Related papers: Quantum Observable Generalized Orthoalgebras
An algebraic formulation of general relativity is proposed. The formulation is applicable to quantum gravity and noncommutative space. To investigate quantum gravity we develop the canonical formalism of operator geometry, after…
We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski…
Quasi-set theory was proposed as a mathematical context to investigate collections of indistinguishable objects. After presenting an outline of this theory, we define an algebra that has most of the standard properties of an orthocomplete…
We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliar operator. This algebra describes quantum systems having eigenvalues of the Hamiltonian depending on the eigenvalues of the two…
We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by $\mathcal{H}_q (f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as…
Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z=L(H), the algebra of bounded linear operators on H. We represent A as a disjoint union of subsets M of Z and, using the…
The problem of ordering operators has afflicted quantum mechanics since its foundation. Several orderings have been devised, but a systematic procedure to move from one ordering to another is still missing. The importance of establishing…
We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of a Batalin-Vilkovisky algebra. While such an operator of order 2 defines a Gerstenhaber (Lie) algebra structure on A,…
In this paper, we show that every pair of absolutely compatible Hilbert space effects are coexistent and exhibit a partial orthogonality property. We introduce the notion of partially ortho-coexistence. We generalize absolute compatibility…
We consider the algebra $\mathcal O(\mathsf M)$ of observables and the (formally) versal morphism $\eta: A \to \mathcal O(\mathsf M)$ defined by the noncommutative deformation functor $\mathsf{Def}_{\mathsf M}$ of a family $\mathsf M = \{…
In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain…
Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ also form a convex effect algebra, they have a more…
Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This…
We give a generalization of the Hodge operator to spaces $(V,h)$ endowed with a hermitian or symmetric bilinear form $h$ over arbitrary fields, including the characteristic two case. Suitable exterior powers of $V$ become free modules over…
As established by Sol\`er, Quantum Theories may be formulated in real, complex or quaternionic Hilbert spaces only. St\"uckelberg provided physical reasons for ruling out real Hilbert spaces relying on Heisenberg principle. Focusing on this…
We develop two approaches to Quantum (or Non-commutative) Graphs based on arbitrary von Neumann algebras $M\subseteq\mathcal B(H)$: one looking at operator bimodules of Hilbert--Schmidt (instead of bounded) operators, and the second looking…
Using a representation of the q-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space…
Starting with a Lie algebroid ${\cal A}$ over a space $M$ we lift its action to the canonical transformations on the affine bundle ${\cal R}$ over the cotangent bundle $T^*M$. Such lifts are classified by the first cohomology $H^1({\cal…
A noncommutative *-algebra that generalizes the canonical commutation relations and that is covariant under the quantum groups SOq(3) or SOq(1,3) is introduced. The generating elements of this algebra are hermitean and can be identified…
We initiate the study of a new notion of duality defined with respect to the module Haagerup tensor product. This notion not only recovers the standard operator space dual for Hilbert $C^*$-modules, it also captures quantum group duality in…