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} ] = G₀[{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 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 G_2n with arbitrarily chosen n.

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• Electronic copy of the preprint: arXiv:math-ph/0206006
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• Misprints, errata, addenda (PDF / Postscript):
  • On p. 5421, the indices l and m on the r.h.s. of the eqs. (31)-(33) should be interchanged. Also, on p. 5443 the indices l and m on the r.h.s. of the eq. (A.2) should be interchanged. For the correct display of these equations see this sheet: PDF / Postscript.
  • P. 5438, eqs. (190), (191), for the correct display of these equations see this sheet: PDF / Postscript.
  • P. 5449, in the second line of ref. 147 the sequence "pp. 354-392" should read correctly: pp. 355-392.

© K. Scharnhorst . Document last modified: June 8, 2023