urs schreiber
Nov4-06, 03:27 PM
Dirk Kreimer and Alain Connes have developed a Hopf algebraic
description of the process of renormalization in quantum field theory.
Their starting point is the definition of a certain algebra structure
on the space of all (Feynman-)graphs. The definition can be found, for
instance, in equation (6) of http://arxiv.org/abs/hep-th/0510202 .
I suspect that this algebra is most elegantly expressed as the
"multicategory algebra" of the obvious multicategory of
(Feynman-)graphs. This multicategory should be the obvious slight
generalization of example 4.2.14 in Tom Leinster's book
http://arxiv.org/abs/math.CT/0305049.
Here by "multicategory algebra" I mean a generalization to
multicategories of the concept "category algebra" or "path algebra" of
an ordinary category. I.e. the algebra generated by the morphisms in
the category with the product derived from the composition law in the
category.
(Hence, in particular I am *not* referring to the concept "algebra
*for* a multicategory" as in "algebra for an operad", discussed in
section 4.3 of Leinster's book.)
I would think such a concept of a multicategory(-path-)algebra should
exist and should be given by the obvious generalization of formula (6)
in the above cited paper.
This concept seems to be so natural that I expect it must have been
considered before. If so, could anyone point me to relevant sources?
P.S.
I have more details on what I have in mind here:
http://golem.ph.utexas.edu/string/archives/000755.html
description of the process of renormalization in quantum field theory.
Their starting point is the definition of a certain algebra structure
on the space of all (Feynman-)graphs. The definition can be found, for
instance, in equation (6) of http://arxiv.org/abs/hep-th/0510202 .
I suspect that this algebra is most elegantly expressed as the
"multicategory algebra" of the obvious multicategory of
(Feynman-)graphs. This multicategory should be the obvious slight
generalization of example 4.2.14 in Tom Leinster's book
http://arxiv.org/abs/math.CT/0305049.
Here by "multicategory algebra" I mean a generalization to
multicategories of the concept "category algebra" or "path algebra" of
an ordinary category. I.e. the algebra generated by the morphisms in
the category with the product derived from the composition law in the
category.
(Hence, in particular I am *not* referring to the concept "algebra
*for* a multicategory" as in "algebra for an operad", discussed in
section 4.3 of Leinster's book.)
I would think such a concept of a multicategory(-path-)algebra should
exist and should be given by the obvious generalization of formula (6)
in the above cited paper.
This concept seems to be so natural that I expect it must have been
considered before. If so, could anyone point me to relevant sources?
P.S.
I have more details on what I have in mind here:
http://golem.ph.utexas.edu/string/archives/000755.html