## 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?

1. I think,it should be Yang-Mills, not Young-Mills if you are talking about the famous field theorists.

1. Oopsy... thanks

2. Dear Florin, as Kashyap observed, you misspelled (twice, so no typo) Yang-Mills as Young-Mills. (Yes, C.N. Yang had the quote about the books.)

You are trying to moralize and claim that physicists are wrong about some basic opinions on the Standard Model and Beyond the Standard Model physics. But you think that gauge theory was written down by Young and Mills.

Try to be a bit sensible. How plausible is such an assertion that a person who can't distinguish Yang (Chinese) from Young (English) has something relevant to say to experts about cutting-edge particle physics?

It's about as plausible as the assumption that the people who write about Einstien revolutionize relativity or those who refer to Hawkins revolutionize black holes. You seem to be like a classic crank. Have you ever calculated your Baez crackpot score?

1. Is this the best you come up with? Typos? Pathetic.

2. It's not a typo - you've written it twice. It's a proof that you're absolutely unfamiliar with basic literature and terminology in theoretical physics.

When one does any physics of the sort, he is writing or reading the phrase Yang-Mills so frequently that it's simply impossible for him to misspell the names.

I am just saying that your big-mouth pronouncements "On the other hand, writing off noncommutative geometry as a mathematical fantasy without physics merits is arrogant" are absolutely unbacked by any content.

Concerning the last question:

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?

Yes, of course, everyone who is working on cutting-edge physics is ignoring the recent developments in maths unless he's forced to incorporate them by the physical arguments. You recommend the exact opposite attitude: to force some content of recent math papers onto physics. Natural science doesn't work and cannot work in this way.

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3. I have more of a question. What bearing does noncommutative geometry have with the locality or nonlocality of the field theory. Your last post was on a noncommutative geometry approach to the standard model, following Connes. I am not certain about the viability of this approach, but if one does field theory this way does the theory have the same locality (equal time commutators of fields etc) that one has with standard field theory?

1. Lawrence,

NCG SM is simply a reformulation of usual quantum field theory in the language of the spectral triple. The concept of locality is different in NCG because the very notion of distance and coordinates is generalized. For example, the gauge degrees of freedom are seen as "inner fluctuations of the metric". There is also a generalized concept of dimensionality called KO-dimension which tells apart different kinds of discrete sets of zero normal dimension.

2. I have Connes' book on noncommutative geometry. It has been a while since I studied it. This does lead me then to another question. If you have a noncommutative geometry it means in generality that

[x_i, y_j] = iħθ_{ij}

where the θ_{ij} is in the standard case a delta function. The Planck constant ħ is an intertwiner in quantum mechanics between uncertainty in momentum and position. It serves a role similar to the speed of light in spacetime that converts spatial distance to time. Can we then do something similar with noncommutative geometry one does with spacetime

In particular with twistor theory the light cone origin is a blow up of a point that is mapped to a Riemann sphere and a light ray to a point. We in effect have correspondence maps between objects in spacetime and objects in projective geometry. In this way a sort of correspondence between standard QFT and NCG QFT might then exist.

This gets to the question of locality and nonlocality. There is this correspondence between gravity in the bulk of an anti-de Sitter spacetime AdS_n and the conformal field theory on the boundary CFT_{n=1}. In addition there is a sort of duality between locality and nonlocality, where if gravity on the bulk is more local then the CFT on the boundary is more nonlocal, and visa versa. This is why a correspondence would potentially be interesting between the two geometries, such as in your table.

3. "The Planck constant ħ is an intertwiner in quantum mechanics between uncertainty in momentum and position. It serves a role similar to the speed of light in spacetime that converts spatial distance to time."

c and h have different origins. Planck's constant is categorical in nature and is responsible for a distinct way of composing two physical systems into a larger one. NCG is categorical in nature as well: it is category theory which provide the algebra-geometry duality.

"Can we then do something similar with noncommutative geometry one does with spacetime"

Maybe, I don't know.

"In particular with twistor theory the light cone origin is a blow up of a point that is mapped to a Riemann sphere and a light ray to a point."

