Related papers: Curves and The Photon
The entanglement of open curves in 3-space appears in many physical systems and affects their material properties and function. A new framework in knot theory was introduced recently, that enables to characterize the complexity of…
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical…
The fractional quantum and statistical mechanics have been developed via new path integrals approach.
We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully…
The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and…
Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to…
We study integral points on affine surfaces by means of a new method, relying on the Subspace Theorem. Under suitable assumptions on the divisor at infinity, we prove that the integral points are contained in a curve. As a corollary, we…
This paper is a survey of author's mathematical and logical study of the problem of quantization of fields.
We demonstrate in theory and experiment the strict equivalence between nonclassical polarization and the entanglement of indistinguishable photons, thereby unifying these two phenomena that appear dissimilar at first sight. This allows us…
The concepts of geometric phase and wave-particle duality are interlinked to several fundamental phenomena in quantum physics, but their mutual relationship still forms an uncharted open problem. Here we address this question by studying…
A permutation may be represented by a collection of paths in the plane. We consider a natural class of such representations, which we call tangles, in which the paths consist of straight segments at 45 degree angles, and the permutation is…
In this paper, we give the quantum wave equations of single photon when it is in the free or medium space. With these equations, we can study light interference and diffraction with quantum approach. Otherwise, they can be applied in…
A reinterpretation of noncommutativity as a mapping of paths is proposed at the level of quantum mechanics.
Photonic quantum computation refers to quantum computation that uses photons as the physical system for doing the quantum computation. The field is largely divided between discrete-variable (DV) and continuous-variable (CV) photonic quantum…
Improved measurements of the proton's structure are now possible thanks to significant technical advances that allow us to probe the proton with polarized photons. These measurements have shown that the proton is not as simple as previously…
We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.
We reconsider the Euclidean version of the photon number integral introduced in ref 1. This integral is well defined for any smooth non-self-intersecting curve in $\R^N$. Besides studying general features of this integral (including it s…
The quantum statistics of particles is determined by both the spins and the indistinguishability of quantum states. Here we studied the quantum statistics of partially distinguishable photons by defining the multi-photon…
In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small…
In the past 20 years, compactifications of the families of curves in algebraic varieties X have been studied via stable maps, Hilbert schemes, stable pairs, unramified maps, and stable quotients. Each path leads to a different enumeration…