Quantum Geometry in Canonical Quantum Gravity |
In General
> s.a. 3D quantum gravity; phenomenology.
* Twisted geometries:
A discrete version of spacetime geometry that generalizes Regge triangulations
by allowing torsion of the Ashtekar-Barbero connection on the 3D space slices.
@ Reviews: Ashtekar gq/99,
gq/01-conf;
Barbero AIP(08)-a0804;
Bojowald AIP(09)-a0910;
Sahlmann JPCS(12)-a1112 [new ideas];
García-Islas RMF-a1902.
@ General references: Bojowald PRD(01)gq [inverse scale factor operator];
Corichi & Zapata IJMPD(08) [loopy and fuzzy];
Bahns et al CMP(11)-a1005;
Tibrewala CQG(14)-a1311 [loop quantum gravity corrections, constraint algebra and general covariance];
Schliemann PRD(14)-a1408 [quantum polyhedra].
@ Twisted geometries:
Freidel & Speziale PRD(10)-a1001,
PRD(10)-a1006;
Rovelli & Speziale PRD(10)-a1005 [and Regge geometries];
Charles & Livine CQG(15)-a1501 [generalization to a q-deformed gauge group, and
CQG+];
Freidel & Livine GRG(19)-a1810 [bubble networks].
@ Discreteness issue: Dittrich & Thiemann JMP(09)-a0708,
comment Rovelli a0708;
Kamiński et al CQG(08)-a0709 [dynamical sector].
@ In 3D theory: Carlip CQG(91) [geometry from holonomies];
Carbone et al CQG(02)gq/01,
Pierri gq/02 [volume];
Budd & Loll CQG(09)-a0906 [no evidence of discreteness].
> Related topics:
see gravitational thermodynamics.
Area Operators > s.a. 3D quantum
gravity; canonical quantum gravity [covariant lqg].
* Area operator: For a surface
\(S\) it can be written as
\(\hat A_S = {1\over2}\, l_{\rm P}^2 \sum_{v\in S} (-O_{v,S})^{1/2}\;,\) with Ov,S = ∑I,J κI,J X iI X iJ = −[2 (J d, iv,S)2 + 2 (J u, iv,S)\(^2\) − (J d+u, iv,S)\(^2\)] .
* Area eigenvalues: For a general state in the kinematical Hilbert space,
aS = \(1\over2\)lP2 ∑v [2j dv (j dv+1) + 2j uv (j uv+1) − j d+uv (j d+uv+1)]1/2,
where all js are (consistent) half-integers; Thus, for a gauge-invariant state with no tangential edges to S,
aS = lP2 ∑v [j(j+1)]1/2 ;
A spin network edge contributes 8πγ
lP2
[j(j+1)]1/2 to the area of
a surface it intersects transversally.
* Consequences: One can calculate
the area of a black hole horizon and relate it to thermodynamical properties of
black holes, as well as the Immirzi-parameter and SU(2)-vs-SO(3) ambiguities.
@ General references: Rovelli in(93);
Rovelli & Smolin NPB(95)gq;
De Pietri & Rovelli PRD(96)gq;
Ashtekar & Lewandowski CQG(97)gq/96;
Frittelli et al CQG(96)gq;
Loll CQG(97)gq/96;
Krasnov CQG(98)gq/97,
CQG(98)gq;
Amelino-Camelia MPLA(98)gq [observability];
Jiménez & Pérez PRD(08)-a0711 [effect of theta-parameter ambiguity];
Engle & Pereira CQG(08)-a0710 [in new spin-foam model];
Barbero et al PRD(09)-a0905 [new definition for spacetimes with inner boundary];
Lim AHP(17)-a1705.
@ And fermions: Montesinos & Rovelli CQG(98)gq;
Ross GRG(01) [torsion and spin].
@ Spectrum:
Helesfai & Bene gq/03 [numerical];
Corichi RMF(05)gq/04;
Asato CQG(16)-a1506 [restriction from condition on cutting spin networks];
Barbero et al CQG(18)-a1712 [eigenvalue distribution].
@ Related topics: Khatsymovsky PLB(94)gq/93 [areas of timelike triangles, from Regge calculus];
Bojowald & Kastrup CQG(00)ht/99 [spherical symmetry];
Khriplovich PLB(02)gq/01 [and black-hole entropy];
Livine & Terno gq/06 [renormalization and entanglement];
Amelino-Camelia et al PLB(09)-a0812 [in Moyal non-commutative plane];
Medved a1005
[quantum black holes and a universal area gap];
Adelman et al CQG(15)-a1401 [quantum volume and length fluctuations].
