Discrete Differential Geometry and Lattice Field Theory

We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our approach has the advantage with respect to other attempts [Ref. 2-6] that the Lorentz invariance is automatically preserved as it can be seen explicitely in the Maxwell, Klein-Gordon and Dirac equations on the lattice. 1 A difference calculus of several independent variables 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 can define the forward and backward promediate operator ∆̃f(x) ≡ 1 2 {f(x+∆x) + f(x)} , ∇̃f(x) ≡ 1 2 {f(x−∆x) + f(x)} Hence the difference or promediate of the product of two functions follows: ∆ {f(x)g(x)} = ∆f(x)∆̃g(x) + ∆̃f(x)∆g(x) (1.1) ∆̃ {f(x)g(x)} = ∆̃f(x)∆̃g(x) + 1 4 ∆f(x)∆g(x) (1.2) This calculus can be enlarged to functions of several independent variables. We use the following definitions: ∆xf(x, y) ≡ f(x+∆x, y)− f(x, y) ∆yf(x, y) ≡ f(x, y +∆y)− f(x, y) ∆̃xf(x, y) ≡ 1 2 {f(x+∆x, y) + f(x, y)} ∆̃yf(x, y) ≡ 1 2 {f(x, y +∆y) + f(x, y)} ∆f(x, y) ≡ f(x+∆x, y +∆y)− f(x, y) ∆̃f(x, y) ≡ 1 2 {f(x+∆x, y +∆y) + f(x, y)} These definitons can be easily generalized to more independent variables but for the sake of brevity we restrict ourselves to two independent variables. From the last definitions it can be proved the following identities: ∆f(x, y) = ∆x∆̃yf(x, y) + ∆̃x∆yf(x, y) (1.3) ∆̃f(x, y) = ∆̃x∆̃yf(x, y) + 1 4 ∆x∆yf(x, y) (1.4) We can also construct the difference calculus for composite functions. For the sake of simplicity we restrict ourselves to functions of two dependent variables and two independent ones, f (u(x, y), v(x, y)). We define: ∆uf ≡ f(u+∆u, v)− f(u, v) ∆vf ≡ f(u, v +∆v)− f(u, v) ∆̃uf ≡ 1 2 {f(u+∆u, v) + f(u, v)} ∆̃vf ≡ 1 2 {f(u, v +∆v) + f(u, v)} ∆xf ≡ f (u(x+∆x, y), v(x+∆x, y))− f (u(x, y), v(x, y)) ∆yf ≡ f (u(x, y +∆y), v(x, y +∆y))− f (u(x, y), v(x, y)) ∆̃xf ≡ 1 2 {f (u(x+∆x, y), v(x+∆x, y)) + f (u(x, y), v(x, y))} ∆̃yf ≡ 1 2 {f (u(x, y +∆y), v(x, y +∆y)) + f (u(x, y), v(x, y))} from which the following identities can be proved: ∆f = ∆u∆̃vf + ∆̃u∆vf = ∆x∆̃yf + ∆̃x∆yf ∆̃f = ∆̃u∆̃vf + 1 4 ∆u∆vf = ∆̃x∆̃yf + 1 4 ∆x∆yf We can define also the difference operators ∆uxf ≡ f(u+∆xu, v)− f(u, v) ∆uyf = f(u+∆yu, v)− f(u, v) ∆vxf = f(u, v +∆xv)− f(u, v) ∆vyf = f(u, v +∆yv)− f(u, v) and similarly for the promediate operator ∆̃uxf = 1 2 {f(u+∆xu, v) + f(u, v)} ∆̃uyf = 1 2 {f(u+∆yu, v) + f(u, v)} ∆̃vxf = 1 2 {f(u, v +∆xv) + f(u, v)} ∆̃vyf = 1 2 {f(u, v +∆yv) + f(u, v)} From which we deduce the following identities: ∆xf = ∆ux∆̃vxf + ∆̃ux∆vxf (1.5) ∆yf = ∆uy∆̃vyf + ∆̃uy∆vyf (1.6) ∆̃xf = ∆̃ux∆̃vxf + 1 4 ∆ux ∆vxf (1.7) ∆̃yf = ∆̃uy∆̃vyf + 1 4 ∆uy ∆vyf (1.8) These formulas can easily be applied to vector-valued functions: ~u = (u1(x), u2(x), . . . , un(x)) = ~u(x) and its “tanget vector” ∆~u ∆x = ( ∆u1 ∆x , ∆u2 ∆x , . . . , ∆un ∆x ) ≡ ~v(x) An inmediate application is the four-position and four-velocity vectors in special relativity: x(τ) ≡ ( x(τ), x(τ), x(τ), x(τ) ) V (τ) ≡ ( ∆x ∆τ , ∆x ∆τ , ∆x ∆τ , ∆x ∆τ ) These vector-valued vector can be expressed as ~u = u~ea for a given set of orthonormal vectors ~ea 2 Discrete differential forms Given a vectorial space V n over Z we can define a real-valued linear function over Z f (u) ≡ 〈ω, u〉 u ∈ V n (2.1) The forms ω constitue a vectorial linear space (dual space) V , and can be expanded in terms of a basis ω ω = σαω α The basis eβ of V n and ω of V n can be contracted in the following way 〈ω, eβ〉 δ α β (2.2) hence 〈ω, ea〉 = σα , 〈 ω, u 〉 = u , 〈ω, u〉 = σαu α (2.3) If we take ω = ∆x as coordinate basis for the linear forms we can construct discrete differential forms (a discrete version of the continuous differential forms) A particular example of this discrete form is the total difference operator (1,3) of a function of several discrete variables written in the following way: ∆f(x, y) = ( ∆x∆̃yf ∆x ) ∆x+ ( ∆̃x∆yf ∆y ) ∆y (2.4) For these discrete forms we can define the exterior product of two form σ and δ