International Journal of Geometric Methods in Modern Physics

A relativistic version of the correspondence principle, a limit in which classical electrodynamics may be derived from quantum electrodynamics (QED), has never been clear, especially when including gravitational mass. Here we introduce a novel classical field theory formulation of electromagnetism, and then show that it approximates QED in the limit of a quantum state which corresponds to a classical charged continua. Our formulation of electromagnetism features a Lagrangian which is gauge invariant, includes a classical complex field from which a divergenceless four-current may be derived, and reproduces all aspects of the classical theory of charged massive continua without any quantum effects. Taking a geometric approach, we identify the four-current as being in the direction of extremal phase velocity of the classical field; the field equations of motion determine this phase velocity as being equal to the mass, which makes the rest density proportional to the squared modulus of the field.

In this paper, we investigate Ricci–Yamabe solitons (RYSs) and gradient Ricci–Yamabe solitons (gradient RYSs) in generalized Robertson–Walker (GRW) spacetimes. At first, we prove that if a GRW spacetime admits a RYS, then it becomes a perfect fluid spacetime (PFS) and the divergence of the Weyl tensor vanishes. Also, a GRW spacetime admitting a RYS is of Petrov type $I$, $D$ or $O$ and in case of four dimension, the spacetime turns into a Robertson–Walker spacetime. Next, we show that if a GRW spacetime of constant scalar curvature admits a gradient RYS, then it becomes a PFS and the divergence of the Weyl tensor vanishes.

This paper investigated the anisotropic quintessence stars in the $f(R,\varphi )$ theory of gravity, where $R$ is the Ricci scalar and $\varphi$ is the scalar potential. We investigated the field equations of the $f(R,\varphi )$ theory of gravity by using the Krori–Barua technique. We have determined that all the obtained solutions are free from central singularity and potentially stable. The observed values of mass and radius of the different strange stars Her X1, SAXJ1808.4-3658, and 4U1820-30 have been used to calculate the values of unknown constants in the Krori and Barua metric. The physical phenomena of stars, such as energy density, pressure components, anisotropy, sound speeds, equation of state parameters, energy conditions and redshift have been investigated in detail.

In this paper, we review some geometrical aspects pertaining to the world of monotone quantum metrics in finite dimensions. Particular emphasis is given to an unfolded perspective for quantum states that is built out of the spectral theorem and is naturally suited to investigate the comparison with the classical case of probability distributions.

We start with a self-contained brief review of the construction of non-standard Lagrangian and Hamiltonian structures for the Liénard equations satisfying Chiellini condition, we apply it to a special polynomial class of Liénard equation and explore the integrability structure. Then after a brief exposition of the contact geometry and its connection with the non-standard Hamiltonian structures, we describe the time evolution of the contact Hamiltonian and Liénard equation. We present the formulation of the Liénard equation in terms of General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) method and also study the gradient-type flow. Finally, we describe the generalized Liénard equation by using the conformal Hamiltonian mechanics and illustrate our construction using spatio-temporal and autocatalysis systems.

In this paper, we review the main features of modified theories of gravity containing higher-order curvature invariants in the action. After summarizing the main features of these theories, we consider their applications to cosmology, pointing out the differences with respect to general relativity that can eventually solve issues exhibited by the latter at low and high energy scales. Specifically, we explore a gravitational action that incorporates both the Ricci scalar $R$ and the topological Gauss–Bonnet term, denoted as ${𝒢}$. Our investigation revolves around the cosmological properties of a specific category of modified gravity theories, chosen upon symmetry considerations. Within the framework of a spatially flat, homogeneous and isotropic cosmic background, we demonstrate that it is possible to account for the presently observed acceleration of the Universe by means of the extra geometric terms carried by the selected $f(R,{𝒢})$ model. This approach offers a way to address the issues associated with the cosmological constant. To achieve this, we first examine the energy conditions and find that, under certain choices for the values of the cosmographic parameters, these conditions are all violated. In the second part of the work, to assess the feasibility of the selected $f(R,{𝒢})$ model, we place observational constraints on its free parameters through a Bayesian Monte Carlo technique applied to late-time cosmic data. Our findings reveal that the $f(R,{𝒢})$ model can effectively reproduce observations at low redshifts, providing an alternative to the standard $\mathrm{\Lambda }$CDM scenario.

A covariant, global, variational framework for perturbations in field theories is presented. Perturbations are obtained as vertical vector fields on the configuration bundle and they drag, exactly, solution into solutions.

