Related papers: Dynamical inverse problem of representation theory…
Based on the representation theory of the $q$-deformed Lorentz and Poincar\'e symmeties $q$-deformed relativistic wave equation are constructed. The most important cases of the Dirac-, Proca-, Rarita-Schwinger- and Maxwell- equations are…
It is shown that the geometry of quantum theory can be derived from geometrical structure that may be considered more fundamental. The basic elements of this reconstruction of quantum theory are the natural metric on the space of…
In contrast to classical physics, the language of quantum mechanics involves operators and wave functions (or, more generally, density operators). However, in 1932, Wigner formulated quantum mechanics in terms of a distribution function…
We propose the method for obtaining invariants of arbitrary representations of Lie groups that reduces this problem to known problems of linear algebra. The basis of this method is the idea of a special extension of the representation…
A pedagogical and self-contained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the Weyl-Wigner…
Our starting point is Mumford's conjecture, on representations of Chevalley groups over fields, as it is phrased in the preface of "Geometric Invariant Theory". After extending the conjecture appropriately, we show that it holds over an…
A quiver is an oriented graph. Quiver mutation is an elementary operation on quivers. It appeared in physics in Seiberg duality in the nineties and in mathematics in the definition of cluster algebras by Fomin-Zelevinsky in 2002. We show,…
We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by…
Starting from a self-dual formulation of gravity, we obtain a noncommutative theory of pure Einstein theory in four dimensions. In order to do that, we use Seiberg-Witten map. It is shown that the noncommutative torsion constraint is solved…
New status in quantum mechanics is connected with recent achievements in the inverse problem. With its help instead of about ten exactly solvable models which serve as a basis of the contemporary education there are infinite (!) number,…
A dynamical theory of hypersurface deformations is presented. It is shown that a (n+1)-dimensional space-time can be always foliated by pure deformations, governed by a non zero Hamiltonian. Quantum deformations states are defined by…
A study of fundamental geometrical interactions shows that the Dirac electron can be represented as a conformal wave. A Riemannian space is used, having coordinates that transform locally as spinors. The wave function becomes a gradient.…
Quantum theory can be regarded as a non-commutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the…
We pose and solve an inverse problem for the classical field equations that arise in the Standard Model of particle physics. Our main result describes natural conditions on the representations, so that it is possible to recover all the…
In this paper we construct a geometric analogue of the Weil representation over a finite field. Our construction is principally invariant, not choosing any specific realization. This eliminates most of the unpleasant formulas that appear in…
Recently Verlinde proposed that gravity can be described as an emergent phenomena arising from changes in the information associated with the positions of material bodies. By using noncommutative geometry as a way to describe the…
The purpose of the paper is to study the foundations of the main axioms of Quantum Mechanics. From a general study of the mathematical properties of the models used in Physics to represent systems, we prove that the states of a system can…
A general theory of dynamics is formulated with the aim of its application in emergent quantum mechanics. In such a framework it is argued that the fundamental dynamics of emergent quantum mechanics must be non-reversible.
A physical theory is proposed that obeys both the principles of special relativity and of quantum mechanics. As a key feature, the laws are formulated in terms of quantum events rather than of particle states. Temporal and spatial…
This is mainly a lecture note taken by myself following Weinberg's book, but also contains some corrections to the abuse of mathematical treatment. This article discusses projective unitary representations of Poincare group on the single…