Geometry of Non-Hausdorff Spaces and Its Significance for Physics

Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be far-reaching and illuminating. Examples of situations in which the Hausdorff relation is of the total type, i.e., when it identifies all points of the considered space, are the space of Penrose tilings and space-times of some cosmological models with strong curvature singularities. With every Hausdorff relation a groupoid can be associated, and a convolutive algebra defined on it allows one to analyze the space that otherwise would remain intractable. The regular representation of this algebra in a bundle of Hilbert spaces leads to a von Neumann algebra of random operators. In this way, a probabilistic description (in a generalized sense) naturally takes over when the concept of point looses its meaning. In this situation counterparts of the position and momentum operators can be defined, and they satisfy a commutation relation which, in the suitable limiting case, reproduces the Heisenberg indeterminacy relation. It should be emphasized that this is neither an additional assumption nor an effect of a quantization process, but simply the consequence of a purely geometric analysis.Comment: 13 LaTex pages, no figure

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma$. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi$ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi$ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.Comment: 13 pages, LaTe

Anatomy of Malicious Singularities

As well known, the b-boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation $\rho$, defined on the Cauchy completed total space $\bar{E}$ of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class $[p_0]$ related to the singularity remains in close contact with all other equivalence classes, i.e., if $p_0 \in \mathrm{cl}[p]$ for every $p \in E$. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on $\bar{E}$, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) of malicious singularities. Our main result is that, in the noncommutative regime, even the strongest singularities are probabilistically irrelevant.Comment: 16 pages in LaTe

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{{\cal A}} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{{\cal A}}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{{\cal A}}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{{\cal A}}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.Comment: 28 LaTex pages. Substantially enlarged version. Improved definition of generalized Einstein's field equation