The flow of a perturbation drags solutions into solutions and the dragged perturbed solutions can be expanded in a series with respect to the flow parameter, hence it contains perturbations at any order.

Mechanics is included as a special case. As a simple application, we recover the well-known discussion about stability of geodesics on a sphere $S_2$.

In this work, a new method to determine and to describe the evolution of physical systems (in classical or quantum regimes) is proposed. The main idea is based in the observation that the complete determination of dynamics of the system is connected to the nonlinear realization (NLR) of the symmetry group associated with the symmetry of the Hamiltonian $H(t)$ of the system. The Maurer–Cartan forms, when they are parametrically associated to the projection on the hypersurface induced by the group in question via pullback, of the Hamiltonian components (based on the generators of the symmetry group by which it is defined) give rise to a fundamental system of equations by which the precise evolution can be determined. Such a system of equations under certain conditions that we will specify in this paper coincides with the system obtained through the Wei–Norman method, thus showing explicitly that our proposal is more general due the universal character of equations from the NLRs approach. As example, the important problem of the Caldirola–Kanai inverted Hamiltonian is completely solved, showing at the same time a new interpretation of what NLRs refer to.

It is remarkable that Heisenberg’s position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg’s uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time.

Geometrical methods in quantum information are very promising for both providing technical tools and intuition into difficult control or optimization problems. Moreover, they are of fundamental importance in connecting pure geometrical theories, like GR, to quantum mechanics, like in the AdS/CFT correspondence. In this paper, we first make a survey of the most important settings in which geometrical methods have proven useful to quantum information theory. Then we lay down a geometric theory of entanglement by a principle of action, discussing a simple example with two qubits and consequences for a quantum theory of space-time.

In this expository paper, we present a brief introduction to the geometrical modeling of some quantum computing problems. After a brief introduction to establish the terminology, we focus on quantum information geometry and $ZX$-calculus, establishing a connection between quantum computing questions and quantum groups, i.e. Hopf algebras.

In this paper, we explore some cosmological solutions of the Friedmann–Lemaître–Robertson–Walker (FLRW) closed universe with nonlocal de Sitter gravity $\sqrt{dS}$ and nonlocal scalar field which has its origin in $p$-adic string theory. In this case, we have that both geometrical and matter sectors of equations of motion (EoM) are nonlocal. The cosmological constant $\mathrm{\Lambda }$ plays a role of dark energy (DE) and is related to the mass of scalar field as $m=\frac{\hslash }{c}\sqrt{\frac{\mathrm{\Lambda }}{6}}=1.5\times 10^{-69}\text{kg}$. We found that the cosmological constant $\mathrm{\Lambda }$ depends on prime number $p$ as a discrete parameter and we conjectured that $\mathrm{\Lambda }$ is not constant during the universe evolution.

In this study, we investigate the dynamics of the Universe during the observed late-time acceleration phase within the framework of the Weyl-type $f(Q,T)$ theory. Specifically, we consider a well-motivated model with the functional form $f(Q,T)=\alpha Q+\frac{\beta }{6{\kappa }^2}T$, where $Q$ represents the scalar of non-metricity and $T$ denotes the trace of the energy–momentum tensor. In this context, the non-metricity $Q_{\mu \alpha \beta }$ of the space-time is established by the vector field $w_{\mu }$. The parameters $\alpha$ and $\beta$ govern the gravitational field and its interaction with the matter content of the Universe. By considering the case of dust matter, we obtain exact solutions for the field equations and observe that the Hubble parameter $H(z)$ follows a power-law behavior with respect to redshift $z$. To constrain the model parameters, we analyze various datasets including the Hubble, Pantheon datasets, and their combination. Our results indicate that the Weyl-type $f(Q,T)$ theory offers a viable alternative to explain the observed late-time acceleration of the Universe avoiding the use of dark energy.

The equivalence principle is considered in the framework of metric-affine gravity. We show that it naturally emerges as a Noether symmetry starting from a general non-metric theory. In particular, we discuss the Einstein equivalence principle and the strong equivalence principle showing their relations with the non-metricity tensor. Possible violations are also discussed pointing out the role of non-metricity in this debate.

This paper is devoted to $N$-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators $L$, whose potentials $Q(x,t)$ tend to constants $Q_{\pm }$ for $x\to \pm \infty$. For special choices of $Q_{\pm }$, we outline the spectral properties of $L$, the direct scattering transform and construct its fundamental analytic solutions. We generalize Wronskian relations for the case of CBC — this allows us to analyze the mapping between the scattering data and the $x$-derivative of the potential $Q_x$. Next, using the Wronskian relations, we derive the dispersion laws for the $N$-wave hierarchy and describe the NLEE related to the given Lax operator.

