We extend the Born-Oppenheimer type of approximation scheme for the Wheeler-DeWitt equation of canonical quantum gravity to arbitrary orders in the inverse Planck mass squared. We discuss in detail the origin of unitarity violation in this scheme and show that unitarity can be restored by an appropriate modification which requires back reaction from matter onto the gravitational sector. In our analysis, we heavily rely on the gauge aspects of the standard Born-Oppenheimer scheme in molecular physics.
The Wheeler-DeWitt equation is based on the use of canonical quantization rules that may be inconsistent for constrained dynamical systems, such as minisuperspaces subject to Einstein's equations. The resulting quantum dynamics has no classical limit and it suffers from the infamous ``problem of time.'' In this article, it is shown how a dynamical time (an internal clock) can be constructed by means of a Hamilton-Jacobi formalism, and then used for a consistent canonical quantization, with the correct classical limit.
We can solve the Wheeler-DeWitt equation of the small universe enough to metric becomes diagonal and take a Gaussian normal coordinate. Our previous works are concerning to this paper. In this paper, we only write how to solve the Wheeler-DeWitt equation of such universe. Our motivation is simple, that is to solve the Wheeler-DeWitt equation. Even if the Wheeler-DeWitt equation is solved, quantum gravity does not complete yet. However, this work may be one of the first step to quantum gravity.
We give an explicit, rigorous framework for calculating quantum probabilities in a model theory of quantum gravity. Specifically, we construct the decoherence functional for the Wheeler-DeWitt quantization of a flat Friedmann-Robertson-Walker cosmology with a free, massless, minimally coupled scalar field, thus providing a complete decoherent histories formulation for this quantum cosmological model. The decoherence functional is applied to study predictions concerning the model's Dirac (relational) observables; the behavior of semiclassical states and superpositions of such states; and to stu…
In contrast to other approaches to (2+1)-dimensional quantum gravity, the Wheeler-DeWitt equation appears to be too complicated to solve explicitly, even for simple spacetime topologies. Nevertheless, it is possible to obtain a good deal of information about solutions and their interpretation. In particular, strong evidence is presented that Wheeler-DeWitt quantization is not equivalent to reduced phase space quantization.
We study multidimensional cosmology to obtain the wavefunction of the universe using wormhole dominance proposal. Using a prescription for time we obtain the Schroedinger-Wheeler-DeWitt equation without any reference to WD equation and WKB ansatz for WD wavefunction. It is found that the Hartle-Hawking or wormhole-dominated boundary conditions serve as a seed for inflation as well as for Gaussian type ansatz to Schroedinger-Wheeler-DeWitt equation.
In this paper we present the Wheeler-DeWitt equation of $f(R,L_m)$ gravity in flat FRW universe, which is the first step of the study of quantum $f(R,L_m)$ cosmology. In the minisuperspace spanned by the FRW scale factor $a$ and the Ricci scalar $R$, we explicitly examine and verify the equivalence of the reduced action. We then canonical quantize the $f(R,L_m)$ model and derive the corresponding Wheeler-DeWitt equation.
We present the Hamiltonian formulation of a relativistic point-particle coupled to Einstein gravity and its canonical quantization à la Wheeler-DeWitt. In the resulting quantum theory, the wave functional is a function of the particle coordinates and the 3-metric. It satisfies a particular Hamiltonian and diffeomorphism constraint, together with a Klein-Gordon-type equation. As usual in the Wheeler-DeWitt theory, the wave function is time-independent. This is also reflected in the Klein-Gordon-type equation, where the time derivative is absent. Before considering gravity, we consider the coupl…
Commutator anomalies obstruct solving the Wheeler-DeWitt constraint equation in Dirac quantization of quantum gravity-matter theory. When the obstruction is removed, there result quantal modifications to the constraints. The same classical theory gives rise to different quantum theories when different procedures for overcoming anomalies are implemented.
The asymptotic of the solution of the discrete Wheeler-DeWitt equation is found in the vicinity of small scale factors. It is shown that this problem is equivalent to the solution of the stationary Schrödinger equation in the (super) space of negative constant curvature. The minimum positive eigenvalue is found from which a continuous spectrum begins.
In this paper we study how all the physical "constants" vary in the framework described by a model in which we have taken into account the generalize conservation principle for its stress-energy tensor. This possibility enable us to take into account the adiabatic matter creation in order to get rid of the entropy problem. We try to generalize this situation by contemplating multi-fluid components. To validate all the obtained results we explore the possibility of considering the variation of the"constants" in the quantum cosmological scenario described by the Wheeler-DeWitt equation. For this…
We make a direct connection between the construction of three dimensional topological state sums from tensor categories and three dimensional quantum gravity by noting that the discrete version of the Wheeler-DeWitt equation is exactly the pentagon for the associator of the tensor category, the Biedenharn-Elliott identity. A crucial role is played by an asymptotic formula relating 6j-symbols to rotation matrices given by Edmonds.
