## CHSH inequality and the rejection of realism

I will start now a series on Bell's theorem and its importance to quantum foundations. Today I will talk about the Clauser, Horne, Shimony, and Holt inequality and its implication.

Quantum mechanics is a probabilistic theory which does not make predictions of the outcome of individual experiments, but makes statistical predictions instead. This opens the door to consider "subquantic" or "hidden variable" theories which would be able to restore full determinism. However, even with the statistical nature of quantum mechanics predictions there is something more which can be investigated: correlations. What distinguishes quantum from classical mechanics is how the observables of a composite system are related to the observables of the individual systems. In the quantum case there is an additional term related to the generators of the Lie algebras of each individual systems and this in turn prevents the neat factorization of the Hermitean observables of the composite system. It is this lack of factorization which prevents in general the factorization of the quantum states. In literature this goes under the (bad) name nonlocality.

Now suppose we have two spatially separated laboratories which receive from a common source pairs of photons. The "left lab" L chooses to measure the polarization of the photons on two directions $$\alpha$$ an $$\gamma$$, while the "right lab" R chooses to measure the polarization of the photons on two directions $$\beta$$ an $$\delta$$, Let's call the outcome of the experiments: $$a, b, c, d$$ for the directions of measurement $$\alpha, \beta\, \gamma, \delta$$, respectively. The values $$a, b, c, d$$ can take are +1 or -1.

Now let us compute the following expression:

$$C=(a+c)b-(a-c)d$$

Suppose $$(a+c) = 0$$, then $$(a-c)=\pm 2$$ and so $$C=\pm 2$$
Similarly if $$(a-c) = 0$$, then $$(a+c)=\pm 2$$ and again $$C=\pm 2$$

Either way $$C=\pm 2$$

Now suppose we have many runs of the experiment and for each run $$i$$we get:

$$a_i b_i + b_i c_i + c_i d_i - d_i a_i = \pm 2$$

from which we deduce on average that:

$$|\langle ab\rangle + \langle bc\rangle + \langle cd\rangle - \langle da\rangle|\leq 2$$

This is the CHSH famous inequality. Now under appropriate circumstances nature violates this inequality:

in an experiment with photons the average correlation between measurements on two distinct directions $$\alpha, \beta$$ is: $$\cos 2(\alpha - \beta)$$ and the inequality to be obeyed is:

$$| \cos 2(\alpha - \beta) + \cos 2(\beta - \gamma) + \cos 2(\gamma - \delta) - \cos 2(\delta - \alpha)| \leq 2$$

but if the angle differences are at 22.5 degrees we get that $$2\sqrt{2} \leq 2$$ so what is going on here?

A natural first objection is that not all 4 measurements can be simultaneously be performed and so we are reasoning counterfactually. But in N runs of the experiment we get 2N experimental results and there is a finite number of ways we can fill in the missing 2N data and in each counterfactual way of filling in the unmeasured data the CHSH inequality is still obeyed.

A second potential objection is that there is no free will and there is a conspiracy going on which prevents an unbiased choice of the 4 directions. There is no counteragument for this objection except that I know I have free will. If free will does not exists then mankind has much deeper troubles than explaining quantum mechanics: try to explain morality and justify the existence of the judicial system.

The introduction of bias can affect correlations and if the detection rate depends on the angle, then for appropriate dependencies one can obtain the quantum correlations. This is the so-called detection loophole, However, if such a dependency exists, it can be tested in additional experiments and the introduction of angle dependency only for Bell test experiments is indefensible. Loophole free Bell experiments while important to push the boundary of experimental technology have no scientific importance and they count only towards experimentalist's bragging rights.

Another way to obtain correlations above 2 is by appealing to contextuality: for example the value of $$a$$ when measured by lab L when lab R measures $$b$$ may not be the same when lab R measures $$d$$. While quantum mechanics is contextual, in this case such an argument means that the the choice lab R makes influences the result of measurement at lab L which is spatially separated!!!

