Free fermions on the lattice in one and two dimensions

Lattice fermions, their symmetries, and their continuum properties are issues central to lattice studies of dense and hot matter. It is a subtle subject because there are fundamental restrictions on the number of fermion species, their handedness, and gauge invariance. To see what the challenges are, we will illustrate lattice forms of the Dirac equation in various settings.

First consider the free Klein–Gordon equation and free Dirac equation on a spatial lattice with continuum time variable. In a Hamiltonian lattice-gauge theory, one would have a three-dimensional lattice and a continuum of time. Excitations in the system could hop from site to site given the rules of the Hamiltonian, the discrete version of the spatial derivatives in the energy, etc. Although Hamiltonian lattice-gauge theory is an important subject, our emphasis here continues to be on the Euclidean version of the theory. However, a short look at Hamiltonian methods is very elementary and enlightening. The details of the problems of lattice versions of the Dirac equation are different depending on whether time is treated as a continuum variable or a discrete one. Euclidean lattice fermions will be discussed in detail below after our introduction.

Let there be a free boson field φ(x, t) in 1 + 1 dimensions, namely one discrete spatial axis and one continuum temporal axis.