International Journal of Geometric Methods in Modern Physics

The functional Schrödinger representation of a scalar field on an $n$-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical quantization based on the space-time symmetric De Donder–Weyl Hamiltonian theory. The functional Schrödinger representation emerges from the precanonical quantization when the ultraviolet parameter $\varkappa$ introduced by precanonical quantization is replaced by $\underline{\gamma }{}_0{\delta }^{\mathrm{\text{inv}}}(0)$, where $\underline{\gamma }{}_0$ is the time-like tangent space Dirac matrix and ${\delta }^{\mathrm{\text{inv}}}(0)$ is an invariant spatial $(n-1)$-dimensional Dirac’s delta function whose regularized value at $x=0$ is identified with the cutoff of the volume of the momentum space. In this limiting case, the Schrödinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration and the canonical functional derivative Schrödinger equation is derived from the manifestly covariant Dirac-like precanonical Schrödinger equation which is independent of a choice of a codimension-one foliation.

In this work, we study inextensible flows of tangent bimagnetic particles in space. Moreover, we obtain the inextensible flows of electric field by Lorentz equation. Finally, we obtain new results for inextensible flows of tangent bimagnetic particles in space.

The paper aims to apply the complex-sedenions to explore the field equations of four fundamental interactions, which are relevant to the classical mechanics and quantum mechanics, in the curved spaces. Maxwell was the first to utilize the quaternions to describe the property of electromagnetic fields. Nowadays, the scholars introduce the complex-octonions to depict the electromagnetic and gravitational fields. And the complex-sedenions can be applied to study the field equations of the four interactions in the classical mechanics and quantum mechanics. Further, it is able to extend the field equations from the flat space into the curved space described with the complex-sedenions, by means of the tangent-frames and tensors. The research states that a few physical quantities will make a contribution to certain spatial parameters of the curved spaces. These spatial parameters may exert an influence on some operators (such as, divergence, gradient, and curl), impacting the field equations in the curved spaces, especially, the field equations of the four quantum-fields in the quantum mechanics. Apparently, the paper and General Relativity both confirm and succeed to the Cartesian academic thought of ‘the space is the extension of substance’.

In this study, we obtain the special type of magnetic trajectories associated with a magnetic field ${{𝒱}}_b$ defined on a 3D Riemannian manifold. We investigate a new representation of binormal spherical indicatrices of magnetic curves. Thus, we study $B_b$-magnetic curves terms of inextensible flows. Furthermore, we give some new characterizations of curvatures in terms of some partial differential equations. Finally, we examine some geometrical and physical features of the moving charged particle corresponding to the $B_b$-magnetic curves. Namely, we compute uniformity of the $B_b$-magnetic curves.

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we define the notion of a weighted$A$-connection on a graded bundle. In a natural sense weighted $A$-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear $A$-connections are adapted to the structure of a vector bundle. This notion generalizes directly to multi-graded bundles and in particular we present the notion of a bi-weighted$A$-connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles.

We propose a new representation of the Schrödinger-like equation for scalar field Friedmann cosmology where the scalar field is the argument, and the Hubble parameter is the analogue to the wave function. Such an approach gives us the possibility to use the Schrödinger potential as a generating function which leads to generalization of known exact solutions. Further, we find a simple transformation of the Hubble parameter which generates new solutions from the Schrödinger-like equation. Several examples are identified where exact forms for the scale factor, Hubble parameter and scalar potential can be written in closed form. Earlier results are regained in our approach.

The present work deals with the dynamical evolution of LRS Bianchi type I (LRS BI) cosmological model with quadratic dark sector coupling. We investigate the phase-plane analysis of LRS BI model with dark energy, when dark energy is modeled as exponential quintessence, and is coupled to dark matter via energy exchange. The evolution of cosmological solutions are studied by using dynamical systems method. Stability and viability of the models are discussed for four different choices of quadratic dark sector coupling parameters. In each model, we have obtained a late-time accelerating fixed point (future attractor), which is not a scaling solution.

In the present paper, the work of Moreas et al. [P. H. R. S. Moraes, G. Ribeiro and R. A. C. Correa, A transition from a decelerated to an accelerated phase of the universe expansion from the simplest non-trivial polynomial function of T in the $f(R,T)$ formalism, Astrophys. Space Sci. 361 (2016) 227–231] is extended to study the FRW model in $f(R,T)$ gravity. The expressions for deceleration and Hubble parameters are determined in terms of redshift. The age of the universe is established using $42$ high-redshift type Ia supernovae data from the Supernova cosmology project and 15 low-redshift type Ia supernovae data from the Calán/Tolono Supernova survey [S. Permutter et al., Measurements of Omega and Lambda from 42 High-Redshift Supernovae, Astrophys. J. 517 (1999) 565–585]. For these redshifts, the data of observed apparent magnitude and luminosity distance are used for the comparison with the obtained theoretical values.

