Quantum theory of geometry: I. Area operators

  title={Quantum theory of geometry: I. Area operators},
  author={Abhay Ashtekar and Jerzy Lewandowski},
  journal={Classical and Quantum Gravity},
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 purely discrete, indicating that the underlying quantum geometry is far from what the continuum picture might suggest. Indeed… 
A new realization of quantum geometry
We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is
Further results on geometric operators in quantum gravity
Comparison with the structure of the discretized theory leads to conclude that anomalous commutators may be a general feature of operators constructed along similar lines within a continuum loop representation of quantum general relativity; the validity of the lattice approach remains unaffected.
Further results on geometric operators in quantum gravity
We investigate some properties of geometric operators in canonical quantum gravity in the connection approach à la Ashtekar, which are associated with the volume, area and length of spatial regions.
Discrete quantum geometries and their effective dimension
The challenge of coherently combining general relativity and quantum field theory into a quantum theory of gravity is one of the main outstanding tasks in theoretical physics. In several related
Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance
This series of lectures gives an introduction to the non-perturbative and background-independent formulation for a quantum theory of gravitation which is called loop quantum gravity. The Hilbert
Quantum Geometry from the Formalism of Loop Quantum Gravity
Introducing Quantum Geometry as a consequence of the quantisation procedure of loop quantum gravity. By recasting general relativity in terms of 1 2 -flat connections, specified by the Holst’s
Quantum theory of geometry: III. Non-commutativity of Riemannian structures
The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures - such as triad and area operators
Quantum Cauchy surfaces in canonical quantum gravity
For a Dirac theory of quantum gravity obtained from the refined algebraic quantization procedure, we propose a quantum notion of Cauchy surfaces. In such a theory, there is a kernel projector for the
Are the spectra of geometrical operators in Loop Quantum Gravity really discrete
One of the celebrated results of Loop Quantum Gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area and volume operators. This is an indication that Planck
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


Representations of the holonomy algebras of gravity and nonAbelian gauge theories
Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a
Coherent State Transforms for Spaces of Connections
Abstract 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.
Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories
We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the
Reality conditions inducing transforms for quantum gauge field theory and quantum gravity
The algebraic form of the Hamiltonian or Hamiltonian constraint of various (field) theories simplifies considerably if one uses certain complex-valued phase space variables. We show, for a general
Representation Theory of Analytic Holonomy C* Algebras
Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of
Weaving a classical metric with quantum threads.
It is shown that there exist quantum states which approximate a given metric at large scales, but such states exhibit a discrete structure at the Planck scale.
Spin network states in gauge theory
Abstract Given a real-analytic manifoldM, a compact connected Lie groupGand a principalG-bundleP→M, there is a, canonical “generalized measure” on the space A / G of smooth connections onPmodulo