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E. R. Livine and S. Speziale, Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
Original
1915
Earlier work this paper cites.
F. Conrady and J. Hnybida, A spin foam model for general Lorentzian 4-geometries
Original
1959
Earlier work this paper cites.
Elsevier, 2014 (1st Ed. Pergamon Press, 1964)
M. A. Naimark, Linear representations of the Lorentz group · 1964
Earlier work this paper cites.
S. Ström, On the matrix elements of a unitary representation of the homogeneous lorentz group
1965
Earlier work this paper cites.
D. vong Duc and N. van Hieu, On the theory of unitary representations of the sl(2c) group
1967
Earlier work this paper cites.
W. A. Benjamin, 1970
W. Ruhl, The Lorentz Group and Harmonic Analysis · 1970
Earlier work this paper cites.
R. L. Anderson, R. Raczka, M. A. Rashid and P. Winternitz, Clebsch-gordan coefficients for the coupling of sl(2,c) principal-series representations
1970
Earlier work this paper cites.
R. L. Anderson, R. Raczka, M. A. Rashid and P. Winternitz, Recursion and symmetry relations for the clebsch-gordan coefficients of the homogeneous lorentz group
1970
Earlier work this paper cites.
A. M. Gavrilik, Clebsch-gordan coefficients for sl(2, c)
1972
Earlier work this paper cites.
I. A. Verdiev, G. A. Kerimov and Y. A. Smorodinsky, “Clebsch-Gordan Coefficients of the Lorentz Group,” Yad. Fiz. 21
1975
Earlier work this paper cites.
G. A. Kerimov and I. A. Verdiev, Clebsch-Gordan Coefficients of the SL(2,c) Group
1978
Earlier work this paper cites.
M. A. Rashid, Boost Matrix Elements Of The Homogeneous Lorentz Group
1979
Earlier work this paper cites.
World Scientific, Singapore, 1988
D. A. Varshalovich, A. N. Moskalev and V. K. Khersonsky, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols · 1988
Earlier work this paper cites.
J. W. Barrett and L. Crane, A Lorentzian signature model for quantum general relativity
2000
Earlier work this paper cites.
J. C. Baez and J. W. Barrett, Integrability for relativistic spin networks
2001
Earlier work this paper cites.
E. R. Livine, Projected spin networks for Lorentz connection: Linking spin foams and loop gravity
2002
Earlier work this paper cites.
J. Engle, R. Pereira and C. Rovelli, The Loop-quantum-gravity vertex-amplitude
Original
2007
Earlier work this paper cites.
E. R. Livine and S. Speziale, A New spinfoam vertex for quantum gravity
Original
2007
Earlier work this paper cites.
E. Alesci and C. Rovelli, The Complete LQG propagator. I. Difficulties with the Barrett-Crane vertex
Original
2007
Earlier work this paper cites.

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J. Engle, E. Livine, R. Pereira and C. Rovelli, LQG vertex with finite Immirzi parameter
Original
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Then

M. Dupuis, L. Freidel, E. R. Livine and S. Speziale, Holomorphic Lorentzian Simplicity Constraints
Original
2012
Later among the works it cites.
M. Han and M. Zhang, Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory
Original
2013
Later among the works it cites.
F. Hellmann and W. Kaminski, Holonomy spin foam models: Asymptotic geometry of the partition function
Original
2013
Later among the works it cites.
V. Bonzom and E. R. Livine, Generating Functions for Coherent Intertwiners
Original
2013
Later among the works it cites.
A. Riello, Self-energy of the Lorentzian Engle-Pereira-Rovelli-Livine and Freidel-Krasnov model of quantum gravity
Original

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2016

Boosting Wigner's nj-symbols

Speziale, Simone

Understand

We study the SL(2,C) Clebsch-Gordan coefficients appearing in the lorentzian EPRL spin foam amplitudes for loop quantum gravity.

We show how the amplitudes decompose into SU(2) nj-symbols at the vertices and integrals over boosts at the edges.
The integrals define edge amplitudes that can be evaluated analytically using and adapting results in the literature, leading to a pure state sum model formulation.
This procedure introduces virtual representations which, in a manner reminiscent to virtual momenta in Feynman amplitudes, are off-shell of the simplicity constraints present in the theory, but with the integrands that peak at the on-shell values.

Built on

E. R. Livine and S. Speziale, Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
Original
1915
Earlier work this paper cites.
F. Conrady and J. Hnybida, A spin foam model for general Lorentzian 4-geometries
Original
1959
Earlier work this paper cites.
Elsevier, 2014 (1st Ed. Pergamon Press, 1964)
M. A. Naimark, Linear representations of the Lorentz group · 1964
Earlier work this paper cites.
S. Ström, On the matrix elements of a unitary representation of the homogeneous lorentz group
1965
Earlier work this paper cites.
D. vong Duc and N. van Hieu, On the theory of unitary representations of the sl(2c) group
1967
Earlier work this paper cites.
W. A. Benjamin, 1970
W. Ruhl, The Lorentz Group and Harmonic Analysis · 1970
Earlier work this paper cites.
R. L. Anderson, R. Raczka, M. A. Rashid and P. Winternitz, Clebsch-gordan coefficients for the coupling of sl(2,c) principal-series representations
1970
Earlier work this paper cites.
R. L. Anderson, R. Raczka, M. A. Rashid and P. Winternitz, Recursion and symmetry relations for the clebsch-gordan coefficients of the homogeneous lorentz group
1970
Earlier work this paper cites.
A. M. Gavrilik, Clebsch-gordan coefficients for sl(2, c)
1972
Earlier work this paper cites.
I. A. Verdiev, G. A. Kerimov and Y. A. Smorodinsky, “Clebsch-Gordan Coefficients of the Lorentz Group,” Yad. Fiz. 21
1975
Earlier work this paper cites.
G. A. Kerimov and I. A. Verdiev, Clebsch-Gordan Coefficients of the SL(2,c) Group
1978
Earlier work this paper cites.
M. A. Rashid, Boost Matrix Elements Of The Homogeneous Lorentz Group
1979
Earlier work this paper cites.
World Scientific, Singapore, 1988
D. A. Varshalovich, A. N. Moskalev and V. K. Khersonsky, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols · 1988
Earlier work this paper cites.
J. W. Barrett and L. Crane, A Lorentzian signature model for quantum general relativity
2000
Earlier work this paper cites.
J. C. Baez and J. W. Barrett, Integrability for relativistic spin networks
2001
Earlier work this paper cites.
E. R. Livine, Projected spin networks for Lorentz connection: Linking spin foams and loop gravity
2002
Earlier work this paper cites.
J. Engle, R. Pereira and C. Rovelli, The Loop-quantum-gravity vertex-amplitude
Original
2007
Earlier work this paper cites.
E. R. Livine and S. Speziale, A New spinfoam vertex for quantum gravity
Original
2007
Earlier work this paper cites.
E. Alesci and C. Rovelli, The Complete LQG propagator. I. Difficulties with the Barrett-Crane vertex
Original
2007
Earlier work this paper cites.