Bianchi I Quantum Cosmology in the Bergmann–Wagoner Theory

The Wheeler–DeWitt equation is considered in the context of generalized scalar-tensor theories of gravitation for the Bianchi type I cosmology. Exact solutions are found for two selfinteracting potentials and arbitrary coupling function. The WKB wavefunctions are obtained and a family of solutions satisfying the Hawking–Page regularity conditions of wormholes are found.     

Emergent dark energy in classical channel gravity with matter

Motivated by the recent increased interest in energy non-conserving models in cosmology, we extend the analysis of the cosmological consequences of the Classical Channel Model of Gravity (CCG). This model is based on the classical–quantum interaction between a test particle and a metric (classical) and results in a theory with a modified Wheeler–deWitt equation that in turn leads to non conservation of energy. We show that CCG applied to a cosmological scenario with primordial matter leads to an emergent dark fluid that at late times behaves as a curvature term in the Friedmann equations, showing that the late time behaviour is always dominated by the vacuum solution. We discuss possible observational constraints for this model and that—in its current formulation—CCG eludes any meaningful constraints from current observations.     

Wormholes, Classical Limit and Dynamical Vacuum in Quantum Cosmology

First a Friedmann-Robertson-Walker (FRW)universe filled with dust and a conformally invariantscalar field is quantized. For the closed model we finda discrete set of wormhole quantum states. In the case of flat spacelike sections we find states withclassical behaviour at small values of the scale factorand quantum behaviour for large values of the scalefactor. Next we study a FRW model with a conformally invariant scalar field and a nonvanishingcosmological constant dynamically introduced byregarding the vacuum as a perfect fluid with equation ofstate p = –. The ensuing Wheeler-DeWittequation turns out to be a bona fide Schrodinger equation, andwe find that there are realizable states with a definitevalue of the cosmological constant. Once again we findfinite-norm solutions to the Wheeler-DeWitt equation with definite values of thecosmological constant that represent wormholes,suggesting that in quantum cosmological models with asimple matter content wormhole states are a commonoccurrence.     

Cosmological singularities and higher-order gravitational lagrangians

General relativity is modified by adding terms proportional to R² and R^μνR_μν to the Lagrangian. One class of solutions of the modified field equations is free of singularities but does not lead to asymptotic behaviour (for large time) of the Friedmann type. A second class, which shows the correct asymptotic behaviour, does contain the usual singularities of Friedmann universes, collapse being modulated by small oscillations only. The quantum effects considered here are thus unable to prevent the occurrence of cosmological singularities under physically reasonable conditions.     

Quantum Noether identities for non-local transformationsin higher-order derivatives theories

Based on the phase-space generating functional of the Green function for a system with a regular/singular higher-order Lagrangian, the quantum canonical Noether identities (NIs) under a local and non-local transformation in phase space have been deduced, respectively. For a singular higher-order Lagrangian, one must use an effective canonical action I_eff^P in quantum canonical NIs instead of the classical I^P in classical canonical NIs. The quantum NIs under a local and non-local transformation in configuration space for a gauge-invariant system with a higher-order Lagrangian have also been derived. The above results hold true whether or not the Jacobian of the transformation is equal to unity or not. It has been pointed out that in certain cases the quantum NIs may be converted to conservation laws at the quantum level. This algorithm to derive the quantum conservation laws is significantly different from the quantum first Noether theorem. The applications of our formulation to the Yang-Mills fields and non-Abelian Chern-Simons (CS) theories with higher-order derivatives are given, and the conserved quantities at the quantum level for local and non-local transformations are found, respectively.Received: 12 February 2002, Revised: 16 June 2003, Published online: 25 August 2003Z.-P. Li: Corresponding authorAddress for correspondence: Department of Applied Physics, Beijing Polytechnic University, Beijing 100022, P.R. China     

On the possibility of dark energy from corrections to the Wheeler–DeWitt equation

We present a method for approximating the effective consequence of generic quantum gravity corrections to the Wheeler–DeWitt equation. We show that in many cases these corrections can produce departures from classical physics at large scales and that this behaviour can be interpreted as additional matter components. This opens up the possibility that dark energy (and possible dark matter) could be large scale manifestations of quantum gravity corrections to classical general relativity. As a specific example we examine the first order corrections to the Wheeler–DeWitt equation arising from loop quantum cosmology in the absence of lattice refinement and show how the ultimate breakdown in large scale physics occurs.     

Noncommutative Bianchi Type II Quantum Cosmology

In this paper we present the noncommutative Bianchi Class A cosmological models coupled to barotropic perfect fluid. The commutative and noncommutative quantum solution to the Wheeler–DeWitt equation for any factor ordering, to the anisotropic Bianchi type II cosmological model are found, using a stiff fluid (γ=1). In our toy model, we introduce noncommutative scale factors, is say, we consider that all minisuperspace variables q ⁱ does not commute, so the simplectic structure was modified.     

Quantization in generalized coordinates III-Lagrangian formulation

It is shown that if one incorporates the generalized coordinate quantum velocitiesQ ¹ as given byQ ¹=l[H,Q ¹](h=1) into the generalized classical Lagrangian for a free particle (the total energy),L=1/2Q ¹ g _tk Q ^k one does not obtain (no matter what ordering of the operatorsq ^l ,q ^k andg _lkwe choose the correct quantum Lagrangian operator which is a transformation from -1/2V² to generalized coordinates (Gruber, 1971, 1972).q ^l as given byq ^l=i[H,q ^l] turns out to be the Hermitian part of a more generaiized operator which we call the total generalized velocity operator similar to the notation in ear previous articles (Gruber, 1971, 1972). This total velocity operator really determines the fundamental structure governing our system in the Lagrangian formulation. We show that ft is through the total velocity operator that we make the transition from classical to quantum mechanics and through our procedure we arrive at the correct quantum Lagrangian operator.     

