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.
Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented bounded in one orientation, e.g., a strictly positive coordinate. We open discussion using canonical and affine quantizations for two simple problems so each procedure can be understood. That analysis opens a modest treatment of quantum gravity gleaned from some typical features that exhibit the profound differences between aspects of seeking the quantum treatment of Einstein's gravity.
We examine the possibility of dealing with gravitational singularities on a quantum level through the use of coherent state or wavelet quantization instead of canonical quantization. We consider the Robertson-Walker metric coupled to a perfect fluid. It is the simplest model of a gravitational collapse and the results obtained here may serve as a useful starting point for more complex investigations in future. We follow a quantization procedure based on affine coherent states or wavelets built from the unitary irreducible representation of the affine group of the real line with positive dilati…
We present the affine coherent state quantization of the Bianchi I model. As in our previous paper on quantum theory of Friedmann models, we employ a variable associated with a perfect fluid to play a role of clock. Then we deparameterize the model. A distinctive feature, absent in isotropic models, is an extra non-holonomic constraint, which survives the deparameterization and constrains the range of physical variables. The appearance of the constraint reflects the `amplification' of singularity due to anisotropy. The quantization smoothes the extra constraint and allows quantum contracting t…
Affine variables, which have the virtue of preserving the positive-definite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations.
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.
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…
Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to the common goals of any quantization, such as Schroedinger's representation and Schroedinger's equation. Careful attention is paid toward seeking favored classical variables, which are those that should be promoted to the principal quantum operators. This effort leads toward classical variables that have a constant positive, zero, or negative curvature, which typically characteriz…
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…
We present a rigorous and functorial quantization scheme for affine field theories, i.e., field theories where local spaces of solutions are affine spaces. The target framework for the quantization is the general boundary formulation, allowing to implement manifest locality without the necessity for metric or causal background structures. The quantization combines the holomorphic version of geometric quantization for state spaces with the Feynman path integral quantization for amplitudes. We also develop an adapted notion of coherent states, discuss vacuum states, and consider observables and …
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.
The Affine Coherent State Quantization procedure is applied to the case of a FRLW universe in the presence of a cosmological constant. The quantum corrections alter the dynamics of the system in the semiclassical regime, providing a potential barrier term which avoids all classical singularities, as already suggested in other models studied in the literature. Furthermore the quantum corrections are responsible for an accelerated cosmic expansion. This work intends to explore some of the implications of the recently proposed "Enhanced Quantization" procedure in a simplified model of cosmology.
Affine coherent states are generated by affine kinematical variables much like canonical coherent states are generated by canonical kinematical variables. Although all classical and quantum formalisms normally entail canonical variables, it is shown that affine variables can serve equally well for many classical and quantum studies. This general purpose analysis provides tools to discuss two major applications: (1) the completely successful quantization of a nonrenormalizable scalar quantum field theory by affine techniques, in complete contrast to canonical techniques which only offer trivial…
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 …
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 employ the framework of affine covariant quantization and associated semiclassical portrait to address two main issues in the domain of quantum gravitational systems: (i) the fate of singularities and (ii) the lack of external time. Our discussion is based on finite-dimensional, symmetry-reduced cosmological models. We show that the affine quantization of the cosmological dynamics removes the classical singularity and univocally establishes a unitary evolution. The semiclassical portrait based on the affine coherent states exhibits a big bounce replacing the big-bang singularity. As a parti…
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.
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 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 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.
In this work we investigate the thermodynamical properties and modified dynamics of black hole solutions in the semiclassical regime of affine coherent state quantization. Using the weak correspondence principle we build a semiclassical effective action that we use to study thermodynamics in the canonical ensemble and to compare classical analytical solutions with the corresponding semiclassical ones. We determine a strong compatibility of these affine black holes with the thermodynamical properties of the classical counterparts, with only minor restrictions on the free parameters of the model…
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 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.
An ultralocal form of any classical field theory eliminates all spatial derivatives in its action functional, e.g., in its Hamiltonian functional density. It has been applied to covariant scalar field theories and even to Einstein's general relativity, by Pilati, as an initial term in a perturbation series that aimed to restore all omitted derivatives. Previously, the author has quantized ultralocal scalar fields by affine quantization to show that these non-renormalizanle theories can be correctly quantized by affine quantization; the story of such scalar models is discussed in this paper. Th…
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…
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…
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.
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…
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.
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 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…
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…
