International Journal of Geometric Methods in Modern Physics

In this paper, DeWitt’s formalism for field theories is presented; it provides a framework in which the quantization of fields possessing infinite-dimensional invariance groups may be carried out in a manifestly covariant (non-Hamiltonian) fashion, even in curved space-time. Another important virtue of DeWitt’s approach is that it emphasizes the common features of apparently very different theories such as Yang–Mills theories and General Relativity; moreover, it makes it possible to classify all gauge theories in three categories characterized in a purely geometrical way, i.e. by the algebra which the generators of the gauge group obey; the geometry of such theories is the fundamental reason underlying the emergence of ghost fields in the corresponding quantum theories, too. These “tricky extra particles”, as Feynman called them in 1964, contribute to a physical observable such as the stress-energy tensor, which can be expressed in terms of Feynman’s Green function itself. Therefore, an entire section is devoted to the study of the Green functions of the neutron scalar meson: in flat space-time, the choice of a particular Green’s function is the choice of an integration contour in the “momentum” space; in curved space-time the momentum space is no longer available, and the definition of the different Green functions requires a careful discussion itself. After the necessary introduction of bitensors, world function and parallel displacement tensor, an expansion for the Feynman propagator in curved space-time is obtained. Most calculations are explicitly shown.

In this study, we provide a brief description of rotational surfaces in 4-dimensional (4D) Galilean space using a curve and matrices in $G_4$. That is, we provide different types of rotational matrices, which are the subgroups of $M$ by rotating a selected axis in $E^4$. Hence, we choose two parameter matrices groups of rotations and we give the matrices of rotation corresponding to the appropriate subgroup in Galilean 4-space and we generate rotated surfaces.

A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction, and a key to quantum gravity. Inspired by Souriau’s symplectic generalization of the Maxwell–Boltzmann–Gibbs equilibrium in Lie group thermodynamics, we investigate a space–time-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated with a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description.

Collapsing process is studied in special type of inhomogeneous spherically symmetric space-time model (known as IFRW model), having no time-like Killing vector field. The matter field for collapse dynamics is considered to be perfect fluid with anisotropic pressure. The main issue of this investigation is to examine whether the end state of the collapse to be a naked singularity or a black hole. Finally, null geodesics is studied near the singularity.

We study Tsallis holographic dark energy model with generalized ghost as an infrared cutoff in the context of Dvali–Gabadadze–Porrati braneworld and loop quantum cosmology with infrared cutoff for flat Friedmann–Robertson–Walker (FRW) universe. In these directions, we develop some cosmological parameters such as equation of state, deceleration parameter, squared speed of sound and $Om$ (combination of Hubble and cosmos redshift parameter) diagnostic to analyze the expansion of the universe. The cosmic planes such as ${\omega }_d-{\omega }_d^{\prime }$ and $r-s$ are also being established. We also compare these parameters and planes with recent Planck data 2018 which represents the consistency in most of the cases.

We study solitons of almost pseudo symmetric Kählerian space-time manifold. It is considered that different curvature tensors like projective, conharmonic and conformal curvature tensors in almost pseudo symmetric Kählerian space-time manifolds are flat. It is shown that solitons are steady, expanding or shrinking under different relations of isotropic pressure, the cosmological constant, energy density and gravitational constant.

We construct the thermodynamic geometry of ($2+1$)-dimensional normal (exotic) BTZ black hole regarding the fluctuation of cosmological constant. We argue that while the thermodynamic geometry of black hole without fluctuation of cosmological constant is a two dimensional flat space, the three-dimensional space of thermodynamics parameters including the cosmological constant as a fluctuating parameter is curved. Some consequences of the fluctuation of cosmological constant will be investigated. We show that such a fluctuation leads to a thermodynamic curvature which is singular at the critical surface. Also, we consider the validity of first thermodynamics law regarding the fluctuation of the cosmological constant.

In this paper, we analyze the thermodynamic stability of Schwarzschild Modified Gravity (MOG) black holes in a non-commutative framework. We show that, unlike a commutative MOG black hole, in the coherent state picture of non-commutativity MOG black holes are thermodynamically stable. At the final stage of evaporation a stable remnant with zero temperatures and finite entropy is left in this non-commutative framework. Also, we consider the Parikh–Wilczek tunneling mechanism of massive particles from non-commutative MOG black holes and demonstrate that information leaks out of non-commutative MOG black holes in the form of some non-thermal correlations.

In this paper, we use the generalized Dirac structure beyond the linear regime of graphene. This is probed using the a deformation of the Dirac structure in graphene by the generalized uncertainty principle. Here, the Planck length is replaced by the graphene lattice spacing. As the graphene sheet is bounded by two boundaries, we analyze this system with suitable boundary conditions. We solve the perturbed Hamiltonian and derive the wave function for this system. We observe that the energy of this system gets corrected due to this deformation. We explicitly calculate these corrections to the energy of this system.

