Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

  title={Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras},
  author={Jerzy Lewandowski and Andrzej Okoł{\'o}w and H. Sahlmann and Thomas Thiemann},
  journal={Communications in Mathematical Physics},
Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one… 
A new look at Lorentz-Covariant Loop Quantum Gravity
In this work, we study the classical and quantum properties of the unique commutative Lorentz-covariant connection for loop quantum gravity. This connection has been found after solving the
Loop quantum gravity without the Hamiltonian constraint
We show that under certain technical assumptions, including the existence of a constant mean curvature (CMC) slice and strict positivity of the scalar field, general relativity conformally coupled to
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
Invariant Connections and Symmetry Reduction in Loop Quantum Gravity
The intention of this thesis is to provide general tools and concepts that allow to perform a mathematically substantiated symmetry reduction in (quantum) gauge field theories. Here, the main focus
Ju l 2 00 9 Loop Quantum Gravity à la Aharonov-Bohm
The state space of Loop Quantum Gravity admits a decomposition into orthogonal subspaces associated to diffeomorphism equivalence classes of graphs. In this paper I discuss the possibility of
Loop Quantum Gravity à la Aharonov–Bohm
The state space of Loop Quantum Gravity admits a decomposition into orthogonal subspaces associated to diffeomorphism equivalence classes of spin-network graphs. In this paper I investigate the
Continuous formulation of the loop quantum gravity phase space
In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of
Flux formulation of loop quantum gravity: Classical framework
We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay
Algebras of Quantum Variables for Loop Quantum Gravity. II. A new formulation of the Weyl C*-algebra
In this article a new formulation of the Weyl C∗-algebra, which has been invented by Fleischhack [7], in terms of C∗-dynamical systems is presented. The quantum configuration variables are given by
On the uniqueness of kinematics of loop quantum cosmology
The holonomy-flux algebra of loop quantum gravity is known to admit a natural representation that is uniquely singled out by the requirement of covariance under spatial diffeomorphisms. In the


On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories
Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant
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
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
Combinatorial Space from Loop Quantum Gravity
The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero
Diffeomorphism covariant representations of the holonomy-flux ⋆-algebra
Recently, Sahlmann (2002 Preprint gr-qc/0207111) proposed a new, algebraic point of view on the loop quantization. He brought up the issue of a -algebra underlying that framework, studied the algebra
Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown
Some Comments on the Representation Theory of the Algebra Underlying Loop Quantum Gravity
Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized
Separable Hilbert space in loop quantum gravity
We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a
Loop constraints: A habitat and their algebra
This work introduces a new space 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