Classical and Quantum Wormholes in a Flat Λ-Decaying Cosmology

We study the classical and quantum wormholes for a flat Euclidean Friedmann-Robertson-Walker metric with a perfect fluid including an ordinary matter source plus a source playing the role of dark energy (decaying cosmological term). It is shown that classical wormholes exist for this model and the quantum version of such wormholes are consistent with the Hawking-Page conjecture for quantum wormholes as solutions of the Wheeler-DeWitt equation. This work has been financially supported by Research Institute for Fundamental Sciences, Tabriz, Iran.     

Nonminimal Derivative Couplings and Inflation in Generalized Theories of Gravity

We study extended theories of gravity where nonminimal derivative couplings of the form R^klϕ,_kϕ,_l are present in the Lagrangian. We show how and why the other couplings of similar structure may be ruled out and then deduce the field equations and the related cosmological models. Finally, we get inflationary solutions which do follow neither from any effective scalar field potential nor from a cosmological constant introduced “by hand”, and we show the de Sitter space‐time to be an attractor solution.     

Spinors, Inflation, and Non-Singular Cyclic Cosmologies

We consider toy cosmological models in which a classical, homogeneous, spinor field provides a dominant or sub-dominant contribution to the energy-momentum tensor of a flat Friedmann-Robertson-Walker universe. We find that, if such a field were to exist, appropriate choices of the spinor self-interaction would generate a rich variety of behaviors, quite different from their widely studied scalar field counterparts. We first discuss solutions that incorporate a stage of cosmic inflation and estimate the primordial spectrum of density perturbations seeded during such a stage. Inflation driven by a spinor field turns out to be unappealing as it leads to a blue spectrum of perturbations and requires considerable fine-tuning of parameters. We next find that, for simple, quartic spinor self-interactions, non-singular cyclic cosmologies exist with reasonable parameter choices. These solutions might eventually be incorporated into a successful past- and future-eternal cosmological model free of singularities. In an Appendix, we discuss the classical treatment of spinors and argue that certain quantum systems might be approximated in terms of such fields.     

Constraints on operator ordering from third quantization

In this paper, we analyse the Wheeler–DeWitt equation in the third quantized formalism. We will demonstrate that for certain operator ordering, the early stages of the universe are dominated by quantum fluctuations, and the universe becomes classical at later stages during the cosmic expansion. This is physically expected, if the universe is formed from quantum fluctuations in the third quantized formalism. So, we will argue that this physical requirement can be used to constrain the form of the operator ordering chosen. We will explicitly demonstrate this to be the case for two different cosmological models.     

Generalised Scalar-tensor Theory in the Bianchi Type I Model

We use a conformal transformation to find solutions to the generalised scalar-tensor theory, with a coupling constant dependent on a scalar field, in an empty Bianchi type I model. We describe the dynamical behaviour of the metric functions for three different couplings: two exact solutions to the field equations and a qualitative one are found. They exhibit non-singular behaviours and kinetic inflation. Two of them admit both General Relativity and string theory in the low-energy limit as asymptotic cases.     

Ermakov Approach for Minisuperspace Oscillators

The Wheeler–DeWitt equation of arbitrary Hartle–Hawking factor ordering for several minisuperspace universe models, such as the pure gravity Friedmann–Robertson–Walker and Taub ones, is mapped onto the dynamics of corresponding classical oscillators. The latter ones are studied by the classical Ermakov invariant method, which is a natural approach in this context. For the more realistic case of a minimally coupled massive scalar field, one can study, within the same type of approach, the corresponding squeezing features as a possible means of describing cosmological evolution. Finally, we comment on the analogy with the accelerator physics.     

The Causal Interpretation of Conformally Coupled Scalar Field Quantum Cosmology

We apply the causal interpretation of quantum mechanics to homogeneous and isotropic quantum cosmology, where the source of the gravitational field is a conformally coupled scalar field, and the maximally symmetric hypersurfaces have positive curvature. In order to simplify the system of coupled equations studied and study the quantum behavior near the singularity, we restricted ourselves to the cases where the scale factor is small. In this case, the general solution of the Wheeler–DeWitt equation is a discrete superposition of Hermitian polynomials multiplied by complex exponentials. Superpositions with up to two parcels are studied, and the phase diagrams of their corresponding Bohmian trajectories are analyzed in detail. Nonsingular periodic quantum solutions are found. We also find that singular quantum solutions present an inflationary era in the begining of the Universe. Numerical calculations indicates that these results remain valid for general superpositions.     

Quantum action principle in curved space

Schwinger's action principle is formulated for the quantum system which corresponds to the classical system described by the LagrangianL _c( , x)=(M/2)g_ij(x) ^{i j}–v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider onlyc-number variations of coordinates and time. The Euler-Lagrange equation, the canonical commutation relations, and the canonical equations of motion are derived from this principle in a consistent manner. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant. The judicious choice of the quantal Lagrangian is essential in our formulation. A quantum mechanical analog of Noether's theorem, which relates the invariance of the quantal action with a conservation law, is established. The ambiguities in the quantal Lagrangian are also discussed and it is pointed out that the requirement of invariance is not sufficient to determine uniquely the quantal Lagrangian and the Hamiltonian.