5,153 research outputs found
Causal Fermion Systems as a Candidate for a Unified Physical Theory
The theory of causal fermion systems is an approach to describe fundamental
physics. Giving quantum mechanics, general relativity and quantum field theory
as limiting cases, it is a candidate for a unified physical theory. We here
give a non-technical introduction.Comment: 19 pages, LaTeX, minor improvements (published version
Fermionic Matrix Models
We review a class of matrix models whose degrees of freedom are matrices with
anticommuting elements. We discuss the properties of the adjoint fermion one-,
two- and gauge invariant D-dimensional matrix models at large-N and compare
them with their bosonic counterparts which are the more familiar Hermitian
matrix models. We derive and solve the complete sets of loop equations for the
correlators of these models and use these equations to examine critical
behaviour. The topological large-N expansions are also constructed and their
relation to other aspects of modern string theory such as integrable
hierarchies is discussed. We use these connections to discuss the applications
of these matrix models to string theory and induced gauge theories. We argue
that as such the fermionic matrix models may provide a novel generalization of
the discretized random surface representation of quantum gravity in which the
genus sum alternates and the sums over genera for correlators have better
convergence properties than their Hermitian counterparts. We discuss the use of
adjoint fermions instead of adjoint scalars to study induced gauge theories. We
also discuss two classes of dimensionally reduced models, a fermionic vector
model and a supersymmetric matrix model, and discuss their applications to the
branched polymer phase of string theories in target space dimensions D>1 and
also to the meander problem.Comment: 139 pages Latex (99 pages in landscape, two-column option); Section
on Supersymmetric Matrix Models expanded, additional references include
Solving Generalized Equations with Bounded Variables and Multiple Residuated Operators
This paper studies the resolution of sup-inequalities and sup-equations with bounded variables such that the sup-composition is defined by using different residuated operators of a given distributive biresiduated multi-adjoint lattice. Specifically, this study has analytically determined the whole set of solutions of such sup-inequalities and sup-equations. Since the solvability of these equations depends on the character of the independent term, the resolution problem has been split into three parts distinguishing among the bottom element, join-irreducible elements and join-decomposable elements
Reflectionless analytic difference operators II. Relations to soliton systems
This is the second part of a series of papers dealing with an extensive class
of analytic difference operators admitting reflectionless eigenfunctions. In
the first part, the pertinent difference operators and their reflectionless
eigenfunctions are constructed from given ``spectral data'', in analogy with
the IST for reflectionless Schr\"odinger and Jacobi operators. In the present
paper, we introduce a suitable time dependence in the data, arriving at
explicit solutions to a nonlocal evolution equation of Toda type, which may be
viewed as an analog of the KdV and Toda lattice equations for the latter
operators. As a corollary, we reobtain various known results concerning
reflectionless Schr\"odinger and Jacobi operators. Exploiting a
reparametrization in terms of relativistic Calogero--Moser systems, we also
present a detailed study of -soliton solutions to our nonlocal evolution
equation
Supersymmetric KP Hierarchy: ``Ghost'' Symmetry Structure, Reductions and Darboux-Backlund Solutions
This paper studies Manin-Radul supersymmetric KP hierarchy (MR-SKP) in three
related aspects: (i) We find an infinite set of additional (``ghost'') symmetry
flows spanning the same (anti-)commutation algebra as the ordinary MR-SKP
flows; (ii) The latter are used to construct consistent reductions of the
initial unconstrained MR-SKP hierarchy which involves a nontrivial modification
for the fermionic flows; (iii) For the simplest constrained MR-SKP hierarchy we
show that the orbit of Darboux-Backlund transformations lies on a
supersymmetric Toda lattice being a square-root of the standard one-dimensional
Toda lattice, and also we find explicit Wronskian-ratio solutions for the
super-tau function.Comment: Minor corrections in few equations. LaTeX, 12 pg
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