An approximation theorem for non-decreasing functions on compact posets
In this short note we prove a theorem of the Stone-Weierstrass sort for subsets of the cone of non-decreasing continuous functions on compact partially ordered sets.
math.CA↗
arXiv subjects
Fabien Besnard
Publications and source records attributed to Fabien Besnard.
Time of Philosophers, Time of Physicists, Time of Mathematicians
Is presentism compatible with relativity ? This question has been much debated since the argument first proposed by Rietdijk and Putnam. The goal of this text is to study the implications of relativity and quantum mechanics on presentism, possibilism, and eternalism. We put the emphasis on the implicit metaphysical preconceptions underlying each of these different approaches to the question of time. We show that there exists a unique version of presentism which is both non-trivial, in the sense that it does not reduce the present to a unique event, and compatible with special relativity and quantum mechanics: the one in which the present of an observer at a point is identified with the backward light cone of that point. However, this compatibility is achieved at the cost of a renouncement to the notion of an objective, observer-independent reality. We also argue that no non-trivial version of presentism survives in general relativity, except if some mechanism forbids the existence of closed timelike curves, in which case precisely one version of possibilism does survive. We remark that the above physical theories force the presentist/possibilist's view of reality to shrink and break up, whereas the eternalist, on the contrary, is forced to grant the status of reality to more and more entities. Finally, we identify mathematics as the "deus ex machina" allowing the eternalist to unify his vision of reality into a coherent whole, and offer to him an "idealist deal": to accept a mathematical ontology in exchange for the assurance of surviving any physical theory.
physics.hist-ph↗
Simultaneity in Minkowski spacetime: from uniqueness to arbitrariness
In 1977, Malament proved a certain uniqueness theorem about standard synchrony, also known as Poincaré-Einstein simultaneity, which has generated many commentaries over the years, some of them contradictory. We think that the situation called for some cleaning up. After reviewing and discussing some of the literature involved, we prove two results which, hopefully, will help clarifying this debate by filling the gap between the uniquess of Malament's theorem, which allows the observer to use very few tools, and the complete arbitrariness of a time coordinate in full-fledged Relativity theory. In the spirit of Malament's theorem, and in opposition to most of its commentators, we emphasize explicit definability of simultaneity relations, and give only constructive proofs. We also explore what happens when we reduce to "purely local" data with respect to an observer.
math-ph↗
A noncommutative view on topology and order
In this paper we put forward the definition of particular subsets on a unital C*-algebra, that we call isocones, and which reduce in the commutative case to the set of continuous non-decreasing functions with real values for a partial order relation defined on the spectrum of the algebra, which satisfies a compatibility condition with the topology (complete separateness). We prove that this space/algebra correspondence is a dual equivalence of categories, which is in fact only a mild generalization of the Gelfand-Naimark duality. Thus we can expect that general isocones could serve to define a notion of noncommutative ordered spaces. We also explore some basic algebraic constructions involving isocones, and classify those which are defined in M_2(C).
math.OA↗
Canonical quantization and the spectral action, a nice example
We study the canonical quantization of the theory given by Chamseddine-Connes spectral action on a particular finite spectral triple with algebra $M_2(\Cset)\oplus\Cset$. We define a quantization of the natural distance associated with this noncommutative space and show that the quantum distance operator has a discrete spectrum. We also show that it would be the same for any other geometric quantity. Finally we propose a physical Hilbert space for the quantum theory. This spectral triple had been previously considered by Rovelli as a toy model, but with a different action which was not gauge-invariant. The results are similar in both cases, but the gauge-invariance of the spectral action manifests itself by the presence of a non-trivial degeneracy structure for our distance operator.
gr-qc↗
Number Operator Algebras and generalizations of supersymmetry
Several quantum systems have been used in the last few years to extend supersymmetry. In this paper we show all this systems fit into the picture of what we call "Number Operator Algebras".
math-ph↗
Number Operator Algebras and deformations of epsilon-algebras
It is well known that the Lie-algebra structure on quantum algebras gives rise to a Poisson-algebra structure on classical algebras as the Planck constant goes to 0. We show that this correspondance still holds in the generalization of super- algebra introduced by Scheunert, called epsilon-algebra. We illustrate this with the example of Number Operator Algebras, a new kind of object that we have defined and classified under some assumptions.
math-ph↗
Number Operator Algebras
Under some hypotheses (symmetry, confluence), we enumerate all quadratically presented algebras, generated by creation and destruction operators, in which number operators exist. We show that these are algebras of bosons, fermions, their immediate generalizations that we call pseudo-bosons and pseudo-fermions, and also matrix algebras, in the finitely generated case. We then recover q-bosons (and pseudo-q-bosons) by a completion operation.
math-ph↗