72,771 research outputs found
A Grassmann integral equation
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.Comment: 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
Nonlinear Integral-Equation Formulation of Orthogonal Polynomials
The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is
investigated. It is shown that for a given function w(x) the equation admits an
infinite set of polynomial solutions P(x). For polynomial solutions, this
nonlinear integral equation reduces to a finite set of coupled linear algebraic
equations for the coefficients of the polynomials. Interestingly, the set of
polynomial solutions is orthogonal with respect to the measure x w(x). The
nonlinear integral equation can be used to specify all orthogonal polynomials
in a simple and compact way. This integral equation provides a natural vehicle
for extending the theory of orthogonal polynomials into the complex domain.
Generalizations of the integral equation are discussed.Comment: 7 pages, result generalized to include integration in the complex
domai
Shear viscosity in theory from an extended ladder resummation
We study shear viscosity in weakly coupled hot theory using the CTP
formalism . We show that the viscosity can be obtained as the integral of a
three-point function. Non-perturbative corrections to the bare one-loop result
can be obtained by solving a decoupled Schwinger-Dyson type integral equation
for this vertex. This integral equation represents the resummation of an
infinite series of ladder diagrams which contribute to the leading order
result. It can be shown that this integral equation has exactly the same form
as the Boltzmann equation. We show that the integral equation for the viscosity
can be reexpressed by writing the vertex as a combination of polarization
tensors. An expression for this polarization tensor can be obtained by solving
another Schwinger-Dyson type integral equation. This procedure results in an
expression for the viscosity that represents a non-perturbative resummation of
contributions to the viscosity which includes certain non-ladder graphs, as
well as the usual ladders. We discuss the motivation for this resummation. We
show that these resummations can also be obtained by writing the viscosity as
an integral equation involving a single four-point function. Finally, we show
that when the viscosity is expressed in terms of a four-point function, it is
possible to further extend the set of graphs included in the resummation by
treating vertex and propagator corrections self-consistently. We discuss the
significance of such a self-consistent resummation and show that the integral
equation contains cancellations between vertex and propagator corrections.Comment: Revtex 40 pages with 29 figures, version to appear in Phys. Rev.
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