scholarly journals CANONICAL QUANTIZATION OF THE BELINSKIĬ-ZAKHAROV ONE-SOLITON SOLUTIONS

1995 ◽  
Vol 04 (06) ◽  
pp. 749-766
Author(s):  
NENAD MANOJLOVIC ◽  
GUILLERMO A. MENA MARUGÁN
Keyword(s):  
Phase Space ◽  
Quantum Dynamics ◽  
Canonical Quantization ◽  
Soliton Solutions ◽  
Irreducible Unitary Representation ◽  
Einstein Equations ◽  
Classical Solutions ◽  
Hamiltonian Constraint ◽  
Dynamical Equations ◽  
Complete Set

We apply the algebraic quantization programme proposed by Ashtekar to the analysis of the Belinskiĭ-Zakharov classical spacetimes, obtained from the Kasner metrics by means of a generalized soliton transformation. When the solitonic parameters associated with this transformation are frozen, the resulting Belinskiĭ-Zakharov metrics provide the set of classical solutions to a gravitational minisuperspace model whose Einstein equations reduce to the dynamical equations generated by a homogeneous Hamiltonian constraint and to a couple of second-class constraints. The reduced phase space of such a model has the symplectic structure of the cotangent bundle over R+×R+. In this reduced phase space, we find a complete set of real observables which form a Lie algebra under Poisson brackets. The quantization of the gravitational model is then carried out by constructing an irreducible unitary representation of that algebra of observables. Finally, we show that the quantum theory obtained in this way is unitarily equivalent to that which describes the quantum dynamics of the Kasner model.

RELATIVISTIC PARTICLE WITH CURVATURE IN AN EXTERNAL ELECTROMAGNETIC FIELD

1991 ◽  
Vol 06 (22) ◽  
pp. 3989-3996 ◽  
Author(s):  
V.V. NESTERENKO
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Phase Space ◽  
Dirac Equation ◽  
Relativistic Particle ◽  
Canonical Quantization ◽  
Hamiltonian Formalism ◽  
External Electromagnetic Field ◽  
Operator Form ◽  
Complete Set

A model of a relativistic particle with curvature interacting with an external electromagnetic field in a “minimal way” is investigated. The generalized Hamiltonian formalism for this model is constructed. A complete set of the constraints in the phase space is obtained and then divided into first- and second-class constraints. On this basis the canonical quantization of the model is considered. A wave equation in the operator form, resembling the Dirac equation in an external electromagnetic field, is obtained. The massless version of this model is briefly discussed.

scholarly journals A PURE DIRAC'S METHOD FOR HUSAIN–KUCHAR THEORY

2013 ◽  
Vol 10 (07) ◽  
pp. 1350030 ◽  
Author(s):  
ALBERTO ESCALANTE ◽  
J. BERRA
Keyword(s):  
Phase Space ◽  
Canonical Quantization ◽  
Mathematical Structure ◽  
Algebra Structure ◽  
Dirac Algebra ◽  
Gauge Transformations ◽  
Full Phase Space ◽  
Extended Action ◽  
Complete Set ◽  
Weighted Measure

A pure Dirac's canonical analysis, defined in the full phase space for the Husain–Kuchar (HK) model is discussed in detail. This approach allows us to determine the extended action, the extended Hamiltonian, the complete constraint algebra and the gauge transformations for all variables that occur in the action principle. The complete set of constraints defined on the full phase space allow us to calculate the Dirac algebra structure of the theory and a local weighted measure for the path integral quantization method. Finally, we discuss briefly the necessary mathematical structure to perform the canonical quantization program within the framework of the loop quantum gravity approach.

Machine learning phase space quantum dynamics approaches

2021 ◽  
Vol 154 (18) ◽  
pp. 184104
Author(s):  
Xinzijian Liu ◽  
Linfeng Zhang ◽  
Jian Liu
Keyword(s):  
Machine Learning ◽  
Phase Space ◽  
Quantum Dynamics ◽  
Learning Phase

scholarly journals ON CANONICAL QUANTIZATION OF THE GAUGED WZW MODEL WITH PERMUTATION BRANES

2011 ◽  
Vol 26 (26) ◽  
pp. 4647-4660
Author(s):  
GOR SARKISSIAN
Keyword(s):  
Riemann Surface ◽  
Phase Space ◽  
Boundary Conditions ◽  
Canonical Quantization ◽  
Simons Theory ◽  
Gauge Fields ◽  
Time Line ◽  
Chern Simons ◽  
Special Case ◽  
Chern Simons Theory

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the N-fold product of the gauged WZW model G/H on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern–Simons theory on a sphere with N holes times the time-line with G and H gauge fields both coupled to two Wilson lines. For the special case of the topological coset G/G we arrive at the conclusion that the phase space of the N-fold product of the topological coset G/G on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern–Simons theory on a Riemann surface of the genus N-1 times the time-line with four Wilson lines.

scholarly journals Dirac’s Method for the Two-Dimensional Damped Harmonic Oscillator in the Extended Phase Space

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 180 ◽  
Author(s):  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Canonical Quantization ◽  
Extended Phase Space ◽  
Two Dimensional ◽  
Class Constraint ◽  
Extended Phase ◽  
Damped Harmonic Oscillator ◽  
Points Of View ◽  
Space Description

