PHYSICAL REVIEW D VOLUME 29, NUMBER 4 15 FEBRUARY 1984 Link fermions in Euclidean lattice gauge theory R. Brower Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138 R. Giles Center for Theoretical Physics, Laboratory for Nuclear Science, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 G. Maturana SCIPP-Physics Department, University of California, Santa Cruz, California 95064 (Received 6 September 1983) The representation of the Wilson lattice fermion propagator as a sum over classical particle tra- jectories is discussed. A simple generalization of this path sum leads to an extended set of fermion theories characterized by one (or more) additional parameters. Such theories are nonlocal when written in terms of the usual four-component Dirac field. They are more naturally characterized by a local action functional whose degrees of freedom are those of a set of two-component Fermi fields defined on directed links of the lattice. Such lattice fields correspond to the direct product of a four-vector and Dirac spinor. For a suitable choice of parameters, the extended fermion theory offers a precocious approach to the continuum dispersion relation as the lattice spacing goes to zero and is therefore of interest for numerical studies of QCD. I. INTRODUCTION The problem of adequately representing fermionic fields in the context of lattice theories is a complex and chal- lenging one. This is due in no small part to the fact that there is apparently no lattice representation which em- bodies all aspects expected and desired of the continuum theory. In particular, local chiral symmetry is incon- sistent with the absence of state doubling. One is there- fore in the position of attempting to define a lattice ver- sion of the theory which embodies explicitly only a subset of the symmetries of the continuum theory and which be- comes equivalent to it only in the zero-lattice-spacing lim- it. Were, we approach this old problem from a new point of view. In this paper, we present a generalization of the ~ i l s o n ' action which is motivated by the representation of the lat- tice fermion propagator in a background gauge field as a sum over contributions from single-particle paths. In Sec. I1 we discuss this Feynman "path sum" representation in detail and show that the spin structure for the Wilson theory can be considerably simplified along each individu- al spacetime path. In fact, though spin factors of the Wil- son theory appear initially as products of 4x4 projection matrices along links of a path through the lattice, they can equivalently be represented as products of cubic group ele- ments located at its corners. This is a vast simplification from the practical point of view of numerical calculations using path-sum methods such as the hopping-parameter expansion2 or stochastic path-sampling techniquesS3Fur- ther, this representation gives us another insight into the mechanism by which the Wilson theory avoids chiral dou- bling: the spectrum of the theory is really determined by the properties of a two-component path sum. Finally, we note that this reduction of the Wilson theory of four- component spinors with spin structure on links to a two- component spinor theory with structure at comers has a precise analogy in the two-dimensional Ising model. In the Ising model, the Feynman ~ a c - w a r d 4 form of the partition function as a sum over closed paths with phases at comers can be directly related to a two-component free fermion theory. We discuss the Ising model briefly in Ap- pendix B. Our generalization of the Wilson theory, which we refer to as a theory of "link fermions," is presented in Sec. 111. It arises from a simple modification of the path sum for the fermion propagator which adds an extra relative weight associated with corners of a lattice path. This modified theory, though simple in terms of paths, corre- sponds to a nonlocal Dirac equation and therefore to a nonlocal fermion action. The theory can be described in terms of a local action involving auxiliary fermion fields associated with directed links of the lattice and having both spin-+ and spin-+ components. Section IV discusses the energy spectrum of the modi- free fermion theory. The zero-lattice-spacing or "continu- um" limit corresponds to a curve, rather than to a point, in the parameter space. This is the curve along which the dimensionless "lattice" mass of the fermion vanishes. Al- though each point on this curve describes a continuum theory of free fermions, various points can be dis- tinguished from one another by the rapidity with which the continuum relativistic form of the single-particle spec- trum is approached. It is shown that a suitable choice of parameters results in an energy spectrum that is close to that of a relativistic free particle even for moderately large lattice momenta. This result suggests that the modified theory may serve as a better starting point for lattice cal- 704 @ 1984 The American Physical Society