An informal outline of Noncommutative Geometry (Part III)

The fuzzy sphere is a really nice example of how the Noncommutative Gel’fand-Naimark Theorem (NCGNT) can be used to define Noncomutative Geometry (NCG) in terms of a Noncommutative Algebra (NCA). Now we will talk about how Alain Connes deepened the understanding in this correspondence. Before doing that, we would like to point out that Connes did not share the motivation of the physicists we mentioned in the previous entry. In other words, his principal motivation was not the geometric measurement problem, he actually wanted to write the distance function in algebraic terms. This might sound easy, but believe me, it is not an easy task at all. For the rest of this post, we will introduce something known as spectral calculus, which is the generalisation of ordinary calculus. Turns out that in order to do this, we also need to define the noncommutative generalisation of a distance function, thus obtaining for free spectral geometry.

Let us start by fixing a Hilbert space ${\mathscr{H}}$, and saying that any complex variable will correspond to a operator in ${\mathscr{H}}$. Moreover, if that variable happens to be real, then the operator must be self-adjoint. This is very similar to what it is usually done to go from classical to quantum mechanics and it should not sound that far-fetched. Fine, so those are our variables, let’s get to do some calculus, shall we?

Suppose that ${\mathscr{H}}$ is separable into a direct sum of two orthogonal closed subspaces. If we specify a self-adjoint operator ${F}$ such that ${F^2=I}$, this decomposition is given by ${{v\in \mathscr{H}|Fv=\pm v}}$ and allows us to introduce the notion of smallness for an operator ${T}$. That is, for any ${0<\varepsilon}$ there is a finite dimensional subspace ${E\subset\mathscr{H}}$ such that ${||T_{E_{\perp}}||<\varepsilon}$, where ${E_{\perp}}$ denotes its orthogonal complement. This condition will hold for compact operators ${T}$ of ${\mathscr{H}}$, which we will denote by ${\mathscr{K}(\mathscr{H})}$. As infinitesimals correspond to “small” quantities, we make use of this notion to introduce them:

Definition 1 Let ${T\in \mathscr{K}(\mathscr{H})}$ and ${\mu_n}$ be the eigenvalues of ${|T|=\sqrt{T^*T}}$, if

$\displaystyle \exists C<\infty:\;\;\; \mu_n(|T|)\leq Cn^{-\alpha},\;\;\;\forall n\geq 1. \ \ \ \ \ (1)$

Then we say that ${T}$ is an infinitesimal of order ${\alpha}$.

Next, we want to define a differential ${d}$ so that when it acts on an operator it makes it an infinitesimal of order one. This can be achieved by setting

$\displaystyle da=[F,a]\qquad ( \text{ for all }a\in \mathscr{A}), \ \ \ \ \ (2)$
where ${\mathscr{A}}$ is the involutive algebra of operators in ${\mathscr{H}}$, we note that this automatically satisfies ${d^2=0}$. We need to guarantee that ${[F,a]\in \mathscr{K}(\mathscr{H})}$ for all ${a\in\mathscr{A}}$, this is tantamount to specifying wanted properties of any representation of ${\mathscr{A}}$ in ${\mathscr{H}}$. Turns out there is something that gets the job done, the Fredholm module.

Definition 2 (Fredholm module) An odd Fredholm module over ${\mathscr{A}}$ is given by
1. An involutive representation ${\pi}$ of ${\mathscr{A}}$ in ${\mathscr{H}}$.
2. An ${F=F^*}$, ${F^2=\mathbb{I}}$ s.t. ${[F,\pi(a)]\in\mathscr{K}}$ for all ${a\in\mathscr{A}}$
An even Fredholm module is given as above plus a ${\mathbb{Z}_2}$ grading ${\varepsilon=\varepsilon^*}$, ${\varepsilon^2=\mathbb{I}}$ s.t. ${[\varepsilon,\pi(a)]=0,\;\; \forall a\in\mathscr{A},\;\;\; \{\varepsilon,F\}=0}$.

Let us now introduce a fundamental concept if we are intending to do NCG (or any geometry for that matter), the distance.

