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J. Math. Phys. 44, 5415 (2003); http://dx.doi.org/10.1063/1.1612896 (35 pages)
A Grassmann integral equation
(Received 15 May 2003; accepted 21 June 2003)
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Add to FacebookThe 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[{
},{Ψ}] = G0[{λ},{λΨ}]+const, 0<λ ∊ R (k, 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.© 2003 American Institute of Physics.
},{Ψ}] = G0[{λ},{λΨ}]+const, 0<λ ∊ R (k, 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.© 2003 American Institute of Physics. © 2003 American Institute of Physics
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