# An informal outline of Noncommutative Geometry (Part I)

General Relativity might be one of the most beautiful and powerful theories produced by mankind to this day. However, it is known in the common lore that the theory has singularities–that is, infinitely strong gravitational interactions. These are enough elements to feel dismayed, but this only worsens when we realise that the Friedmann-Lemaître-Robertson-Walker, Schwarzschild and Kerr spacetimes–all of them being physically meaningful solutions to the Einstein Field Equations (EFE)–may contain singularities. The first describes the dynamics of our observable universe under the assumptions of homogeneity and isotropy. The other two, describe the exterior of a spherically symmetry object, and singularities are reached when considering black holes. In that case, Schwarzschild corresponds to a static black hole while Kerr to a rotating one. This might lead one to think that for all practical purposes, a well-posed solution (a Lorentzian manifold without singularities) to the EFE is missing.

There have been many proposals on how to fix this, among these, we can find Noncommutative Geometry (NCG). Its main idea is that the description of spacetime as a continuum is neither physically nor mathematically well-defined. This can be heuristically motivated by pondering between General Relativity and Quantum Mechanics via the so-called Geometrical Measurement Problem. This problem arises when one takes the Uncertainty Principle and the Schwarzschild Radius into consideration at the same time. Let’s say I want to measure a particular spacetime point with arbitrary precision–Planck’s length and beyond. The Geometrical Measurement Problem implies that this is not possible due to the fact that for probing smaller distances we need higher energies and at some point (around Planck’s length) the energy involved in measuring could create a black hole. This leads us to conclude that the precision of the measurement is bounded by the Planck’s length from below. Thus, shorter distances do not appear to have operational value. This indicates we must drop the continuum description of spacetime, which can be attained by considering a Noncommutative spacetime.

As of now, we haven’t found a better description for spacetime than the one given by geometry, this indicates that maybe we should search for a noncommutative version of geometry. By geometry, we mean differential geometry, which indicates that we need to build a new (noncommutative) calculus. This approach seems to have the advantage of incorporating the bedrock of quantum theory (noncommutativity) into the construction of geometry from the very start. In order to venture into NCG, we will show that the usual commutative geometry can be portrayed in algebraic terms. This is contained in two very powerful theorems:

Theorem 1 (Gel’fand-Naimark) Let ${\mathcal{A}}$ be a commutative ${C^*}$ algebra. Then ${\mathcal{A}}$ is isometrically ${\ast}$isomorphic to ${C^0(\mathcal{M})}$ for some Hausdorff space ${\mathcal{M}}$.

We know that if ${\mathcal{A}}$ is a commutative algebra, then there is a way to construct a Hausdorff space ${\mathcal{M}}$ such that ${\mathcal{A}}$ is isomorphic to the algebra of complex valued continuous functions ${C^0(\mathcal{M})}$. As for the isomorphism, ${\ast:\mathcal{A}\rightarrow C(\mathcal{A})}$ satisfies ${||x^\ast||=||x||}$. Roughly speaking, this means that all information about the topology of ${\mathcal{M}}$ is encoded into a commutative ${C^*}$-algebra. Thus, we can forget about the topological spaces and just consider the algebras instead.

Theorem 2 (Serre-Swan) Let ${\mathcal{M}}$ be a compact finite dimensional manifold. A ${C^\infty(\mathcal{M})}$-module ${\mathcal{E}}$ is isomorphic to a module ${\Gamma(E,\mathcal{M})}$ of smooth sections of a vector bundle ${E \rightarrow \mathcal{M}}$, if and only if it is finite projective.

A vector bundle is completely characterised by its smooth sections. The space of sections can be thought of as a (right) module over the algebra ${C^\infty(\mathcal{M})}$. Therefore, locally trivial, finite-dimensional complex vector bundles over a compact Hausdorff space ${\mathcal{M}}$ correspond to finite projective modules over the algebra ${C^\infty(\mathcal{M})}$. This thick mind blender can be understood as: all the information about the vector bundles over ${\mathcal{M}}$ is encoded into a module over ${C^\infty(\mathcal{M}).}$ Thus, we can forget about the vector bundles and just consider the modules instead. This can be formalised in terms of category theory, as stated in nLab: we have an equivalence of categories between that of finite rank smooth vector bundles over ${X}$ and finitely generated projective modules over ${C^\infty(X)}$.

Intuitively, these theorems recast the topology and vector bundle of a fibre bundle in algebraic terms. Let us recall fibre bundle can be reconstructed if we have the base space, fibre, structure group, the set of coverings and the transition functions. All of these things are encoded into the algebra and the modules over it, as granted by the aforementioned theorems. Then, this provides enough data to find the projection, total space and local trivialisations. Therefore, a fibre bundle can be obtained from an algebra and the modules over it.

This means that thanks to Gel’fand, Naimark, Serre and Swan, we can stop thinking about geometry and just do algebra instead. This is like so sweet. However, a warning is foreordained

So far, we have seen that one can do commutative geometry with commutative algebra. Next, we will see that we can do noncommutative algebra and get the noncommutative analogue of geometry. The cornerstone of NCG is contained within the following theorem:

Theorem 3 (The Non Commutative Gel’fand-Naimark theorem) Let ${\mathcal{A}}$ be a ${C^*}$ algebra. Then ${\mathcal{A}}$ is isometrically ${\ast}$-isomorphic to a norm-closed ${\ast}$-subalgebra of bounded operators ${\mathscr{B}(\mathscr{H})}$ for some Hilbert space ${\mathscr{H}}$.

This theorem suggest that to find a Noncommutative analogue of geometry we must:

1. Take a non commutative algebra.
2. Look at the resulting algebra of bounded operators ${\mathscr{B}(\mathcal{H})}$.
3. Consider the projective modules over this algebra.
4. Claim that this defines a geometry.

This geometry will not be a manifold in general and this is where one of the main conceptual challenges of NCG arises: how to make geometrical sense of something that is not a manifold. Some may argue that understanding this is not needed for NCG to be useful and describe spacetime beyond Planck’s length, to which I agree. However, from a mathematical viewpoint, this in general is rather opaque. Nevertheless, in my next post, we will talk about the fuzzy sphere, one of the clearest and easiest examples–when in doubt, always go for the sphere.

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