Klaus Scharnhorst - Publications

K. Scharnhorst: A Grassmann integral equation. Journal of Mathematical Physics 44(2003)5415-5449 [math-ph/0206006] (DOI: 10.1063/1.1612896). [SPIRES 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 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[{BarΨ},{Ψ} ] = G0[{λ BarΨ},{λΨ} ] + const., 0 < λ (- R

(BarΨk, 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 ≠ 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.

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