

Categorical and analytic invariants in Algebraic geometry 1
September 16, 2015 11:50–12:50, Moscow, Steklov Mathematical Institute






From Riemann to Feynman geometry in Feynman approach to QFT
A. S. Losev^{} 
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Abstract:
In the Feynmann approach to QFT correlators are given as an integrals
over the space of fields. Since the space of fields is
infinitedimensional the integral has no mathematical sense, and in
practical applications this definition is accompanied by the special
rules of computation of the integral.
These rules (called renormalization procedures) differ from theory to
theory, and people are not even trying to prove that different rules
lead to
equivalent definitions.
We propose the following radical change of the situation. The
Riemannian
geometry with infinitedimensional space of fields should be replaced
by
socalled Feymnan
geometry where the space is replaced by an Ainfinity structure that is
either finite dimensional(strong Feymnan geometry) or
infinitedimensional with operations belong to
the traceclass (weekly Feynman) geometry. We give an example of such
geometries: lattice Ainfinity geometry and noncommutative fuzzy
geometry as examples of strongly Feynmann geometries,
and Costello Ainfinity geometry as an example of weekly Feynman
geometry, and discuss a program of reformulation of QFT, in which
standard infinite dimensional integral should be replaced by a
limit of the existing integrals over fields in Feynman geometries.
Joint work with Sen Hu, USTC
Language: English

