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

  title={Krawtchouk matrices from the Feynman path integral and from the split quaternions},
  author={Jerzy Kocik},
  journal={arXiv: Group Theory},
  • Jerzy Kocik
  • Published 1 April 2016
  • Mathematics, Physics
  • arXiv: Group Theory
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. 
Lucas, Fibonacci, and Chebyshev polynomials from matrices
A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.
Spinors and Descartes configurations of disks
We define spinors for pairs of tangent disks in the Euclidean plane and prove a number of theorems, one of which may be interpreted as a "square root of Descartes Theorem". In any Apollonian diskExpand


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 shufflingExpand
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 valuesExpand
The spectrum of symmetric Krawtchouk matrices
Abstract Symmetric Krawtchouk matrices are introduced as a modification of Krawtchouk matrices, whose entries are values of the Krawtchouk polynomials. Of particular interest are spectral properties.
Clifford Algebras and Euclid’s Parametrization of Pythagorean Triples
Abstract.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 R21, whoseExpand
On Krawtchouk polynomials
The aim of this article is to fill in the gap in detailed development of krawtchouk polynomials in coding theory and graph theory. Expand
Krawtchouk Polynomials and Finite Probability Theory
Some general remarks on random walks and martingales for finite probability distributions are presented. Orthogonal systems for the multinomial distribution arise. In particular, a class ofExpand
Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces
  • V. Levenshtein
  • Mathematics, Computer Science
  • IEEE Trans. Inf. Theory
  • 1995
Universal bounds for the cardinality of codes in the Hamming space F/sub r//sup n/ with a given minimum distance d and/or dual distance d' are stated. A self-contained proof of optimality of theseExpand
A Rosetta Stone for Quantum Mechanics with an Introduction to Quantum Computation
The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to beginExpand
Orthogonal Polynomials
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
The Theory of Error-Correcting Codes
Linear Codes. Nonlinear Codes, Hadamard Matrices, Designs and the Golay Code. An Introduction to BCH Codes and Finite Fields. Finite Fields. Dual Codes and Their Weight Distribution. Codes, DesignsExpand