47 Results Found

The process algebra model has been suggested as an alternative mathematical framework for non-relativistic quantum mechanics (NRQM). It appears to reproduce the wave functions of non-relativistic quantum mechanics to a high degree of accuracy. It pos...

We present a formal reconstruction of the conventional number systems, including integers, rationals, reals, and complex numbers, based on the principle of relational finitude over a finite field ${\mathbb{F}}_{\mathsf{p}}$. Rather than assuming actual infinity, we define ar...

Real Clifford algebras for arbitrary numbers of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real, complex or...

In this review, we give a basic introduction to the $\kappa$-deformed relativistic phase space and free quantum fields. After a review of the $\kappa$-Poincaré algebra, we illustrate the construction of the $\kappa$-deformed phase space of a classical relativistic partic...

We present a concise and self-contained extension of the Finite Ring Continuum (FRC) program, showing that symmetry-complete prime shells ${\mathbb{F}}_{\mathsf{p}}$ with $\mathsf{p}=4\mathsf{t}+1$ exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot...

An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general...

The Feynman path integral plays a central role in quantum mechanics, linking classical action to propagators and relating quantum electrodynamics (QED) to Feynman diagrams. However, the path-integral formulations used in non-relativistic quantum mech...

Following the work of Giulio Racah and others from the 1940s onward, the rotational symmetry of atoms and ions, e.g., the conservation of angular momentum, has been utilized in order to efficiently predict atomic behavior, from their level structure...

Recent work showed that $\kappa$-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincaré algebra of special-relativistic symmetries,...

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Max...

Two novel spacetimes are introduced in this work as anisotropic preferred reference frames tailored for applications in relativistic quantum information systems. The resulting anisotropic geometry arises intrinsically from the underlying algebraic st...

Till now, kinetic theory and statistical mechanics of either free or interacting point particles were well defined only in non-relativistic inertial frames in the absence of the long-range inertial forces present in accelerated frames. As shown in th...

Process algebras have been developed within computer science and engineering to address complicated computational and manufacturing problems. The process algebra described herein was inspired by the Process Theory of Whitehead and the theory of combi...

I report results suggesting that it is possible to introduce laws of relativistic kinematics endowing different types of particles with suitably different deformed-Lorentz-symmetry properties. I also consider some possible applications of these resul...

In this paper, a generalized form of relativistic dynamics is presented. A realization of the Poincaré algebra is provided in terms of vector fields on the tangent bundle of a simultaneity surface in ${\mathbb{R}}^4$ . The construction of this reali...

Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that...

I report, emphasizing some key open issues and some aspects that are particularly relevant for phenomenology, on the status of the development of “doubly-special” relativistic (“DSR”) theories with both an observer-independent high-velocity scale and...

We emphasize the differences between the chirality concept applied to relativistic fermions and the ususal chirality concept in Euclidean spaces. We introduce the gamma groups and we use them to classify as direct or indirect the symmetry operators e...

This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum an...

We establish the correspondence between two apparently unrelated but in fact complementary approaches of a relativistic deformed kinematics: the geometric properties of momentum space and the loss of absolute locality in canonical space-time, which c...

The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the quantumness...

The objective of this work is to derive the structure of Minkowski spacetime using a Hermitian spin basis. This Hermitian spin basis is analogous to the Pauli spin basis. The derived Minkowski metric is then employed to obtain the corresponding Loren...

Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) a...

In this article, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Section 2), “it is shown that certain mathematical...

We present one of the main lines of development of Poincaré sphere representation in polarization optics, by using largely some of our contributions in the field. We refer to the action of deterministic devices, specifically the diattenuators,...

Can the holographic principle be extended beyond the well-known AdS/CFT correspondence? During the last couple of years, there has been a substantial amount of research trying to find answers for this question. In this work, we provide a review of re...

Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmon...

A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup G, we offer a new way to construct a gyrogroup...

We review a non-relativistic current algebra symmetry approach to constructing the Bogolubov generating functional of many-particle distribution functions and apply it to description of invariantly reduced Hamiltonian systems of the Boltzmann type ki...

It is known that, in quantum field theory, localized operations, e.g., given by unitary operators in local observable algebras, may lead to non-causal, or superluminal, state changes within their localization region. In this article, it is shown that...

For further insight into the perturbation technique within the framework of the asymptotic iteration method (PAIM), we suggest this method to be used as an alternative method to the traditional well-known perturbation techniques. We show by means of...

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in th...

This paper investigates the validity of the long wavelength approximation in the calculation of two-photon decay of $2s_{1/2}$ level in hydrogen-like ions with nuclear charge $Z=1-100$ based on time-dependent second-order perturbation theory and angul...

We derive generalised uncertainty relations (GURs) for orbital angular momentum and spin in the recently proposed smeared-space model of quantum geometry. The model implements a minimum length and a minimum linear momentum and recovers both the gener...

The algebras of the symmetry operators for the Klein–Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an an...

A method is proposed for describing the dynamics of systems of interacting particles in terms of an auxiliary field, which in the static mode is equivalent to given interatomic potentials, and in the dynamic mode is a classical relativistic composite...

The aim of this review is threefold. On the one hand, we intend it to serve as a gentle introduction to the Interacting Quantum Atoms (IQA) methodology for those unfamiliar with it. Second, we expect it to act as an up-to-date reference of recent dev...

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of...

Wigner’s little groups are the subgroups of the Poincaré group whose transformations leave the four-momentum of a relativistic particle invariant. The little group for a massive particle is SO(3)-like, whereas for a massless particle, it...

Whether an algebraic or a geometric or a phenomenological prescription is applied, the first fundamental form is unambiguously related to the modeling of the curved spacetime. Accordingly, we assume that the possible quantization of the first fundame...

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic...

Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin,...

In this work we study the spectral dimensionality of spacetime around a radiating Schwarzschild black hole using a recently introduced formalism of quantum gravity, where the alterations of the gravitational field produced by the radiation are repres...

In non-relativistic quantum mechanics, the dynamics of closed quantum systems is described by Hamiltonians which are self-adjoint in appropriately chosen Hilbert spaces. For PT-symmetric quantum systems, the Hamiltonians are, in general, no longer se...

A toy model (suggested by Klauder) was analyzed from the perspective of first-class and second-class Dirac constrained systems. First-class constraints are often associated with the existence of important gauge symmetries in a system. A comparison wa...

A Lorentz transformation group SO$(m, n)$ of signature $(m, n)$, $m, n \in \mathbb{N}$, in m time and n space dimensions, is the group of pseudo-rotations of a pseudo-Euclidean space of signature $(m, n)$. Accordingly, the Lorentz gro...

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (...