Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics

TL;DR

This paper explores the mathematical properties of pseudo-unitary operators in quantum systems, including their structure, exponential relations, and applications to quantum dynamics and classical mechanics.

Contribution

It provides a characterization of pseudo-unitary operators, links them to pseudo-Hermitian matrices, and applies these concepts to quantum and classical systems.

Findings

Characterization theorem for block-diagonalizable pseudo-unitary operators

Every pseudo-unitary matrix is the exponential of i times a pseudo-Hermitian matrix

Analysis of 2x2 pseudo-unitary matrices and their quantum system applications

Abstract

We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of $i=\sqrt{-1}$ times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudo-unitary matrices. In particular, we present a thorough treatment of $2\times 2$ pseudo-unitary matrices and discuss an example of a quantum system with a $2\times 2$ pseudo-unitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group $Sp(2n)$ with the real subgroup of a matrix group that is isomorphic to the pseudo-unitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudo-unitary dynamical groups.