Related papers: Basics of Quantum Mechanics, Geometrization and so…
Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the…
In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in…
Here is discussed application of the Weyl pair to construction of universal set of quantum gates for high-dimensional quantum system. An application of Lie algebras (Hamiltonians) for construction of universal gates is revisited first. It…
This work explores the intersection of quantum mechanics and curved spacetime by employing the Wigner formalism to investigate quantum systems in the vicinity of black holes. Specifically, we study the quantum dynamics of a probe particle…
It is shown that the geometry of quantum theory can be derived from geometrical structure that may be considered more fundamental. The basic elements of this reconstruction of quantum theory are the natural metric on the space of…
We present a phase space formulation of quantum mechanics in the Schr\"odinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard…
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…
The phenomenon of quantum entanglement is thoroughly investigated, focussing especially on geometrical aspects and on bipartite systems. After introducing the formalism and discussing general aspects, some of the most important separability…
Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum…
A new version of hidden variables theory founded on the generalisation of world's geometry is proposed. The quantum-mechanical motion as the motion in some "inner space", which has a structure of the integrable Weyl space is examined.…
I describe a constructive foundation for Quantum Mechanics, based on the discreteness of the degrees of freedom of quantum objects and on the Principle of Relativity. Taking Einstein's historical construction of Special Relativity as a…
In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques…
Four formulations of quantum mechanics on noncommutative Moyal phase spaces are reviewed. These are the canonical, path-integral, Weyl-Wigner and systematic formulations. Although all these formulations represent quantum mechanics on a…
Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold $M$ is a homogeneous Riemannian space with the given action of isometry transformation…
In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also…
In this tenth paper of the series we aim at showing that our formalism, using the Wigner-Moyal Infinitesimal Transformation together with classical mechanics, endows us with the ways to quantize a system in any coordinate representation we…
This paper addresses the question why quantum mechanics is formulated in a unitary Hilbert space, i.e. in a manifestly complex setting. Investigating the linear dynamics of real quantum theory in a finite-dimensional Euclidean Hilbert space…
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this…
Five physical assumptions are proposed that together entail the general qualitative results, including the Born rule, of non-relativistic quantum mechanics by physical and information-theoretic reasoning alone. Two of these assumptions…
We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…