# A Lorentzian signature model for quantum general relativity

@article{Barrett2000ALS, title={A Lorentzian signature model for quantum general relativity}, author={John W. Barrett and Louis Crane}, journal={Classical and Quantum Gravity}, year={2000}, volume={17}, pages={3101-3118} }

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra.
We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028.
As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are…

## 317 Citations

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## References

SHOWING 1-10 OF 66 REFERENCES

Relativistic spin networks and quantum gravity

- Physics
- 1998

Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2)×SU(2). Relativistic quantum spins are related to the geometry of the two-dimensional faces of a…

An Introduction to spin foam models of quantum gravity and BF theory

- Physics
- 1999

In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of ‘spin foam’ is intended to serve as a similar…

Harmonic Analysis on the quantum Lorentz group

- Mathematics
- 1997

This work begins with a review of complexification and realification of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of different classes of functions…

'Sum over surfaces' form of loop quantum gravity

- Physics
- 1997

We derive a spacetime formulation of quantum general relativity from (hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the 3-geometry in proper time. We…

Quantum models of space-time based on recoupling theory

- Physics
- 1984

Models of geometry that are intrinsically quantum-mechanical in nature arise from the recoupling theory of space-time symmetry groups. Roger Penrose constructed such a model from SU(2) recoupling in…

Harmonic Analysis on the Quantum Lorentz Group

- Mathematics
- 1999

Abstract:Using a continuation of 6j symbols of SUq(2) with complex spins, we give a new description of the unitary representations of SLq(2 ℂ)ℝ and find explicit expressions for the characters of…

On relativistic spin network vertices

- Physics
- 1999

Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called “relativistic spin networks” which satisfy…

Knots and quantum gravity: Progress and prospects

- Physics
- 1994

Recent work on the loop representation of quantum gravity has revealed previously unsuspected connections between knot theory and quantum gravity, or more generally, 3-dimensional topology and…

Quantum deformations of the Lorentz group

- Physics
- 2002

Three properties characteristic for the Lorentz group are selected and all quantum groups with the same properties are found. As a result, a number of one, two and three parameter quantum…