Introduction and Overview

Noncommutative geometry asks: ``What is the geometry of the Quantum World?''

Quantum field theory considers aggregates of ``particles'', which are of two general species, ``bosons'' and ``fermions''. These are described by solutions of (relativistic) wave equations:

• Bosons: Klein-Gordon equation, $(\square + m^2)\phi(x) = \rho_b(x)$ - ``source term'';
• Fermions: Dirac equation, $(i\delslash - m)\psi(x) = \rho_f(x)$ - ``source term'';

where $x = (t, \vec{x}) = (x^0, x^1, x^2, x^3)$; $\square = - \del^2/\del t^2 + \del^2/\del\vec{x}^2$; and $\delslash = \sum_{\mu=0}^3 \gamma^{\mu} \del/\del x^{\mu}$. In order that $\delslash$ be a ``square root of $\square$'', we need $(\gamma^0)^2 = -1$, $(\gamma^j)^2 = +1$ for $j = 1,2,3$ and $\gamma^{\mu}\gamma^{\nu} = -\gamma^{\nu}\gamma^{\mu}$ for $\mu \neq \nu$. Thus, the $\gamma^{\mu}$ must be matrices; in fact there are four ($4 \x 4$) matrices satisfying these relations.

Point-like measurements are often ruled out by quantum mechanics; thus we replace points $x \in M$ by coordinates $f \in C(M)$. The metric distance on a Riemannian manifold $(M, g)$ can be computed in two ways:
$$\begin{align*} d_g(p,q) &:= \inf\set{\operatorname{length}(\gamma \: [0,1]\to M... ...\vert f(p) - f(q)\vert : f \in C(M), \Vert\Dslash, f\Vert \leq 1}, \end{align*}$$
where $\Dslash$ is a Dirac operator with positive-definite signature (all $(\gamma^{\mu})^2 = +1$) if it exists, so the Dirac operator specifies the metric. $\Dslash$ is an (unbounded) operator on a Hilbert space $\sH = L^2(M,S)$ of ``square-integrable spinors'' and $\Coo(M)$ also acts on $\sH$ by multiplication operators with $\Vert[\Dslash, f]\Vert = \Vert\grad f\Vert _\infty$.

Noncommutative geometry generalizes $(\Coo(M), L^2(M,S), \Dslash)$ to a spectral triple of the form $(\sA, \sH, D)$, where $\sA$ is a ``smooth'' algebra acting on a Hilbert space $\sH$, $D$ is an (unbounded) selfadjoint operator on $\sH$, subject to certain conditions: in particular that $[D, a]$ be a bounded operator for each $a \in \sA$. The tasks of the geometer are then:

1. To describe (metric) differential geometry in an operator language.
2. To reconstruct (ordinary) geometry in the operator framework.
3. To develop new geometries with noncommutative coordinate algebras.

The long-term goal is to geometrize quantum physics at very high energy scales, but we are still a long way from there.

The general program of these lectures is as follows.

(A)

The classical theory of spinors and Dirac operators in the Riemannian case.

(B)

The operational toolkit for noncommutative generalization.

(C)

Reconstruction: how to recover differential geometry from the operator framework.

(D)

Examples of spectral triples with noncommutative coordinate algebras.