Last, if the values of $$a, b, c, d$$ do not exist prior to measurement, this decouples again the value of $$a$$ when lab R measures $$b$$ from the value of $$a$$ when lab R measures $$d$$.

Assuming free will is true, we have only two choices at out disposal to be able to obtain correlations above 2:
• measurement in a spatially separated lab affects the outcome on the remote lab
• the outcome of measurement does not exist before measurement.
The first choice is taken by dBB theory because the quantum potential changes instantaneously and the second option is advocated by the Copenhagen camp. (I am excluding the MWI proposal because in it there is no valid derivation of Born rule. I am also excluding collapse models because they are a departure from quantum mechanics and experiments will soon be able to reject them).

Now here is the catch: the two labs need not be spatially separated and one experiment in lab L can unambiguously happen before the experiment in lab R. When the R lab measurement takes place it cannot affect the outcome in the L lab because that is in the past and already happened!

But can the first measurement affect the second one? In dBB this is possible as long as the first particle and its quantum potential is still around to "guide" the second particle. However, if after the first measurement the first particle is annihilated by its antiparticle then its quantum potential vanishes. The behavior of quantum potential after annihilation  is a reason why a relativistic second quantization dBB theory is not possible: either the quantum potential sticks around and messes up subsequent measurements, or vanishes and then the correlations cannot occur in the case above. (dBB supporters pin their hopes on a "future to be discovered" relativistic dBB quantum field theory which never materialized and cannot exists for several reasons.)

So from the two choices above only one remains valid:

the outcome of measurement does not exist before measurement

Realism is rejected by Bell's theorem. However in literature Bell's result is presented instead as a rejection of locality. But this is an abuse of language: locality=state factorization. Nature and quantum mechanics are incompatible with a state factorization. State factorization is just factorization, not locality. Rejection of realism is the only viable option left.

## Can you generate entangled particles which never interacted?

This post is the continuation of the last one because (1) it attracted a lot of attention and (2) no consensus was reached by the viewers. In particular, Lubos Motl was insisting on the fact that you cannot generate entangled particles which never interacted. So I am making one last attempt to convince him of the contrary.

The setting is from last time: entanglement swapping. This time I will not write Latex and instead I will explain the picture above (please excuse my poor MSPaint abilities). For the sake of argument, I am using photons and I show their worldliness in black going at 45 degree angles. Alice and Bob have the red resonant cavities which capture the 1 and 4 photons (feel free to replace the cavities with long enough optical fiber loops). The cavities have a release or absorption mechanism which is activated by Charlie upon obtaining the result of a projective measurement on photons 2 and 3. On average Charlie obtains his desired output 25% of the time in which case he sends the signal to release the 1 and 4 photons to the outside world. The other 75% of the time Charlie sends the signal to absorb the 1 and 4 photons.

From the outside Alice, Bob, Charlie, and 1 and 4 photons are inside of a green box. At random times out of the green box comes out two entangled photons which never interacted in the past.

Now here is the key Lubos statement:

"Excellent. Now, the filtering is caused by the decision in Charlie's brain which is in the intersection of the two past light cones of events -measurements of particles 1,4, right?

So you haven't found any counterexample to my statement, have you?"

Now here is why Lubos is wrong: See the picture below. He contends it is the purple area which is important and the fact that Charlies decision is made inside it  (again excuse my lack of precision drawing 45 degree lines).

Why is the purple area irrelevant to the discussion? Because when the photons 1 and 4 are in their respective red cavities they do not interact with each other!

Viewed from outside of the green box at random times comes out two entangled photons which never interacted in the past and could not have interacted in the past because first they were spatially separated, and second they got trapped in an isolation cavity long enough for the signal from Charlie to reach their cavity and release them.

It does not matter that Charlie's brain is in the intersection of the two past light cones of events measurements of particles 1,4 at the exit of the green box (the purple area). It matters that  Charlie's brain is not in the yellow area of the intersection of the two past light cones of the events of particles 1,4 entering the holding red cavities.