In this paper, we study the changes of quantum effects of a growing universe by using Wheeler-DeWitt equation (WDWE) together with de Broglie-Bohm quantum trajectory approach. From WDWE, we obtain the quantum modified Friedmann equations which have additional terms called quantum potential compared to standard Friedmann equations. The quantum potential governs the behavior of the early universe, providing energy for inflation, while it decreases rapidly as the universe grows. The quantum potential of the grown-up universe is much smaller than that required for accelerating expansion. This indi…
The paper is devoted to fundamental problems of the Wheeler - DeWitt quantum geometrodynamics, which was the first attempt to apply quantum principles to the Universe as a whole. Our purpose is to find out the origin of these problems and follow up their consequences. We start from Dirac generalized Hamiltonian dynamics as a cornerstone on which the Wheeler - DeWitt theory is based. We remind the main statements of the famous DeWitt's paper of 1967 and discuss the flaws of the theory: the well-known problem of time, the problem of Hilbert space and others. In the concluding part of the paper w…
The Wheeler-DeWitt equation for the Bianchi Class A cosmological models is expressed generally in terms of the second-order differential equation like the Klein-Gordon equation. To obtain the positive-definite probability density, a new method extending the Dirac-Square-Root formalism, which factorizes the Wheeler-DeWitt equation into the first-order differential equation using the Pauli matrices, is investigated. The solutions to the Dirac type equation thus obtained are expressed in terms of two-component spinor form. The probability density defined by the solution is positive-definite and t…
In this paper we present the quantum LRS Bianchi IX models or, as they may be better known, the Taub-NUT models in addition to the quantum LRS Bianchi VIII models, both coupled to a stiff matter source. We solve their Wheeler DeWitt equations for an arbitrary Hartle-Hawking ordering parameter using separation of variables. Afterwards we construct a superposition of their respective wave functions and discuss some of the interesting qualitative properties they possess.
We re-examine the quantum geometrodynamical approach within the Eddington-inspired-Born-Infeld theory of gravity, which was first proposed in our previous work [1]. A thorough analysis of the classical Hamiltonian with constraints is carried and the correctness and self-consistency of the modified Wheeler-DeWitt equation (WDW) is studied. We find that based on the newly obtained WDW equation derived with the use of the Dirac brackets, the conclusion reached in Ref.[1] can be corroborated. The big rip singularity present in the classical theory, and induced by a phantom perfect fluid, is expect…
A new asymptotic expansion method is developed to separate the Wheeler-DeWitt equation into the time-dependent Schrödinger equation for a matter field and the Einstein-Hamilton-Jacobi equation for the gravitational field including the quantum back-reaction of the matter field. In particular, the nonadiabatic basis of the generalized invariant for the matter field Hamiltonian separates the Wheeler-DeWitt equation completely in the asymptotic limit of $m_p^2$ approaching infinity. The higher order quantum corrections of the gravity to the matter field are found. The new asymptotic expansion meth…
We study the quantum theory of the Einstein-Maxwell action with a cosmological term in the spherically symmetric space-time, and explored quantum black hole solutions in Reissner-Nordstrom-de Sitter geometry. We succeeded to obtain analytic solutions to satisfy both the energy and momentum constraints.
The main result of this paper is the 4-dimensional supermetric version of the Wheeler-DeWitt equation, that uses only one time variable for the both roles - as internal time and for the ADM split, as Hamiltonian evolution parameter. We study the ADM split with respect to the scalar massless field serving as internal time. The 4-dimensional hyper-surfaces $Σ_{ϕ= const}$ span the 5-dimensional space with the scalar field being the fifth coordinate. As a result we obtain the analog of the Wheeler-DeWitt equation for the 4-dimensional supermetric. We compare the ADM action with the non-compactifie…
The equation for canonical gravity produced by Wheeler and DeWitt in the late 1960s still presents difficulties both in terms of its mathematical solution and its physical interpretation. One of these issues is, notoriously, the absence of an explicit time. In this short note, we suggest one simple and straightforward way to avoid this occurrence. We go back to the classical equation that inspired Wheeler and DeWitt (namely, the Hamilton--Jacobi--Einstein equation) and make explicit, before quantization, the presence of a known, classically meaningful notion of time. We do this by allowing Ham…
In this paper, we will analyse the consequences of deforming the canonical commutation relations consistent with the existence of a minimum length and a maximum momentum. We first generalize the deformation of first quantized canonical commutation relation to second quantized canonical commutation relation. Thus, we arrive at a modified version of second quantization. A modified Wheeler-DeWitt equation will be constructed by using this deformed second quantized canonical commutation relation. Finally, we demonstrate that in this modified theory the big bang singularity gets naturally avoided.
