Related papers: On the physical and mathematical foundations of qu…
The Feynman path integral is defined over the space $\mathbb{R}^T$ of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration…
These notes were inspired by the course ''Quantum Field Theory from a Functional Integral Point of View'' given at the University of Zurich in Spring 2017 by Santosh Kandel. We describe Feynman's path integral approach to quantum mechanics…
This paper is a survey of a new family of Banach spaces ${KS}^2$ and $SD^2$ that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the $L^p$ spaces provide for the Lebesgue integrable functions. These spaces…
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 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More…
In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. (The theory is constructive in that, operators acting at different times, actually commute.) We first develop an operator version of the…
One of the key elements of Feynman's formulation of non-relativistic quantum mechanics is a so-called Feynman path integral. It plays an important role in the theory, but it appears as a postulate based on intuition rather than a…
Finsler geometry motivates a generalization of the Riemannian structure of spacetime to include dependence of the spacetime metric and associated invariant tensor fields on the four-velocity coordinates as well as the spacetime coordinates…
We introduce and investigate the Henstock-Kurzweil (HK) integral for Riesz-space-valued functions on time scales. Some basic properties of the HK delta integral for Riesz-space-valued functions are proved. Further, we prove uniform and…
Feynman proposed a postulate or a method of quantization in his celebrated paper in 1948. Applying Feynman's postulate to temporally continuous quantum measurements of the positions of particles, Mensky proposed the restricted Feynman path…
Feynman path integrals are now a standard tool in quantum physics and their use in differential geometry leads to new mathematical insights. A logical treatment of quantum phenomena seems to require a sustained mathematical analysis of path…
We formulate quantum mechanics in spacetimes with real-order fractional geometry and more general factorizable measures. In spacetimes where coordinates and momenta span the whole real line, Heisenberg's principle is proven and the…
The Feynman path integral approach to quantum mechanics is examined in the case where the configuration space is curved. It is shown how the ambiguity that is present in the choice of path integral measure may be resolved if, in addition to…
Feynman's Lagrangian path integral was an outgrowth of Dirac's vague surmise that Lagrangians have a role in quantum mechanics. Lagrangians implicitly incorporate Hamilton's first equation of motion, so their use contravenes the uncertainty…
This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of the Feynman path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional…
One of several possibilities to construct a quantum theory of gravity is employing the Feynman path integral. This approach is plagued by some problems: the integration measure is not uniquely defined, the Einstein-Hilbert action unbounded,…
Discretizations of the Feynman-Kac path integral representation of the quantum mechanical density matrix are investigated. Each infinite-dimensional path integral is approximated by a Riemann integral over a finite-dimensional function…
The general boundary formulation of quantum field theory is applied to a massive scalar field in two dimensional Rindler space. The field is quantized according to both the Schr\"odinger-Feynman quantization prescription and the holomorphic…
Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these…
Quantum theory and functional analysis were created and put into essentially their final form during similar periods ending around 1930. Each was also a key outcome of the major revolutions that both physics and mathematics as a whole…