It is hard to explain all this in words without pictures. I tried to have a crude drawing last time in the comment section but formatting mangled it. I was suspecting Lubos was only pretending not to understand my argument, but now a week after the exchange I tend think he had a genuine misunderstanding because while he was thinking of the purple area I was talking about the yellow one.

Talking about other comments, last time a statement from Andrei caught my attention:

""S1 does not imply nonlocality, but nonrealism"

It does not imply non-locality for the particle that is measured, but implies non-locality for the distant particle. You measure one particle here and you create the value of the spin for both entangled particles (including the one that is far away)."

I will address this in a future post because a quick reply does not make justice to the topic. Is is true that "You measure one particle here and you create the value of the spin for both entangled particles (including the one that is far away)"? This is deeply related with Einstein's realism criteria:

"If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding lo this physical quantity."

This is a very natural criteria given a classical intuition, but it is however false. And showing why and how it is false is not a trivial matter. And if Einstein realism criteria is false then this is not true either: "you measure one particle here and you create the value of the spin for both entangled particles (including the one that is far away)."

Measurement does change something about the remote particle: it's state. But if the states are related to epistemology as opposed to ontology then there is no nonlocality problem. The easiest way to understand this is in the Bayesian paradigm where I change my degrees of belief: local measurement changes my local degree of belief about the remote particle.

## take 2

Finally I am ready to discuss the topic I promised: entanglement swapping. This s not a hard topic, and in fact it is usually given as a homework problem but since Lubos insists on doubling and tripling down on the fact that quantum correlations can arise only due to prior interaction:

"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."

"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."

I want to work out the problem in detail for anyone interested to see under what conditions two particles can become entangled even though they never interacted.

Let me start with the usual Bell states:

$$|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle)+|11\rangle)$$
$$|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle)-|11\rangle)$$
$$|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle)+|10\rangle)$$
$$|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle)-|10\rangle)$$

and have two pairs of Bell particles (which do originate as Lubos demands from a common origin in the past). It does not matter which Bell states we start with and for the sake of the example let's pick the following 4 particles in the state below:

$$|1234\rangle = |\Psi^-\rangle_{12}|\Psi^-\rangle_{34}= \frac{1}{2}(|0101\rangle-|0110\rangle-|1001\rangle+|1010\rangle)$$

Now suppose that the first pair  $$|\Psi^-\rangle_{12}$$ is split between Alice and Charlie, and the second pair $$|\Psi^-\rangle_{34}$$ is split between Charlie and Bob such that Alice has particle 1, Charlie has particles 2 and 3, and Bob has particle 4. Particles 1 and 2 share a common origin, and the same is true for particles 3 and 4, but particles 1 and 4 NEVER interacted in the past.

Now Charlie does a projective  measurement on say $$\Phi^+$$ on particles 2 and 3:

$$P_{\Phi^+} = \frac{1}{2}(|00\rangle+|11\rangle)(\langle00|+\langle11|)$$
$$P_{\Phi^+} = \frac{1}{2}(|00\rangle\langle00| + |00\rangle\langle11| + |11\rangle\langle00| + |11\rangle\langle11|)$$

This means that we apply the projector:

$$I_1 \otimes P_{\Phi^{+}_{23}}\otimes I_4$$ on $$|1234\rangle$$:

Ignoring the overall normalization factors for simplicity sake this means we need to compute:

$$|00\rangle_{23}\langle00| + |00\rangle_{23}\langle11| + |11\rangle_{23}\langle00| + |11\rangle_{23}\langle11|)$$
$$(|0101\rangle-|0110\rangle-|1001\rangle+|1010\rangle)=$$

$$-|1001\rangle - |0000\rangle - |1111\rangle - |0110\rangle=$$

$$-(|11\rangle+|00\rangle)_{14}|00\rangle_{23} - (|11\rangle+|00\rangle)_{14}|11\rangle_{23} =$$

$$-(|11\rangle+|00\rangle)_{14} (|11\rangle+|00\rangle)_{23} = - |\Phi^+\rangle_{14}|\Phi^+\rangle_{23}$$

and now particles 1 and 4 are entangled despite never interacting in the past.

So how does Lubos explain this?

"In quantum information, this is a part of the "LOCC" ("local operations and classical computation" do not create or increase entanglement) principle"

True but irrelevant. LOCC cannot increase entanglement overall, but now two particles which never interacted became entangled.

"Entanglement swapping is surely not a counterexample of LOCC, there can't be any counterexample. It's just swapping."

Again true but irrelevant.

"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."

This is fuzzy handwaving talk. It is true one can do correlation swapping in the classical world: suppose Alice has left and right gloves and she puts them in a bag. Bob does the same thing with his gloves and Charlie picks up a glove from Alice's bag and one from Bob's bag. Upon inspection of what he extracted he know how the gloves left in the two bags are correlated. However in the quantum case things are qualitatively different because of the active role of the observer and the fact that the result of measurement does not exist before measurement. I think I know where Lubos is coming from. I am speculating that he truly believes that quantum correlations are like Beltramann' socks, and if so his position makes perfect sense. However this is an unsustainable position and I plan to show why in subsequent posts. Lubos war on Bell is unsustainable as well. If you want to criticize Bell you have to do it on his genuine fault: the idea of beables. But this is a topic for another day, Let's continue:

"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."

This is a crackpot statement. The introduction of the speed of light arguments illustrates a fundamental misunderstanding. Basically here Lubos attempts to come up with a handwaving argument of sliding the light cones of particles 1 and 4 until overlap for the purpose of generating correlation. Collapse in the classical and the quantum world is the result of information being revealed. There is no such thing as a propagation of collapse and/or correlation at the speed of light or slower. Any talk of propagation of collapse/correlation is nonsense.

I am pretty sure Lubos will counter this and  attempt to clean the handwaving but if he is precise he will have to pick between two unacceptable options: (1) introducing considerations of propagation of correlations or (2) explain the 1 and 4 particle entanglement similar with my gloves example above and contradicting the role of the observer. If I were to make a prediction he is going to pick option 2 and argue that quantum correlations are precisely like Beltramann' socks.

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

Here I talk about entanglement swapping, not teleportation. True they are closely related, and true, entanglement is not created by LOCC. Again the contention is on this 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"
which is proven false by the computation above.

It is always dangerous to make grandiose statements using "all", "any", etc.

I won't get into the ER=EPR argument, but at some point in the future I'll explain why it is irrelevant in the quantum foundation/interpretation area.

"Feel free to write another completely wrong blog post - you have already written dozens of those - this URL is one giant pseudointellectual dumping ground."

On quantum interpretation, both me and Lubos are in the Copenhagen camp and I (civilly and without burning bridges) disagree more with the Bohmian or the MWI positions than with Lubos. From 10,000 feet if you do not care about subtle points and excluding his vitriol Lubos is basically correct in his quantum mechanics intuition.

However there is a fundamental difference between us: I am (neo) Copenhagen because my work proved to me this is the correct point of view while Lubos is a self-appointed defender of quantum orthodoxy. Lubos shoots from the hip, is not aware of subtle points in quantum mechanics, and hides his ignorance by ad hominem attacks. I do not care about him being a jerk any more than I care about being criticized for not wearing pink slippers at work. I do care very much however about being right and if I am proven wrong I do acknowledge my mistakes (nobody is perfect). Like Trump, Lubos never acknowledges mistakes and doubles and triples down. And I just could not resist calling the emperor naked when this happens.

## Impressions from California

I just got back from a one week vacation in California, and although I wanted to complete last week post, I could not concentrate on physics, so I will postpone that post one more week and today I want to share my impressions from the trip.

