Related papers: Differential structure of Greechie logics
Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete…
A new class of noncommutative $k$-algebras (for $k$ an algebraically closed field) is defined and shown to contain some important examples of quantum groups. To each such algebra, a first order theory is assigned describing models of a…
The aim of this lecture is to give a pedagogical explanation of the notion of a Poisson Lie structure on the external algebra of a Poisson Lie group which was introduced in our previous papers. Using this notion as a guide we construct…
Quantum information and computation may serve as a source of useful axioms and ideas for the quantum logic/quantum structures project of characterizing and classifying types of physical theories, including quantum mechanics and classical…
We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,1)-category (like spectra or chain complexes) gives rise to a differential cohomology diagram and a homotopy formula, which are common features of…
We introduce a model of noncommutative geometry that gives rise to the uncertainty relations recently derived from the discussion of a quantum clock. We investigate the dynamics of a free particle in this model from the point of view of…
Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be…
We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie…
For differential calculi over certain right coideal subalgebras of quantum groups the notion of quantum tangent space is introduced. In generalization of a result by Woronowicz a one to one correspondence between quantum tangent spaces and…
A brief review of the construction and classifiaction of the bicovariant differential calculi on quantum groups is given.
The fact that quantum theory is non-differentiable, while general relativity is built on the assumption of differentiability sources an incompatibility between quantum theory and gravity. Higher order geometry addresses this issue directly…
We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for…
Products and tensor products are linked by a universal property. Imposing the invariance of the laws of Nature under tensor composition along with Leibniz identity determines quantum and classical mechanics algebraic structure through the…
We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the…
This paper reveals a categorical equivalence connecting two distinct quantum logic structures. The first is the orthomodular lattice, an algebraic system designed to formalize the properties of quantum systems. The second is a finitary…
In classical physics, a single measurement can in principle reveal the state of a system. However, quantum theory permits numerous non-equivalent measurements on a physical system, each providing only limited information about the state.…
A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are…
Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum…