Odd Dimensional Translation between Complex and Quaternionic Mechanics

TL;DR

This paper develops a comprehensive method for translating between complex and quaternionic quantum mechanics, addressing previous limitations and resolving issues with matrix reduction and eigenstates.

Contribution

It introduces a new reduction technique for complex matrices to quaternionic form, enhancing the translation rules between CQM and QQM.

Findings

Resolved reduction of complex matrix groups in quaternionic QM.

Avoided anomalous eigenstates in quaternionic representations.

Provided a complete translation framework between CQM and QQM.

Abstract

We complete the rules of translation between standard complex quantum mechanics (CQM) and quaternionic quantum mechanics (QQM) with a complex geometry. In particular we describe how to reduce ($2n$+$1$)-dimensional complex matrices to {\em overlapping\/} ($n$+$1$)-dimensional quaternionic matrices with generalized quaternionic elements. This step resolves an outstanding difficulty with reduction of purely complex matrix groups within quaternionic QM and avoids {\em anomalous} eigenstates. As a result we present a more complete translation from CQM to QQM and viceversa.