K. Scharnhorst: A Grassmann integral equation. Journal of Mathematical Physics 44:11(2003)5415-5449 (DOI: 10.1063/1.1612896) [arXiv:math-ph/0206006]. [INSPIRE record]
Abstract: 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 G_{m} (i.e., a sought after element of the Grassmann algebra G_{m}). 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 G_{2n}, 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[{BarΨ},{Ψ} ] = G_{0}[{λ BarΨ},{λΨ} ] + const., 0 < λ (- R
(BarΨ_{k}, Ψ_{k}, k = 1,...,n, are the generators of the Grassmann algebra G_{2n}), between the finite-dimensional analogues G_{0} 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 ≠ 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 G_{2n} with arbitrarily chosen n.