Related papers: Is there a Jordan geometry underlying quantum phys…
An algebra A not encountered in either the usual algebraic varieties or supervarieties is introduced. A is a graded and deformed version of the quaternions, with structure similar to that of a Jordan-Lie superalgebra as defined by Okubo and…
These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key…
Over a field of characteristic $0$ we give a concrete, computation--ready description of Jordan algebra structures and their low--order deformation theory. The Jordan identity is quartic in the elements and cubic in the multiplication, and…
The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this…
In this paper, we classify four-dimensional Jordan algebras over an algebraically closed field of characteristic different of two. We establish the list of 73 non-isomorphic Jordan algebras.
Standard particle theory is based on quantized matter embedded in a classical geometry. Here, a complementary model is proposed, based on classical matter -- massive bodies, without quantum properties -- embedded in a quantum geometry. It…
The apparent impossibility of extending non-relativistic quantum mechanics to a relativistic quantum theory is shown to be due to the insufficient structural richness of the field of complex numbers over which quantum mechanics is built. A…
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…
Quantum deformations of sets of points of the real and the complexified projective line are constructed. These deformations depend on the deformation parameter q and certain further parameters \lambda_{ij}. The deformations for which the…
Quantum bits can be isolated to perform useful information-theoretic tasks, even though physical systems are fundamentally described by very high-dimensional operator algebras. This is because qubits can be consistently embedded into…
The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical…
We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.
Advances in quantum computing over the last two decades have required sophisticated mathematical frameworks to deepen the understanding of quantum algorithms. In this review, we introduce the theory of Lie groups and their algebras to…
We demonstrate how one can see quantization of geometry, and quantum algebraic structure in supersymmetric gauge theory.
We briefly describe our application of a version of noncommutative differential geometry to the 3-dim quantum space covariant under the quantum group of rotations $SO_q(3)$ and sketch how this might be used to determine the correct physical…
We review the main features of a mathematical framework encompassing some of the salient quantum mechanical and geometrical aspects of Hall systems with finite size and general boundary conditions. Geometrical as well as algebraic…
We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical…
The measurement process for hidden-configuration formulations of quantum mechanics is analysed. It is shown how a satisfactory description of quantum measurement can be given in this framework. The unified treatment of hidden-configuration…
A general noncommutative-geometric theory of principal bundles is presented. Quantum groups play the role of structure groups. General quantum spaces play the role of base manifolds. A differential calculus on quantum principal bundles is…
Geometries with horizons offer insights into relationships between general relativity and quantum physics. Quantum mechanics constrains relationships between kinematic parameters and the coordinates describing the dynamics. Example quantum…