Related papers: Commuting and noncommuting infinitesimals
In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.…
We elaborate on the existing notion that quantum mechanics is an emergent phenomenon, by presenting a thermodynamical theory that is dual to quantum mechanics. This dual theory is that of classical irreversible thermodynamics. The linear…
By considering (non-relativistic) quantum mechanics as it is done in practice in particular in condensed-matter physics, it is argued that a deterministic, unitary time evolution within a chosen Hilbert space always has a limited scope,…
In this treatise on the theory of the continuum of the surreal numbers of J.H. Conway, is proved ,that the three different techniques and hierarchies of the continuums of the transfinite real numbers of Glayzal A. (1937) defined through…
It was generally believed throughout the 20-th century that irreversibility is a purely classical event without operator counterpart. However, a classical irreversible system cannot be consistently decomposed into a finite number of…
The noncommutativity concept has wide range of applications in physical and mathematical theories. Noncommutativity in the position-time coordinates concerns the microscale structure of space-time. the noncommutativity is an intrinsic…
There are only a few invariants one classically associates with precompact translation surfaces, among them certain numberfields, i.e. fields which are finite extensions of the field Q of rational numbers. These fields are closely related…
Classifying the nonflat hypersurfaces in Euclidean space $f\colon M^n\to\mathbb{R}^{n+1}$ that locally admit smooth infinitesimal deformations that preserve the Gauss map infinitesimally was a problem only considered by Schouten \cite{Sc}…
The well known relation between extended supersymmetry and complex geometry in the non-linear sigma-models is reviewed, and some recent developments related to the introduction of the non-anti-commutativity, in the context of the…
We study subfields of surreal numbers, called hyperseries fields, that are suited to be equipped with derivations and composition laws. We show how to define embeddings on hyperseries fields that commute with transfinite sums and all…
From antiquity the conceptual perception of space changed painfully and at a relatively slow pace. It went through mythological descriptions, religious beliefs, metaphysical worldviews and cosmological models with a mechanistic structure,…
We survay some nice result concerning the irrationals with a metric space point of view.Here is ofcourse nothing new may be or an expert in this field.
The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to…
In recent publications in physics and mathematics, concerns have been raised about the use of real numbers to describe quantities in physics, and in particular about the usual assumption that physical quantities are infinitely precise. In…
Given a finite quiver, its double may be viewed as its non-commutative "cotangent" space, and hence is a non-commutative symplectic space. Crawley-Boevey, Etingof and Ginzburg constructed the non-commutative reduction of this space while…
Let $A$ and $B$ be two Morita equivalent finite dimensional associative algebras over a field $\Bbbk$. It is well known that Hochschild cohomology is invariant under Morita equivalence. Since infinitesimal deformations are connected with…
Contrary to the widespread belief, the problem of the emergence of classical mechanics from quantum mechanics is still open. In spite of many results on the $\h \to 0$ asymptotics, it is not yet clear how to explain within standard quantum…
We consider "unphysical", kinematic observables that do not commute with the constraints of a gauge system in the context of an extension of the system. We show that these observables, while not predictable, can nevertheless be said to have…
Equivalence in physics is discussed on the basis of experimental data accompanied by experimental errors. It is pointed out that the introduction of the equivalence being consistent with the mathematical definition is possible only in…
Classical limits of quantum systems are shown to lead to different conceptions of spaces different from the classical one underlying the process of quantization of such systems. The accent is put in situations where traces of…