In this article, I present a volume average regularization for the second functional derivative operator that appears in the metric-basis Wheeler-DeWitt equation. Naively, the second functional derivative operator in the Wheeler-DeWitt equation is infinite, since it contains terms with a factor of a delta function or derivatives of the delta function. More precisely, the second functional derivative contains terms that are only well defined as a distribution---these terms only yield meaningful results when they appear within an integral. The second functional derivative may, therefore, be regu…
In the Wheeler-DeWitt framework, by a gauge fixing procedure, we set up a scheme to recover a Schrödinger type equation, living in the orbits space with the {\it lapse} function as evolution parameter. By means of the associated stationary equation, we have the possibility of calculating quantum corrections to classical quantities. The Schwarzschild wormhole case is discussed as an example of application.
It is shown that the Wheeler DeWitt constraint of canonical gravity coupled to Klein Gordon scalar fields and expressed in terms of Ashtekar's variables admits formal solutions which are parametrized by loops in the three dimensional hypersurface and which are extensions of the well known Wilson loop solutions found by Jacobson, Rovelli and Smolin.
We justify the way of the direct quantization which means immediate quantization of a conservation law. It is shown that this approach is equivalent to introducing the super Hamiltonian on a minisuperspace in spirit of the Wheeler-DeWitt's approach. Then we will have: (i) all values are observable and have an obvious physical meaning and well-defined application domain; (ii) wave function is well-defined without time slicing and often can be exactly obtained; (iii) we can take off major mathematical troubles, and therefore, more complicate models can be considered exactly without the perturbat…
One of the unsolved issues in the quantum gravity comes from the Wheeler-DeWitt equation, which is second order functional derivative equation. In this paper, we introduce a new method to solve the Wheeler-DeWitt equation. Usually one treats the state functional of the space of 3-dimensional metrics, which do not contain timelike metrics. However we can expand this state to the state which has support on the space of spacetime metrics with using additional constraint which requires the recovery of 4-dimensional quantum gravity. Enlarging the support of the state functional of the spacetime met…
In this letter, we discuss the Wheeler-DeWitt equation with an ordering parameter in the Friedmann-Robertson-Walker universe. The solutions when the universe was very small and at the end of the expansion are obtained in terms of Bessel and Heun functions, respectively. We also obtain a boundary condition which should be satisfied by the ordering parameter, namely, $0 \leq p \leq 2$. We investigate the minimum value of the scale factor with respect to the maximum value of the probability density.
Exact solutions of the Wheeler-DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of th…
We present a chaplygin gas Friedmann-Robertson-Walker quantum cosmological model. In this work the Schutz's variational formalism is applied with positive, negative, and zero constant spatial curvature. In this approach the notion of time can be recovered. These give rise to Schrödinger-Wheeler-DeWitt equation for the scale factor. We use the eigenfunctions in order to construct wave packets for each case. We study the time dependent behavior of the expectation value of the scale factor, using the many-worlds interpretations of quantum mechanics.
The Wheeler-DeWitt equation is derived from the bosonic sector of the heterotic string effective action assuming a toroidal compactification. The spatially closed, higher dimensional Friedmann-Robertson-Walker (FRW) cosmology is investigated and a suitable change of variables rewrites the equation in a canonical form. Real- and imaginary-phase exact solutions are found and a method of successive approximations is employed to find more general power series solutions. The quantum cosmology of the Bianchi IX universe is also investigated and a class of exact solutions is found.
It is argued heuristically -- using an ${\bf S}^3 \times {\bf S}^6$ minisuperspace model -- that there might be a fundamental quantum gravity effect stabilizing internal spaces with non-vanishing Ricci curvature.
