# Differential geometry on the space of connections via graphs and projective limits

@article{Ashtekar1995DifferentialGO, title={Differential geometry on the space of connections via graphs and projective limits}, author={Abhay Ashtekar and Jerzy Lewandowski}, journal={Journal of Geometry and Physics}, year={1995}, volume={17}, pages={191-230} }

## 291 Citations

Coherent State Transforms for Spaces of Connections

- Mathematics
- 1994

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.…

Some Comments on the Representation Theory of the Algebra Underlying Loop Quantum Gravity

- Mathematics
- 2002

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…

Quantization of diffeomorphism invariant theories with fermions

- Mathematics
- 1998

We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P→Σ be a principal G-bundle over space and let F be a…

Quantum gravity kinematics from extended TQFTs

- Physics
- 2017

In this paper, we show how extended topological quantum field theories (TQFTs) can be used to obtain a kinematical setup for quantum gravity, i.e. a kinematical Hilbert space together with a…

Quantum theory of geometry: I. Area operators

- Mathematics
- 1996

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…

Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance

- Physics
- 1999

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…

Some results concerning the representation theory of the algebra underlying loop quantum gravity

- Mathematics
- 2011

Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of (kinematical) observables and of a representation of A on a measure space over the space of…

A new realization of quantum geometry

- Physics, Mathematics
- 2015

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…

Flux formulation of loop quantum gravity: Classical framework

- Physics
- 2014

We recently introduced a new representation for loop quantum gravity (LQG), 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…

Ju l 2 00 9 Loop Quantum Gravity à la Aharonov-Bohm

- Physics
- 2009

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…

## References

SHOWING 1-10 OF 35 REFERENCES

Coherent State Transforms for Spaces of Connections

- Mathematics
- 1994

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.…

Representation Theory of Analytic Holonomy C* Algebras

- Mathematics
- 1993

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…

Diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations

- Mathematics
- 1993

The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical…

Representations of the holonomy algebras of gravity and nonAbelian gauge theories

- Mathematics
- 1992

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…

Generalized measures in gauge theory

- Mathematics
- 1994

LetP →M be a principalG-bundle. We construct well-defined analogs of Lebesgue measure on the spaceA of connections onP and Haar measure on the groupG of gauge transformations. More precisely, we…

Projective techniques and functional integration for gauge theories

- Mathematics
- 1995

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…

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

- Mathematics
- 1995

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…