相关论文: Geometric Calculus and the Fibre Bundle descriptio…
Some elements of classical mechanics and classical statistical mechanics are formulated in terms of fibre bundles. In the bundle approach the dynamical and distribution functions are replaced by liftings of paths in a suitably chosen…
We show how the basic operations of quantum computing can be expressed and manipulated in a clear and concise fashion using a multiparticle version of geometric (aka Clifford) algebra. This algebra encompasses the product operator formalism…
These notes give an introduction to the quantization procedure called geometric quantization. It gives a definition of the mathematical background for its understanding and introductions to classical and quantum mechanics, to differentiable…
In this paper we will extend the notion of tangent bundle to a $\z$ graded tangent bundle. This graded bundle has a Lie algebroid structure and we can develop notions semi-riemannian metrics, Levi-civita connection, and curvature, on it. In…
In this paper we investigate quantum circle bundles from the point of view of compact quantum metric spaces. The raw input data is a circle action on a unital $C^*$-algebra together with a quantum metric of spectral geometric origin on the…
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles (`quantum bundles') over the spacetime manifold yield a quantum ``formalism'' along…
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…
In this paper, algebroid bundle associated to affine metrics provide an structure for unification of gravity and electromagnetism and, geometrization of matter.
Clifford algebras are important structures in Geometric Algebra and Quantum Mechanics. They have allowed a formalization of the primitive operators in Quantum Theory. The algebras are built over vector spaces with dimension a power of 2…
Geometric quantization is an attempt at using the differential-geometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory. Generally, the process of geometric…
Following the B. Hiley belief that unresolved problems of conventional quantum mechanics could be the result of a wrong mathematical structure, an alternative basic structure is suggested. Critical part of the structure is modification of…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schr\"odinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally,…
It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of…
Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
The underlying mathematical structures of gauge theories are known to be geometrical in nature and the local and global features of this geometry have been studied for a long time in mathematics under the name of fibre bundles. It is now…
It is well--known that if one is given a principal $G$--bundle with a principal connection, then for every unitary finite--dimensional linear representation of $G$ one can induce a linear connection and a Hermitian structure on the…
The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas:…
In this paper we investigate the possibility of constructing a complete quantization procedure consisting of geometric and deformation quantization. The latter assigns a noncommutative algebra to a symplectic manifold, by deforming the…