Division Algebras and Quantum Theory

TL;DR

This paper explores how quantum theory formulated over real, complex, and quaternionic Hilbert spaces can be unified through Dyson's three-fold way, clarifying internal issues and linking to physical symmetries.

Contribution

It presents a unified framework for real, complex, and quaternionic quantum theories using Dyson's three-fold way, resolving internal problems and illuminating physical symmetries.

Findings

Unified structure explains differences in quantum theories

Clarifies the role of time reversal symmetry

Connects Hilbert space types with physical representations

Abstract

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly "complex" representations), those that are self-dual thanks to a symmetric bilinear pairing (which are "real", in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are "quaternionic", in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds - real, complex and quaternionic - can be seen as Hilbert spaces of the other kinds, equipped with extra structure.