Geodesic
Loop quantum gravity vacuum with nondegenerate geometry
Symmetry, integrability and geometry: methods and applications, Tome 8 (2012) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In loop quantum gravity, states of the gravitational field turn out to be excitations over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also important to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.
Keywords: loop quantum gravity, representations, geometric condensate. 
@article{SIGMA_2012_8_a25,
     author = {Tim Koslowski and Hanno Sahlmann},
     title = {Loop quantum gravity vacuum with nondegenerate geometry},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2012},
     volume = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a25/}
}
TY  - JOUR
AU  - Tim Koslowski
AU  - Hanno Sahlmann
TI  - Loop quantum gravity vacuum with nondegenerate geometry
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2012
VL  - 8
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a25/
LA  - en
ID  - SIGMA_2012_8_a25
ER  - 
%0 Journal Article
%A Tim Koslowski
%A Hanno Sahlmann
%T Loop quantum gravity vacuum with nondegenerate geometry
%J Symmetry, integrability and geometry: methods and applications
%D 2012
%V 8
%U http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a25/
%G en
%F SIGMA_2012_8_a25
Tim Koslowski; Hanno Sahlmann. Loop quantum gravity vacuum with nondegenerate geometry. Symmetry, integrability and geometry: methods and applications, Tome 8 (2012). http://geodesic.mathdoc.fr/item/SIGMA_2012_8_a25/

[1] Ashtekar A., Lewandowski J., “Differential geometry on the space of connections via graphs and projective limits”, J. Geom. Phys., 17 (1995), 191–230 ; arXiv: hep-th/9412073 | DOI | MR | Zbl

[2] Ashtekar A., Lewandowski J., “Projective techniques and functional integration for gauge theories”, J. Math. Phys., 36 (1995), 2170–2191 ; arXiv: gr-qc/9411046 | DOI | MR | Zbl

[3] Ashtekar A., Lewandowski J., “Quantum theory of geometry. I. Area operators”, Classical Quantum Gravity, 14 (1997), A55–A81 ; arXiv: gr-qc/9602046 | DOI | MR | Zbl

[4] Ashtekar A., Lewandowski J., “Relation between polymer and Fock excitations”, Classical Quantum Gravity, 18 (2001), L117–L127 ; arXiv: gr-qc/0107043 | DOI | MR | Zbl

[5] Ashtekar A., Lewandowski J., Marolf D., Mourão J., Thiemann T., “Quantization of diffeomorphism invariant theories of connections with local degrees of freedom”, J. Math. Phys., 36 (1995), 6456–6493 ; arXiv: gr-qc/9504018 | DOI | MR | Zbl

[6] Ashtekar A., Rovelli C., Smolin L., “Weaving a classical metric with quantum threads”, Phys. Rev. Lett., 69 (1992), 237–240 ; arXiv: hep-th/9203079 | DOI | MR | Zbl

[7] Bahr B., Thiemann T., “Gauge-invariant coherent states for loop quantum gravity. I. Abelian gauge groups”, Classical Quantum Gravity, 26 (2009), 045011, 22 pp. ; arXiv: 0709.4619 | DOI | MR | Zbl

[8] Bahr B., Thiemann T., “Gauge-invariant coherent states for loop quantum gravity. II. Non-Abelian gauge groups”, Classical Quantum Gravity, 26 (2009), 045012, 45 pp. ; arXiv: 0709.4636 | DOI | MR | Zbl

[9] Bianchi E., Magliaro E., Perini C., “Coherent spin-networks”, Phys. Rev. D, 82 (2010), 024012, 7 pp. ; arXiv: 0912.4054 | DOI

[10] Dziendzikowski M., Okolów A., “New diffeomorphism invariant states on a holonomy-flux algebra”, Classical Quantum Gravity, 27 (2010), 225005, 24 pp. ; arXiv: 0912.1278 | DOI | MR | Zbl

[11] Fleischhack C., Construction of generalized connections, arXiv: math-ph/0601005

[12] Fleischhack C., “Irreducibility of the Weyl algebra in loop quantum gravity”, Phys. Rev. Lett., 97 (2006), 061302, 4 pp. ; arXiv: math-ph/0407006 | DOI

