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Standard quantum mechanics is viewed as a limit of a cut system with artificially restricted dimension of a Hilbert space. Exact spectrum of cut momentum and coordinate operators is derived and the limiting transition to the infinite…
The relational version of the modal interpretation offers both a consistent quantum ontology and solution for quantum paradoxes within the framework of nonrelativistic quantum mechanics. In the present paper this approach is generalized for…
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…
Spin 1 particle is investigated in 3-dimensional curved space of constant negative curvature. An extended helicity operator is defined and the variables are separated in a tetrad-based 10-dimensional Duffin--Kemmer equation in quasi…
We study a relativistic quantum particle in cosmic string spacetime in the presence of a uniform magnetic field and a Coulomb-type scalar potential. It is shown that the radial part of this problem possesses the $su(1,1)$ symmetry. We…
The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) linearized to vector spaces, particularly in Hilbert spaces. That is,…
A precise physical description and understanding of the classical dual content of quantum theory is necessary in many disciplines today: from concepts and interpretation to quantum technologies and computation. In this paper we investigate…
We argue that a conformally invariant extension of general relativity coupled to the Standard Model is the fundamental theory that needs to be quantized. We show that it can be treated by loop quantum gravity techniques. Through a gauge…
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…
Although classical mechanics and quantum mechanics are separate disciplines, we live in a world where Planck's constant \hbar>0, meaning that the classical and quantum world views must actually {\it coexist}. Traditionally, canonical…
In loop quantum cosmology the quantum dynamics is well understood. We approximate the full quantum dynamics in the infinite dimensional Hilbert space by projecting it on a finite dimensional submanifold thereof, spanned by suitably chosen…
We derive an exact expression for the quantumness of a Hilbert space (defined in quant-ph/0302092), and show that in composite Hilbert spaces the signal states must contain at least some entangled states in order to achieve such a…
A consistent theory, which describes the incoherent scattering of classically moving relativistic particles by the nuclei of crystal planes without any phenomenological parameter is presented. The basic notions of quantum mechanics are…
This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…
We examine canonical quantization of relativistic field theories on the forward hyperboloid, a Lorentz-invariant surface of the form $x_\mu x^\mu = \tau^2$. This choice of quantization surface implies that all components of the 4-momentum…
A new class of state transformations that are quantum mechanically prohibited is introduced. These can be seen as the generalization of the universal-NOT transformation which, for all pure inputs state of a given Hilbert space produces pure…
Basic theoretical issues relating to the response of confined relativistic particles are considered including the scaling of the response in spacelike and timelike regions of momentum transfer and the role of final state interactions. A…
Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface $\Sigma$, which is the…
This paper can be seen as an exercise in how to adapt quantum mechanics from a strict relativistic perspective while being respectful and critical towards the experimental achievements of the contemporary theory. The result is a fully…
Quantum mechanics is sensitive to the geometry of the underlying space. Here, we present a framework for quantum scattering of a non-relativistic particle confined to a two-dimensional space. When the motion manifold hosts localized…