Related papers: On the physical and mathematical foundations of qu…
Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive…
The paper is the second part of the work devoted to the problem of time in quantum cosmology. Here we consider in detail two approaches within the scope of Feynman path integration scheme: The first, by Simeone and collaborators, is…
The existence of a quantum field theory over the "field with one element" was first addressed in 2012 by Bejleri and Marcolli, where it was shown that wonderful compactifications of the graph configuration spaces that appear in the…
We define and develop a framework to understand functional integrals as countable families of Banach-valued Haar integrals on locally compact topological groups. The definition forgoes the goal of constructing a genuine measure on an…
Well defined quantum field theory (QFT) for the electroweak force including quantum electrodynamics (QED) and the weak force is obtained by considering natural unitary representations of a group $K\subset U(2,2)$, where $K$ is locally…
Normally we quantize along the space dimensions but treat time classically. But from relativity we expect a high level of symmetry between time and space. What happens if we quantize time using the same rules we use to quantize space? To do…
We re-use some original ideas of de~Broglie, Schr\"odiger, Dirac and Feynman to revise the ensemble interpretation of wave function in quantum mechanics. To this end we introduce coherence (auto-concordance) of ensembles of quantum…
We propose a general theoretical approach to quantum measurements based on the path (histories) summation technique. For a given dynamical variable A, the Schr\"odinger state of a system in a Hilbert space of arbitrary dimensionality is…
Many interesting physical theories have analytic classical actions. We show how Feynman's path integral may be defined non-perturbatively, for such theories, without a Wick rotation to imaginary time. We start by introducing a class of…
The Feynman Path Integral is extended in order to capture all solutions of a quantum field theory. This is done via a choice of appropriate integration cycles, parametrized by M in SL(2,C), i.e., the space of allowed integration cycles is…
By adapting Feynman's sum over paths method to a quantum mechanical system whose phase space is a torus, a new proof of the Landsberg-Schaar identity for quadratic Gauss sums is given. In contrast to existing non-elementary proofs, which…
In the Feynman-Kac[1] path integral approach the eigenvalues of a quantum system can be computed using Wiener measure which uses Brownian particle motion. In our previous work[2-3] on such systems we have observed that the Wiener process…
This paper summarizes and generalizes a recently proposed mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time…
Ralph Henstock (1923 - 2007) worked in non-absolute integration, including the Riemann-complete or gauge integral which, independently, Jaroslav Kurzweil also discovered in the 1950's. As a Cambridge undergraduate Henstock took a course of…
Picard--Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its…
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…
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this…
The theory of integration over R is rich with techniques as well as necessary and sufficient conditions under which integration can be performed. Of the many different types of integrals that have been developed since the days of Newton and…
We describe how to construct and compute unambiguously path integrals for particles moving in a curved space, and how these path integrals can be used to calculate Feynman graphs and effective actions for various quantum field theories with…
This article begins with a review of quantum measure spaces. Quantum forms and indefinite inner-product spaces are then discussed. The main part of the paper introduces a quantum integral and derives some of its properties. The quantum…