Canonical forms of Self-Adjoint Operators in Minkowski Space-time

TL;DR

This paper classifies the Jordan canonical forms of self-adjoint operators in Minkowski space-time and provides a method to obtain a canonical basis that simplifies both the operator and the metric.

Contribution

It introduces a classification of Jordan forms for self-adjoint operators in indefinite inner product spaces and details how to construct a canonical basis for these forms.

Findings

Complete classification of Jordan forms for self-adjoint operators in Minkowski space-time

Method to construct a canonical basis that simplifies the metric and operator

Framework applicable to pseudo-Euclidean spaces

Abstract

We classify the possible Jordan canonical forms of self-adjoint operators in Minkowski space-time (in fact in pseudo-Euclidean space, i.e. an indefinite inner product space) and we show how to obtain a Jordan canonical basis which also puts the metric in a canonical form.