A $(2m+1)$-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological term $\mathrm{\Lambda }$ is considered. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters $H>0$ and $h$, corresponding to two factor spaces of dimension $m>2$ and obeying $H\ne \pm h$. We prove that the solutions, obeying $h/H\ne -\frac{m-1}{m-2},-\frac{m-2}{m-1}$, are stable (in a class of cosmological solutions with diagonal metrics) for $h>-H$ and they are unstable for $h<-H$. A subclass of solutions with small enough variation of the effective gravitational constant $G$ is considered. It is shown that all solutions from this subclass are stable.

An interesting theoretical question is the probability of information loss in the black hole due to the transition point. The presented study is an attempt to investigate this subject. When the black hole is exhausted due to Hawking radiation, the entropy carried away by all emissions and string ball is not equal to the entropy in the original black hole. We conclude that the total entropy is not conserved due to the transition point. This means that some information is lost at the corresponding point. Using the Gottesman and Preskill method, the amount of information loss in the black hole at the correspondence point is obtained.

We employ the Cartan–Karlhede algorithm in order to completely characterize the class of Gödel-like spacetimes for three-dimensional gravity. By examining the permitted Segre types (or P-types) of the Ricci tensor, we present the results of the Cartan–Karlhede algorithm for each subclass in terms of the algebraically independent Cartan invariants at each order. Using this smaller subset of Cartan invariants, we express the scalar polynomial curvature invariants for the Gödel-like spacetimes in terms of this subset of Cartan invariants and generate a minimal set of scalar polynomial curvature invariants that uniquely characterize metrics in the class of Gödel-like spacetimes and identify the subclasses in terms of the P-types of the Ricci tensor.

In this paper, we establish some general Kastler–Kalau–Walze type theorems for any dimensional manifolds with boundary which generalize the results in [J. Wang and Y. Wang, The Kastler–Kalau–Walze type theorem for six-dimensional manifolds with boundary, J. Math. Phys. 56 (2015), Article ID: 052501, 14 pp.].

The rotational embedded submanifold was first studied by Kuiper as a submanifold in ${𝔼}^{n+d}$. The generalized Beltrami submanifolds and toroidal submanifold are the special examples of these kind of submanifolds. In this paper, we consider $3$-dimensional rotational embedded submanifolds in Euclidean $5$-space ${𝔼}^5$. We give some basic curvature properties of this type of submanifolds. Further, we obtain some results related with the scalar curvature and mean curvature of these submanifolds. As an application, we give an example of rotational submanifold in ${𝔼}^5$.

We study the phantom fluid in the late universe, thus assuming the equation of state parameter $w$ to be less than $-1$. The fluid is assumed to consist of two components, one laminar component $\rho$ and one turbulent component ${\rho }_T$, the latter set proportional to $\rho$ as well as to the Hubble parameter, ${\rho }_T=3\tau H\rho$ with $\tau$ a positive constant associated with the turbulence. The effective energy density is taken to be ${\rho }_e=\rho +{\rho }_T$, and the corresponding effective pressure is $p_e=w{\rho }_e$, with $w$ constant. These basic assumptions lead to a Big Rip universe; the physical quantities diverging during a finite rip time $t_s$. We then consider the mass accretion of a black hole in such a universe. The most natural assumption of setting the rate $dM/dt$ proportional to $M^2$ times the sum ${\rho }_e+p_e$ leads to a negative mass accretion, where $M(t)$ goes to zero linearly in $(t_s-t)$ near the singularity. The Hubble parameter diverges as ${(t_s-t)}^{-1}$, whereas ${\rho }_e$ and $p_e$ diverge as ${(t_s-t)}^{-2}$. We also discuss other options and include, for the sake of comparison, some essential properties of mass accretion in the early (inflationary) universe.

There are two types of warped product pseudo-slant submanifolds, $M_{𝜃}{\times }_f{M}_{\perp }$ and $M_{\perp }{\times }_f{M}_{𝜃}$, in a nearly Kaehler manifold. We derive an optimization for an extrinsic invariant, the squared norm of second fundamental form, on a nontrivial warped product pseudo-slant submanifold $M_{\perp }{\times }_f{M}_{𝜃}$ in a nearly Kaehler manifold in terms of a warping function and a slant angle when the fiber $M_{𝜃}$ is a slant submanifold. Moreover, the equality is verified for depending on what $M_{𝜃}$ and $M_{\perp }$ are, and also we show that if the equality holds, then $M_{\perp }{\times }_f{M}_{𝜃}$ is a simply Riemannian product. As applications, we prove that the warped product pseudo-slant submanifold has the finite Kinetic energy if and only if $M_{\perp }{\times }_f{M}_{𝜃}$ is a totally real warped product submanifold.

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.