We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the Engle, Pereira, Rovelli and Livine (EPRL) model. The q-deformed models are based on the representation theory of two copies of $U_q(\mathfrak {su}(2))$ at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models, we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties, and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.
REFERENCES
1.
N.
Reshetikhin
and V. G.
Turaev
, Invent. Math.
103
, 547
(1991
).2.
V. G.
Turaev
and O. Y.
Viro
, Topology
31
, 865
(1992
).3.
H. H.
Andersen
, Commun. Math. Phys.
149
, 149
(1992
).4.
5.
G.
Ponzano
and T.
Regge
, “Semiclassical limit of Racah coefficients
,” Spectroscopic and Group Theoretical Methods in Physics
(North-Holland
, Amsterdam
, 1968
).6.
7.
L.
Crane
, L.
Kauffman
, and D.
Yetter
, J. Knot Theory Ramif.
6
, 177
(1997
).8.
9.
J. D.
Roberts
, Topology
34
, 771
(1995
).10.
L.
Freidel
, K.
Krasnov
, and R.
Puzio
, Adv. Theor. Math. Phys.
3
, 1289
(1999
).11.
J. W.
Barrett
and L.
Crane
, Class. Quantum Grav.
17
, 3101
(2000
).12.
J. W.
Barrett
and L.
Crane
, J. Math. Phys.
39
, 3296
(1998
).13.
D. N.
Yetter
, e-print arXiv:math/9801131v1.14.
K.
Noui
and P.
Roche
, Class. Quantum Grav.
20
, 3175
(2003
).15.
J.
Engle
, E.
Livine
, R.
Pereira
, and C.
Rovelli
, Nucl. Phys. B
799
, 136
(2008
).16.
L.
Freidel
and K.
Krasnov
, Class. Quantum Grav.
25
, 125018
(2008
).17.
J.
Engle
, R.
Pereira
, and C.
Rovelli
, Phys. Rev. Lett.
99
, 161301
(2007
).18.
J.
Engle
, R.
Pereira
, and C.
Rovelli
, Nucl. Phys. B
798
, 251
(2008
).19.
R.
Pereira
, Class. Quantum Grav.
25
, 085013
(2008
).20.
J. W.
Barrett
, R. J.
Dowdall
, W. J.
Fairbairn
, H.
Gomes
, and F.
Hellmann
, J. Math. Phys.
50
, 112504
(2009
).21.
J. W.
Barrett
, W. J.
Fairbairn
, and F.
Hellmann
, Int. J. Mod. Phys. A
25
, 2897
(2010
).22.
J. W.
Barrett
, R. J.
Dowdall
, W. J.
Fairbairn
, F.
Hellmann
, and R.
Pereira
, Class. Quantum Grav.
27
, 165009
(2010
).23.
J. W.
Barrett
, R. J.
Dowdall
, W. J.
Fairbairn
, H.
Gomes
, F.
Hellmann
, and R.
Pereira
, to appear in the Proceedings for the 2nd Corfu summer school and workshop on quantum gravity and quantum geometry
.24.
F.
Conrady
and L.
Freidel
, Phys. Rev. D
78
, 104023
(2008
).25.
F.
Conrady
and L.
Freidel
, Class. Quantum Grav.
25
, 245010
(2008
).26.
C.
Perini
, C.
Rovelli
, and S.
Speziale
, Phys. Lett. B
682
, 78
(2009
).27.
C.
Rovelli
, Class. Quant. Grav.
28
, 114005
(2011
).28.
E.
Buffenoir
and P.
Roche
, ‘Commun. Math. Phys.
207
, 499
(1999
).29.
E.
Buffenoir
and P.
Roche
, J. Math. Phys.
41
, 7715
(2000
).30.
P.
Podles
and S. L.
Woronowicz
, Commun. Math. Phys.
130
, 381
(1990
).31.
32.
V.
Chari
and A.
Pressley
, A Guide to Quantum Groups
(Cambridge University Press
, Cambridge, England
, 1994
).33.
A.
Klimyk
and K.
Schmudgen
, Quantum Groups and Their Representations
(Springer
, Berlin
, 1997
).34.
W.
Pusz
, Commun. Math. Phys.
152
, 591
(1993
).35.
S.
Majid
, Foundations of Quantum Group Theory
(Cambridge University Press
, Cambridge, England
, 1995
).36.
J. C.
Baez
and J. W.
Barrett
, Class. Quantum Grav.
18
, 4683
(2001
).37.
J.
Engle
and R.
Pereira
, Phys. Rev. D
79
, 084034
(2009
).38.
E.
Bianchi
, D.
Regoli
, and C.
Rovelli
, Class. Quantum Grav.
27
, 185009
(2010
).39.
D.
Arnaudon
, e-print arXiv:hep-th/9203011.40.
L.
Alvarez
-Gaume, C.
Gomez
, and G.
Sierra
, Nucl. Phys. B
330
, 347
(1990
).41.
B.
Bakalov
and A.
Kirillov
, Lectures on Tensor Categories and Modular Functors
, AMS University Lecture Series
Vol. 21
(American Mathematical Society
, Providence, Providence
, 2001
).42.
R.
Oeckl
, J. Geom. Phys.
46
, 308
(2003
).43.
S.
Mizoguchi
and T.
Tada
, Phys. Rev. Lett.
68
, 1795
(1992
).44.
E.
Witten
, Nucl. Phys. B
311
, 46
(1988
).45.
E.
Witten
, Commun. Math. Phys.
121
, 351
(1989
).46.
R.
Borissov
, S.
Major
, and L.
Smolin
, Class. Quantum Grav.
13
, 3183
(1996
).47.
L.
Smolin
, e-print arXiv:hep-th/0209079.48.
E. R.
Livine
and S.
Speziale
, Phys. Rev. D
76
, 084028
(2007
).49.
M.
Han
, J. Math. Phys.
52
, 072501
(2011
).50.
The antipode of a Hopf algebra Hop with the opposite coproduct is the inverse of the antipode of the Hopf algebra H.
51.
Note however that the summation label arising from the expression of the R-matrix is fixed to K2.
52.
An indecomposable representation has dimension 2r and is characterised by a half-integer I such that 1 ⩽ 2I + 1 ⩽ r, see, for instance, Refs. 3, 4, 39, and 40, and 32.
53.
Note that the construction differs from that of the quantum Barrett-Crane model (Ref. 13) where q′ = q−1.
54.
Note that the notion of 15j symbol used here is closely related to but not equivalent to the standard definition. The standard
$U_q^{(res)}(\mathfrak {su}(2))$
15j symbol is obtained by composing five Clebsch-Gordan morphisms for $U_q^{(res)}(\mathfrak {su}(2))$
and closing the resulting expression with a quantum trace. As it is our aim to explicitly exhibit the similarities with the Lorentzian model, we work with a different expression, which is the analogue of the quantities called 15j symbols there and in the classical EPRL models.© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.
