An informal outline of Noncommutative Geometry (Part II)

Disclaimer: This post is part of a series about Noncommutative Geometry. I do not pretend to be rigorous nor thorough. The main idea is just to cover (in a rather informal way) the main concepts.

In the first part of this series, I mentioned that Noncommutative Geometry (NCG) can be rather opaque because the notion of a manifold is lost. However, I also mentioned that there is an example that showcases the features that are gained when going from a commutative to a noncommutative setting. That example is the fuzzy sphere, first studied by Madore, from which this post has been heavily inspired.

The algebra of functions on a fuzzy sphere differs from that of the ordinary sphere because it’s not commutative. If we are talking about functions on the sphere, spherical harmonics ${Y_{\ell,j}}$ are certainly a good starting point. These labels obey ${\ell\leq j}$ , and the product rule for the harmonics is given by

$\displaystyle Y_{a,\alpha }\left(\theta ,\varphi \right)Y_{b,\beta }\left(\theta ,\varphi \right) =\mu\sum _{c\leq\gamma }\left(-1\right)^{\gamma }{\sqrt {2c+1}}\left({\begin{array}{ccc}a&b&c\\\alpha &\beta &-\gamma \end{array}}\right)\left({\begin{array}{ccc}a&b&c\\0&0&0\end{array}}\right)Y_{c,\gamma }\left(\theta ,\varphi \right), \ \ \ \ \ (1)$

where ${\mu}$ is a constant that depends on on ${a}$ and ${b}$ and the quantities in parenthesis are the Wigner ${3j}$ -symbols. The truncation in the sum has a lot of importance, it replaces an infinite-dimensional commutative algebra by a ${\gamma^{2}}$ -dimensional noncommutative ${C^\ast}$ algebra. Furthermore, there is an isometric ${\ast}$ -isomorphism between spherical harmonics and the ${su(2)}$ algebra. If you take a look at the noncommutative Gel’fand-Naimark Theorem, you can see that we have met all of the conditions. This means that we can think think of the generators of ${su(2)}$ as the building blocks for a noncommutative geometry, that of the fuzzy sphere.

That being said, to study the fuzzy sphere in two dimensions, we begin by considering ${\mathbb{R}^3}$ with coordinates ${x^a}$ ( ${a=1,\ldots,3}$ ) and metric ${\delta_{ab}}$ (this is the Kronecker delta), the radius is defined as usual

$\displaystyle r^2=\delta_{ab}x^ax^b. \ \ \ \ \ (2)$

Any any complex valued continuous function on the sphere ${\mathbb{S}^2}$ can be (formally) expressed as

$\displaystyle f=f_0+f_ax^a+\frac{1}{2}f_{ab}x^ax^b+\cdots \in C^0(\mathbb{S}^2). \ \ \ \ \ (3)$

If we truncate all the terms up to the constant ${f_0}$ , the commutative algebra ${C^0(\mathbb{S}^2)}$ is reduced to the algebra of complex numbers, which we will denote by algebra ${\mathcal{A}_1}$ . On the other hand, this truncation reduces the geometry of the sphere to that of a point. Nothing really surprising here, we basically trivialised the problem, and since we algebra of the complex numbers is commutative, we can hardly see where the Noncommutativity will emerge. Bear with me comrades.

Let’s do the same up to the linear term ${f_a}$ , this defines a four dimensional vector space, which means that the geometry of the sphere now has been reduced to four points. This vector space can be seen as the algebra of ${M_{2\times2}}$ matrices under the map ${x^a\mapsto \mathchar'26\mkern-9mu k \sigma^a}$ , where ${\sigma^a}$ are the Pauli matrices and ${\mathchar'26\mkern-9mu k}$ a constant with dimensions of lenght. Let us denote this algebra by ${\mathcal{A}_2}$ . The Pauli matrices correspond to the 2-dimensional ( ${j=1/2}$ ) irreducible representation of the ${su(2)}$ algebra, allowing for ${\ell}$ , the eigenvalue of ${\sigma_3}$ , to take just two values: ${\ell \in \{-1/2,+1/2\}}$ . This means that among these four points that constitute our truncated geometry, we can just distinguish two–hence the name fuzzy. Due to our lack of imagination, we shall name this distinguishable points north and south pole. Finally, we note that (2) does not introduces any constraint but does establishes a relation between the radius and the constant ${\mathchar'26\mkern-9mu k }$ , namely