I visited San Francisco, Lake Tahoe, and Yosemite National Park. I have been to San Francisco before but this time I had more time to explore the city and surroundings. Besides the regular attractions like Fisherman's Wharf, Ghirardelli square, the cable car rides, and the sea lions on pier 39, I went to see the giant redwoods in Muir forest. The view is nice if you can find a parking space: there were cars lined the side of the road for 4 miles next to the entrance. Also coming back to San Francisco I had to cross back on the Golden Gate bridge which just happens to be a toll road in this direction. An ominous sign scare you with a \$500 fine for not paying the toll, but when you get to the toll booth there is nobody there.  At the toll plaza they take a picture of your license plate and they chase you after. When you google the toll information, you get to the official site which lets you pay the toll within 48 hours. However the idiotic site first asks you to enter the time you pass the toll and then it informs you that from that time forward up to the end of the day you were allowed one crossing. So if you missed the crossing time by one minute you wasted the payment and you are still a toll violator. Also on the negative side, the weather in San Francisco is incredibly cold. The sun is shining brightly causing quick sunburns, and yet the air temperature requires a decent coat in the middle of summer.

Next stop was Lake Tahoe-a beautiful four seasons resort. There you can either spend time at the beaches enjoying an incredibly clean water which goes knee deep for about half a mile,

 Emerald Bay Lake Tahoe

or you can take a gondola up the mountain for zip lines, high ropes climbing, mountain coaster rides, and rock climbing activities.

The pinnacle of the trip was the Yosemite National Park. This is a big place of breathtaking beauty choke full of tourists (about 50% were Europeans). The traffic and tourist density surpasses that of downtown Washington DC at the Smithsonian museums. The chipmunks are almost domesticated and they beg you for food every time you start eating anything.

The main attraction is the valley from where you can hike to Vernal and Nevada Falls

and also if you apply for a permit some time in advanced to Half Dome peak (this is a 2 day hike)

The picture above is from the Glacier Point and you see the half dome and the two waterfalls on the center right.  The hike to the Nevada fall is a strenuous 6 hours hike (2000 feet elevation difference) and you need to carry 2 liter of water per person to do it.

The places to stay inside the park require one year booking in advance, and the first hotel outside the park is about 30 miles away from the center of the valley. The first day was dedicated exploring the valley and hiking to the waterfalls, the second day I went to the glacier point to see the entire thing, and the last day I drove on the north road of the park to some beautiful views of the high mountains, lakes, and meadows. Everywhere there were long lines: 1 hour to enter the park, 90 minutes to take the free bus to glacier point, two hour drive to the north of the park which is in the far mountains in the picture above.  All in all I was deeply impressed by the beauty of the Yosemite which surpasses everything else I saw in the US.

## Correlations and Entanglement swapping

For this week I want to do a pedagogical presentation of entanglement swapping, but I got busy with other things and tomorrow I'll go on vacation for a week in San Francisco, Lake Tahoe, and Yosemite National Park. Normally I would stay late to write the post, but this involves a lot of LaTeX and so I have to postpone this post. Sorry for the delay, I'll be back in a week.

## 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.

## What is Noncommutative Geometry?

Noncommutative geometry is not well known and is even less understood by the physics community. Part of the problem is its abstract advanced mathematics which requires a sizable effort to learn, and part of the problem is the lack of down to earth explanations of its basic ideas. I think it was Yang (the Yang from Yang-Mills) who said something like: there are two kinds of mathematical books: the ones you cannot read past the first page, and the ones you cannot read past the first sentence.

Now let's talk quantum mechanics. One thing everyone agrees with is that in quantum mechanics one encounters both discrete and continuous spectra. Another thing which almost everyone agrees with is that the notion of trajectory for elementary particles does not exist. So is there a unified body of mathematics where continuous and discrete naturally coexist, and where the notion of trajectory is not used?

Riemannian geometry is based on the concept of metric and line element but can those ideas be somehow generalized? The starting point in understanding noncommutative geometry is to consider the concept of a real variable x. Classically this is usually expressed as a function from a subset of R into R: $$f:X\rightarrow R$$. So what is wrong with this? The problem is the coexistence of the discrete with the continuum: if x has the cardinality of the continuum then the multiplicity of a discrete variable would also have the cardinality of the continuum and then we would have problems to define measure theory.

