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?