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Jerzy Kocik

Publications and source records attributed to Jerzy Kocik.

35 records · Page 2Linked to original sources
arXiv2018

Skein relations for spin networks, modified

An alternative framework underlying connection between tensor ${\rm sl}_2$-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet it preserves the standard chromatic graph evaluations. In contrast with the standard formulation, the background space is that of symmetric tensor spaces, which seems to be in accordance with the representation theory of ${\rm SL}(2)$. An example of Apollonian disk packing is shown to be a source of spin networks. The graph labeling is extended to non-integer values, resulting in the complex-values of chromatic evaluations.

math-ph↗
arXiv2018

Coupled-Ring Resonance and Unitary Groups

The group-theoretic content of photonic coupled microrings resonance phenomena is shown, in particular, an interesting emergence of pseudo-unitary group. The application to the resonance condition in a tri-microring configuration is solved exactly. A practical application of this work will be in the resonance frequency tuning based on the coupling coefficient, in particular, the Mach-Zehnder interferometer approach was analyzed for the coupling modulation.

physics.optics↗
arXiv2017

Casimir energy of Sierpinski triangles

Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. The Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is introduced and the Berry-Weyl conjecture is discussed for these geometries. We propose that for a class of fractals, comprising of compartmentalized cavities, it is possible to establish a finite value to the Casimir energy even while the Casimir energy of the individual cavities consists of divergent terms.

hep-th↗
arXiv2016

Cromlech, menhirs and celestial sphere: an unusual representation of the Lorentz group

We present a novel representation of the Lorentz group, the geometric version of which uses "reversions" of a sphere while the algebraic version uses pseudounitary 2x2 matrices over complex numbers and quaternions, and Clifford algebras in general. A remarkably simple formula for relativistic composition of velocities and an accompanying geometric construction follows. The method is derived from the diffeomorphisms of the celestial sphere induced by Lorentz boost.

math-ph↗
arXiv2016

Krawtchouk matrices from the Feynman path integral and from the split quaternions

An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the role of Krawtchouk matrices in different areas, including quantum information.

math.GR↗
arXiv2016

Relativistic observer and Maxwell equations

The Ehresmann connection on a fiber bundle that is not compatible with a (possible) Lie group structure is illustrated by the geometry of a general anholonomic observer in the Minkowski space. The 3D split of Maxwell's equations induces geometric terms that are the (generalized) curvature and torque of the connection. The notion of torque is introduced here as a Lie coalgebra-valued endomorphism field and measures the deviation of a connection from being principal.

math-ph↗
arXiv2014

A porism concerning cyclic quadrilaterals

We present a geometric theorem on a porism about cyclic quadrilaterals, namely the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as the circle.

math.MG↗
arXiv2014

Geometric diagram for relativistic addition of velocities

A geometric diagram that allows one to visualize the Poincaré formula for relativistic addition of velocities in one dimension is presented. An analogous diagram representing the angle sum formula for trigonometric tangent is given.

physics.hist-ph↗
arXiv2012

Clifford Algebras and Euclid's Parameterization of Pythagorean Triples

We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra $\mathbb{R}_{2,1}$, whose minimal version may be conceptualized as a 4-dimensional real algebra of "kwaternions." We observe that this makes Euclid's parameterization the earliest appearance of the concept of spinors. We present an analogue of the "magic correspondence" for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.

math.NT↗
arXiv2012

Duplex numbers, diffusion systems, and generalized quantum mechanics

We show that the relation between the Schrödinger equation and diffusion processes has an algebraic nature and can be revealed via the structure of "duplex numbers." This helps one to clarify that quantum mechanics cannot be reduced to diffusion theory. Also, a generalized version of quantum mechanics where $\C$ is replaced by a normed algebra with a unit is proposed.

math-ph↗
arXiv2012

Harmonic evolutions on graphs

We define the harmonic evolution of states of a graph by iterative application of the harmonic operator (Laplacian over $Z_2$). This provides graphs with a new geometric context and leads to a new tool to analyze them. The digraphs of evolutions are analyzed and classified. This construction can also be viewed as a certain topological generalization of cellular automata.

math.CO↗
arXiv2012

Canonical endomorphism field on a Lie algebra

We show that every Lie algebra is equipped with a natural $(1,1)$-variant tensor field, the "canonical endomorphism field", naturally determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We show its relevance for classical mechanics, in particular for Lax equations. We show that the space of Lax vector fields is closed under Lie bracket and we introduce a new bracket for vector fields on a Lie algebra. This bracket defines a new Lie structure on the space of vector fields.

math-ph↗
arXiv2007

Lens sequences

A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious ``underground'' sequences underlying them are discovered in this paper.

math.NT↗
arXiv2007

A theorem on circle configurations

A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.

math.HO↗
arXiv2007

Krawtchouk matrices from classical and quantum random walks

Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.

quant-ph↗
arXiv2007

Krawtchouk polynomials and Krawtchouk matrices

Krawtchouk matrices have as entries values of the Krawtchouk polynomials for nonnegative integer arguments. We show how they arise as condensed Sylvester-Hadamard matrices via a binary shuffling function. The underlying symmetric tensor algebra is presented. Our approach is used to solve Kac' formulation of the Ehrenfest urn model. Connections with quantum and classical random walks are shown as well as various extensions of the classical polynomials.

quant-ph↗