Towards an Euclidean (=axiomatic) non-commutative geometry.

I first noted a non-commutative Riemann surface in my Ph.D thesis in 1972 when using extensions of central valuations to pseudo-vbaluations on non-commutative algebras.

During my visit at Cambridge (UK) in 1973 I defined the prime spectrum and a Zariski like topology on non-commutative rings attemptying to arrive at a non-comutative sheaf theory (published with D.Murdoch in 1975) and then started to apply this to PI rings leading to several papers and the first book on non-commutative Algebraic Geometry in 1978 published in the Springer Lecture Notes series (with my student A. Verschoren who died not so long ago).

The PI case yields a nc-theory with a dense central part, so I started to generalise this to a category theoretical setting using torsion theories, noting that the intersection of open sets in the topology corresponded to a composition of localization functors with the composition being a localization functor again exactly when the functors commuted, leading to the idea of a nc-geometry as a deformation of its commutative shadow which related to idempotency of the topological open sets!

This started my idea of an axiomatic theory for non-commutaive topology, with special cases of skew topology and virtual topology, where topology was replaced by some non-commutative lattices (and then related to nc-Heyting algebras). In the book Virtual Topology and Functor Geometry (Marcel Dekker Publ.) I reached a satisfactory abstract axiomatic theory, for example the definition of a nc-Grothendieck topology which turned out to be given by a perfect symmetrization of the definition given by A. Grothendieck and that highlighted the fact that non-commutativity had the effect of having everything ordered, so I realised the quantum nc-logic should be a totally ordered one (like language is!).

Meanwhile M.Artin (MIT) who visited Antwerp a lot got into that nc-geometry of algebras, leading to an interesting cooperation between him and my student M.Van den Bergh, but he decided the geometrical language could be virtually applied to algebraic properties of the algebras and I disagreed with that, I wanted an intrinsic notion of non-commutative space.

Also A.Connes (Field medaillist) stuck to the notion of nc-geometry as a virtual language talking about algebraic properties of algebras of operators on some Hilbert spaces, and also there I needed a real geometric object defined axiomatically, like Euclid did to avoid the use of numbers and algebra.

I had already encountered “observables” as some associated graded aspect of filtrations on spectral series (mentioned in the Virtual Topology book) and everything related to some lattices on Hilbert spaces, unexpectedly I also found a surprising relation with the pseudo-places I defined in my Ph.D thesis and even today I am concvinced there is some stringent deeper relation between those concepts hiding a newer model for reality; afterall discrete valuation rings are an algebraic analogon for points in Algebraic Geometry so the nc-version of pseudo-places (pseudo-valuation riongs) should have some geometric meaning too.

Sheaves over lattices were called quantum sheaves by some mathematicians and I got convinced the notion of sheaf had to be replaced by a more general notion of étale “space”. Ater I gave some lectures at Physics gatherings in Sweden and UK and I was asked how real events could fit in such geometry, I realized real events do not fit in any geometry because that is only (!) an abstract construction of a model in our mind, so I realised it has to be a dynamic process and an étale “space” over the total order topology of Time, just being a totally ordered set ordering the momentary states of the universe.

A state in a moment being unobservable and non-existing even — if existing is only possible in a non-trivial time interval — places and “space” only could make sense when dynamic as a series of places of non-existing potentials in states connected by the gluing correspondences between states. So the “geometry “ I needed was one spread out over the order topology of time coming from non-commutative topologies defined abstractly in the states and then yielding dynamic places and dynamic topologies on something we view as “space”.

Reality may then be viewed as embedded in the processes of momentary potentials etalated over that dynamic “space”, as everything existing is a process over time. The paradigm-shifts this entails for physics ,science and philosophy I started to analyse in the book “Time Hybrids” published a year ago.

So in my spacetime model there is a much more intrinsic mixing of the notions of “space” and time, in fact the series of momentary places approximated by some “averaging” of a series of them over some existence interval of an object was not a place at all, I called it a “dynamic place”. This makes the Heisenberg incertainty relation into a simplified approximation and a distortion of the real unobservability going on in reality.

Now the structures I am facing are abstract but completely new,and open for future investigation. For example to investigate applicability ( in a measurable way !) to reality it would be nice to please the physicists and reintroduce some measuring by real numbers — thereby sacrificing the genericity of the model — but since the new paradigms would persist, it would be a worthwhile enterprise.

I guess I am the father of the generic nc-geometry, the restriction to algebraic theory of special algebras appearing in Physiscs is worthwhile but not something a mathematician should strive for — physicists can do that — even if it gets much more attention. For the physical meaning of my model it is necesary to dive into the “void” where everything existing is being processed and prepared to exist later; I prepared some structure there and I hope some mathematicians will go further with it. For me, Time is running out.