# A Grassmann integral equation

Source: [A Grassmann integral equation](<https://www.sota2.com/research/paper/68fa23252c2a376019393352>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T05:59:47.306Z

Authors: K. Scharnhorst

Published: 2002-06-06T14:31:34.000Z

## 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 integrations and which is to be obeyed by an unknown function over a (finite-dimensional) 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\[{\\bar\\Psi},{\\Psi}\] = G\_0\[{\\lambda\\bar\\Psi}, {\\lambda\\Psi}\] + const., 0\<\\lambda\\in R (\\bar\\Psi\_k, \\Psi\_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 \\lambda \\not= 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 \\lambda=1, the considered Grassmann integral equation 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/math-ph/0206006>)

[PDF](<https://arxiv.org/pdf/math-ph/0206006.pdf>)