$\displaystyle r^2=3 \mathchar'26\mkern-9mu k^2. \ \ \ \ \ (4)$

This was way more interesting than the first truncation, the noncommutativity emerges from the Pauli matrices, which are the most iconic trio ever, even if some people will dare to say that it is the Kardashians.

The next truncation is up to quadratic terms, and since (2) is quadratic as well, it will set the dimension of our vector space to be nine. Therefore, now the geometry of the sphere got reduced to the geometry of nine points. As in the previous case, we use ${x^a\mapsto \mathchar'26\mkern-9mu k J^a }$ to map this vector space to the algebra of ${3\times 3}$ matrices ${M_{3\times 3}}$ which we will now call ${\mathcal{A}_3}$ . Here, ${J^a}$ are the basis of the 3-dimensional ( ${j=1}$ ) irreducible representation of the ${su(2)}$ algebra, in this case the ${\ell}$ the eigenvalue of ${J_3}$ can take the values ${{-1,0,+1}}$ . This means that now we have three distinguishable points, the poles and a new addition that we shall call the equator–thus fuzzy, but less than the previous one. The radial relation under this representation becomes

$\displaystyle r^2=8\mathchar'26\mkern-9mu k^2. \ \ \ \ \ (5)$

We could repeat this over and over again, for each ${n\in\mathbb{N}}$ we would find:

The commutative algebra ${C^0(\mathbb{S}^2)}$ gets reduced to the noncommutative algebra ${\mathcal{A}_n}$ of ${n\times n}$ matrices.
The map ${x^a\mapsto \mathchar'26\mkern-9mu k J^a }$ is valid for any ${n}$ . In each case ${J^a}$ conforms the basis of the ${n}$ -dimensional irreducible representation of the ${su(2)}$ algebra.
The geometry of the sphere ${\mathbb{S}^2}$ gets reduced to the geometry of ${n^2}$ points.
Under this representation, the constraint (2) becomes: ${r^2=(j^2-1)\mathchar'26\mkern-9mu k}$ . For large ${n}$ we have ${ \mathchar'26\mkern-9mu k\sim r/n}$ .

We note that ${J^a}$ obey

$\displaystyle [J^a,J^b]=i( \mathchar'26\mkern-9mu k/r)\varepsilon^{ab}_{\;\;\;c}J^c, \ \ \ \ \ (6)$

which commutes for large ${n}$ , as expected. Also, in this same limit we recover the commutative algebra of complex valued continuous functions on the sphere and we recover all of the points, all them distinguishable at this point.

Let us recapitulate: our starting point was a ${C^*}$ algebra of functions, the spherical harmonics. Then, we mapped it to a ${*}$ -algebra of bounded operators, as the Noncommutative Gel’fand-Naimark Theorem (NGMT) entails. No surprises so far, but now let us pretend we are really smart and we thought of this the other way around, that is: our starting point is the algebra of operators for a given ${n}$ , then use the NGMT to obtain fuzzy spheres that will approximate the usual sphere as ${n}$ grows large. Obtaining geometries from algebras of bounded operators is one of the main goals of NCG and one of the main motivation of Alain Connes to study it. Since of the main consequences of General Relativity is that geometry is physics, and we have obtained a “noncommutative geometry” from a noncommutative algebra, here are some words of encouragement for those who feel lost

Mathematics can lead us in a direction we would not take if we only followed up physical ideas by themselves.
P. A. M. Dirac

However, be sure to be extra careful, not of all of us have the intuition and talent that characterised Dirac.

We will see more about the work of Connes on our next entry, where we will mainly talk about the spectral triple, which might be the most iconic trio of them all.

If you have any questions or more iconic trios, feel free to drop me a Tweet!

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