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A question to George Musser

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and

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Algebra of coordinates vs. quantum mechanics number system

As I was preparing to start writing the weekly blog post I noticed a spike in readership from Lubos' blog and this seemed very odd: usually those happen after I write something and Lubos counters it, not before.

So it turned out to be a guest

post by George Musser where he touched on a thorny issue: nonlocality. Now here is what he stated:

*Lubos defines nonlocality as a violation of relativistic causality - an ability to signal at spacelike separation [...] In our present understanding of physics, this is impossible [...] At times, physicists and popularizers of physics have been guilty of leaving the impression that quantum correlations are nonlocal in this sense, and Lubos is right to take them to task (for instance here, here, and here).*

**So here is my question to Mr. Musser: where exactly I left that impression in my post?**

For the record, if I did leave this particular impression it was not my intention and in that case mea culpa: I accept that I wrote a bad post. However my gut feeling is that Mr. Musser did not took the time to understand what I was saying.

Puzzled by this allegation I start reading the comments and (as I expected) it went downhill:

*Thanks, George, for the remarks. You were telling me that you agreed about the key points but I think that your blog post makes it spectacularly clear that you misunderstand these issues just like all others whom I have criticized concerning this topic over the years.*

Dear George, can we please stop this exchange that can't lead anywhere? By now, you have repeated 100% of the idiocies that are commonly said about these issues. You haven't omitted a single one. I've erased last traces of doubts on whether you are a 100% anti-quantum zealot. You surely are one.

But enough is enough of nonlocality and Lubos, Let's come back to the topic of the week.

My interest in noncommutative geometry started from a side problem, the study of Connes' toy model

\(A = C^{\infty}(M) \otimes M_n (C)\)

when \(n=2\). This is an interesting problem in itself unrelated to the spectral triple. One way to understand this is to decouple \(C^{\infty}(M) \) from \(M_2 (C)\) and treat \(M_2 (C)\) as a number system for quantum mechanics. But can it be done?

Here is the motivation. The algebraic structure of quantum mechanics can be derived in the framework of category theory because of an universal property linking products with the tensor product. As such any physical principles we impose on the tensor product induces mathematical constraints on the algebras involved. **The physical principle in question is the invariance of the laws of nature under composition. **This is a natural principle because the laws of nature do not change by adding additional degrees of freedom. From this one derives the Jordan algebra of observables, the Lie algebra of generators, and a compatibility condition which yields in the end Noether's theorem.

Now on the Lie algebra part one can use Cartan's beautiful theory of

classification of Lie algebras and obtain the four infinite series along with the five exceptional cases. So

**what happens to this classification if one imposes the additional compatibility condition?**

It turns out that **there is an exceptional cases of interest. **This correspond to \(SO(2,4)\), and we may have found an nonphysical case because this is isomorphic with \(SU(2,2)\) **which violates positivity. **But can this be cured?

Positivity is an additional distinct property/axiom of quantum mechanics, so there is at least a hope it can be done. In a generalized sense we can restore positivity when we consider a constraint case in the

BRST formalism. However, something is lost and something is gained. What we gain is a new number system for quantum mechanics: \(M_2 (C)\), but what we lost is the Hilbert space which needs to be replaced by a Hilbert module. Physically this means that to any experimental question we ask nature we do not attach a probability like in ordinary quantum mechanics, but

**we attach a 4-vector current probability density respecting a ****continuity ****equation**. The resulting theory contains Dirac's theory of the electron and is intimately related to

Hodge decomposition.

So we did not gain anything physical in the end, but \(A = C^{\infty}(M) \otimes M_n (C)\) sits at the intersection of Connes' theory of the spectral triple with the theory of the number systems for quantum mechanics and with generalizations of the concept of norm and Hilbert spaces. It was the investigation of this toy model which made me put the effort to understand noncommutative geometry. **Can the algebra of the Standard Model in the noncommutative geometry formalism be understood as a number system for quantum mechanics?** **The answer is no.** To qualify to be a number system for quantum mechanics requires the invariance of the formalism under system composition. **Only complex quantum mechanics respects this. **The physical explanation is that two fermions cannot be considered another fermion for example.