Chern-Simons Theory, Vassiliev Invariants, Loop Quantum Gravity and Functional Integration Without Integration

TL;DR

This paper explores the mathematical relationship between Chern-Simons theory and Vassiliev invariants, presenting a measure-free approach that enhances the rigor of functional integral concepts and discusses implications for loop quantum gravity.

Contribution

It introduces a measure-free, rigorous framework for understanding Chern-Simons functional integrals and their connection to knot invariants, with applications to loop quantum gravity.

Findings

Establishes a measure-free conceptualization of the Chern-Simons functional integral.

Links Vassiliev invariants to Chern-Simons theory without traditional measure theory.

Discusses implications for the mathematical foundations of loop quantum gravity.

Abstract

This paper is an exposition of the relationship between Witten's Chern-Simons functional integral and the theory of Vassiliev Invariants of knots and links in three dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory. This approach makes it possible to discuss the mathematics intrinsic to the functional integral rigorously and without functional integration. Applications to loop quantum gravity are discussed. We thank the organizers of the Conference on 60 Years of Yang-Mills Gauge Field Theories (25 to 28 May 2015) for the invitation and opportunity to speak about these ideas in Singapore.