Related papers: Basics of Quantum Mechanics, Geometrization and so…
Geometric quantum mechanics aims to express the physical properties of quantum systems in terms of geometrical features preferentially selected in the space of pure states. Geometric characterisations are given here for systems of one, two,…
Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is…
A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding…
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…
The manifold of pure quantum states is a complex projective space endowed with the unitary-invariant geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given…
The purpose of this contribution is to give an introduction to quantum geometry and loop quantum gravity for a wide audience of both physicists and mathematicians. From a physical point of view the emphasis will be on conceptual issues…
We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the…
In this paper and a companion paper, we show how the framework of information geometry, a geometry of discrete probability distributions, can form the basis of a derivation of the quantum formalism. The derivation rests upon a few…
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of…
Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate…
The use of geometric methods has proved useful in the hamiltonian description of classical constrained systems. In this note we provide the first steps toward the description of the geometry of quantum constrained systems. We make use of…
Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the…
A scheme for constructing quantum mechanics is given that does not have Hilbert space and linear operators as its basic elements. Instead, a version of algebraic approach is considered. Elements of a noncommutative algebra (observables) and…
The properties which give quantum mechanics its unique character - unitarity, complementarity, non-commutativity, uncertainty, nonlocality - derive from the algebraic structure of Hermitian operators acting on the wavefunction in complex…
Recent research in the geometric formulation of quantum theory has implied that Weyl Geometry can be used to merge quantum theory and general relativity consistently as classical field theories. In the Weyl Geometric framework, it seems…
The basic principles of the quantum mechanics in the K-field formalism are stated in the paper. The basic distinction of this theory arises from that the quantum theory equations (including well-known Schrodinger, Klein-Gordon and quadratic…
We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard K\"ahler framework, we introduce an extension in which the symplectic structure is allowed…
The quantum mechanical formalism for position and momentum of a particle in a one dimensional cyclic lattice is constructively developed. Some mathematical features characteristic of the finite dimensional Hilbert space are compared with…
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…
In this short letter we review Schwinger's formulation of Quantum Mechanics and we argue that the mathematical structure behind Schwinger's "Symbolism of Atomic Measurements" is that of a groupoid. In this framework, both the Hilbert space…