Related papers: Algebraic geometry informs perturbative quantum fi…
In this paper we discuss some questions about geometry over the field with one element, motivated by the properties of algebraic varieties that arise in perturbative quantum field theory. We follow the approach to F1-geometry based on…
The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas:…
We explain how to tropicalize scalar quantum field theory and show that tropicalized massive scalar quantum field theory is exactly solvable. This exact solution manifests as a non-linear recursion equation fulfilled by the expansion…
This model is one of the possible geometrical interpretations of Quantum Mechanics where found to every image Path correspondence the geodesic trajectory of classical test particles in the random geometry of the stochastic fields…
A Feynman period is a particular residue of a scalar Feynman integral which is both physically and number theoretically interesting. Two ways in which the graph theory of the underlying Feynman graph can illuminate the Feynman period are…
We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms,…
Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical…
We reformulate the analysis of singularities of Feynman integrals in a way that can be practically applied to perturbative computations in the Standard Model in dimensional regularization. After highlighting issues in the textbook treatment…
A recursive algebraic method which allows to obtain the Feynman or Schwinger parametric representation of a generic L-loops and (E+1) external lines diagram, in a scalar $\phi ^{3}\oplus \phi ^{4}$ theory, is presented. The representation…
We calculate 3-loop master integrals for heavy quark correlators and the 3-loop QCD corrections to the $\rho$-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be…
In this review, we summarize the main ideas of perturbative algebraic quantum field theory, which is a rigorous framework combining some of the Haag-Kastler axioms with perturbative methods involving formal power series. It allows for the…
We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a…
Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular,…
Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have achieved remarkable…
The physical phase space in gauge systems is studied. Effects caused by a non-Euclidean geometry of the physical phase space in quantum gauge models are described in the operator and path integral formalisms. The projection on the Dirac…
In this paper, we study structure theorems of algebras of symmetric functions. Based on a certain relation on elementary symmetric polynomials generating such algebras, we consider perturbation in the algebras. In particular, we understand…
Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of…
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…
The paper puts together some loosely connected observations, old and new, on the concept of a quantum field and on the properties of Feynman amplitudes. We recall, in particular, the role of (exceptional) elementary induced representations…
Present and future high-precision tests of the Standard Model and beyond for the fundamental constituents and interactions in Nature are demanding complex perturbative calculations involving multi-leg and multi-loop Feynman diagrams.…