# Are the spectra of geometrical operators in Loop Quantum Gravity really discrete

@article{Dittrich2009AreTS, title={Are the spectra of geometrical operators in Loop Quantum Gravity really discrete}, author={Bianca Dittrich and Thomas Thiemann}, journal={Journal of Mathematical Physics}, year={2009}, volume={50}, pages={012503-012503} }

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 scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present, because the afore mentioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli’s partial and…

## Figures from this paper

## 85 Citations

### Algebraic quantum gravity (AQG): IV. Reduced phase space quantization of loop quantum gravity

- Physics
- 2010

We perform a canonical, reduced phase space quantization of general relativity by loop quantum gravity (LQG) methods. The explicit construction of the reduced phase space is made possible by the…

### Loop quantum gravity without the Hamiltonian constraint

- Physics
- 2013

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…

### Quasi-Hopf Symmetry in Loop Quantum Gravity with Cosmological constant and Spinfoams with timelike surfaces

- Physics
- 2018

In this thesis we study two separate problems concerning improvements to the Loop quantum gravity and spinfoam approach to quantum gravity. In the first part we address the question about the origin…

### Loop quantum cosmology with complex Ashtekar variables

- Physics
- 2014

We construct and study loop quantum cosmology (LQC) when the Barbero–Immirzi parameter takes the complex value γ = ± i ?> . We refer to this new approach to quantum cosmology as complex LQC. This…

### A pr 2 01 8 Loop quantum gravity and the continuum

- Physics
- 2018

In this paper, we will make an attempt to clarify the relation between three-dimensional euclidean loop quantum gravity with vanishing cosmological constant and quantum field theory in the continuum.…

### The status of quantum geometry in the dynamical sector of loop quantum cosmology

- Physics, Mathematics
- 2008

This paper is motivated by the recent papers by Dittrich and Thiemann and, respectively, Rovelli discussing the status of quantum geometry in the dynamical sector of quantum geometry. Since the…

### Black holes in loop quantum gravity

- PhysicsReports on progress in physics. Physical Society
- 2017

This is a review of results on black hole physics in the context of loop quantum gravity, finding the discreteness of geometric quantities at the Planck scale predicted by this approach to quantum gravity to be key.

### The status of Quantum Geometry in the dynamical sector of Loop Quantum Cosmology

- Physics, Mathematics
- 2009

This letter is motivated by the recent papers by Ditrich and Thiemann and, respectively, by Rovelli discussing the status of Quantum Geometry in the dynamical sector of Loop Quantum Geometry. Since…

### Observables in gravity : a review 3 2 Complete observables for Hamiltonian constraint systems

- Physics
- 2011

We present an overview on observables in gravity mainly from a loop quantum gravity perspective. The gauge group of general relativity is the diffeomorphism group of the underlying manifold.…

## References

SHOWING 1-10 OF 65 REFERENCES

### Simplification of the spectral analysis of the volume operator in loop quantum gravity

- Physics
- 2004

The volume operator plays a crucial role in the definition of the quantum dynamics of loop quantum gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only…

### Properties of the volume operator in loop quantum gravity: I. Results

- Physics
- 2008

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in loop quantum gravity, which is the quantum analog of the classical volume expression for regions in…

### Quantum spin dynamics (QSD) V: Quantum Gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories

- Physics
- 1998

It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague…

### When do measures on the space of connections support the triad operators of loop quantum gravity

- Mathematics
- 2011

In this work we investigate the question under what conditions Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian…

### 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…

### A length operator for canonical quantum gravity

- Mathematics
- 1998

We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which an SU(2) connection is diagonal and it is…

### Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

- Mathematics
- 2006

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:…

### On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories

- Mathematics
- 2003

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…