.Comment: 3 pages, Lattice2003(chiral

Topological structures and phases in U(1) gauge theory

We show that topological properties of minimal Dirac sheets as well as of currents lines characterize the phases unambiguously. We obtain the minimal sheets reliably by a suitable simulated-annealing procedure.Comment: 6 pages, 3 figures, uuencoded postscript file. Contribution to LATTICE 9

Phase transition and dynamical-parameter method in U(1) gauge theory

Monte Carlo simulations of the 4-dimensional compact U(1) lattice gauge theory in the neighborhood of the transition point are made difficult by the suppression of tunneling between the phases, which becomes very strong as soon as the volume of the lattice grows to any appreciable size. This problem can be avoided by making the monopole coupling a dynamical variable. In this manner one can circumvent the tunneling barrier by effectively riding on top of the peaks in the energy distribution which meet for sufficiently large monopole coupling. Here we present an efficient method for determining the parameters needed for this procedure, which can thus be implemented at low computational cost also on large lattices. This is particularly important for a reliable determination of the transition point. We demonstrate the working of our method on a 16^4 lattice. We obtain an equidistribution of configurations across the phase transition even for such a relatively large lattice size.Comment: 11 pages, latex, 2 figures included, uuencode

General chiral gauge theories on the lattice

We still extend the large class of Dirac operators decribing massless fermions on the lattice found recently, only requiring that such operators decompose into Weyl operators. After deriving general relations and constructions of operators, we study the basis representations of the chiral projections. We then investigate correlation functions of Weyl fermions for any value of the index, stressing the related conditions for basis transformations and their consequences, and getting the precise behaviors under gauge transformations and CP transformations. Various further developments include considerations of the explicit form of the effective action and of a representation of the general correlation functions in terms of alternating multilinear forms. For comparison we also consider gauge-field variations and their respective applications. Finally we compare with continuum perturbation theory.Comment: 35 pages; v2: Section 9.3 replaced by new Section 10, version to appear in Nucl. Phys.