Related papers: Projective Techniques and Functional Integration
Integration is the final key step when turning an infinitesimal argument into a result applicable to quantities of finite size. Conceptually, it is about combining infinitesimal contributions to a finite whole. We make a first step towards…
We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces,…
We study finite-dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick's theorem allows…
Finite frames, or spanning sets for finite-dimensional Hilbert spaces, are a ubiquitous tool in signal processing. There has been much recent work on understanding the global structure of collections of finite frames with prescribed…
Construction and classification of 2D superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of…
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory…
This short review is devoted to measures on infinite dimensional spaces. We start by discussing product measures and projective techniques. Special attention is paid to measures on linear spaces, and in particular to Gaussian measures.…
Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection. Motivated by Givental's work on mirror symmetry, we use gauge theory techniques and the Frobenius Integrability Theorem to find flat sections for…
Graphical functions have emerged as a powerful framework for evaluating multi-loop Feynman integrals in perturbative quantum field theory. Defined as massless three-point position-space integrals, they reveal rich analytic structures and…
The algebraic approach to quantum mechanics has been vital to the development of quantum theory since its inception, and it has evolved into a mathematically rigorous $C^\ast$-algebraic formulation of the theory's axioms. Conversely, the…
In quantum mechanics, one can express the evolution operator and other quantities in terms of functional integrals. The main goal of this paper is to prove corresponding results in the geometric approach to quantum theory. We apply these…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
We consider the class of profinite diffeological spaces, that is, diffeological spaces which diffeologies are deduced by pull-back of diffeologies on finite-dimensional manifolds through a system of projection mappings. This class includes…
Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method…
The quantum gravity problem of N point particles interacting with the gravitational field in 2+1 dimensions is approached working out the phase-space functional integral. The maximally slicing gauge is adopted for a non compact open…
We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean)…
Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge…
Learning depth from spherical panoramas is becoming a popular research topic because a panorama has a full field-of-view of the environment and provides a relatively complete description of a scene. However, applying well-studied CNNs for…
Many generative models originally developed in finite-dimensional Euclidean space have functional generalizations in infinite-dimensional settings. However, the extension of rectified flow to infinite-dimensional spaces remains unexplored.…
Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply…