The gravitational field is usually neglected in the calculation of atomic energy levels as its effect is much weaker than the electromagnetic field, but that is not the case for a particle orbiting a black hole. In this work, the canonical quantization of a massive and massless particles under gravitational field exerted by this tiny but very massive object, both neutral and charged, is carried out. By using this method, a very rare exact result of the particle's quantized energy can be discovered. The presence of a very strong attractive field and also the horizon make the energy complex valued and force the corresponding wave function to be a quasibound state. Moreover, by taking the small scale limit, the system becomes a gravitational atom in the sense of Hydrogenic atoms energy levels and its wave function can be rediscovered. Moreover, analogous to electronic transitions, the transition of the particle in this case emits a graviton which carries a unique fingerprint of the black hole such as black hole's mass and charge.
A canonical quantization of two-dimensional gravity minimally coupled to real scalar and spinor Majorana fields is presented. The physical state space of the theory is completely described and calculations are also made of the average value of the metric tensor relative to states close to the ground state
I investigate a new idea of perturbation theory in covariant canonical quantization. I present preliminary results for a toy model of a harmonic oscillator with a quartic perturbation, and show that this method reproduces the quantized spectrum of standard quantum theory. This result indicates that when the exact solutions to classical equations are not known, covariant canonical quantization via perturbation theory could be a viable approximation scheme for finding observables, and suggests a physically interesting way of extending the scope of covariant canonical quantization in quantum grav…
In this paper we have carried out a transformation from Ashtekar's theory of GR into a reduced theory where the physical degrees of freedom are explicit. We have performed the canonical analysis, computed the classical dynamics and have performed a quantization on this reduced space, constructing a Hilbert space of states for vanishing cosmological constant. Finally, we have clarified the canonical structure of the dual theory in relation to the original Ashtekar theory.
We find the canonical formulation of the Poincare BFCG theory in terms of the spatial 2-connection and its canonically conjugate momenta. We show that the Poincare BFCG action is dynamically equivalent to the BF action for the Poincare group and we find the canonical transformation relating the two. We study the canonical quantization of the Poincare BFCG theory by passing to the Poincare-connection basis. The quantization in the 2-connection basis can be then achieved by performing a Fourier transform. We also briefly discuss how to approach the problem of constructing a basis of spin-foam st…
Quantization of gravity is discussed in the context of field quantization based on an analogue of canonical formalism (the De Donder-Weyl canonical theory) which does not require the space+time decomposition. Using Horava's (1991) De Donder-Weyl formulation of General Relativity we put forward a covariant generalization of the Schrödinger equation for the wave function of space-time and metric variables and a supplementary ``bootstrap condition'' which self-consistently incorporates the classical background space-time geometry as a quantum average and closes the system of equations. Some open …
We examine the possibility of a constraint-free quantization of linearized gravity, based on the Teukolsky equation for black hole perturbations. We exhibit a simple quadratic (but complex) Lagrangian for the Teukolsky equation, leading to the interpretation that the elementary excitations (gravitons bound to the Kerr black hole) are unstable.
The groupoid approach to noncommutative unification of general relativity with quantum mechanics is compared with the canonical gravity quantization. It is shown that by restricting the corresponding noncommutative algebra to its (commutative) subalgebra, which determines the space-time slicing, an algebraic counterpart of superspace (space of 3-metrics) can be obtained. It turns out that when this space-time slicing emerges the universe is already in its commutative regime. We explore the consequences of this result.
We investigate the canonical quantization of the electromagnetic field on the Kerr background. We give new expressions for the expectation value of the electromagnetic stress-energy tensor in various vacua states and give a physical interpretation of the separate terms appearing in them. We numerically calculate the luminosity in these states. We also study the form of the renormalized stress-energy tensor close to the horizon when the electromagnetic field is in the past Boulware state.
We perform the canonical quantization of a relativistic spinless particle moving in a curved and static spacetime. We show that the classical theory already describes at the same time both particle and antiparticle. The analyses involves time-depending constraints and we are able to construct the two-particle Hilbert space. The requirement of a static spacetime is necessary in order to have a well defined Schrödinger equation and to avoid problems with vacuum instabilities. The severe ordering ambiguities we found are in essence the same ones of the well known non-relativistic case.
The present article summarizes the work of the papers \cite{1} dealing with the quantization of pure gravity and gravity coupled to a Maxwell field and a cosmological constant in presence of spherical symmetry. The class of models presented is intended as an interesting testing ground for the quantization of full 3+1 gravity. We are working in Ashtekar's self-dual representation.
A formalism previously introduced by the author using tesselated Cauchy surfaces is applied to define a quantized version of gravitating point particles in 2+1 dimensions. We observe that this is the first model whose quantum version automatically discretizes time. But also spacelike distances are discretized in a very special way.
We present two results on the recently proposed new spin foam models. First, we show how a (slightly modified) restriction on representations in the EPRL model leads to the appearance of the Ashtekar-Barbero connection, thus bringing this model even closer to LQG. Second, we however argue that the quantization procedure used to derive the new models is inconsistent since it relies on the symplectic structure of the unconstraint BF theory.
This is a PhD thesis on a categorical generalization of the $BF$ model called the $BFCG$ theory. The new theory is equivalent to General Relativity at the classical level but, at the quantum level, seeks to address the problem of the coupling of spinfoams to fermions and, the question of the semiclassical limit of a spinfoam model. Please see enclosed abstract for further details.
We suggest a modification of the Barrett-Crane spin foam model of 4-dimensional Lorentzian general relativity motivated by the canonical quantization. The starting point is Lorentz covariant loop quantum gravity. Its kinematical Hilbert space is found as a space of the so-called projected spin networks. These spin networks are identified with the boundary states of a spin foam model and provide a generalization of the unique Barrette-Crane intertwiner. We propose a way to modify the Barrett-Crane quantization procedure to arrive at this generalization: the B field (bi-vectors) should be promot…
The Ashtekar variables have been use to find a number of exact solutions in quantum gravity and quantum cosmology. We investigate the origin of these solutions in the context of a number of canonical transformations (both complex and real) of the basic Hamiltonian variables of general relativity. We are able to present several new solutions in the minisuperspace (quantum cosmology) sector. The meaning of these solutions is then discussed.
We consider the quantum dynamics of both open and closed two- dimensional universes with ``wormholes'' and particles. The wave function is given as a sum of freely propagating amplitudes, emitted from a network of mapping class images of the initial state. Interference between these amplitudes gives non-trivial scattering effects, formally analogous to the optical diffraction by a multidimensional grating; the ``bright lines'' correspond to the most probable geometries.
One of the hardest problems to tackle in the dynamics of canonical approaches to quantum gravity is that of the Hamiltonian constraint. We investigate said problem in the context of formal geometric quantization. We study the implications of the non uniqueness in the choice of the vector field which satisfies the presymplectic equation for the Hamiltonian constraint, and study the implication of the same in the quantization of the theory. Our aim is to show that this non uniqueness in the choice of said vector field, which really stems from refoliation invariance leads to a very ambiguous noti…
The paper is the first of two parts of a work reviewing some approaches to the problem of time in quantum cosmology, which were put forward last decade, and which demonstrated their relation to the problems of reparametrization and gauge invariance of quantum gravity. In the present part we remind basic features of quantum geometrodynamics and minisuperspace cosmological models, and discuss fundamental problems of the Wheeler - DeWitt theory. Various attempts to find a solution to the problem of time are considered in the framework of the canonical approach. Possible solutions to the problem a…
We present a toy model approach to the canonical non-perturbative quantization of the spatially-flat Robertson-Walker Universes with cosmological constant, based on the fact that such models are exactly solvable within the framework of a simple Lagrangian formulation. The essential quantum dynamical metric-field and the corresponding Hamiltonian, explicitly derived in terms of annihilation and creation operators, point out that the Wheeler - DeWitt equation is a natural (quantum) generalization of the $G_{44}$ - Einstein equation for the classical De Sitter spacetime and selects the physical s…
Canonical quantization (CQ) is built around $[Q,P]=i\hbar1\!\!1$, while affine quantization (AQ) is built around $[Q,D]=i\hbar\,Q$, where $D\equiv(PQ+QP)/2$. The basic CQ operators must fit $-\infty< P, Q <\infty$, while the basic AQ operators can fit $-\infty<P<\infty$ and $ 0<Q<\infty$, $-\infty <Q<0$, or even $-\infty<Q\neq0<\infty$. AQ can also be the key to quantum gravity, as our simple outline demonstrates.
This is a summary of a talk delivered at the workshop ``Quantum gravity in the Southern Cone II''. We present a very brief review of current results on canonical quantization of general relativity using Ashtekar's variables and loop quantization.
We employ the familiar canonical quantization procedure in a given cosmological setting to argue that it is equivalent to and results in the same physical picture if one considers the deformation of the phase-space instead. To show this we use a Probabilistic Evolutionary Process (PEP) to make the solutions of these different approaches comparable. Specific model theories are used to show that the independent solutions of the resulting Wheeler-DeWitt equation are equivalent to solutions of the deformation method with different signs for the deformation parameter. We also argued that since the …
We carry out the canonical quantization of the Levi-Cività family of static and cylindrical solutions. The reduced phase space of this family of metrics is proved to coincide with that corresponding to the Kasner model, including the associated symplectic structures, except for that the respective domains of definition of one of the phase space variables are not identical. Using this result, we are able to construct a quantum model that describes spacetimes of both the Levi-Cività and the Kasner type, and in which the three-dimensional spatial topology is not uniquely fixed. Finally, we quanti…
The canonical formalism for expanding metrics scenarios is presented. Non-unitary time evolution implied by expanding geometry is described as a trajectory over unitarily inequivalent representations at different times of the canonical commutation relations. Thermal properties of inflating Universe are also discussed.
We discuss the relation between canonical and Schrodinger quantization of the CGHS model. We also discuss the situation when background charges are added to cancel the Virasoro anomaly. New physical states are found when the square of the background charges vanishes.
The canonical quantization of the essentially nonlinear midisuperspace model describing cylindrically symmetric gravitational waves with two polarizations is presented. A Fock space type representation is constructed. It is based on a complete set of quantum observables. Physical expectation values may be calculated in arbitrary excitations of the vacuum. Our approach provides a non-linear generalization of the quantization of the collinearly polarized Einstein-Rosen gravitational waves.
We propose a general construction of quantum states for linear canonical quantum fields on a manifold, which encompasses and generalizes the "standard" procedures existing in textbooks. Our method provides pure and mixed states on the same footing. A large class of examples finds a simple and unified treatment in our approach. Applications discussed here include thermodynamical equilibrium states for Minkowski fields and quantum field theory in the Rindler's and in the open de Sitter universes. Our approach puts the above examples into perspective and unravels new possibilities for quantizatio…
A formulation of Covariant Canonical Quantization is discussed, which works on an extended Hilbert space and reduces to conventional canonical quantization when constraining to the solution of the field equation a priori. From the formal point of view it may be seen as a formalism between the canonical operator and the functional integral approach. A covariant number operator and two symmetric vacua are constructed. By that means, certain well-known quantities like the LSZ formula are rederived via a projection limit. The time-ordering operator can be replaced by taking into account the mirror…
A previously used quantization mechanism is applied to the continuous states of the shielded strong gravity scenario (hep-th/0602183), yielding two types of spectra for uncharged black hole scalars. Each yields the general morphology for states expected in this scenario at LHC and at arbitrarily higher energies, once the parameters are determined by the two lowest-lying scalar states. A particularized example for the preferred type of quantization is numerically evaluated.
The problem of defining and constructing representations of the Canonical Commutation Relations can be systematically approached via the technique of {\it algebraic quantization}. In particular, when the phase space of the system is linear and finite dimensional, the `vertical polarization' provides an unambiguous quantization. For infinite dimensional field theory systems, where the Stone-von Neumann theorem fails to be valid, even the simplest representation, the Schroedinger functional picture has some non-trivial subtleties. In this letter we consider the quantization of a real free scalar…
Canonical quantization of gravitational systems is obstructed by the problem of time. Due to diffeomorphism symmetry the Hamiltonian vanishes: dynamics with respect to a background time parameter appears "frozen." Two strategies towards the quantization of such systems are the identification of a clock degree of freedom before quantization (deparametrization), and quantization on a kinematical Hilbert space which is subject to constraints (Dirac quantization). The usual canonical quantization in quantum field theory is analogous to deparametrization. Here we introduce a frozen formalism and Di…
This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the three-dimensional context, where the two approaches are in good agreement, we show how the canonical quantization à la Witten of Riemannian gravity with a positive cosmological constant is related to the Turaev-Viro spin foam model, and how the Ponzano-Regge amplitudes are related to the physical scalar product of Riemannian loop quantum gravity without cosmological constant. In the four-dimensional case, we recall a Lorentz-covariant…
We present a quantization of the Hamiltonian and diffeomorphism constraint of canonical quantum gravity in the spin network representation. The novelty consists in considering a space of wavefunctions based on the Vassiliev knot invariants. The constraints are finite, well defined, and reproduce at the level of quantum commutators the Poisson algebra of constraints of the classical theory. A similar construction can be carried out in 2+1 dimensions leading to the correct quantum theory.
In this paper is considered a generalized quantization principle for the gravitational field in canonical quantum gravity, especially with respect to quantum geometrodynamics. This assumption can be interpreted as a transfer from the generalized uncertainty principle in quantum mechanics, which is postulated as generalization of the Heisenberg algebra to introduce a minimal length, to a corresponding quantization principle concerning the quantities of quantum gravity. According to this presupposition there have to be given generalized representations of the operators referring to the observabl…
In cosmology minisuperspace models are described by nonlinear time-reparametrization invariant systems with a finite number of degrees of freedom. Often these models are not explicitly integrable and cannot be quantized exactly. Having this in mind, we present a scheme for the (approximate) quantization of perturbed, nonintegrable, time-reparametrization invariant systems that uses (approximate) gauge invariant quantities. We apply the scheme to a couple of simple quantum cosmological models.
We construct phase-space structure of a typical higher-order theory of gravity, in the background of anisotropic Bianchi-1 mini-superspace, following `Modified Horowitz Formalism' as well as applying `Dirac Algorithm' (after taking care of the divergent terms), and establish equivalence. Canonical quantization, and semiclassical approximation are performed to expatiate the fact that such a quantum theory transits successfully to a classical de-Sitter universe. Inflation has thereafter been studied. The numerical values of the inflationary parameters show excellent agreement with the latest rel…
We propose a manifestly covariant canonical method of field quantization based on the classical De Donder-Weyl covariant canonical formulation of field theory. Owing to covariance, the space and time arguments of fields are treated on an equal footing. To achieve both covariance and consistency with standard noncovariant canonical quantization of fields in Minkowski spacetime, it is necessary to adopt a covariant Bohmian formulation of quantum field theory. A preferred foliation of spacetime emerges dynamically owing to a purely quantum effect. The application to a simple time-reparametrizatio…
We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete can…
The relation between a recently proposed path integral for minisuperspaces and different canonical quantizations is established. The step of the procedure where a choice between non equivalent theories is made is identified. Coordinates avoiding such a choice are found for a class of homogeneous cosmologies.
The canonical quantization of a Schwarzschild black hole yields a picture of the black hole that is shown to be equivalent to a collection of oscillators whose density of levels is commensurate with that of the statistical bootstrap model. Energy eigenstates of definite parity exhibit the Bekenstein mass spectrum, $M \sim \sqrt{\cal N} M_p$, where ${\cal N} \in {\bf N}$. From the microcanonical ensemble, we derive the statistical entropy of the black hole by explicitly counting the microstates corresponding to a macrostate of fixed total energy.
Ambiguities arising in different approaches (canonical, quasiclassical, path integration) to quantization are discussed by an example of the mechanics of a point-like particle in the Riemannian space (the geodesic dynamics). A way to select a single rule of quantization is proposed by requirement of a consistency of the quantum mechanics' following from the canonical and quasiclassical approaches. This rule selects also a definition of the path integration. A geometric interpretation of the noncovariance of the canonical Hamilton operator of a particle with respect to the diffeomorphisms of th…
We study the canonical quantization of the theory given by Chamseddine-Connes spectral action on a particular finite spectral triple with algebra $M_2(\Cset)\oplus\Cset$. We define a quantization of the natural distance associated with this noncommutative space and show that the quantum distance operator has a discrete spectrum. We also show that it would be the same for any other geometric quantity. Finally we propose a physical Hilbert space for the quantum theory. This spectral triple had been previously considered by Rovelli as a toy model, but with a different action which was not gauge-i…
In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant $Λ>0$ in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection $A$ and the canonically conjugate triad field $e$. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of $A+=A + \sqrtΛe$. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection $A_λ=A+λe$ on the kinematical Hilbert spa…
We consider the form of the path integral that follows from canonical quantization and apply it to the first order form of the Einstein-Hilbert action in $d > 2$ dimensions. We show that this is inequivalent to what is obtained from applying the Faddeev-Popov (FP) procedure directly. Due to the presence of tertiary first class constraints, the measure of the path integral is found to have a substantially different structure from what arises in the FP approach. In addition, the presence of second class constraints leads to non-trivial ghosts, which cannot be absorbed into the normalization of t…
