1,920 research outputs found
The Epstein-Glaser approach to pQFT: graphs and Hopf algebras
The paper aims at investigating perturbative quantum field theory (pQFT) in
the approach of Epstein and Glaser (EG) and, in particular, its formulation in
the language of graphs and Hopf algebras (HAs). Various HAs are encountered,
each one associated with a special combination of physical concepts such as
normalization, localization, pseudo-unitarity, causality and an associated
regularization, and renormalization. The algebraic structures, representing the
perturbative expansion of the S-matrix, are imposed on the operator-valued
distributions which are equipped with appropriate graph indices. Translation
invariance ensures the algebras to be analytically well-defined and graded
total symmetry allows to formulate bialgebras. The algebraic results are given
embedded in the physical framework, which covers the two recent EG versions by
Fredenhagen and Scharf that differ with respect to the concrete recursive
implementation of causality. Besides, the ultraviolet divergences occuring in
Feynman's representation are mathematically reasoned. As a final result, the
change of the renormalization scheme in the EG framework is modeled via a HA
which can be seen as the EG-analog of Kreimer's HA.Comment: 52 pages, 5 figure
Ringel-Hall algebras of cyclic quivers
These are notes from my mini-course at ICRA13, Sao Paulo, 200
Coherence Constraints for Operads, Categories and Algebras
Coherence phenomena appear in two different situations. In the context of
category theory the term `coherence constraints' refers to a set of diagrams
whose commutativity implies the commutativity of a larger class of diagrams. In
the context of algebra coherence constrains are a minimal set of generators for
the second syzygy, that is, a set of equations which generate the full set of
identities among the defining relations of an algebraic theory.
A typical example of the first type is Mac Lane's coherence theorem for
monoidal categories, an example of the second type is the result of Drinfel'd
saying that the pentagon identity for the `associator' of a quasi-Hopf algebra
implies the validity of a set of identities with higher instances of this
associator.
We show that both types of coherence are governed by a homological invariant
of the operad for the underlying algebraic structure. We call this invariant
the (space of) coherence constraints. In many cases these constraints can be
explicitly described, thus giving rise to various coherence results, both
classical and new.Comment: 29 pages, LaTeX209, article 12pt + leqno style. A substantially
revised versio
- …