Exponentiation and decomposition formulas for common operators 1: Classical applications

TL;DR

This paper provides a comprehensive tutorial on exponentiation and decomposition formulas for key matrix groups in physics, highlighting their mathematical properties, isomorphisms, and applications in classical dynamics and optics.

Contribution

It introduces explicit formulas and relations for matrix exponentiation and decomposition in symplectic, unitary, and orthogonal groups, emphasizing their isomorphisms and practical utility.

Findings

Sp(2) is isomorphic to SU(1,1) and SO(1,2)

SU(2) is isomorphic to SO(3)

Decomposition formulas facilitate understanding of physical systems

Abstract

In this tutorial, exponentiation and factorization (decomposition) formulas are derived and discussed for common matrix operators that arise in studies of classical dynamics, linear and nonlinear optics, and special relativity. To understand the physical properties of systems of common interest, one first needs to understand the mathematical properties of the symplectic group Sp(2), the special unitary groups SU(2) and SU(1,1), and the special orthogonal groups SO(3) and SO(1,2). For these groups, every matrix can be written as the exponential of a generating matrix, which is a linear combination of three fundamental matrices (generators). For Sp(2), SU(1,1) and SO(1,2), every matrix also has a Schmidt decomposition, in which it is written as the product of three simpler matrices. The relations between the entries of the matrix, the generator coefficients and, where appropriate, the Schmidt-decomposition parameters are described in detail. It is shown that Sp(2) is isomorphic to (has the same structure as) SU(1,1) and SO(1,2), and SU(2) is isomorphic to SO(3). Several examples of these isomorphisms (relations between Schmidt decompositions and product rules) are described, which illustrate their usefulness (complicated results can be anticipated or derived easily). This tutorial is written at a level that is suitable for senior undergraduate students and junior graduate students.