Search Results (50)

We investigate pseudo-complex General Relativity (pcGR)—a coordinate-extended formulation of General Relativity (GR)—within the framework of Hořava-Lifshitz gravity, a regularized theory featuring anisotropic scaling. The pcGR framework bridges GR with modified gravitational theories through the introduction of a minimal length scale. Focusing on Schwarzschild black holes, we derive the Wheeler-deWitt equation, obtaining a quantized description of pcGR. Using perturbative methods and semi-classical approximations, we analyze the solutions of the equations and their physical implications. A key finding is the avoidance of the central singularity due to nonlinear interaction terms in the Hořava-Lifshitz action. Notably, extrinsic curvature (kinetic energy) contributions prove essential for singularity resolution, even in standard GR. Furthermore, the theory offers new perspectives on dark energy, proposing an alternative mechanism for its accumulation. Full article

In this work, we apply fractional calculus to study quantum cosmology. Specifically, our Wheeler-DeWitt (WDW) equation includes a Friedman-Robertson-Walker (FRW) geometry, a radiation fluid, a positive cosmological constant ($\mathsf{\Lambda }$), and an ad-hoc potential. We employ the Riesz fractional derivative, which introduces a parameter $\alpha$, where $1<\alpha \le 2$, in the WDW equation. We investigate numerically the tunneling probability for the Universe to emerge using a suitable WKB approximation. Our findings are as follows. When we decrease the value of $\alpha$, the tunneling probability also decreases, suggesting that if fractional features could be considered to ascertain among different early universe scenarios, then the value $\alpha =2$ (meaning strict locality and standard cosmology) would be the most likely. Finally, our results also allow for an interesting discussion between selecting values for $\mathsf{\Lambda }$ (in a non-fractional conventional set-up) versus balancing, e.g., both $\mathsf{\Lambda }$ and $\alpha$ in the fractional framework. Full article

We recently proved that in our model of quantum gravity, the solutions to the quantized version of the full Einstein equations or to the Wheeler–DeWitt equation could be expressed as products of spatial and temporal eigenfunctions, or eigendistributions, of self-adjoint operators acting in corresponding separable Hilbert spaces. Moreover, near the big bang singularity, we derived sharp asymptotic estimates for the temporal eigenfunctions. In this paper, we show that, by using these estimates, there exists a complete sequence of unitarily equivalent eigenfunctions which can be extended past the singularity by even or odd mirroring as sufficiently smooth functions such that the extended functions are solutions of the appropriately extended equations valid in $\mathbb{R}$ in the classical sense. We also use this phenomenon to explain the missing antimatter. Full article

Within the framework of symmetric teleparallel $f\left(Q\right)$-gravity, using a connection defined in the non-coincidence gauge, we derive the Wheeler–DeWitt equation of quantum cosmology. The gravitational field equation in $f\left(Q\right)$-gravity permits a minisuperspace description, rendering the Wheeler–DeWitt equation a single inhomogeneous partial differential equation. We use the power-law $f\left(Q\right)=f_0{Q}^{\mu }$ model, and with the application of linear quantum observables, we calculate the wave function of the universe. Finally, we investigate the effects of the quantum correction terms in the semi-classical limit. Full article

Considering the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and the Einstein scalar field system as an underlying gravitational model to construct fractional cosmological models has interesting implications in both classical and quantum regimes. Regarding the former, we just review the most fundamental approach to establishing an extended cosmological model. We demonstrate that employing new methodologies allows us to obtain exact solutions. Despite the corresponding standard models, we cannot use any arbitrary scalar potentials; instead, it is determined from solving three independent fractional field equations. This article concludes with an overview of a fractional quantum/semi-classical model that provides an inflationary scenario. Full article

We find a novel phenomenon in the solution to the Wheeler–DeWitt equation by solving numerically the equation assuming $O\left(4\right)$-symmetry and imposing the Hartle–Hawking wave function as a boundary condition. In the slow-roll limit, as expected, the numerical solution gives the most dominant steepest-descent that describes the probability distribution for the initial condition of a universe. The probability is consistent with the Euclidean computations, and the overall shape of the wave function is compatible with analytical approximations, although there exist novel differences in the detailed probability computation. Our approach gives an alternative point of view for the no-boundary wave function from the wave function point of view. Possible interpretations and conceptual issues of this wave function are discussed. Full article

