## 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.

1. I have not understood Lubos Motl's claims against nonlocality. He seems to equate nonlocality with signalling or some superluminal influence. Of course we know that no information or qubits travel on spacelike intervals. I also have not argued with him on this, for much the same reason I prefer not to put my hand down holes in the ground made by rattlesnakes.

LC

1. Can you please tell us in what sense signalling could be different than influence - so that one could be superluminal and the other couldn't?

Signalling is nothing else than an influence that has some purpose, that is subjectively considered helpful by a human - because he sends or gets information by it. But what's happening in signaling is nothing else than influence.

If influence may travel faster than light, then can signals - it's the same influences that are just exploited. If signals travel faster than light, then influences do because a signal is just an example of an influence.

Do you disagree? How can you?

2. A quantum entanglement means the only knowledge accessible is the composite system. There does not exist any information about the two quantum systems, 2 for bipartite and 3 for tripartite etc, that make it up. As such there is no "subquantal information" being exchanged --- it does not exist. Quantum mechanics as a representation in space or spacetime, but a quantum state is independent of spacetime. A nonlocal influence is not then due to signals traveling faster than light, but due to the fact that the two parts of an EPR pair (again bipartite) are simply the same thing.

3. You haven't answered my question - you haven't even tried to use the same words (signaling, influence). What you wrote instead is incoherent, too, but it really makes no sense to try to get more of this stuff from you so please be aware that I am not asking anything.

2. "for much the same reason I prefer not to put my hand down holes in the ground made by rattlesnakes." LOL

3. Lubos, quick question:

in http://motls.blogspot.com/2012/03/most-of-research-of-nonlocality-is.html you state:

"Whenever there's some correlation in the world – in our quantum world – it's a consequence of the two subsystems' interactions (or common origin) in the past."

Do you still stand by this statement? Can you have say two entangled electrons who never interacted with each other?

1. small correction:

In some ways what Lubos says is true. It does not have to be in the past however. There is the Hanbury Brown Twiss effect that results in entangled photons not at the source but at the detector. It is a sort of form of the Wheeler Delayed Choice Experiment.

There does have to be some sort of interaction that sets up an entanglement. The easiest to understand is the quantum decay or tunneling where a neutral particle spin = 0 produces a pair of spin particles in a singlet entangled state. The hydronium atom of e plus e^+ is a good example. In this case an interaction such as QED between the two produces two entangled photons with opposite polarization.

A part of what Lubos is saying is that QFT is formulated by local operators, such as the Wrightman equal time commutators of operators on a spatial surface, and interactions are then local. Lubos then says that the results must be local; entanglement is not magically created. That is true, gravitation or any gauge field can't create entanglement out of nothing or convert a bipartite entanglement into a tripartite or GHZ entangled state.

What is the problem? It is with QFT. QFT I think is an approximation that is such that we can assume there exists an infinite number of oscillators and that field amplitudes are all completely local. However, in AdS/CFT formalism it turns out that if the CFT on the boundary is local then it is not possible to locally identify gravitation information in the AdS bulk, and visa versa. So something else is going on. What is going on is that basic quantum physics is on the verge of casting off this straight jacket that has been fitted around it in standard QFT.

In effect it might be that the equivalency of gravitation and QM, such as spacetime built up from entanglements, will demand that QFT or CFT no longer be strictly local. If YM gauge theory is equivalent to gravity, the gravity as a quantum system must be nonlocal. A propagator defines how fields of gravity, eg spatial information, is transported in spacetime. It is not as possible to impose these locality conventions.

Big changes are coming, and they will not come by just denying nonlocality in quantum physics.

2. I don't deny it has a kernel of truth: entanglement does not appear out of thin air and interaction is required. However, the question is for Lubos to clarify if he still stands by 100% to this particular statement.

3. While waiting for Lubos' reply, perhaps I can repeat a well known fact. The whole confusion arose from Bell's unfortunate use of the word 'non-locality' for non-factorizability of the wave function psi(r1,r2) for the two particles. As long as this description is correct, there is nothing to explain. No superluminal or even sub luminal communication is required. If you want to call the description by a joint wave function when the particles are miles apart,non-local that is just a matter of semantics. But Crowell has raised an interesting point. If I understand, he means that when quantum gravity becomes important, this description by a joint wave function will break down. Do you agree?

