Klaus Scharnhorst  Publications
K. Scharnhorst: A Grassmann integral equation.
Journal of Mathematical Physics 44:11(2003)54155449
(DOI: 10.1063/1.1612896) [arXiv:mathph/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 (finitedimensional) 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 lowdimensional 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
finitedimensional 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 ( R
(BarΨ_{k}, Ψ_{k},
k = 1,...,n,
are the generators of the Grassmann algebra G_{2n}),
between the finitedimensional 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 ≠ 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 nonGaussian integral.
The investigation sheds light on the structures to be met
for Grassmann algebras G_{2n} with
arbitrarily chosen n.
The article is cited in:

M. Ostilli: On the probabilistic approach for Gaussian Berezin integrals.
Annals of Physics (New York) 308(2003)555577
(DOI: 10.1016/S00034916(03)001775)
[arXiv:condmat/0301462].

B. Pioline: Cubic free field theory.
arXiv:hepth/0302043,
version 2, 4 pp.. Version 1 is published in:
L. Baulieu, E. Rabinovici, J. Harvey, B. Pioline, P. Windey (Eds.):
Progress in String, Field and Particle Theory,
Proceedings of the NATO Advanced Study Institute, Cargèse,
Corsica, France, May 27  June 8, 2002.
NATO Science Series II: Mathematics, Physics and Chemistry, Vol. 104.
Kluwer Academic Publishers, Dordrecht, 2003, pp. 453456.

J. Feinberg: Fredholm's minors of arbitrary order: their
representations as a determinant of resolvents and in terms of free
fermions and an explicit formula for their functional derivativ.
Journal of Physics A: Mathematical and General 37(2004)62996310
(DOI: 10.1088/03054470/37/24/008)
[arXiv:mathph/0402029].

B.G. Giraud, R. Peschanski:
On positive functions with positive Fourier transforms.
Acta Physica Polonica B37(2006)331346
( http://thwww.if.uj.edu.pl/acta/vol37/pdf/v37p0331.pdf )
[arXiv:mathph/0504015].

K. Scharnhorst, J.W. van Holten:
Nonlinear BogolyubovValatin transformations: Two modes.
Annals of Physics (New York) 326(2011)28682933
(DOI: 10.1016/j.aop.2011.05.001) [NIKHEF preprint NIKHEF/2010005, arXiv:1002.2737].
= paper [32]

T. Rashkova:
Matrix algebras over Grassmann algebras and their PIstructure.
Acta Universitatis Apulensis ICTAMI2011(2011)169184
( http://www.emis.de/journals/AUA/ictami2011/Paper12Ictami2011.pdf ).
The article is part of the special issue: D. Breaz, N. Breaz, N. Ularu (Eds.):
Proceedings of the International Conference on Theory and Applications of
Mathematics and Informatics, ICTAMI2011, Alba Iulia, 2124 iulie 2011.
Acta Universitatis Apulensis ICTAMI2011(2011)1566
( https://www.emis.de/journals/AUA/2011.html ).

T. Rashkova:
On the nilpotency in matrix algebras with Grassmann entries.
Serdica Mathematical Journal 38(2012)7990
( http://www.math.bas.bg/serdica/2012/2012079090.pdf ).
The article is part of:
V. Drensky, A. Giambruno, M. Kochetov, P. Koshlukov, M. Zaicev (Eds.):
Proceedings of the International Workshop Polynomial Identities in Algebras. II,
September 26, 2011, the Memorial University of Newfoundland in St. John's, NL, Canada.
Serdica Mathematical Journal 38:13(2012)ixxii, 1506
( http://www.math.bas.bg/serdica/n13_12.html ).

T. Rashkova:
On some properties and special identities in the second order matrix algebra over Grassmann algebras.
Demonstratio Mathematica 46(2013)2936
(DOI: 10.1515/dema20130438).

B.G. Giraud, R. Peschanski:
On the positivity of Fourier transforms.
arXiv:1405.3155, 12 pp..
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Document last modified: October 29, 2018
Document address: http://www.nat.vu.nl/~scharnh/cite30.htm