Linear representations of SU(2) described by using Kravchuk polynomials

TL;DR

This paper introduces the Kravchuk transform, a unitary transform based on Kravchuk polynomials, providing a new way to describe SU(2) representations with applications in quantum mechanics and signal processing.

Contribution

It presents a novel unitary transform using Kravchuk polynomials to describe SU(2) representations, offering deeper structural insights and computational advantages.

Findings

Defined the Kravchuk transform with properties similar to the finite Fourier transform.

Provided an alternative description of SU(2) irreducible representations.

Suggested applications in quantum mechanics and signal processing.

Abstract

We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU(2). Our approach offers a deeper insight into the structure of the linear representations of SU(2) and new possibilities of computation, very useful in applications in quantum mechanics, quantum information, signal and image processing.