Quantum de Sitter geometry is discussed using elementary field operator algebras in Krein space quantization from an observer-independent point of view, i.e., ambient space formalism. In quantum geometry, the conformal sector of the metric becomes a dynamical degree of freedom, which can be written in terms of a massless minimally coupled scalar field. The elementary fields necessary for the construction of quantum geometry are introduced and classified. A complete Krein–Fock space structure for elementary fields is presented using field operator algebras. We conclude that since quantum de Sitter geometry can be constructed by elementary fields operators, the geometry quantum state is immersed in the Krein–Fock space and evolves in it. The total number of accessible quantum states in the universe is chosen as a parameter of quantum state evolution, which has a relationship with the universe’s entropy. Inspired by the Wheeler–DeWitt constraint equation in cosmology, the evolution equation of the geometry quantum state is formulated in terms of the Lagrangian density of interaction fields in ambient space formalism. Full article

Grassmann variables are used to formally transform a system with constraints into an unconstrained system. As a result, the Schrödinger equation arises instead of the Wheeler–DeWitt one. The Schrödinger equation describes a system’s evolution, but a definition of the scalar product is needed to calculate the mean values of the operators. We suggest an explicit formula for the scalar product related to the Klein–Gordon scalar product. The calculation of the mean values is compared with an etalon method in which a redundant degree of freedom is excluded. Nevertheless, we note that a complete correspondence with the etalon picture is not found. Apparently, the picture with Grassmann variables requires a further understanding of the underlying Hilbert space. Full article

In the particular configuration of the scalar field k-essence in the Wheeler–DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is $\beta =\frac{2\alpha }{2\alpha -1}$. This fractional equation belongs to different intervals depending on the value of the barotropic parameter; when ${\omega }_X\in [0,1]$, the order belongs to the interval $1\le \beta \le 2$, and when ${\omega }_X\in [-1,0)$, the order belongs to the interval $0<\beta \le 1$. In the quantum scheme, we introduce the factor ordering problem in the variables $(\mathsf{\Omega },\varphi )$ and its corresponding momenta $({\mathsf{\Pi }}_{\mathsf{\Omega }},{\mathsf{\Pi }}_{\varphi })$, obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time $T\left(\tau \right)$; however, in the dust era we found a closed solution in the gauge time $\tau$. Full article

In this contribution, motivated by the quest to understand cosmic acceleration, based on the theory of Hořava–Lifshitz and on the branch-cut gravitation, we investigate the effects of non-commutativity of a mini-superspace of variables obeying the Poisson algebra on the structure of the branch-cut scale factor and on the acceleration of the Universe. We follow the guiding lines of a previous approach, which we complement to allow a symmetrical treatment of the Poisson algebraic variables and eliminate ambiguities in the ordering of quantum operators. On this line of investigation, we propose a phase-space transformation that generates a super-Hamiltonian, expressed in terms of new variables, which describes the behavior of a Wheeler–DeWitt wave function of the Universe within a non-commutative algebraic quantum gravity formulation. The formal structure of the super-Hamiltonian allows us to identify one of the new variables with a modified branch-cut quantum scale factor, which incorporates, as a result of the imposed variable transformations, in an underlying way, elements of the non-commutative algebra. Due to its structural character, this algebraic structure allows the identification of the other variable as the dual quantum counterpart of the modified branch-cut scale factor, with both quantities scanning reciprocal spaces. Using the iterative Range–Kutta–Fehlberg numerical analysis for solving differential equations, without resorting to computational approximations, we obtained numerical solutions, with the boundary conditions of the wave function of the Universe based on the Bekenstein criterion, which provides an upper limit for entropy. Our results indicate the acceleration of the early Universe in the context of the non-commutative branch-cut gravity formulation. These results have implications when confronted with information theory; so to accommodate gravitational effects close to the Planck scale, a formulation à la Heisenberg’s Generalized Uncertainty Principle in Quantum Mechanics involving the energy and entropy of the primordial Universe is proposed. Full article