We investigate the canonical quantization of gravity coupled to pointlike matter in 2+1 dimensions. Starting from the usual point particle action in the first order formalism, we introduce auxiliary variables which make the action locally Poincaré invariant. A Hamiltonian analysis shows that the gauge group is actually larger than the Poincaré group -- certain additional gauge constraints are present which act on the matter degrees of freedom. These additional constraints are necessary to mimic the diffeomorphism invariance present if the theory is formulated with a spacetime metric. The addit…
We study the Fock quantization of scalar fields of Klein-Gordon type in nonstationary scenarios propagating in spacetimes with compact spatial sections, allowing for different field descriptions that are related by means of certain nonlocal linear canonical transformations that depend on time. More specifically, we consider transformations that do not mix eigenmodes of the Laplace-Beltrami operator, which are supposed to be dynamically decoupled. In addition, we assume that the canonical transformations admit an asymptotic expansion for large eigenvalues (in norm) of the Laplace-Beltrami opera…
We consider spin system defined on the coadjoint orbit with noncompact symmetry and investigate the quantization. Classical spin with noncompact SU(N,1) symmetry is first formulated as a dynamical system and the constraint analysis is performed to reduce the system from the group space to the coadjoint orbit which is a symplectic manifold with Kahler structure. We achieve this by solving the constraint directly. It is shown that the dynamical variables describing the noncompact spins can be written as functions of canonically conjugate variables and canonical quantization is possible on the re…
The canonical quantization in Weyl gauge of gauge fields in static space-times is presented. With an appropriate definition of transverse and longitudinal components of gauge fields, the Gauss law constraint is resolved explicitly for scalar and spinor QED and a complete non-perturbative solution is given for the quantized Maxwell-field coupled to external currents. The formalism is applied in investigations of the electromagnetic field in Rindler spaces. The relation of creation and annihilation operators in Minkowski and Rindler spaces is established and initial value problems associated wit…
This is a collection of lectures given at the University of Heidelberg, especially but not exclusively for people who want to learn something about the canonical approach to quantum gravity, which is however not included in these lectures. They are about Dirac's general method to construct a quantum theory out of a classical theory, which has to be defined in terms of a Lagrangian. The classical Hamiltonian formalism is reviewed, with emphasis on the relation between constraints and gauge symmetries, and quantization is carried out without any kind of gauge fixing. The method is applied to thr…
Conventional canonical quantization procedures directly link various c-number and q-number quantities. Here, we advocate a different association of classical and quantum quantities that renders classical theory a natural subset of quantum theory with \hbar>0, in conformity with the real world wherein nature has chosen \hbar>0 rather than \hbar=0. While keeping the good results of conventional procedures, some examples are presented for which the new procedures offer better results than conventional ones.
Gravity does not naturally fit well with canonical quantization. Affine quantization is an alternative procedure that is similar to canonical quantization but may offer a positive result when canonical quantization fails to offer a positive result. Two simple examples given initially illustrate the power of affine quantization. These examples clearly point toward an affine quantization procedure that vastly simplifies a successful quantization of general relativity.
Canonical quantization of gravity requires knowledge about the representation theory of its constraint algebra, which is physically equivalent to the algebra of arbitrary 4-diffeomorphisms. All interesting lowest-energy representations are projective, making the relevant algebra into a four-dimensional generalization of the Virasoro algebra. Such diffeomorphism anomalies are invisible in field theory, because the relevant cocycles are functionals of the observer's trajectory in spacetime. The multi-dimensional Virasoro algebra acts naturally in the phase space of arbitrary histories, with dyna…
Quantization of electromagnetic fields is investigated in the framework of stochastic variational method (SVM). Differently from the canonical quantization, this method does not require canonical form and quantization can be performed directly from the gauge invariant Lagrangian. The gauge condition is used to choose dynamically independent variables. We verify that, in the Coulomb gauge condition, SVM result is completely equivalent to the traditional result. On the other hand, in the Lorentz gauge condition, SVM quantization can be performed without introducing the indefinite metric. The tem…
We discuss the behaviour of external fields that interact with a Dirac neutral particle with a permanent electric dipole moment in order to achieve relativistic bound states solutions in a noninertial frame and in the presence of a topological defect spacetime. We show that the noninertial effects of the Fermi-Walker reference frame induce a radial magnetic field even in the absence of magnetic charges, which is influenced by the topology of the cosmic string spacetime. We then discuss the conditions that the induced fields must satisfy to yield the relativistic bound states corresponding to t…
