arXiv subjects

Michael Heller

Publications and source records attributed to Michael Heller.

21 records · Page 2Linked to original sources

arXiv2004

Observables in a Noncommutative Unification of Quanta and Gravity. A Final Model

We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in {\it Gen. Rel. Grav.} (2004) {\bf 36}, 111-126. 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.

gr-qc↗

arXiv2004

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $Γ= E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $Γ$. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, ϕ)$, where $ϕ$ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $ϕ$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, ϕ)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $ϕ$ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.

gr-qc↗

arXiv1997

Local and Global Properties of the World

The essence of the method of physics is inseparably connected with the problem of interplay between local and global properties of the universe. In the present paper we discuss this interplay as it is present in three major departments of contemporary physics: general relativity, quantum mechanics and some attempts at quantizing gravity (especially geometrodynamics and its recent successors in the form of various pregeometry conceptions). It turns out that all big interpretative issues involved in this problem point towards the necessity of changing from the standard space-time geometry to some radically new, most probably non-local, generalization. We argue that the recent noncommutative geometry offers attractive possibilities, and gives us a conceptual insight into its algebraic foundations. Noncommutative spaces are, in general, non-local, and their applications to physics, known at present, seem very promising. One would expect that beneath the Planck threshold there reigns a ``noncommutative pregeometry'', and only when crossing this threshold the usual space-time geometry emerges.

gr-qc↗