We review Holst formalism and dynamical equivalence with standard GR (in dimension $4$). Holst formalism is written for a spin coframe field $e_{\mu }^I$ and a $\text{Spin}(3,1)$-connection ${\omega }_{\mu }^{IJ}$ on spacetime $M$ and it depends on the Holst parameter $\gamma \in ℝ-\{0\}$. We show the model is dynamically equivalent to standard GR, in the sense that up to a pointwise $\text{Spin}(3,1)$-gauge transformation acting on (uppercase Latin) frame indices, solutions of the two models are in one-to-one correspondence. Hence, the two models are classically equivalent. One can also introduce new variables by splitting the spin connection into a pair of a $\text{Spin}(3)$-connection $A_{\mu }^i$ and a $\text{Spin}(3)$-valued 1-form $k_{\mu }^i$. The construction of these new variables relies on a particular algebraic structure, called a reductive splitting. A weaker structure than requiring that the gauge group splits as the products of two sub-groups, as it happens in Euclidean signature in the selfdual formulation originally introduced in this context by Ashtekar, and it still allows to deal with the Lorentzian signature without resorting to complexifications. The reductive splitting of $\text{SL}(2,ℂ)$ is not unique and it is parameterized by a real parameter $\beta$ which is called the Immirzi parameter. The splitting is here done on spacetime, not on space as it is usually done in the literature, to obtain a $\text{Spin}(3)$-connection $A_{\mu }^i$, which is called the Barbero–Immirzi connection on spacetime. One obtains a covariant model depending on the fields $(e_{\mu }^I,A_{\mu }^i,k_{\mu }^i)$ which is again dynamically equivalent to standard GR. Usually in the literature one sets $\beta =\gamma$ for the sake of simplicity. Here, we keep the Holst and Immirzi parameters distinct to show that eventually only $\beta$ will survive in boundary field equations.

We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of $p$-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, $p$-adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a $p$-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a $p$-adic Hilbert space — we consider an algebraic definition of physical states in $p$-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a $p$-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the $p$-adic setting, with an affine geometry; therefore, a symmetry transformation of a $p$-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a $p$-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.

In this review, we provide a concrete overview of the Raychaudhuri equation, Focusing Theorem and convergence conditions in a plethora of backgrounds and discuss the consequences. We also present various classical and quantum approaches suggested in the literature that could potentially mitigate the initial big-bang singularity and the black-hole singularity.

In this paper, we explore the characteristics of two novel regular spacetimes that exhibit a nonzero vacuum energy term, under the form of a (quasi) anti-de Sitter phase. Specifically, the first metric is spherical, while the second, derived by applying the generalized Newman–Janis algorithm to the first, is axisymmetric. We show that the equations of state of the effective fluids associated with the two metrics asymptotically tend to negative values, resembling quintessence. In addition, we study test particle motions, illustrating the main discrepancies among our models and more conventional metrics exhibiting non-vanishing anti-de Sitter phase.

The description of atoms is based on 3D time and some relativistic effects about spinning bodies have been published previously. The time displacement of the electrons also plays an important role. While the principal quantum number $n$ refers to the angular momentum $n\hslash$ observed by external observer, the azimuth quantum number $l$ refers to the angular momentum $l\hslash$ observed from the electron itself. The intrinsic angular momentum observed by the electron is $\pm \hslash /2$ according to Stern–Gerlach experiment, but the angular momentum observed by external observer is about ${\alpha }^2\hslash /4$. The magnetic quantum numbers are deduced from the mentioned effects and the trajectories of electrons are non-probabilistic and geometrically well determined. The spin quantum number indicates the time arrow toward the future or toward the past. So, the electrons with opposite time arrows can be grouped in pairs, where the nucleus is in the middle. Descriptions of the dark matter particles and the electrons are given. Using the value $\alpha =1/137.035999166$ suggested from the QED, the $g$-factors of the electron and muon are $g_e=-2.002319304361166$; $g_{\mu }=-2.002331839354934$, which give excellent agreements with the experiments. So, if we equalize the formulas for $g$-factor from QED and this approach, it determines the theoretical value of $\alpha$, without experiment.

