A Royal Road to Quantum Theory (or Thereabouts)

TL;DR

This paper presents a simple framework using Euclidean Jordan algebras to represent finite-dimensional probabilistic systems, unifying real, complex, and quaternionic quantum mechanics with room for exotic theories.

Contribution

It introduces an easy derivation of quantum theory representations via Euclidean Jordan algebras from basic assumptions, unifying different types of quantum mechanics.

Findings

Unified framework for real, complex, and quaternionic quantum mechanics

Representation of probabilistic systems using Euclidean Jordan algebras

Some room for exotic alternative theories

Abstract

This paper fails to derive quantum mechanics from a few simple postulates. But it gets very close --- and it does so without much exertion. More exactly, I obtain a representation of finite-dimensional probabilistic systems in terms of euclidean Jordan algebras, in a strikingly easy way, from simple assumptions. This provides a framework within which real, complex and quaternionic QM can play happily together, and allows some --- but not too much --- room for more exotic alternatives. (This is a leisurely summary, based on recent lectures, of material from the papers arXiv:1206:2897 and arXiv:1507.06278, the latter joint work with Howard Barnum and Matthew Graydon. Some further ideas are also explored.)