From composability two-product algebra to quantum mechanics

Last time we introduced the composability two-product algebra consisting of the Lie algebra \(\alpha\) and the Jordan algebra \(\sigma\) along with their compatibility relationship. This structure was obtained by categorical arguments using two natural principles of nature:

- laws of nature are invariant under time evolution

- laws of nature are invariant under system composition

What we did not obtain were spectral properties. However, in the finite dimensional case, we do not need spectral properties and we can fully recover quantum mechanics in this particular case. The trick is to classify all possible two-product algebras because there are only a handful of them. This is achieved with the help of the Artin-Weddenburn theorem. 

First some preliminary. We need to introduce a Lie-Jordan-Banach (JLB) algebra by augmenting the composability two-product algebra with spectral properties:

-a JLB-algebra is a composability two-product algebra with the following two additional properties:

• \(||x\sigma x|| = {||x||}^{2}\)
• \(||x\sigma x||\leq ||x\sigma x + y\sigma y||\)

Then we can define a C* algebra by compexification of a JLB algebra where the C* norm is:

\(||a+ib|| = \sqrt{{||a||}^{2}+{||b||}^{2}}\)

Conversely from a C* algebra we define a JLB algebra as the self-adjoint part and where the Jordan part is:

\(a\sigma b = \frac{1}{2}(ab+ba)\)

and the Lie part is:

\(a\alpha b = \frac{i}{\hbar}(ab-ba)\)

From C* algebra we recover the usual quantum mechanics formulation by GNS construction which gets for us:

- a Hilbert space H

- a distinguished vector \(\Omega\) on H arising out of the identity of the C* algebra

- a representation \(\pi\) of the algebra as linear operators on H

- a state \(\omega\) on C* represented as \(\omega (A) = \langle \Omega, \pi (A)\Omega\rangle_{H}\)

Conversely, from quantum mechanics a C* algebra arises as bounded operators on the Hilbert space.

The infinite dimensional case is a much harder open problem. Jumping from the Jordan-Banach operator algebra side to the C* and von Neuman algebras is very tricky and this involves characterizing the state spaces of operator algebras. Fortunately all this is already settled by the works of Alfsen, Shultz, Stormer, Topping, Hanche-Olsen, Kadison, Connes.