Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantumatics: A New Mathematics of Superposed Operators and its Equivalence to the Path Integral Formulation

Sakhile Nyembe

Published Aug 11, 2026• 0 citations
Publisher

Abstract

This paper introduces Quantumatics, a superposed operator framework that bypasses Hilbert space notation and is shown to be mathematically equivalent to Feynman's Path Integral Formulation.
DOI
10.5281/zenodo.21892616

Related papers

Operator functionals and path integrals

C. Kourkoumelis, Stephen J. Nettel

In a paper entitled ’’An Operator Calculus Having Applications in Quantum Electrodynamics’’ [Phys. Rev. 84, 108 (1951)], Feynman gives techniques and theorems for working with ordered operators. We believe that the entire development, because of its direct involvement in the transition from the Heisenberg–Schrödinger to Feynman’s path-integral formulation of quantum mechanics, is of general interest. We aim to make these less appreciated early techniques of Feynman more accessible to the general physicist.

American Journal of Physics•Jan 1, 1977•2 citations

Mathematical Feynman Path Integrals and Their Applications

Sonia Mazzucchi

"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--

WORLD SCIENTIFIC eBooks•Nov 7, 2019•3 citations

Mathematical Concepts in Physics

Tepper L. Gill

Abstract The purpose of this paper show how mathematical concepts not designed for physics can muddle understanding and/or prevent progress. We provide the following examples to justify our position: 1. A proof that Minkowski’s (third) postulate is incompatible with Einstein’s first postulate for two or more particles. 2. A proof that the Dirac equation is not physically equivalent to the square root equation but is related by a unitary transformation. 3. The design of a Hilbert space for Feynman’s path integral formulation of quantum theory KS 2[ℝ n ], which is not physically equivalent to L 2[ℝ n ], the Hilbert space for the Schödinger formulation, but is related by a unitary transformation. A major contribution of this paper is that mathematical equivalence is not the same as physical equivalence and that unbiased physical analysis and justification of mathematical concepts should be required before acceptance into physical theory.

Journal of Physics Conference Series•Apr 1, 2025•0 citations

Feynman quantization: An operator approach to path integration over commuting and anticommuting variables

V. Janiš

Czechoslovak Journal of Physics•Aug 1, 1990•1 citation

Quantum Register Algebra: the mathematical language for quantum computing

Jaroslav Hrdina, Dietmar Hildenbrand, Aleš Návrat +5 more

arXiv:2208.02608•Quantum Information Processing•Aug 31, 2023•8 citations

On the physical and mathematical foundations of quantum physics via functional integrals

Giampiero V. M. Esposito, Tepper L. Gill

In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman–DeWitt–Faddeev–Popov ghost fields. Such fields result from the fiber-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock–Kurzweil integral is first introduced, because it makes it possible to integrate functions like [Formula: see text]. The Lebesgue measure on [Formula: see text] is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space [Formula: see text], which contains [Formula: see text] as a continuous dense embedding, and contains both the test functions [Formula: see text] and their dual [Formula: see text], the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to [Formula: see text], where [Formula: see text] is any separable Hilbert space. Last, the existence of a unique universal definition of time, [Formula: see text], that we call historical time, is proven. We use [Formula: see text] as the order parameter for our construction of Feynman’s time ordered operator calculus, which in turn is used to extend the path integral in order to include all time-dependent groups and semigroups with a reproducing kernel representation.

arXiv:2408.16404•International Journal of Modern Physics A•Jun 12, 2025•1 citation
In a paper entitled ’’An Operator Calculus Having Applications in Quantum Electrodynamics’’ [Phys. Rev. 84, 108 (1951)], Feynman gives techniques and theorems for working with ordered operators. We believe that the entire development, because of its direct involvement in the transition from the Heisenberg–Schrödinger to Feynman’s path-integral formulation of quantum mechanics, is of general interest. We aim to make these less appreciated early techniques of Feynman more accessible to the general physicist.
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
Abstract The purpose of this paper show how mathematical concepts not designed for physics can muddle understanding and/or prevent progress. We provide the following examples to justify our position: 1. A proof that Minkowski’s (third) postulate is incompatible with Einstein’s first postulate for two or more particles. 2. A proof that the Dirac equation is not physically equivalent to the square root equation but is related by a unitary transformation. 3. The design of a Hilbert space for Feynman’s path integral formulation of quantum theory KS 2[ℝ n ], which is not physically equivalent to L 2[ℝ n ], the Hilbert space for the Schödinger formulation, but is related by a unitary transformation. A major contribution of this paper is that mathematical equivalence is not the same as physical equivalence and that unbiased physical analysis and justification of mathematical concepts should be required before acceptance into physical theory.
In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman–DeWitt–Faddeev–Popov ghost fields. Such fields result from the fiber-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock–Kurzweil integral is first introduced, because it makes it possible to integrate functions like [Formula: see text]. The Lebesgue measure on [Formula: see text] is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space [Formula: see text], which contains [Formula: see text] as a continuous dense embedding, and contains both the test functions [Formula: see text] and their dual [Formula: see text], the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to [Formula: see text], where [Formula: see text] is any separable Hilbert space. Last, the existence of a unique universal definition of time, [Formula: see text], that we call historical time, is proven. We use [Formula: see text] as the order parameter for our construction of Feynman’s time ordered operator calculus, which in turn is used to extend the path integral in order to include all time-dependent groups and semigroups with a reproducing kernel representation.