With due consideration of reasonable cosmological assumptions within the limit of the present cosmological scenario, we have analyzed a spherically symmetric metric in 5D setting within the framework of Lyra manifold. The model universe is predicted to be a DE model, dominated by vacuum energy. The model represents an oscillating model, each cycle evolving with a big bang and ending at a big crunch, undergoing a series of bounces. The universe is isotropic and undergoes super-exponential expansion. The value of Hubble’s parameter is measured to be $H=67.0691$ which is very close to $H_0=67.36\pm 0.54\mathrm{\text{km}}{\mathrm{\text{s}}}^{-1}{\mathrm{\text{Mpc}}}^{-1}$, the value estimated by the latest Planck 2018 result. A detailed discussion on the cosmological parameters obtained is also presented with graphs.

In this work, the cosmological inflationary parameters in the correspondence of teleparallel gravity for the scalar–tensor theory are investigated. After the review of $f(T)$ and $f(T,B)$ gravity cosmology, we use the slow-roll approximations to study the behavior of the inflationary parameters namely the spectral index $n_s$ and tensor-to-scalar ratio $r$, and a comparison with observational data for different paradigmatic $f(T)$ gravity models such as exponential, Linder and power-law models is considered. We also consider the boundary term $B$ associated with these three models. The obtained behavior of the parameters under consideration shows that it is possible to constrain $f(T)$ and $f(T,B)$ models based on observational data.

In this paper, we introduced and established some group invariant results of $(2+1)$-dimensional mKdV–Calogero–Bogoyavlenskii–Schiff equation. Using the one-parameter Lie-group of transformations, we explored various closed-form solutions. The procedure minimizes the number of independent variables by one in every proceeding stage leading to form a system of the ordinary differential equations. The nature of solutions is investigated both analytically and physically through their evolutionary profiles by considering adequate choices of arbitrary functions and constants. The obtained results have been plotted with the aid of numerical simulation to obtain a significant appearance of the traced results. Simulation is carried out by taking an adequate option of arbitrary constants and functions, applying MATLAB code through progressing profiles. Wave solutions derived here are positons, multiple solitons, negaton and kink types which are shown through graph analysis.

In this paper, we investigate a scalar field Brans–Dicke cosmological model in Lyra’s geometry which is based on the modifications in a geometrical term as well as energy term of Einstein’s field equations. We have examined the validity of the proposed cosmological model on the observational scale by performing statistical analysis from the latest $H(z)$ and SN Ia observational data. We find that the estimated values of Hubble’s constant and matter energy density parameter is in agreement with their corresponding values, obtained from recent observations of Wilkinson Microwave Anisotropy Probe (WMAP) and Plank collaboration. We also derived the deceleration parameter, age of the universe and jerk parameter in terms of red-shift and computed its present values. The dynamics of the deceleration parameter in the derived model of the universe show a signature flipping from positive to a negative value and also indicate that the present universe is in the accelerating phase.

We show that our Universe lives in a topological and non-perturbative vacuum state full of a large amount of hidden quantum hairs, the hairons. We will discuss and elaborate on theoretical evidences that the quantum hairs are related to the gravitational topological winding number in vacuo. Thus, hairons are originated from topological degrees of freedom, holographically stored in the de Sitter area. The hierarchy of the Planck scale over the Cosmological Constant (CC) is understood as an effect of a Topological Memory intrinsically stored in the space-time geometry. Any UV quantum destabilizations of the CC are re-interpreted as Topological Phase Transitions, related to the disappearance of a large ensamble of topological hairs. This process is entropically suppressed, as a tunneling probability from the $N$- to the 0-states. Therefore, the tiny CC in our Universe is a manifestation of the rich topological structure of the space-time. In this portrait, a tiny neutrino mass can be generated by quantum gravity anomalies and accommodated into a large $N$-vacuum state. We will re-interpret the CC stabilization from the point of view of Topological Quantum Computing. An exponential degeneracy of topological hairs non-locally protects the space-time memory from quantum fluctuations as in Topological Quantum Computers.

In the present work, we construct the Tsallis holographic quintessence model of dark energy in $f(R,T)$ gravity with Hubble horizon as infrared (IR) cut-off. In a flat Friedmann–Robertson–Walker (FRW) background, the correspondence among the energy density of the quintessence model with the Tsallis holographic density permits the reconstruction of the dynamics and the potentials for the quintessence field. The suggested Hubble horizon IR cut-off for the Tsallis holographic dark energy (THDE) density acts for two specific cases: (i) THDE 1 and (ii) THDE 2. We have reconstructed the Tsallis holographic quintessence model in the region ${\omega }_{\mathrm{\Lambda }}>-1$ for the equation of state (EoS) parameter for both the cases. we investigate the behavior of several well-known statefinder quantities, like the deceleration parameter, the jerk and the parameter $s$. In addition, the quintessence phase of the THDE models is analyzed with swampland conjecture to describe the accelerated expansion of the Universe.

Several methods use the Fourier transform from momentum space to twistor space to analyze scattering amplitudes in Yang–Mills theory. However, the transform has not been defined as a concrete complex integral when the twistor space is a three-dimensional complex projective space. To the best of our knowledge, this is the first study to define it as well as its inverse in terms of a concrete complex integral. In addition, our study is the first to show that the Fourier transform is an isomorphism from the zeroth Čech cohomology group to the first one. Moreover, the well-known twistor operator representations in twistor theory literature are shown to be valid for the Fourier transform and its inverse transform. Finally, we identify functions over which the application of the operators is closed.