The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac’s canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As a result, the quantum system is simply modeled by the original quantum Hamiltonian.

scholarly journals POLYMER GEOMETRY AT PLANCK SCALE AND QUANTUM EINSTEIN EQUATIONS

1996 ◽  
Vol 05 (06) ◽  
pp. 629-648 ◽  
Author(s):  
ABHAY ASHTEKAR
Keyword(s):  
Quantum Gravity ◽  
Planck Scale ◽  
Coarse Graining ◽  
Space Structure ◽  
Einstein Equations ◽  
Hamiltonian Constraint ◽  
Recent Approach ◽  
Canonical Approach ◽  
Background Fields ◽  
Discrete Spectra

Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge equivalent connections. This calculus does not use any background fields (such as a metric) and thus well-suited to a fully non-perturbative treatment of quantum gravity. Using this framework, quantum geometry is examined. Fundamental excitations turn out to be one-dimensional, rather like polymers. Geometrical observables such as areas of surfaces and volumes of regions are purely discrete spectra. Continuum picture arises only upon coarse graining of suitable semi-classical states. Next, regulated quantum diffeomorphism constraints can be imposed in an anomaly-free fashion and the space of solutions can be given a natural Hilbert space structure. Progress has also been made on the quantum Hamiltonian constraint in a number of directions. In particular, there is a recent approach based on a generalized .Wick transformation which maps solutions to the Euclidean quantum constraints to those of the Lorentzian theory. These developments are summarized. Emphasis is on conveying the underlying ideas and overall pictures rather than technical details.

scholarly journals REVISED CANONICAL QUANTUM GRAVITY VIA THE FRAME FIXING

2004 ◽  
Vol 13 (01) ◽  
pp. 165-186 ◽  
Author(s):  
SIMONE MERCURI ◽  
GIOVANNI MONTANI
Keyword(s):  
Quantum Dynamics ◽  
Fundamental Problem ◽  
Semiclassical Limit ◽  
Momentum Density ◽  
Hamiltonian Constraint ◽  
Additional Energy ◽  
Canonical Quantum Gravity ◽  
Quantum Geometrodynamics ◽  
Canonical Quantum ◽  
Natural Way

We present a new reformulation of the canonical quantum geometrodynamics, which allows one to overcome the fundamental problem of the frozen formalism and, therefore, to construct an appropriate Hilbert space associate to the solution of the restated dynamics. More precisely, to remove the ambiguity contained in the Wheeler–DeWitt approach, with respect to the possibility of a (3+1)-splitting when space–time is in a quantum regime, we fix the reference frame (i.e. the lapse function and the shift vector) by introducing the so-called kinematical action. As a consequence the new super-Hamiltonian constraint becomes a parabolic one and we arrive to a Schrödinger-like approach for the quantum dynamics. In the semiclassical limit our theory provides General Relativity in the presence of an additional energy–momentum density contribution coming from non-zero eigenvalues of the Hamiltonian constraints. The interpretation of these new contributions comes out in natural way that soon as it is recognized that the kinematical action can be recasted in such a way that it describes a pressureless, but, in general, non-geodesic perfect fluid.

Quantum Dynamics in Phase Space

1994 ◽  
pp. 475-484
Author(s):  
A. Smerzi ◽  
V. Kondratyev ◽  
A. Bonasera
Keyword(s):  
Phase Space ◽  
Quantum Dynamics

scholarly journals Time-covariant Schrödinger equation and invariant decay probability: the $$\Lambda $$-Kantowski–Sachs universe

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
Theodoros Pailas ◽  
Nikolaos Dimakis ◽  
Petros A. Terzis ◽  
Theodosios Christodoulakis
Keyword(s):  
Schrödinger Equation ◽  
Wave Packet ◽  
Time Dependence ◽  
Schrodinger Equation ◽  
Canonical Quantization ◽  
Dynamical Equations ◽  
Quadratic Constraint ◽  
Decay Probability ◽  
Flrw Universe ◽  
Arbitrary Time Dependence

AbstractThe system under study is the $$\Lambda $$ Λ -Kantowski–Sachs universe. Its canonical quantization is provided based on a recently developed method: the singular minisuperspace Lagrangian describing the system, is reduced to a regular (by inserting into the dynamical equations the lapse dictated by the quadratic constraint) possessing an explicit (though arbitrary) time dependence; thus a time-covariant Schrödinger equation arises. Additionally, an invariant (under transformations $$t=f({\tilde{t}})$$ t = f ( t ~ ) ) decay probability is defined and thus “observers” which correspond to different gauge choices obtain, by default, the same results. The time of decay for a Gaussian wave packet localized around the point $$a=0$$ a = 0 (where a the radial scale factor) is calculated to be of the order $$\sim 10^{-42}{-}10^{-41}~\text {s}$$ ∼ 10 - 42 - 10 - 41 s . The acquired value is near the end of the Planck era (when comparing to a FLRW universe), during which the quantum effects are most prominent. Some of the results are compared to those obtained by following the well known canonical quantization of cosmological systems, i.e. the solutions of the Wheeler–DeWitt equation.

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