## The algebraic structure of quantum and classical mechanics

Let's recap on what we derived so far. We started by considering time as a continous functor and we derived Leibniz identity from it. Then for a particular kind of time evolution which allows a representation as a product we were able to derive two products $$\alpha$$ and $$\sigma$$ for which we derived the fundamental bipartite relations.

Repeated applications of Leibniz identity resulted in proving $$\alpha$$ as a Lie algebra, and $$\sigma$$ as a Jordan algebra and an associator identity between them:

$$[A,B,C]_{\sigma} + \frac{J^2 \hbar^2}{4}[A,B,C]_{\alpha} = 0$$

where $$J$$ is a map between generators and observables encoding Noether's theorem.

Now we can combine the Jordan and Lie algebra as:

$$\star = \sigma\pm \frac{J \hbar}{2}\alpha$$

and it is not hard to show that this product is associative (pick $$\hbar = 2$$ for convenience):

$$[f,g,h]_{\star} = (f\sigma g \pm J f\alpha g)\star h - f\star(g\sigma h \pm J g\alpha h)=$$
$$(f\sigma g)\sigma h \pm J(f\sigma g)\alpha h \pm J(f\alpha g)\sigma h + J^2 (f\alpha g)\alpha h$$
$$−f\sigma (g\sigma h) \mp J f\sigma (g\alpha h) \mp J f\alpha (g\sigma h) − J^2 f\alpha (g\alpha h) =$$
$$[f, g, h]_{\sigma} + J^2 [f, g, h]_{\alpha} ±J\{(f\sigma g)\alpha h + (f\alpha g)\sigma h − f\sigma (g\alpha h) − f\alpha (g\sigma h)\} = 0$$

because the first part is zero by associator identity and the second part is zero by applying Leibniz identity. In Hilbert space representation the star product is nothing but the complex number multiplication in ordinary quantum mechanics

Now we can introduce the algebraic structure of quantum (and classical) mechanics:

A composability two-product algebra is a real vector space equipped with two bilinear maps $$\sigma$$ and $$\alpha$$ such that the following conditions apply:

- $$\alpha$$ is a Lie algebra,
- $$\sigma$$ is a Jordan algebra,
- $$\alpha$$ is a derivation for $$\sigma$$ and $$\alpha$$,
- $$[A, B, C]_{\sigma} + \frac{J^2 \hbar^2}{4} [A, B, C]_{\alpha} = 0$$,
where $$J \rightarrow (−J)$$ is an involution mapping generators and observables, $$1\alpha A = A\alpha 1 = 0$$, $$1\sigma A = A\sigma 1 = A$$

For quantum mechanics $$J^2 = -1$$. In the finite dimensional case the composability two-product algebra is enough to fully recover the full formalism of quantum mechanics by using the Artin-Wedderburn theorem.

The same structure applies to classical mechanics with only one change: $$J^2 = 0$$.

In classical mechanics case, in phase space, the usual Poisson bracket representation for product $$\alpha$$ can be constructively derived from above:
$$f\alpha g = \{f,g\} = f \overset{\leftrightarrow}{\nabla} g = \sum_{i=1}^{n} \frac{\partial f}{\partial q^i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q^i}$$

and the product $$\sigma$$ is then the regular function multiplication.

In quantum mechanics case in the Hilbert space representation we have the commutator and the Jordan product:

$$A\alpha B = \frac{i}{\hbar} (AB − BA)$$
$$A\sigma B = \frac{1}{2} (AB + BA)$$

or in the phase space representation the Moyal and cosine brackets:

$$\alpha = \frac{2}{\hbar}\sin (\frac{\hbar}{2} \overset{\leftrightarrow}{\nabla})$$
$$\sigma = \cos (\frac{\hbar}{2} \overset{\leftrightarrow}{\nabla})$$

where the associative product is the star product.

Update: Memorial Day holiday interfered with this week's post. I was hoping to make it back home on time to write it today, but I got stuck on horrible traffic for many hours. I'll postpone the next post for a week.