Looking for a Noncommutative Geometry... Again!

Disclaimer: This postal service is part of a series virtually Noncommutative Geometry. I do not pretend to be rigorous nor thorough. The chief idea is simply to comprehend (in a rather breezy style) the principal 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 case that showcases the features that are gained when going from a commutative to a noncommutative setting. That case is the fuzzy sphere, get-go studied by Madore, from which this postal service has been heavily inspired.

The algebra of functions on a fuzzy sphere differs from that of the ordinary sphere considering it's not commutative. If we are talking most functions on the sphere, spherical harmonics ${Y_{\ell,j}}$ are certainly a good starting indicate. These labels obey ${\ell\leq j}$ , and the production rule for the harmonics is given past

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

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 y'all take a look at the noncommutative Gel'fand-Naimark Theorem, you can meet that we take met all of the weather. This ways that we can think call up of the generators of ${su(2)}$ as the edifice blocks for a noncommutative geometry, that of the fuzzy sphere.

That beingness said, to study the fuzzy sphere in two dimensions, we begin past because ${\mathbb{R}^3}$ 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 exist (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 nosotros truncate all the terms up to the abiding ${f_0}$ , the commutative algebra ${C^0(\mathbb{S}^2)}$ is reduced to the algebra of complex numbers, which nosotros volition denote past algebra ${\mathcal{A}_1}$ . On the other mitt, this truncation reduces the geometry of the sphere to that of a signal. Nothing really surprising hither, we basically trivialised the problem, and since nosotros algebra of the circuitous numbers is commutative, nosotros can hardly run into where the Noncommutativity will sally. Bear with me comrades.

Let'due south practice the same upward to the linear term ${f_a}$ , this defines a 4 dimensional vector infinite, which ways that the geometry of the sphere now has been reduced to four points. This vector infinite tin be seen as the algebra of ${M_{2\times2}}$ matrices nether 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 abiding with dimensions of lenght. Permit us denote this algebra past ${\mathcal{A}_2}$ . The Pauli matrices correspond to the two-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, nosotros can merely distinguish 2–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 just 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 beginning truncation, the noncommutativity emerges from the Pauli matrices, which are the almost iconic trio e'er, even if some people will dare to say that information technology is the Kardashians.

The next truncation is up to quadratic terms, and since (2) is quadratic too, information technology will set the dimension of our vector space to exist nine. Therefore, now the geometry of the sphere got reduced to the geometry of nine points. As in the previous example, we utilise ${x^a\mapsto \mathchar'26\mkern-9mu k J^a }$ 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 ground of the 3-dimensional ( ${j=1}$ ) irreducible representation of the ${su(2)}$ algebra, in this instance the ${\ell}$ the eigenvalue of ${J_3}$ can accept the values ${{-1,0,+1}}$ . This ways that now nosotros have three distinguishable points, the poles and a new add-on that we shall call the equator–thus fuzzy, just 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 echo this over and once again, for each ${n\in\mathbb{N}}$ nosotros would find:

Nosotros annotation 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. As well, in this same limit nosotros 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.

Allow us recapitulate: our starting indicate 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 at present let us pretend nosotros are really smart and we thought of this the other way effectually, that is: our starting point is the algebra of operators for a given ${n}$ , and so use the NGMT to obtain fuzzy spheres that volition approximate the usual sphere as ${n}$ grows big. Obtaining geometries from algebras of bounded operators is i of the main goals of NCG and ane of the master motivation of Alain Connes to report information technology. Since of the chief consequences of Full general Relativity is that geometry is physics, and nosotros have obtained a "noncommutative geometry" from a noncommutative algebra, hither are some words of encouragement for those who experience lost

Mathematics tin atomic number 82 us in a direction we would not take if nosotros only followed up concrete ideas past themselves.
P. A. M. Dirac

All the same, be certain to be extra careful, not of all of usa have the intuition and talent that characterised Dirac.

We will see more than most the piece of work of Connes on our adjacent entry, where nosotros will mainly talk about the spectral triple, which might be the almost iconic trio of them all.

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

allensompoo1977.blogspot.com

Source: https://nonsequitur.blog/2020/09/10/an-informal-outline-of-noncommutative-geometry-part-ii/

Looking for a Noncommutative Geometry... Again!

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