An Introduction to Lattice Chiral Fermions

This write-up starts by introducing lattice chirality to people possessing a fairly modern mathematical background, but little prior knowledge about modern physics. I then proceed to present two new and speculative ideas. 1.1 What are Dirac/Weyl fermions ? One can think about (Euclidean) Field Theory as of an attempt to define integrals over function spaces [1]. The functions are of different types and are called fields. The integrands consist of a common exponential factor multiplied by various monomials in the fields. The exponential factor is written as exp(S) where the action S is a functional of the fields. Further restrictions on S are: (1) locality (2) symmetries. Locality means that S can be written as an integral over the base space (space-time) which is the common domain of all fields and the integrand at a point depends at most exponentially weakly on fields at other, remote, space-time points. S is required to be invariant under an all important group of symmetries that act on the fields. In a sense, S is the simplest possible functional obeying the symmetries and generically represents an entire class of more complicated functionals, which are equivalently appropriate for describing the same physics. Dirac/Weyl fields have two main characteristics: (1) They are Grassmann valued, which means they are anti-commuting objects and (2) there is a form of S, possibly obtained by adding more fields, where the Dirac/Weyl fields, ψ, enter only quadratically. The Grassmann nature of ψ implies that the familiar concept of integration needs to be extended. The definition of integration over Grassmann valued fields is algebraic and for an S where the ψ fields enter quadratically, as in S = ¯ ψKψ+...., requires only the propagator, K −1 , and the determinant, det K. Hence, only the linear properties of the operator K come into play, and concepts like a " Grassmann integration measure " are, strictly