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We construct two complete invariants of oriented classical knots in space. The value of each invariant on any knot is a set, infinite for the first invariant and finite for the second. The finite set is computed algorithmically from any…
We present the alternative topological twisting of N=4 Yang-Mills, in which the path integral is dominated not by instantons, but by flat connections of the COMPLEXIFIED gauge group. The theory is nontrivial on compact orientable…
The second-order temporal interference between two independent single-mode continuous-wave lasers is discussed by employing two-photon interference in Feynman's path integral theory. It is concluded that whether the second-order temporal…
We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of…
Some well-known examples of constrained quantum systems commonly quantized via Feynman path integrals are re-examined using the notion of conditional integrators introduced in [1]. The examples yield some new perspectives on old results. As…
We model a quantum walk of identical particles that can change their exchange statistics by hopping across a domain wall in a 1D lattice. Such a "statistical boundary" is transparent to single particles and affects the dynamics only by…
We introduce a new graph polynomial in two variables. This ``interlace'' polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant…
We introduce a technique for recovering a sufficiently smooth function from its ray transform over a wide class of curves in a general region of Euclidean space. The method is based on a complexification of the underlying vector fields…
Our rigorous path integrals costruction for the evolution operators is extended to metric-affine manifolds.
This paper proposes a numerical method using neural networks to solve the path integral problem in quantum mechanics for arbitrary potentials. The method is based on a radial basis function expansion of the interaction term that appears in…
I investigate the relationship between the phase space path integral in (2+1)-dimensional gravity and the canonical quantization of the corresponding reduced phase space in the York time slicing. I demonstrate the equivalence of these two…
The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to…
The Feynman path integrals for the magnetic Schroedinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials…
We study the behaviour of functions of pairs of commuting self-adjoint operators under perturbations by relatively bounded operators. We obtain analogs of our earlier results for functions of a single self-adjoint operator under relatively…
A path-integral approach to non-perturbative topological invariants of knots, links and manifolds of dimension three and four using topological quantum field theory of Schwarz (Chern-Simons) type is presented.
In this paper, we have defined bicomplex valued functions of bounded variations and rectifiable hyperbolic path. We have studied the integration of product-type bicomplex functions over rectifiable hyperbolic path. Also we have established…
Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.
In this paper we develop a structure called Link Algebra, in which we present a Set with two binary operations and an axiom system developed from the study of graph theory and set/antiset theory, sowing main theorems and definitions. Once…
Coupled instantons are introduced by generalizing the double well potential to multiple mutually coupled wells. Physically this corresponds to the simultaneous tunneling of multiple degrees of freedom. A system with four equal minima is…
The transport of energetic particles in a spatially varying magnetic field is described by the focused transport equation. In the past two versions of this equation were investigated. The more commonly used standard form described a…