arXiv subjects

Jerzy Lewandowski

Publications and source records attributed to Jerzy Lewandowski.

At least 127 records · Page 7Linked to original sources

arXiv2000

Generic Isolated Horizons and their Applications

Boundary conditions defining a generic isolated horizon are introduced. They generalize the notion available in the existing literature by allowing the horizon to have distortion and angular momentum. Space-times containing a black hole, itself in equilibrium but possibly surrounded by radiation, satisfy these conditions. In spite of this generality, the conditions have rich consequences. They lead to a framework, somewhat analogous to null infinity, for extracting physical information, but now in the \textit{strong} field regions. The framework also generalizes the zeroth and first laws of black hole mechanics to more realistic situations and sheds new light on the `origin' of the first law. Finally, it provides a point of departure for black hole entropy calculations in non-perturbative quantum gravity.

gr-qc↗

arXiv2000

Space-Times Admitting Isolated Horizons

We characterize a general solution to the vacuum Einstein equations which admits isolated horizons. We show it is a non-linear superposition -- in precise sense -- of the Schwarzschild metric with a certain free data set propagating tangentially to the horizon. This proves Ashtekar's conjecture about the structure of spacetime near the isolated horizon. The same superposition method applied to the Kerr metric gives another class of vacuum solutions admitting isolated horizons. More generally, a vacuum spacetime admitting any null, non expanding, shear free surface is characterized. The results are applied to show that, generically, the non-rotating isolated horizon does not admit a Killing vector field and a spacetime is not spherically symmetric near a symmetric horizon.

gr-qc↗

arXiv1999

2-Form Gravity of the Lorentzian Signature

We introduce a new spinorial, BF-like action for the Einstein gravity. This is a first, up to our knowledge, 2-form action which describes the real, Lorentzian gravity and uses only the self-dual connection. In the generic case, the corresponding classical canonical theory is equivalent to the Einstein-Ashtekar theory plus the reality conditions.

gr-qc↗

arXiv1999

Degenerate Sectors of the Ashtekar Gravity

This work completes the task of solving locally the Einstein-Ashtekar equations for degenerate data. The two remaining degenerate sectors of the classical 3+1 dimensional theory are considered. First, with all densitized triad vectors linearly dependent and second, with only two independent ones. It is shown how to solve the Einstein-Ashtekar equations completely by suitable gauge fixing and choice of coordinates. Remarkably, the Hamiltonian weakly Poisson commutes with the conditions defining the sectors. The summary of degenerate solutions is given in the Appendix.

gr-qc↗

arXiv1999

Diffeomorphism invariant Quantum Field Theories of Connections in terms of webs

In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise smooth finite paths and loops. In particular, we $(i)$ characterize the spectrum of the Ashtekar-Isham configuration space, $(ii)$ introduce spin-web states, a generalization of the spin-network states, $(iii)$ extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism invariant states and finally $(iv)$ extend the 3-geometry operators and the Hamiltonian operator.

gr-qc↗

arXiv1997

Quantum Theory of Geometry II: Volume operators

A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric. It is used to continue the exploration of the quantum Riemannian geometry. Operators corresponding to volume of three-dimensional regions are regularized rigorously. It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator. Both operators can be completely specified by giving their action on states labelled by graphs. The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics. (The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.) The difference between the two operators can be attributed directly to the standard quantization ambiguity. Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.

gr-qc↗

arXiv1997

On the consistency of the constraint algebra in spin network quantum gravity

We point out several features of the quantum Hamiltonian constraints recently introduced by Thiemann for Euclidean gravity. In particular we discuss the issue of the constraint algebra and of the quantum realization of the object $q^{ab}V_b$, which is classically the Poisson Bracket of two Hamiltonians.

gr-qc↗

arXiv1997

Loop constraints: A habitat and their algebra

This work introduces a new space $\T'_*$ of `vertex-smooth' states for use in the loop approach to quantum gravity. Such states provide a natural domain for Euclidean Hamiltonian constraint operators of the type introduced by Thiemann (and using certain ideas of Rovelli and Smolin). In particular, such operators map $\T'_*$ into itself, and so are actual operators in this space. Their commutator can be computed on $\T'_*$ and compared with the classical hypersurface deformation algebra. Although the classical Poisson bracket of Hamiltonian constraints yields an inverse metric times an infinitesimal diffeomorphism generator, and despite the fact that the diffeomorphism generator has a well-defined non-trivial action on $\T'_*$, the commutator of quantum constraints vanishes identically for a large class of proposals.

gr-qc↗

arXiv1996

2+1 Sector of 3+1 Gravity

The rank-2 sector of classical $3+1$ dimensional Ashtekar gravity is considered. It is found that the consistency of the evolution equations with the reality of the volume requires that the 3-surface of initial data is foliated by 2-surfaces tangent to degenerate triads. In turn, the degeneracy is preserved by the evolution. The 2-surfaces behave like $2+1$ dimensional empty spacetimes with a massless complex field propagating along each of them. The results provide some evidence for the issue of evolving a non-degenerate gravitational field into a degenerate sector of Ashtekar's phase space.

