Exact solutions of the Dirac equation and induced representations of the Poincaré group on the lattice

We deduce the structure of the Dirac field on the lattice from the discrete version of differential geometry and from the representation of the integral Lorentz transformations. The analysis of the induced representations of the Poincaré group on the lattice reveals that they are reducible, a result that can be considered a group theoretical approach to the problem of fermion doubling. 1. DISCRETE DIFFERENTIAL GEOMETRY Given a function of one independent variable the forward and backward differences are defined as ∆f (x) = f (x + ∆x) − f (x) ∇f (x) = f (x) − f (x − ∆x) Similarly we define the average operators ∆f (x) = 1 2 {f (x + ∆x) + f (x)} ∇f (x) = 1 2 {f (x) + f (x − ∆x)} This calculus can be enlarged to functions of several variables. In particular, we have for the total difference operator ∆f (x, y) f (x + ∆x, y + ∆y) − f (x, y) = ∆ x ∆ y f (x, y) + ∆ x ∆ y f (x, y) in an obvious notation. The finite increments of the variables ∆x, ∆y · · ·, can be used to introduce a discrete differential form ω = a (x, y) ∆x + b (x, y) ∆y and the usual exterior product, from which we construct the p-form and the dual (n − p)-form in the n-dimensional space. σ = 1 p! σ i 1 ···ip ∆x i 1 ∧ · · · ∧ ∆x ip (* σ) k 1 ···k n−p = 1 p! σ i 1 ···ip ε i 1 ···ipk 1 ···kp From the p-forms we can construct (p+1)-forms with the help of the exterior difference, for instance, ∆ω ≡ ∆a ∧ ∆x + ∆b ∧ ∆y = ∆ x ˜ ∆ y b ∆x − ˜ ∆ x ∆ y a ∆y ∆x ∧ ∆y The exterior calculus can be used to express the laws of physics in terms of difference calculus, such as classical electrodynam-ics, quantum field theory [1]. In particular, for the wave equation we have − * ∆ * ∆φ = 0, or