Related papers: Feynman integrals and motives
Even the uninitiated will know that Quantum Field Theory cannot be introduced systematically in just four lectures. I try to give a reasonably connected outline of part of it, from second quantization to the path-integral technique in…
Motivic homotopy theory is meant to play the role of algebraic topology, in particular homotopy theory, in the context of algebraic geometry. As proved by Oliver Rondigs and Paul Arne Ostvaer, this theory is closely connected to Voevodsky's…
In the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts…
This dissertation explores various nonlinear responses that arise from the rich interplay between quantum geometry, disorder, magnetism and topology in quantum materials. In addition to presenting generalizations of quantum kinetic theory,…
We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.
In seeking to arrive at a theory of ``quantum gravity'', one faces several choices among alternative approaches. I list some of these ``forks in the road'' and offer reasons for taking one alternative over the other. In particular, I…
This work deals with scalar field theories and supersymmetric quantum mechanics. The investigation is inspired by a recent result, which shows how to use the reconstruction mechanism to describe two distinct field theories from the very…
I offer some historical comments about the origins of Feynman's path integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's…
The dual relationship between two n-1 parameter families of quantum field theories based on extended complex numbers is investigated in two dimensions. The non-local conserved charges approach is used. The lowest rank affine Toda field…
In algebraic quantum field theory the spacetime manifold is replaced by a suitable base for its topology ordered under inclusion. We explain how certain topological invariants of the manifold can be computed in terms of the base poset. We…
We give a very concise review of the group field theory formalism for non-perturbative quantum gravity, a higher dimensional generalisation of matrix models. We motivate it as a simplicial and local realisation of the idea of 3rd…
A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general…
We consider coupling an ordinary quantum field theory with an infinite number of degrees of freedom to a topological field theory. On R^d the new theory differs from the original one by the spectrum of operators. Sometimes the local…
The relationship between the sources of physical fields and the fields themselves is investigated with regard to the coupling of topological information between them. A class of field theories that we call topological field theories is…
In this paper, I consider the issue of how two mathematical models of modern physics, the variational principles and the quantum path integral formalism, relate to reality. I assume that the observed phenomena are consistent with the…
This note, in a rather expository manner, serves as a conceptional introduction to the certain underlying mathematical structures encoding the geometric quantization formalism and the construction of Witten's quantum invariants, which is in…
For any given sequence of integers there exists a quantum field theory whose Feynman rules produce that sequence. An example is illustrated for the Stirling numbers. The method employed here offers a new direction in combinatorics and graph…
We discuss the meaning of background independence in quantum theories of gravity where geometry and gravity are emergent and illustrate the possibilities using the framework of quantum causal histories.
Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the…
Two criteria for planarity of a Feynman diagram upon its propagators (momentum flows) are presented. Instructive Mathematica programs that solve the problem and examples are provided. A simple geometric argument is used to show that while…