Quantum theory of geometry: I. Area operators
Abhay Ashtekar,Jerzy Lewandowski +1 more
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In this article, a functional calculus for quantum geometry is developed for a fully nonperturbative treatment of quantum gravity, which is used to begin a systematic construction of a quantum theory of geometry, and Regulated operators corresponding to 2-surfaces are introduced and shown to be self-adjoint on the underlying (kinematical) Hilbert space of states.Abstract:
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, the fundamental excitations of quantum geometry are one dimensional, rather like polymers, and the three-dimensional continuum geometry emerges only on coarse graining. The full Hilbert space admits an orthonormal decomposition into finite-dimensional subspaces 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 three-dimensional geometric operators, e.g. the ones corresponding to volumes of regions.read more
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Discreteness of area and volume in quantum gravity
Carlo Rovelli,Lee Smolin +1 more
TL;DR: In this article, the authors studied the spectrum of the volume in nonperturbative quantum gravity, and showed that the spectrum can be computed by diagonalizing finite dimensional matrices, which can be seen as a generalization of the spin networks.
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TL;DR: An important feature of the new form of constraints is the natural embedding of the constraint surface of the Einstein phase space into that of Yang-Mills phase space, which provides new tools to analyze a number of issues in both classical and quantum gravity.
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Lectures on Non-Perturbative Canonical Gravity
TL;DR: In this article, the authors present an up-to-date account of a non-perturbative, canonical quantization program for gravity, which was highlighted in virtually every major conference in gravitational physics over the past three years.
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Loop space representation of quantum general relativity
TL;DR: In this article, the authors define a new representation for quantum general relativity, in which exact solutions of the quantum constraints may be obtained, by means of a noncanonical graded Poisson algebra of classical observables, defined in terms of Ashtekar's new variables.
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Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
TL;DR: In this article, a quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphicism constraint is solved and the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions.