610 research outputs found
An operator representation for Matsubara sums
In the context of the imaginary-time formalism for a scalar thermal field
theory, it is shown that the result of performing the sums over Matsubara
frequencies associated with loop Feynman diagrams can be written, for some
classes of diagrams, in terms of the action of a simple linear operator on the
corresponding energy integrals of the Euclidean theory at T=0. In its simplest
form the referred operator depends only on the number of internal propagators
of the graph.
More precisely, it is shown explicitly that this \emph{thermal operator
representation} holds for two generic classes of diagrams, namely, the
two-vertex diagram with an arbitrary number of internal propagators, and the
one-loop diagram with an arbitrary number of vertices.
The validity of the thermal operator representation for diagrams of more
complicated topologies remains an open problem. Its correctness is shown to be
equivalent to the correctness of some diagrammatic rules proposed a few years
ago.Comment: 4 figures; references added, minor changes in notation, final version
accepted for publicatio
Amalgams of Inverse Semigroups and C*-algebras
An amalgam of inverse semigroups [S,T,U] is full if U contains all of the
idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra
of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam
of C*(S) and C*(T) over C*(U). Using this result, we describe certain
amalgamated free products of C*-algebras, including finite-dimensional
C*-algebras, the Toeplitz algebra, and the Toeplitz C*-algebras of graphs
The hardness of the iconic must: Can Peirceâs existential graphs assist modal epistemology?
Charles Peirceâs diagrammatic logic - the Existential Graphs - is presented as a tool for illuminating how we know necessity, in answer to Benacerrafâs famous challenge that most âsemantics for mathematicsâ do not âfit an acceptable epistemologyâ. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not just to individual things but to how those things are related in larger structures, certain aspects of which force certain others to be a particular way
Ward identities and combinatorics of rainbow tensor models
We discuss the notion of renormalization group (RG) completion of
non-Gaussian Lagrangians and its treatment within the framework of
Bogoliubov-Zimmermann theory in application to the matrix and tensor models.
With the example of the simplest non-trivial RGB tensor theory (Aristotelian
rainbow), we introduce a few methods, which allow one to connect calculations
in the tensor models to those in the matrix models. As a byproduct, we obtain
some new factorization formulas and sum rules for the Gaussian correlators in
the Hermitian and complex matrix theories, square and rectangular. These sum
rules describe correlators as solutions to finite linear systems, which are
much simpler than the bilinear Hirota equations and the infinite Virasoro
recursion. Search for such relations can be a way to solving the tensor models,
where an explicit integrability is still obscure.Comment: 48 page
Introductory lectures to loop quantum gravity
We give a standard introduction to loop quantum gravity, from the ADM
variables to spin network states. We include a discussion on quantum geometry
on a fixed graph and its relation to a discrete approximation of general
relativity.Comment: Based on lectures given at the 3eme Ecole de Physique Theorique de
Jijel, Algeria, 26 Sep -- 3 Oct, 2009. 52 pages, many figures. v2 minor
corrections. To be published in the proceeding
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