Twistors are related to SU(2,2) which is isomprphic with the conformal group Spin(2,4). The same thing is the at the root of quantionic QM which is a constrained QM on SO(2,4) to obtain positivity (by BRST formalism). The resulting theory is SO(3,1)xSU(2)xU(1) or Connes' toy model for complex 2x2 matrices.

"This gets to the question of locality and nonlocality."

Unlikely, but it may depend on your definition of locality.

"There is this correspondence between gravity in the bulk of an anti-de Sitter spacetime AdS_n and the conformal field theory on the boundary CFT_{n=1}. In addition there is a sort of duality between locality and nonlocality, where if gravity on the bulk is more local then the CFT on the boundary is more nonlocal, and visa versa. This is why a correspondence would potentially be interesting between the two geometries"

This is highly speculative.

Not my table. The credit goes to Connes.

4. The speed of light is the basis for projective geometry of spacetime. A light cone as a projective variety is identical to a Riemann sphere. A point is then dual to a space. Similarly a null curve is a point. This constructs the twistor space. Quantum mechanics has a duality between conjugate variables, which in a discrete form is the reciprocal lattice map such as in solid state physics. Relativity → blow up of a point, QM → reciprocal lattice, point mapped to ∞.

As I indicated the Tsirelson bound and the null interval are isomorphic to each other. In effect quantum entanglement builds up the symmetries of spacetime, such as defined by null Killing vectors. I think QM and GR are the same thing, or that GR is a sort of classicality that emerges from QM entanglements with “large N.”

The SU(2,2):= (4,4) = (3,3)⊕(3,1)⊕(1,3)⊕(1,1), which for SU(2,2) ~ CL(4) → 1, 4, 6, 4, 1 in binomial distribution of 4, 3, 2, 1, 0 chains. The 6 = (3,3) corresponds to G_2(V^4), the (3,1) is CP^4 = CP^3∪C^3, the (1,3) is the dual CP^4 (twistor – ambitwistor) and the (1,1) are scalars. These are the “Higgs” scalars in a Hitchen bundle.

5. "I think QM and GR are the same thing, or that GR is a sort of classicality that emerges from QM entanglements with “large N.”"

I tend to disagree. Let me explain. Cones showed that SM + GR is GR on a "noncommutative space" because it all follows from diffeomorphism invariance in the new space. Now enter exhibit A: supersymmetry.

In SUSY you can have a formulation in the so-called "superspace" which at first sight looks a lot like Cones NCG space. However, this is not the case since SUSY's superspace does not come from a C* algebra and you don't get a spectral triple. So one has to be very careful with the details and make sure the gut feeling intuition is backed up by the math.

Is GR an emergent QM phenomena? Indeed since everything is quantum at origin. Does it have to do with entanglement and large N? Of course. But how to go from one end to the other is a very tricky business and we need the meat of the proposal to pass judgement, otherwise it is only handwaving.

6. I don't know what the relationship between NCG and supersymmetry would be. Superspace built from Grassmannian generators with

{ξ_a, bar-ξ_b'} = -2iσ^μ_{ab'}∂_μ, p_μ = -i∂_μ

has some appearances of being NCG. The primary difference is that this is anticommuting. One the other hand I see no reason why NCG can't be worked in a graded Lie algebraic context with generators g = h + k with rules

[h, h] \subset h, [h, k] \subset k {k, k} \subset h.

7. One should distinguish here between supersymmetry on target spacetime andon the worldvolume. Connes' original axtioms for spectral triples were formulated in terms of generalized Laplace operators, and axiomatized the structure of a bosonic quantum particle (its Hilbert space, Hamiltonian, and subalgebra of smooth poisition operators). Later he realized that the theory works much better if the Hilbert space is taken to be a Z/2Z-graded (a super-Hilbert space) and the generalized Laplace operator is assumed to be the square of a generalized Dirac operator. That gives the now standard axioms for spectral triples. In terms of thinking of them as encoding a quantum mechanical particle, the step means to pass from worldline translation symmetry (generated by the generalized Laplace operator regarded as a Hamiltonian) to worldline supersymmetry (generated by the generalized Dirac operator). In 1 dimension the supersymmetry algebra reduces to the relation H = D^2 between a Hamiltonian H and a corresponding Dirac operator D. In this sense spectral triples in the modern sense are an axiomatization of the ingredients of supersymmetric quantum mechanics.