Definition 3 Let ${ds}$ be the line element for a Riemannian manifold ${\mathcal{M}}$, then the geodesic distance between ${x}$ and ${y}$ is given by

$\displaystyle d(x,y)=\sup \left\{|f(x)-f(y)|; f\in C^\infty(\mathcal{M}), \bigg|\bigg|\frac{df}{ds}\bigg|\bigg|\leq 1\right\}. \ \ \ \ \ (3)$

A generalisation of this is given by introducing the “baby steps” operator ${g}$

Definition 4 Let ${x^\mu \in \mathscr{A}}$

$\displaystyle \mathscr{K}\ni g=(dx^\mu)^*g_{\mu\nu}dx^\nu=([F,x^\mu])^*g_{\mu\nu}[F,x^\nu], \ \ \ \ \ (4)$

we note that ${g}$ is a positive infinitesimal, from which we can make the association ${ds=g^{1/2}}$. Furthermore, we demand that ${[g,F]=0.}$

As we mentioned at the beginning of this post, one of the main objectives of Connes was to write the distance function (3) in purely algebraic terms. This can be done if we replace the points ${x,\,y}$ by pure states ${\xi,~\phi}$ on the ${C^*}$-algebra closure of ${\mathscr{A}}$ and use the evaluation map to get

$\displaystyle \phi(f)=f(x), \;\;\;\; \xi(f)=f(y),\;\;\;\forall f\in\mathscr{A}. \ \ \ \ \ (5)$

We note that we can write ${df/ds=[F/g^{1/2},a]}$, as ${g}$ commutes with ${F}$. Using this and (5), the distance function (3) acquires the following form

$\displaystyle d(\phi,\xi)=\sup \{|\phi(f)-\xi(f)|: f\in \mathscr{A}, \big|\big|[Fg^{-1/2},f]\big|\big|\leq 1 \} \ \ \ \ \ (6)$.
Which suggests introducing the operator

$\displaystyle D=Fg^{-1/2},\quad D^2=g^{-1},\quad D=F|D|,\quad |D|=g^{-1/2} \ \ \ \ \ (7)$
also known as the Dirac operator. Therefore, if we wish to define distance in algebraic terms, we need to prescribe an algebra ${\mathscr{A}}$ over a Hilbert space ${\mathscr{H}}$, a metric ${g}$ and a Fredholm module ${F}$, however, as ${D}$ contains the last two, we just need to provide the so-called spectral triple ${(\mathscr{A},\mathscr{H},D)}$. In our previous post, we said that if anyone challenged us to name a more iconic trio than the Kardashians, we would of course say: the Pauli matrices. However, after seeing that all the data we need to do (Non)commutative differential geometry is contained in the spectral triple, we stand corrected.

Let us continue, we will say that the triple is of dimension ${n}$ if ${|D|^{-n}}$ is an infinitesimal of order ${1/n}$. Having established the notions of differential and metric, one last element is needed: the integral. As in the commutative case, if we want an integral that neglects all infinitesimals of order ${n>1}$, the trace seems to be a fine candidate. Nevertheless, under closer inspection, we realise that trace of an infinitesimal of order 1 diverges as ${\ln(N)}$ (this gets worse for higher orders), which means that said infinitesimal is not on the domain of the trace. Fortunately, Dixmier came out with a solution to this: extract the divergence in a scale-invariant way, so that for any positive operator ${T}$ we have:

Definition 5 (Dixmier Trace)

$\displaystyle \mathrm{Tr}_\omega(T)=\frac{1}{\ln (N)}\sum_{n=0}^{N-1}\mu_n(T). \ \ \ \ \ (8)$

In analogy to the trace, it is linear ${ \mathrm{Tr}_\omega(aT+S)=a\mathrm{Tr}_\omega(T)+\mathrm{Tr}_\omega(S)}$ for complex ${a}$. It is cyclic ${\mathrm{Tr}_\omega(TS)=\mathrm{Tr}_\omega(ST)}$ if ${S}$ is bounded. And ${\mathrm{Tr}_\omega(T)=0}$ when the order of ${T}$ is greater than ${1}$, as we wanted.

Then, using this and the data provided by the spectral triple, we define:

Definition 6 (Spectral integral) For a ${n}$-dimensional spectral triple ${(\mathscr{A},\mathscr{H},D)}$ the integral of ${a\in\mathscr{A}}$ is defined by

$\displaystyle \int a=\frac{1}{C}\mathrm{Tr}_\omega (a|D|^{-n}). \ \ \ \ \ (9)$

Where ${C}$ is just to guarantee scale independence, futhermore, we guarantee tameness by imposing ${\int ab=\int ba,\;\;\;\; \int a^*a \geq 0}$.

All of the above is summarised in the following table

The spectral triple is one of the most powerful and elegant ideas of modern mathematics, we hope that now it is clear how it arises when trying to generalise differential geometry into the noncommutative real.

In the next post, we will use this spectral calculus to recover the usual calculus on manifolds and to explore the fuzzy sphere one more time!

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