Feynman geometry

TL;DR

This paper introduces Feynman geometries, a new framework for defining quantum field theories, distinguishing between strong and weak types based on the dimensionality of $A_ abla$ structures, and constructs families with continuum limits.

Contribution

It defines Feynman geometries and classifies them into strong and weak types, providing a foundation for quantum field theories and constructing continuum limit families.

Findings

Strong Feynman geometries have finite-dimensional $A_ abla$ structures.

Weak Feynman geometries have infinite-dimensional $A_ abla$ structures with trace-class operators.

Constructed families of Feynman geometries approach a continuum limit.

Abstract

In this paper we introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of $A_\infty$ structures is finite dimensional. A weak Feynman geometry is a geometry when the vector space of $A_\infty$ structures is infinite dimensional while the relevant operators are of trace-class. We construct families of Feynman geometries with "continuum" as their limit.