The Wheeler-DeWitt equation for the minimally coupled FRW-massive-scalar-field minisuperspace is written as a two-component Schrödinger equation with an explicitly `time'-dependent Hamiltonian. This reduces the solution of the Wheeler-DeWitt equation to the eigenvalue problem for a non-relativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler-DeWitt equation. Further analysis of the mode expansion shows that in ge…
In the case of a spatially flat Friedmann--Lema\^ıtre--Robertson--Walker universe in $f\left( R\right) $-gravity we write the Wheeler-DeWitt equation of quantum cosmology. The equation depends on the functional form of $f\left( R\right) $. We select to work with four specific functions of $f\left( R\right) $, in which the field equations for the classical models are integrable and solvable through quadratures. For these models, we determine similarity solutions for the Wheeler-DeWitt equation by determine Lie-Bäcklund transformations. In addition we show how the classical limit is recovered by…
One of the central difficulties in the quantization of the gravitational interactions is that they are described by a set of constraints. The standard strategy for dealing with the problem is the Dirac quantization procedure, which leads to the Wheeler-DeWitt equation. However, solutions to the equation are known only for specific symmetry-reduced systems, including models of quantum cosmology. Novel methods, which enable solving the equation for complex gravitational configurations are, therefore, worth seeking. Here, we propose and investigate a new method of solving the Wheeler-DeWitt equat…
Recently, Wang with Stankiewicz (Phys. Lett. B 800 (2020) 135106) opposed the widespread belief that due to quantization, the Big Bang singularity {\em must} necessarily get smeared and replaced by a non-singular "Big Bounce" process. Their claim is supported in a methodically independent manner in which just a highly schematic model (viz., the discrete and drastically simplified Wheeler-DeWitt equation) is considered. Our comparatively elementary argumentation is constructive, based on the assumption that the evolution of the Universe can fully consistently be described using an innovative, h…
This is a pedagogical paper which explains some ideas in cosmology at a level accessible to undergraduate students. It does not use general relativity, but uses the ideas of Newtonian cosmology worked out by Milne and McCrea. The cosmological constant is also introduced within a Newtonian framework. Following standard quantization procedures the Wheeler-DeWitt equation in the minisuperspace approximation is derived for empty and non-empty universes.
Of all the possible ways to describe the behavior of the universe that has undergone a big trip the Wheeler-DeWitt equation should be the most accurate -- provided, of course, that we employ the correct formulation. In this article we start by discussing the standard formulation introduced by González-Díaz and Jimenez-Madrid, and show that it allows for a simple yet efficient method of the solution's generation, which is based on the Moutard transformation. Next, by shedding the unnecessary restrictions, imposed on aforementioned standard formulation we introduce a more general form of the Whe…
We can obtain one solution of the Hamiltonian constraint equation in the local sense. The form of the state is suggested from the up-to-down method in our previous work. The up-to-down method works for different way in treating the general metrics. In the mini-superspace approach there appears additional constraint in the 4-dimensional quantum gravity Hilbert space. However, in the general treatment of the metrics this method works as only solving technique.
We investigate the possibility of computing energy by means of operators associated to the Wheeler-DeWitt equation. By choosing three dimensional wormholes as a framework, we apply such calculation scheme to the black hole pair creation. We compare our results with the recent ones appeared in the literature.
In this paper, the Wheeler-DeWitt (WDW) equation is derived in null-foliated 4D spacetimes. WDW equation written in null-foliated spacetime presents an enormous simplification compared to the spacelike-foliated spacetime as the null-foliations are 2D. These can be solved exactly and uniquely to give the vertex operators of non-critical strings as a solution. A correspondence is established between null surfaces in 4D to string worldsheet geometry. Attempts are made to derive the physical consequences of this correspondence.
We solve the Wheeler-DeWitt equation for {\it four}-dimensional Einstein gravity as an expansion in powers of the Planck mass by means of a heat kernel regularization. Our results suggest that in the universe with a very small radius or with a very large curvature beyond a Planck scale expectation values of operators are reduced to calculations in a path integral representation of {\it three}-dimensional Einstein gravity.
Taking into account the global one-dimensionality conjecture recently proposed by the author, the Cauchy-like analytical wave functional of the Wheeler-DeWitt theory is derived. The crucial point of the integration strategy is canceling of the singular behavior of the effective potential, which is performed through the suitable change of variables introducing the invariant global dimension. In addition, the conjecture is extended onto the wave functionals dependent on both Matter felids as well as the invariant global dimension. Through application of the reduction within the quantum gravity m…
In this paper, we consider the Wheeler-DeWitt equation modified by a deformation of the second quantized canonical commutation relations. Such modified commutation relations are induced by a Generalized Uncertainty Principle. Since the Wheeler-DeWitt equation can be related to a Sturm-Liouville problem where the associated eigenvalue can be interpreted as the cosmological constant, it is possible to explicitly relate such an eigenvalue to the deformation parameter of the corresponding Wheeler-DeWitt equation. The analysis is performed in a Mini-Superspace approach where the scale factor appear…
Lie symmetries are discussed for the Wheeler-De Witt equation in Bianchi Class A cosmologies. In particular, we consider General Relativity, minimally coupled scalar field gravity and Hybrid Gravity as paradigmatic examples of the approach. Several invariant solutions are determined and classified according to the form of the scalar field potential. The approach gives rise to a suitable method to select classical solutions and it is based on the first principle of the existence of symmetries.