We analyze the canonical quantum dynamics of the isotropic Universe with a metric approach by adopting a self-interacting scalar field as relational time. When the potential term is absent, we are able to associate the expanding and collapsing dynamics of the Universe with the positive- and negative-frequency modes that emerge in the Wheeler–DeWitt equation. On the other side, when the potential term is present, a non-zero transition amplitude from positive- to negative-frequency states arises, as in standard relativistic scattering theory below the particle creation threshold. In particular, we are able to compute the transition probability for an expanding Universe that emerges from a collapsing regime both in the standard quantization procedure and in the polymer formulation. The probability distribution results similar in the two cases, and its maximum takes place when the mean values of the momentum essentially coincide in the in-going and out-going wave packets, as it would take place in a semiclassical Big Bounce dynamics. Full article

To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, $R\left(t\right)$) representing another observable. In the language of physics, the information carried by $R\left(t\right)\ne R^{†}\left(t\right)$ opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric $\mathsf{\Theta }\left(t\right)$. From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of $\mathsf{\Theta }\left(t\right)$ is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the “kinematical input” operator $R\left(t\right)$ may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor $a\left(t\right)$ emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of $R\left(t\right)$ mimicking the birth of the Universe (“Big Bang”) at $t=0$. Full article

In this contribution to the Festschrift for Prof. Remo Ruffini, we investigate a formulation of quantum gravity using the Hořava–Lifshitz theory of gravity, which is General Relativity augmented by counter-terms to render the theory regularized. We are then led to the Wheeler–DeWitt (WDW) equation combined with the classical concepts of the branch-cut gravitation, which contemplates as a new scenario for the origin of the Universe, a smooth transition region between the contraction and expansion phases. Through the introduction of an energy-dependent effective potential, which describes the space-time curvature associated with the embedding geometry and its coupling with the cosmological constant and matter fields, solutions of the WDW equation for the wave function of the Universe are obtained. The Lagrangian density is quantized through the standard procedure of raising the Hamiltonian, the helix-like complex scale factor of branched gravitation as well as the corresponding conjugate momentum to the category of quantum operators. Ambiguities in the ordering of the quantum operators are overcome with the introduction of a set of ordering factors $\alpha$, whose values are restricted, to make contact with similar approaches, to the integers $\alpha =[0,1,2]$, allowing this way a broader class of solutions for the wave function of the Universe. In addition to a branched universe filled with underlying background vacuum energy, primordial matter and radiation, in order to connect with standard model calculations, we additionally supplement this formulation with baryon matter, dark matter and quintessence contributions. Finally, the boundary conditions for the wave function of the Universe are imposed by assuming the Bekenstein criterion. Our results indicate the consistency of a topological quantum leap, or alternatively a quantum tunneling, for the transition region of the early Universe in contrast to the classic branched cosmology view of a smooth transition. Full article

Using a particular form of the quantum K-essence scalar field, we show that in the quantum formalism, a fractional differential equation in the scalar field variable, for some epochs in the Friedmann–Lema$\widehat{ı}$tre–Robertson–Walker (FLRW) model (radiation and inflation-like epochs, for example), appears naturally. In the classical analysis, the kinetic energy of scalar fields can falsify the standard matter in the sense that we obtain the time behavior for the scale factor in all scenarios of our Universe by using the Hamiltonian formalism, where the results are analogous to those obtained by an algebraic procedure in the Einstein field equations with standard matter. In the case of the quantum Wheeler–DeWitt (WDW) equation for the scalar field $\varphi$, a fractional differential equation of order $\beta =\frac{2\alpha }{2\alpha -1}$ is obtained. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; that is to say, when ${\omega }_X\in [0,1]$, the order belongs to the interval $1\le \beta \le 2$, and when ${\omega }_X\in [-1,0)$, the order belongs to the interval $0<\beta \le 1$. The corresponding quantum solutions are also given. Full article

This paper shows that the field defined by the Wheeler–DeWitt equation for pure gravity is neither a standard gravitational field nor the field representing a particular universe. The theory offers a unified description of geometry and matter, with geometry being fundamental. The quantum theory possesses gravitational decoherence when the signature of $R^{\left(3\right)}$ changes. The quantum theory resolves singularities dynamically. Application to the FLRW $\kappa =0$ shows the creation of local geometries during quantum evolution. The 3-metric is modified near the classical singularity in the case of the Schwarzschild geometry. Full article