Considering the problem of autonomous development of quantum mechanics in the broader context of solution of Hilbert's sixth problem (which relates to joint axiomatization of physics and probability theory), a formalism is evolved in this two-part work which facilitates the desired autonomous development and satisfactory treatments of quantum-classical correspondence and quantum measurements. T...

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, i...

-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and compositions thereof. However, they have previously been limited by an inability to express families of diagrams involving non-commutative structures which play a central role in algebraic quantum informati...

Introduction: Inspired by quantum mechanics and probability, non-commutative probability has become an independent field of mathematical research. We refer to P.A. Meyer’s exposition [Me], the successive conferences on quantum probability [AvW], the lecture notes by Jajte [Ja1, Ja2] on almost sure and uniform convergence and finally the work of Voiculescu, Dykema, Nica [VDN] and of Biane, Speic...

Quantum heat cycles and quantum refrigerators are analyzed using various systems as their working mediums. For example, to evaluate the efficiency work done of Carnot cycle in regime, one can consider harmonic oscillator a medium. all these well-defined substances (which commutative space structure), engine is not up mark efficiency. So, inevitable question arise, observe catalytic effect on en...

We investigate quantum electrodynamic effects under the influence of an external, time-dependent electromagnetic field, which mediates dynamic modifications of the radiative corrections. Specifically, we consider the quantum electrodynamic vacuum-polarization tensor under the influence of two external background fields: a strong laser field and a nuclear Coulomb field. We calculate the charge a...

A general theory of characteristic classes of quantum principal bundles is sketched, incorporating basic ideas of classical Weil theory into the conceptual framework of non-commutative differential geometry. A purely cohomological interpretation of the Weil homomorphism is given, together with a geometrical interpretation via quantum invariant polynomials. A natural spectral sequence is describ...