We investigate the class of ultralocal metrics on the configuration space of canonical gravity. It is described by a parameter $α$, where $α=0.5$ corresponds to general relativity. For $α$ less than a critical value the signature is positive definite, while for all other values it is indefinite. We show that in the positive definite case gravity becomes repulsive. From the primordial helium abundance we find that $α$ must lie between $0.4$ and $0.55$.
The Quantum Wheeler-DeWitt operator can be derived from an affine commutation relation via the affine group representation formalism for gravity, wherein a family of gauge-diffeomorphism invariant affine coherent states are constructed from a fiducial state. In this article, the role of the fiducial state is played by a regularized Gaussian peaked on densitized triad configurations corresponding to 3-metrics of constant spatial scalar curvature. The affine group manifold consists of points in the upper half plane, wherein each point is labeled by two local gravitational degrees of freedom from…
We compare two different approaches to the treatment of the Wheeler-DeWitt equation and the introduction of time in quantum cosmology. One approach is based on the gauge-fixing procedure in theories with first-class constraints, while the other uses the Born-Oppenheimer method. We apply both to a very simple cosmological model and observe that they give similar predictions. We also discuss the problem of time in non-relativistic quantum mechanics and some questions concerning the correspondence between classical and quantum theories.
We show that the solutions of the Wheeler-DeWitt equation in a homogeneous and isotropic universe are given by triconfluent Heun functions for the spatially closed, flat, and open geometries of the Friedmann-Robertson-Walker universe filled with different forms of energy. In a matter-dominated universe, we find the polynomial solution and the energy density spectrum. In the cases of radiation-dominated and vacuum universes, there are no polynomial solutions as shown.
We consider a Bohmian approach to the Wheeler-DeWitt quantization of the Friedmann-Lemaitre-Robertson-Walker model and investigate the question whether or not there are singularities, in the sense that the universe reaches zero volume. We find that for generic wave functions (i.e., non-classical wave functions), there is a non-zero probability for a trajectory to be non-singular. This should be contrasted to the consistent histories approach for which it was recently shown by Craig and Singh that there is always a singularity. This result illustrates that the question of singularities depends …
The Wheeler-DeWitt Equation represents a tool to study Quantum Gravity and Quantum Cosmology. Its solution in a very general context is, of course, impossible. To this purpose we consider some distortions of General Relativity like Gravity's Rainbow, Varying Speed of Light Cosmology, Generalized Uncertainty Principle deformations and Hořava-Lifshitz gravity which could allow the calculation of some observables like the cosmological constant. For simplicity we consider only the Mini-Superspace approach related to a Friedmann-Lemaître-Robertson-Walker space-time.
We reexamine the relationship between the path integral and canonical formulation of quantum general relativity. In particular, we present a formal derivation of the Wheeler-DeWitt equation from the path integral for quantum general relativity by way of boundary variations. One feature of this approach is that it does not require an explicit 3+1 splitting of spacetime in the bulk. For spacetimes with a spatial boundary, we show that the dependence of the transition amplitudes on spatial boundary conditions is determined by a Wheeler-DeWitt equation for the spatial boundary surface. We find tha…
We consider the quantization of space-times which can possess different topologies within a symmetry reduced version of Wheeler-DeWitt theory. The quantum states are defined from a natural decomposition as an outer-product of a topological state, dictating the topology of the two-surfaces of the space-time, and a geometric state, which controls the geometry and is comprised of solutions to the Wheeler-DeWitt constraints. Within this symmetry reduced theory an eigenvalue equation is derived for the two-volume of spacetime, which for spherical topology is fixed to a value of $4π$. However, for t…
Bohmian Quantum Cosmology from the Wheeler-DeWitt Equation
Academy of Athens（雅典科学院）；National Observatory of Athens（希腊国家天文台）；National Technical University of Athens（雅典国立技术大学）；CAS Key Laboratory for Researches in Galaxies and Cosmology, School of Astronomy and Space Science, University of Science and Technology of China（中国科学院星系与宇宙学重点实验室，中国科学技术大学天文与空间科学学院）；Universidad Católica del Norte（北天主教学会大学）；National & Kapodistrian University of Athens（雅典国立卡波蒂斯坦大学）
