A Grassmann integral equation
| dc.creator | Scharnhorst, K. | |
| dc.date | 2002-06-06 | |
| dc.date | 2003-02-07 | |
| dc.date.accessioned | 2026-07-07T11:32:25Z | |
| dc.date.available | 2026-07-07T11:32:25Z | |
| dc.description | 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[{\barΨ},Ψ] = G_0[{λ\barΨ}, {λΨ}] + const., 0<λ\in 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\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 λ=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. | |
| dc.description | 58 pages LaTeX (v2: mainly, minor updates and corrections to the reference section; v3: references [4], [17]-[21], [39], [46], [49]-[54], [61], [64], [139] added) | |
| dc.identifier | https://arxiv.org/abs/math-ph/0206006 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0206006 | |
| dc.identifier | J.Math.Phys.44:5415-5449,2003 | |
| dc.identifier | doi:10.1063/1.1612896 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/197590 | |
| dc.subject | Mathematical Physics | |
| dc.subject | High Energy Physics - Lattice | |
| dc.subject | High Energy Physics - Theory | |
| dc.title | A Grassmann integral equation | |
| dc.type | text |