A Grassmann integral equation

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann (Berezin) integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra Gm (i.e., a sought after element of the Grassmann algebra Gm). A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G2n, n = 2, 3, 4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{Ψ̄}, {Ψ}] = G0[{λΨ̄}, {λΨ}]+const., 0 < λ ∈ R (Ψ̄k, Ψk, k = 1, . . . , n, are the generators of the Grassmann algebra G2n), between the finite-dimensional analogues G0 and G of the (“classical”) action and effective action functionals, respectively, a special Grassmann integral equation is being established and solved which also is equivalent to a coupled system of nonlinear matrix equations. If λ 6= 1, solutions to this Grassmann integral equation exist for n = 2 (and consequently, also for any even value of n, specifically, for n = 4) but not for n = 3. If λ = 1, the considered Grassmann integral equation (of course) has always a solution which corresponds to a Gaussian integral, but remarkably in the case n = 4 a further solution is found which corresponds to a non-Gaussian integral. The investigation sheds light on the structures to be met for Grassmann algebras G2n with arbitrarily chosen n.