A Royal Road to Quantum Theory (or Thereabouts), Extended Abstract

TL;DR

This paper presents a simplified framework for representing finite-dimensional probabilistic models using Jordan algebras, unifying real, complex, and quaternionic quantum mechanics, and explores their categorical organization.

Contribution

It introduces a streamlined method to derive quantum models from basic assumptions and extends the framework to symmetric monoidal categories with dagger-compact structure.

Findings

Unified treatment of real, complex, and quaternionic quantum mechanics

Simplified derivation of probabilistic models from basic assumptions

Establishment of symmetric monoidal categories with dagger-compact structure

Abstract

A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum mechanics can be treated on an equal footing, and allows some (but not too much) room for other alternatives. This is based on earlier work (arXiv:1206:2897), but the development here is further simplified, and also extended in several ways. I also discuss the possibilities for organizing probabilistic models, subject to the assumptions discussed here, into symmetric monoidal categories, showing that such a category will automatically have a dagger-compact structure. (Recent joint work with Howard Barnum and Matthew Graydon (arXiv:1507.06278) exhibits several categories of this kind.)