However the problem is not encountered in quantum mechanics formalism and the quantum analog of a real variable is a selfadjoint operator in a Hilbert space (there is no ambiguity about which Hilbert space because all infinite dimensional separable Hilbert spaces are isomorphic). What makes the coexistence of the continuum with the discrete possible in the quantum mechanics case is the noncommutativity of operators. Guided by this analogy the task is to extend the usual geometric concepts by using the following prescription:

-identify possible quantum mechanics inspired analogies of the usual mathematical concepts
-verify that in the commutative case they are equivalent with the usual definitions
-introduce noncommutativity and see what we obtain

The next steps comes from the Gelfand-Naimark correspondence: compact Hausdorff spaces correspond to unital C ∗ -algebras:

any commutative C*-algebra (a Banach *-algebra with $$||a||^2 = ||a^* a||$$) is isomorphic with the algebra of continuous functions vanishing at infinity on some topological space.

Connes' theory of spectral triples (A, H, D) extends Gelfand duality beyond topology into differential, homological, and spin aspects:

Riemannian Geometry <==>commutative spectral triple ==> noncommutative spectral triple ==> noncommutative geometry.

Then the following dictionary is obtained:

 Commutative Noncommutative measure space von Neumann algebra locally compact space C∗- algebra complex variable operator on a Hilbert space real variable sefadjoint operator range of a function spectrum of an operator integral trace

and the list continues with many more advanced concepts like de Rham cohomology, Chern Weil theory, index theorems, etc.

So now let's revisit something which I introduced two posts ago: the concept of distance in noncommutative geometry:

$$d(x,y) = Sup \{|f(x)-f(y)|; || [D, f] || \leq 1\}$$

This definition looks strange so let me sketch how one arrives at it in the noncommutative framework. The problem with the Riemannian definition of distance:

$$d(x,y) = Inf \{\int_{\gamma} ds |\gamma~is ~a~path~between~x~and~y\}$$

is the usage of the path concept. If we have a space made out of continuous and a discrete pieces then there is no path possible which link them. So we need something which reduces to the Riemannian definition in the commutative case but which does not use the notion of trajectory.

Enter the Monge-Kantorovich optimal transport theory.

Suppose you own coal mines and factories and want to transport the coal from the mines to your factories. Transport cost money and you want to optimize the total transport price. Then one clever mathematician (Kantorovich) comes with a proposal to you: outsource the shipping problem to him and he will charge you only a loading and an unloading price. Moreover, he proves to you that in his proposal the loading/unloading prices will be lower or equal than the minimum transportation cost you face when you do the shipping yourself. The minimization problem for you became a maximization problem for Kantorovich.

Now to connect this to the distance definition, make your price be proportional with the line element. Under suitable conditions (which are fulfilled by the metric tensor) one can apply a minmax principle and you define the distance as the Kantorovich dual. Additional mathematical manipulations of the Kantorovich dual formula in case of the metric yield the Sup definition from above.  However this definition has a big advantage: it is defined regardless of the notion of a path and works in the noncomutative case as well. The key point now is that we can define the notion of distance, neighborhood and topology in cases containing discrete spaces where you only get trivial topologies by using Riemannian (commutative) geometry.

Mathemathics is about exploring the infinitely rich and connected landscape of math. Noncommutative geometry is a quantum mechanics inspired mathematical paradigm of exploring the landscape of the algebra-geometry duality. Connes took this paradigm further and applied it in physics resulting in an alternative formulation of the Standard Model. There he first made an incorrect prediction of the Higgs boson mass. If the prediction were true, this would have given him a big physics credibility boost. So it is fair to say that the incorrect prediction caused a loss of physics credibility. Later on he had a second look at the model and saw that he overlooked something which brought agreement between theory and experiment. The overlooked element also makes predictions of new physics. Can this prediction be trusted? I would say not yet. Why? because we know the Standard Model is only an approximate description of nature and we expect supersymmetry to be discovered. So the bet of new physics hinges on the validity of the Standard Model itself-a risky strategy. On the other hand, writing off noncommutative geometry as a mathematical fantasy without physics merits is arrogant. If you want to make new contributions in physics, does it make any sense to use the state-of-the-art mathematics from 100 years ago and ignore recent advances in math?