4. "If I understand, he means that when quantum gravity becomes important, this description by a joint wave function will break down. Do you agree?"

Don't quite get the question because those are sensitive terms with a lot of fine points. But I can say this: when quantum gravity is essential and other theories of nature lose validity (like classical general relativity), quantum mechanics still applies 100%. QM is an exact description of nature at all scales and at all energy levels. QM is not an "emergent" theory, or an approximation.

5. My main point is that quantum field theories QFTs are formulated according to local oscillators specified on a spatial surface of 3-dimensions. The commutators of these field amplitudes all commute, which is associated with their being spatially separated.

However, if an interaction generates nonlocality, say a bipartite system of qubits, then if entanglement symmetry is noetherian conserved it implies this nonlocality (entanglement etc) is transferred and not created. If so then our whole perspective on QFT is not fundamentally correct, even if it works pretty well from a pragmatic perspective for particle physics.

BTW, Motl is of course not always right. His stance against global warming is hopelessly wrong, but he prefers his right wing political ideology over a discipline of science he thinks he can dismiss. The evidence in favor of AGW is becoming very solid. He also dismissed the AMPS firewall problem as nonsense, when in fact this has been a very salient and clear problem.

6. Of course I stand by the statement. Causality/locality implies that any entanglement - or any correlation - between objects in two places has to result from their contact or interaction in the intersection of their past light cones.

Among other things, that's why the Big Bang theory had to be extended to inflation to explain the uniform temperatures in the Universe: 2 points in the Universe had no shared past, so they couldn't have arranged their temperatures to the same values. It's not even some detailed microscopic correlation - but even that may be said to be causally impossible.

In quantum information, this is a part of the "LOCC" ("local operations and classical computation" do not create or increase entanglement) principle, see e.g. Section 3.2 of Maldacena-Susskind "Cool horizons for entangled black holes" (1306.0533, ER=EPR).

7. Thanks. Entanglement swapping is a direct counterexample. Next time I will work out in detail the how to entangle two electrons who never interacted with each other. Of course something had to interact with something to create the correlation so your statement is true in spirit, but it overreaches. It is the active role of the observer during measurement which makes this counterexample possible and there is no such thing possible in classical physics.

8. Indeed LOCC does not "create entanglement," but this does not mean interactions do not transfer entanglement. If QFT is nonlocal on some level, then we can reformulate it according to a vast reduction in the number of degrees of freedom. The infinite number of local field amplitudes or HOs is then a fiction and the real degrees of freedom are a far few number of nonlocal fields. This is sort of where noncommutative geometry can enter into QFT.

It is the case that nonlocality or entanglement can emerge from the detection. However, this is an interaction on some fundamental level. This can transfer entanglement or nonlocal phase from the fundamental interaction or QFT involved with this detection to the quantum system in the past. This is a form of the Wheeler Time Delay Experiment. The role of the observer is not that mystical IMO, for detecting quantum states or events is ultimately some form of physical interaction.

9. Florin: Entanglement swapping is surely not a counterexample of LOCC, there can't be any counterexample. It's just swapping. It's like the entanglement is riding on a train A and changes the trains to another train B that happens to meet A at some point. So the pair entangled afterwards is described differently after the trains are switched but the entanglement is preserved and links two places that change continuously and at most by the speed of light.

Why teleportation isn't a counterexample - or a source of nonlocality - was also discussed in detail in Susskind's recent arXiv:1604.02589.

Feel free to write another completely wrong blog post - you have already written dozens of those - this URL is one giant pseudointellectual dumping ground. Quantum physics doesn't introduce any nonlocality - any possibility to create correlations remotely - relatively to classical physics. Quantum mechanics conceptually differs by its non-realism and the influence of observers on the system, but it's still true that the observers only affect the degrees of freedom in their location.

In the ER=EPR language, a remote creation of entanglement would be equivalent to a remote or superluminal "throwing" of a wormhole to a different place. One simply can't do that. A wormhole that doesn't exist forever can only be created from a local effect and gradual stretching afterwards.

4. Entanglement swapping is very interesting. But does it mean any more than- if A and B are in contact and B and C are in contact then A and C are in contact? My guess is that swapping makes a joint wave function psi(r1,r3) from psi (r1,r2) and psi (r2,r3).But I am not clear about the details. Swapping does need more discussion.