## Impressions from FQMT 2017

I just came back from Vaxjo where I had a marvelous time. It does sounds cliche, but this year was the best conference organized by Professor Khrennikov and I got many pleasant and unexpected surprises.

The conference did had one drawback: everyday after the official talks we continue the discussions about quantum mechanics well past midnight at "The Bishops Arms" where we drank too many beers causing me to gained a few pounds :)

At the conference I had a chance to meet and talk with Phillipe Grangier (he worked with Aspect on the famous Bell experiment) and I witness him giving the best cogent comments on all talks: from experimental to theoretical. He even surprised me when he asked at the end of my presentation why I am using time to derive Leibniz identity, where any other symmetry will do? Indeed this is true, but the drawback is that any other symmetry lacks generality later on during composition arguments. Suppose we compose two physical systems: one with with a continuous symmetry and another without, then the composed system will lack that symmetry. The advantage of using time is that it works for all cases where energy is conserved.

Grangier presented his approach on quantum mechanics reconstruction using contextuality and continuity (like in Hardy's 5 reasonable axioms paper). The problem of continuity is that it lacks physical intuition/motivation. Why not impose right away the C* condition: $$||a^* a|| = {||a||}^2$$ and recover everything from it?

Bob Coecke and Aleks Kissinger book on the pictorial formalism: "Picturing Quantum Processes" was finally ready and was advertised at the conference. If you go to www.cambridge.org/pqp you can get with a 20% discount when you enter the code COECKE2017 at the checkout.

Coecke 's talk was about causal theories and his main idea was: "time reversal of any causal theory = eternal noise". This looks deep, but it is really a trivial observation: you can't get anything meaningful and you can't control signals which have an information starting point because the starting point corresponds to the notion of false and anything is derivable from false.

Robert Raussendorf from University of Vancouver had a nice talk about measurement based quantum computations where measurements are used to control the computation and he identified a cohomological framework.

One surprise talk for me was the one given by Marcus Appleby from University of Sydney who presented a framework of equivalence for Quantum Mechanics between finite and infinite dimensional cases. This is of particular importance to me as I recovered quantum mechanics in the finite dimensional case only and I am searching for an approach to handle the infinite dimensional case.

I made new friends there and I got very good advice and ideas - a big thank you. I also got to give many in person presentations of my quantum reconstruction program.

There was one person claiming he solved the puzzles of the many worlds interpretation. I sat next to him at the conference dinner and I invited him to have a guest post at this blog to present his solution. As a disclaimer, I think MWI lacks the proper notion of probability and I am yet to see a solution but I am open to listen to new arguments. What I would like to see is an explanation of how to reconcile the world split of 50-50% when the quantum probabilities are 80-20%? I did not see this explained in his presentation to my satisfaction, but maybe I was not understating the argument properly.

## Jordan-Banach, Jordan-Lie-Banach, C* algebras, and quantum mechanics reconstruction

This a short post written as waiting for my flight at Dulles Airport on my way to Vaxjo Sweden for a physics conference.

First some definitions. a Jordan-Banach algebra is a Jordan algebra with the usual norm properties of a Banach algebra. A Jordan-Lie-Banach algebra is a Jordan-Banach algebra which is a Lie algebra at the same time. A Jordan-Lie algebra is the composability two-product algebra which we obtained using category theory arguments.

Last time I hinted about this week's topic which is the final step in reconstructing quantum using category theory arguments. What we obtain from category theory is a Jordan-Lie algebra which in the finite dimensional case has the spectral properties for free because the spectrum in uniquely defined in an algebraic fashion (things gets very tricky in the infinite dimensional case). So in the finite dimensional case JL=JLB.

But how can we go from Jordan-Banach algebra to C*? In general it cannot be done. C* algebras correspond to quantum mechanics and on the Jordan side we have the octonionic algebra which is exceptional. Thus cannot be related to quantum mechanics because octonions are not associative. However we can define state spaces for both Jordan-Banach and C* algebras and we can investigate their geometry. The geometry is definable in terms if projector elements which obey: $$a*a = a$$. In turn this defines the pure states as the boundary of the state spaces. If the two geometries are identical, we are in luck.

Now the key question is: under what circumstances can we complexify a Jordan-Banach algebra to get a C* algebra?

In nature, observables play a dual role as both observables and generators. In literature this is called dynamic correspondence. Dynamic correspondence is the essential ingredient: any JB algebra with dynamic correspondence is the self-adjoint part of a C* algebra. This result holds in general and can be established by comparing the geometry of the state spaces for JB and C* algebras.

Now for the punch line: a JL algebra comes with dynamic correspondence and I showed that in prior posts. The conclusion is therefore:

in the finite dimensional case: JL is a JLB algebra which gives rise to a C* algebra by complexification and by GNS construction we obtain the standard formulation of quantum mechanics.

Quantum mechanics is fully reconstructed in the finite dimensional case from physical principles using category theory arguments!

By the way this is what I'll present at the conference (the entire series on QM reconstruction).

## 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.