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November 2, 2025

Dynamics in Jordan Algebras

Posted by John Baez

In ordinary quantum mechanics, in the special case where observables are described as self-adjoint n×nn \times n complex matrices, we can describe time evolution of an observable O(t)O(t) using Heisenberg’s equation

ddtO(t)=−i[H,O(t)] \frac{d}{d t} O(t) = -i [H, O(t)]

where HH is a fixed self-adjoint matrix called the Hamiltonian. This framework is great when we want to focus on observables rather than states. But one algebraic framework for working with observables, namely Jordan algebras. In this stripped-down framework all we can do is raise observables to powers and take real linear combinations of them. This lets us define a ‘Jordan product’ of observables:

A∘B=12((A+B) 2−A 2−B 2)=12(AB+BA) A \circ B = \frac{1}{2} ((A + B)^2 - A^2 - B^2) = \frac{1}{2} (A B + B A)

but not commutators and not multiplication by ii. What do we do then?

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