The primary objective of this study is to conduct a thorough examination of relativistic structures within the framework of $f({ℛ},{𝒯})$ gravity, where ${ℛ}$ is the Ricci scalar and ${𝒯}$ is the trace of the energy–momentum tensor. We investigate the emerging characteristics of compact stars, focusing on the ${ℛ}+2\xi {𝒯}$ model, where $\xi$ is the coupling constant. In the initial sections of the paper, we concentrate on the examination of exact interior solutions involving perfect fluid. With this aim, we introduce the metric coefficient $L=S{(1+2a{r}^2)}^{d/2}$, where $a$ and $S$ are arbitrary constants to be computed from the matching conditions and $d$ is a positive odd integer. It is worth noticing that we select a group of 10 significant compact stars from recent studies, namely, Vela $X-1$, EXO $1785-248$, $4U1820-30$, SMC$X-4$, Her$X-1$, PSR $J1614-2230$, PSR$J1903+327$, Cen$X-3$, $4U1538-52$ and LMC$X-4$. To investigate the viability of our model, we conduct various checks, including the analysis of energy density and pressure components, equilibrium, and energy conditions, stability analysis and the adiabatic index. An important part of this work provides the slowly rotating property of stars, accompanied by the derivation of the moment of inertia. The relationship of mass, energy density, radius and moment of inertia for stellar objects are observed. Our findings demonstrate that obtained results are physically acceptable and consistent with our considered model.

In this paper, we have studied the flat Friedmann–Robertson–Walker (FRW) space time with domain wall and holographic dark energy in the context of $f(Q)$ theory of gravity. The exact solutions of the field equations are obtained using two different forms of varying deceleration parameters: (i) $q=-b(t)-1$ and (ii) $q=-lt+m-1$. It is observed that the universe is accelerating and expanding. Furthermore, we have also discussed some physical parameters of the investigated model.

The limited velocity in a geometry of Lorentzian signature seem to prevent the universe to reach thermodynamic equilibrium as suggested by the cosmic microwave background. Thus it was suggested that the universe which was initially minuscule has reached a more considerable radius in a short duration by a process known as cosmic inflation. However, to drive such a process have led to the suggestion of an ad-hoc scalar field the inflaton, which has no purpose in nature other than driving the cosmic inflation field and then stopping it once the universe reached the right size. In a recent paper it was shown that rapid expansion can occur without postulating an inflation by following Hawking’s suggestion and assuming that primordially the metric of the universe had an Euclidean signature, in which case velocity is not limited and thermalization and rapid expansion are derived without the need to assume an ad-hoc field. However, while in the previous work emphasis was put on the dynamics and physical statistics of the particles in a Euclidean space versus Lorentzian space in which both spaces were given. No mathematical model was given regarding the development of current Lorentzian space-time from the early Euclidean space-time, and how fundamental problems such as the space-time singularity and the homogeneity of the CMB can be solved in the hybrid Euclidean–Lorentzian picture. This lacuna is to be rectified in this paper.

In this paper, we have presented bulk viscous cosmological model of the universe in the modified gravity in which the Lagrangian of the gravitational action contains a general function $F(T)$, where $T$ denotes torsion, for flat Friedmann–Robertson–Walker spacetime with isotropic fluid. We have studied cosmological solution in $F(T)$ theory of gravity, especially of a particular preference $F(T)=T+f(T)=(1+\gamma )T+\alpha T^2$, where $\gamma$ and $\alpha$ are coupling constants, in the presence of bulk viscosity described by Eckart theory, Truncated Israel Stewart theory and Full Israel Stewart theory. The physical and geometrical prospectives of the models in Eckart, Truncated Israel Stewart theory and Full Israel Stewart theory are deliberated in details. The investigation of the variation of bulk viscous pressure, energy density, scale factor, Hubble parameter and deceleration parameter with cosmic growth is furnished in the relevant theories. The models are analyzed by comparison with recent observational data. The cosmological models are well-matched with observations.

In this paper, we investigate the Friedmann–Lemaitre–Robertson–Walker cosmological models within the framework of Rastall gravity incorporating particle creation. The modified field equations for Rastall gravity are derived and exact solutions are obtained under various assumptions of the scale factor. The qualitative behavior of our solutions depends on the Rastall coupling parameter $\psi =k\lambda$. Following Akarsu et al. [Ö. Akarsu et al., Rastall gravity extension of the standard $\mathrm{\Lambda }$ CDM model: Theoretical features and observational constraints, Eur. Phys. J. C 80 (2020) 1050], we have restricted the Rastall coupling parameter $\psi (k=1)$ to the range $-0.0001<\psi <0.0007$ at 68% CL from CMB+BAO data. Further, we have discussed the distinct physical behavior of the derived models in detail.