gr-qc↗

arXiv1996

Quantum Theory of Gravity I: Area Operators

A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states. It is shown that their spectra are {\it purely} discrete indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed, the fundamental excitations of quantum geometry are 1-dimensional, rather like polymers, and the 3-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite dimensional sub-spaces which can be interpreted as the spaces of states of spin systems. Using this property, the complete spectrum of the area operators is evaluated. The general framework constructed here will be used in a subsequent paper to discuss 3-dimensional geometric operators, e.g., the ones corresponding to volumes of regions.

gr-qc↗

arXiv1996

Quantum Field Theory of Geometry

Over the past five years, there has been significant progress on the problem of quantization of diffeomorphism covariant field theories with {\it local} degrees of freedom. The absence of a background space-time metric in these theories gives rise to a host of conceptual and technical difficulties because most of the familiar methods from axiomatic, constructive and perturbative quantum field theory are no longer applicable. Perhaps the most striking examples of these problems arise in the construction of a quantum field theory of geometry. We show that these problems can be tackled using new non-perturbative methods. In particular, one can rigorously define certain geometric operators and show that their spectrum is discrete. Thus, there is a precise sense in which the geometry is quantized at the Planck scale and the continuum picture is only a coarse-grained approximation.

hep-th↗

arXiv1996

Volume and Quantizations

The aim of this letter is to indicate the differences between the Rovelli-Smolin quantum volume operator and other quantum volume operators existing in the literature. The formulas for the operators are written in a unifying notation of the graph projective framework. It is clarified whose results apply to which operators and why.

gr-qc↗

arXiv1996

Differential Geometry on the Space of Connections via Graphs and Projective Limits

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, $\agb$, of the space $\ag$ of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. $\agb$ is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, $\agb$ is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ``floating lattices'' in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on $\agb$: differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although $\agb$ is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well-suited for diffeomorphism invariant theories such as quantum general relativity.

hep-th↗

arXiv1995

Quantization of diffeomorphism invariant theories of connections with local degrees of freedom

Quantization of diffeomorphism invariant theories of connections is studied. A solutions of the diffeomorphism constraints is found. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain-Kuchař model. The main results also pave way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to combined in an appropriate fashion with a coherent state transform to incorporate complex connections.

gr-qc↗

arXiv1994

Coherent State Transforms for Spaces of Connections

The Segal-Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group $G$ with its normalized Haar measure $μ_H$, the Hall transform is an isometric isomorphism from $L^2(G, μ_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, ν)$, where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of holomorphic functions on $G^{\Co}$, and $ν$ an appropriate heat-kernel measure on $G^{\Co}$. We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group $G$ by (a certain extension of) the space ${\cal A}/{\cal G}$ of connections modulo gauge transformations. The resulting ``coherent state transform'' provides a holomorphic representation of the holonomy $C^\star$ algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4-dimensions.

gr-qc↗

arXiv1994

Projective Techniques and Functional Integration

A general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out integration over the non-linear, infinite dimensional spaces of connections modulo gauge transformations. This method of evaluating functional integrals can be used either in the Euclidean path integral approach or the Lorentzian canonical approach. A number of measures discussed are diffeomorphism invariant and therefore of interest to (the connection dynamics version of) quantum general relativity. The account is pedagogical; in particular prior knowledge of projective techniques is not assumed. (For the special JMP issue on Functional Integration, edited by C. DeWitt-Morette.)

gr-qc↗

arXiv1994

Topological Measure and Graph-Differential Geometry on the Quotient Space of Connections

(This is a report for the Proceedings of ``Journees Relativistes 1993'' written in September 1993. Containes a short description of the results published elsewhere in the joint paper with A. Ashtekar) Integral calculus on the space of gauge equivalent connections is developed. By carring out a non-linear generalization of the theory of cylindrical measures on topological vector spaces, a faithfull, diffeomorphism invariant measure is introduced on a suitable completion of the quotient space. The strip (i.e. momentum) operators are densely-defined in the resulting Hilbert space and interact with the measure correctly, to become essentially self adjoint operators.

gr-qc↗

arXiv1993

Landau Level Ground--State Degeneracy, and Its Relevance for a General Quantization Procedure

The quantum dynamics of a two-dimensional charged spin $1/2$ particle is studied for general, symmetry--free curved surfaces and general, nonuniform magnetic fields that are, when different from zero, orthogonal to the defining two surface. Although higher Landau levels generally lose their degeneracy under such general conditions, the lowest Landau level, the ground state, remains degenerate. Previous discussions of this problem have had less generality and/or used supersymmetry, or else have appealed to very general mathematical theorems from differential geometry. In contrast our discussion relies on simple and standard quantum mechanical concepts. The mathematical similarity of the physical problem at hand and that of a phase-space path integral quantization scheme of a general classical system is emphasized. Adopting this analogy in the general case leads to a general quantization procedure that is invariant under general coordinate transformations-- completely unlike any of the conventional quantization prescriptions -- and therefore generalizes the concept of quantization to new and hitherto inaccesible situations. In a complementary fashion , the so-obtained picture of general quantization helps to derive useful semiclassical formulas for the Hall current in the case of a filling factor equal to one for a general surface and magnetic field.

gr-qc↗