Zdzisław Odrzygóźdź

Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model

AbstractWe construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid Γ given by the action of a finite group on a space E. We define the algebra

State vector reduction as a shadow of a noncommutative dynamics

\mathcal{A}

Structure of Malicious Singularities

of smooth complex valued functions on Γ, with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time geometry. The quantum sector of the model is explored in terms of the regular representation of the algebra

Gravitational Aharonov-Bohm Effect

\mathcal{A}

Application Supporting Commercial Bank's Capital Management

, and its correspondence with the standard quantum mechanics is established.

Practical aspects of the use of the central management information system for management by objectives in a large universal comercial bank

We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in Gen. Rel. Grav. (36, 111–126 (2004)). Generalized symmetries of the model are defined by a groupoid Γ given by the action of a finite group on a space E. The geometry of the model is constructed in terms of suitable (noncommutative) algebras on Γ. We investigate observables of the model, especially its position and momentum observables. This is not a trivial thing since the model is based on a noncommutative geometry and has strong nonlocal properties. We show that, in the position representation of the model, the position observable is a coderivation of a corresponding coalgebra, “coparallelly” to the well-known fact that the momentum observable is a derivation of the algebra. We also study the momentum representation of the model. It turns out that, in the case of the algebra of smooth, quickly decreasing functions on Γ, the model in its “quantum sector” is nonlocal, i.e., there are no nontrivial coderivations of the corresponding coalgebra, whereas in its “gravity sector” such coderivations do exist. They are investigated.