This is, incidentally, why modern spectral triples arise as the point particle limit of 2d superconformal field theories, instead of 2d bosonic conformal field theories (see below).

Now, all this is still only worldline supersymmetric, not target space supersymmetric.

8. You are mentioning something with the worldline SUSY that I have thought of. We can include this with a string world sheet in three dimensions. The geometry of the world lines is then dual to the string world sheets. The worldlines form knots and braids, which has a duality to string world sheets. The 2-space plus time (2, 1) metric means there is a Lorentz transformation in this space SO(2,1). This has π_1(SO(2,1)) = Z, which is different from the Z_2 = Z/2Z of higher dimensional Lorentz group. The spin (2,1) is then not a universal cover and PSO(2,1) and the exchange of two particle paths is obstructed. This obstruction leads to both bosonic and fermionic statistics.

I like the ncatlab website BTW

4. I would say that one encounters only discrete spectra in quantum mechanics, but given a large number of discrete events, it is often convenient to use continuum math.

Likewise, trajectories are always discrete particle actions and the notion of continuous trajectories in space and time is likewise a convenience of dealing with a very large number of events.

Gravity then becomes a pairing of discrete yet complementary photon events whose now very, very large number represents the continuum of space and time trajectories very well.

It seems to me that you already have the pieces of this puzzle and just need to assemble them correctly...

5. Connes' spectral triples arise as the point particle limit of 2d (super-)conformal field theories. A commented list of references on this relation is here: ncatlab.org/nlab/show/2-spectral%20triple#References

Under this relation, the way spectral triples encode an effective target space geometry as seen by a quantum particle is analogous to how a 2d (S)CFT encodes an effective target space geomtry as seen by a quantum (super-)string.

The first rigorous account of this is due to

D. Roggenkamp, K. Wendland, "Limits and Degenerations of Unitary Conformal Field Theories" arXiv:hep-th/0308143

summarized in

D. Roggenkamp, K. Wendland, "Decoding the geometry of conformal field theories" arXiv:0803.0657

Yan Soibelman used this relation of 2d SCFT to Connes spectral triple in order to approach the analysis of aspects of the landscape of string vacua:

Y. Soibelman, "Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry" (pdf), in Sati et al. (eds.), "Mathematical Foundations of Quantum Field and Perturbative String Theory", Proceedings of Symposia in Pure Mathematics, AMS (2001)

For a kind of reverse construction, in

S. Carpi, R. Hillier, Y. Kawahigashi, R. Longo, "Spectral triples and the super-Virasoro algebra" (arXiv:0811.4128)

the authors realize 2d SCFTs essentially as local nets (in the sense of AQFT) of spectral triples.

1. There is now an exposition at "PhysicsForums Insights" of the close relation between spectral triples and string backgrounds encoded by 2d SCFTs. See here:
www.physicsforums.com/insights/spectral-standard-model-string-compactifications/

6. @Urs, one of the strengths of NCG is the contrains it puts on model building (mostly the scalar sector but not only). Do the 2d super-conformal field theories having spectral triples as point particle limit provide the same type of constraints?

1. Lifting a spectral triple to a 2d CFT means asking for much stronger constraints, even, since there is much more data in the 2d CFT than just its point particle limit reflected in the spectral triple.

This is a version of the fact that there are much stronger constraints on a string background to be consistent (anomaly free) than on a random QFT.

The article by Soibelman referenced above means to make use of this for saying something about the landscape of perturbative string vacua. A perturbative string vacuum is a 2d SCFT of central charge 15, which in addition satisfies modularity and sewing constraints. While the moduli space of all 2d SCFTs is hard to analyze, the shadow that it throws, via the point particle limit, in the space of spectral triples is more tractable. And Soibelman's article analyzes this shadow space.

In this context one cannot help but notice the following coincidence:

The spectral triples arising from 2d SCFTs of central charge 15 in string theory have, famously, KO-dimension 4+6 (mod 8). Now this is precisely the KO dimension that Connes claims in

A. Connes, "Noncommutative Geometry and the standard model with neutrino mixing", JHEP0611:081 (hep-th/0608226)

is necessary to get a viable standard model-like effective theory from a spectral triple.

Maybe it's a coincidence. Or maybe it points to something deep.