Related papers: Differential equations and Feynman integrals
We discuss the differential equation method for establishing dynamic concentration of discrete random processes. We present several relatively simple examples of it and aim to make the method understandable to the unfamiliar reader who has…
In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form…
This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.
Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…
In this talk we discuss mathematical structures associated to Feynman graphs. Feynman graphs are the backbone of calculations in perturbative quantum field theory. The mathematical structures -- apart from being of interest in their own…
This paper recalls the development of gauge theory culminating in Yang-Mills theory, and the application of differential geometry including connections on fiber bundles to field theory. Finally, we see how the preceding is used to explain…
Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms -- whose properties as…
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be…
Integration By Parts (IBP) is an important method for computing Feynman integrals. This work describes a formulation of the theory involving a set of differential equations in parameter space, and especially the definition and study of an…
We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.
This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a…
Based on the Baikov representation, we present a systematic approach to compute cuts of Feynman Integrals, appropriately defined in $d$ dimensions. The information provided by these computations may be used to determine the class of…
We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.
We provide a new method to calculate the full microlocal description of singularities of Feynman integrals. This is done by associating a unique constructible function to the system of partial differential equations (PDEs) annihilating the…
This talk reviews recent developments in the field of analytical Feynman integral calculations. The central theme is the geometry associated to a given Feynman integral. In the simplest case this is a complex curve of genus zero (aka the…
We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The…
In this paper we introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of…
We reformulate differential equations (DEs) for Feynman integrals to avoid doubled propagators in intermediate steps. External momentum derivatives are dressed with loop momentum derivatives to form tangent vectors to unitarity cut…
Some well-known examples of constrained quantum systems commonly quantized via Feynman path integrals are re-examined using the notion of conditional integrators introduced in [1]. The examples yield some new perspectives on old results. As…
In this paper, we revisit the diffusive representations of fractional integrals established in \cite{diethelm2023diffusive} to explore novel variants of such representations which provide highly efficient numerical algorithms for the…