[13] Fleischhack C., “Representations of the Weyl algebra in quantum geometry”, Comm. Math. Phys., 285 (2009), 67–140 ; arXiv: math-ph/0407006 | DOI | MR | Zbl

[14] Freidel L., Livine E.R., “$\mathrm U(N)$ coherent states for loop quantum gravity”, J. Math. Phys., 52 (2011), 052502, 21 pp. ; arXiv: 1005.2090 | DOI | MR

[15] Koslowski T.A., Cosmological sectors in loop quantum gravity, Ph.D. thesis, Universität Würzburg, 2008

[16] Koslowski T.A., Dynamical quantum geometry (DQG programme), arXiv: 0709.3465

[17] Lewandowski J., “Volume and quantizations”, Classical Quantum Gravity, 14 (1997), 71–76 ; arXiv: gr-qc/9602035 | DOI | MR | Zbl

[18] Lewandowski J., Okolów A., Sahlmann H., Thiemann T., “Uniqueness of diffeomorphism invariant states on holonomy-flux algebras”, Comm. Math. Phys., 267 (2006), 703–733 ; arXiv: gr-qc/0504147 | DOI | MR | Zbl

[19] Okolów A., Lewandowski J., “Automorphism covariant representations of the holonomy-flux $\ast$-algebra”, Classical Quantum Gravity, 22 (2005), 657–679 ; arXiv: gr-qc/0405119 | DOI | MR | Zbl

[20] Okolów A., Lewandowski J., “Diffeomorphism covariant representations of the holonomy-flux $\star$-algebra”, Classical Quantum Gravity, 20 (2003), 3543–3567 ; arXiv: gr-qc/0302059 | DOI | MR | Zbl

[21] Oriti D., Sindoni L., “Toward classical geometrodynamics from the group field theory hydrodynamics”, New J. Phys., 13 (2011), 025006, 44 pp. ; arXiv: 1010.5149 | DOI

[22] Sahlmann H., “On loop quantum gravity kinematics with a non-degenerate spatial background”, Classical Quantum Gravity, 27 (2010), 225007, 17 pp. ; arXiv: 1006.0388 | DOI | MR | Zbl

[23] Sahlmann H., “Some results concerning the representation theory of the algebra underlying loop quantum gravity”, J. Math. Phys., 52 (2011), 012502, 9 pp. ; arXiv: gr-qc/0207111 | DOI | MR

[24] Sahlmann H., “When do measures on the space of connections support the triad operators of loop quantum gravity?”, J. Math. Phys., 52 (2011), 012503, 14 pp. ; arXiv: gr-qc/0207112 | DOI | MR

[25] Sahlmann H., Thiemann T., “Irreducibility of the Ashtekar–Isham–Lewandowski representation”, Classical Quantum Gravity, 23 (2006), 4453–4471 ; arXiv: gr-qc/0303074 | DOI | MR | Zbl

[26] Sahlmann H., Thiemann T., On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories, arXiv: gr-qc/0302090

[27] Thiemann T., “Complexifier coherent states for quantum general relativity”, Classical Quantum Gravity, 23 (2006), 2063–2117 ; arXiv: gr-qc/0206037 | DOI | MR | Zbl

[28] Thiemann T., “Gauge field theory coherent states (GCS). I. General properties”, Classical Quantum Gravity, 18 (2001), 2025–2064 ; arXiv: hep-th/0005233 | DOI | MR | Zbl

[29] Thiemann T., “Quantum spin dynamics (QSD)”, Classical Quantum Gravity, 15 (1998), 839–873 ; arXiv: gr-qc/9606089 | DOI | MR | Zbl

[30] Varadarajan M., “Fock representations from $\mathrm U(1)$ holonomy algebras”, Phys. Rev. D, 61 (2000), 104001, 13 pp. ; arXiv: gr-qc/0001050 | DOI | MR

[31] Varadarajan M., “Towards new background independent representations for loop quantum gravity”, Classical Quantum Gravity, 25 (2008), 105011, 17 pp. ; arXiv: 0709.1680 | DOI | MR | Zbl