Volume Operators
> s.a. chaos in classical gravity.
* Idea: A suitably regularized version of
V(R) = ∫R |det E|1/2 , det E = \(1\over3!\)εabc εijk Eai Ebj Eck .
* Ambiguity: There are two
regularizations (internal, A&L; and external/loop, R&S), that can
be resolved probably looking at the relationship with lengths and areas.
* Eigenvalues: Non-trivial
only from at least 4-valent vertices; Type-(1,1,1,1) vertices contribute
l03
(31/2/8)1/2.
* Remark: The function
VΣ is the generating functional
of the co-triad; VΣ \(\mapsto\)
eia
by functional differentiation.
@ General references: Rovelli & Smolin NPB(95)gq;
De Pietri & Rovelli PRD(96)gq;
Ashtekar & Lewandowski JGP(95)ht/94;
Loll CQG(97)gq/96;
Lewandowski CQG(97)gq/96 [Rovelli-Smolin vs others];
Ashtekar & Lewandowski ATMP(97)gq;
Giesel & Thiemann CQG(06)gq/05,
CQG(06)gq/05 [consistency check];
Hari Dass & Mathur CQG(07)gq/06 [matrix elements in loop basis];
Flori & Thiemann a0812 [semiclassical analysis];
Ding & Rovelli CQG(10)-a0911 [in covariant quantum gravity];
Yang & Ma PRD(16)-a1602 [new volume and inverse volume operators];
Astuti et al a1603 ["volume entropy"];
Perlov a1806 [new approach];
Ariwahjoedi et al GRG(19)-a1810 [Hermiticity];
Kamiński a1906
[ill-defined evolution of volume expectation values];
Ling et al PLB(19)-a1907
[quantum entanglement of boundary states and quantum geometry in the bulk].
@ Spectrum: Thiemann JMP(98)gq/96;
Brunnemann & Thiemann CQG(06)gq/04;
Meissner CQG(06)gq/05;
Brunnemann & Rideout CQG(06)gq-MGXI,
CQG(08)-a0706,
CQG(08)-a0706;
Brunnemann & Rideout CQG(10)-a1003 [and matroids];
Bianchi & Haggard PRD(12)-a1208 [Bohr-Sommerfeld quantization];
Aquilanti et al JPA(13)-a1301 [hidden symmetries and spectrum];
Yang & Ma a1505 [graphical method].
@ Lattice approach: Loll PRL(95)gq,
NPB(96)gq/95,
NPB(97)gq.
@ Special cases: Bojowald & Swiderski CQG(04)gq [spherical symmetry];
Neville PRD(06)gq/05,
PRD(06)gq/05 [planar or cylindrical symmetry].
Other Operators and Quantities
@ General references: Ariwahjoedi et al CQG(15)-a1404 [nodes, links, spins and observables];
Alesci et al PRD(15)-a1507 [coherent state operators];
Freidel & Pérez a1507 [2D surface boundaries of Cauchy slices];
Goeller & Livine CQG(18)-a1805 [quadrupole moment operator];
Becker & Pagani PRD(19)-a1810 [in the Asymptotic Safety scenario];
Long & Ma PRD(20)-a2003 [in all dimensions].
@ Length: Loll CQG(97)gq/96;
Thiemann JMP(98)gq/96;
Bianchi NPB(09)-a0806;
Ma et al PRD(10)-a1004;
Lecian a1708 [semiclassical].
@ Angles: Major CQG(99)gq;
Seifert gq/01-ug;
Major & Seifert CQG(02)gq/01,
Major CQG(10)-a1005 [atoms of geometry].
@ Curvature: Alesci et al PRD(14)-a1403 [3D curvature operator];
Ariwahjoedi et al IJGMP(15)-a1503 [2+1 lqg, curvatures and discrete Gauss-Codazzi equation];
Lim RPMP(18)-a1803 [quantized curvature];
Nemoul & Mebarki IJGMP(19)-a1803 [3D Ricci scalar curvature and edge length operators];
Brunekreef & Loll a2011 [curvature profile].
@ Spectral dimension: Modesto CQG(09)-a0812,
a0905,
CQG(09) [fractal, scale-dependent spectral dimension];
Calcagni et al CQG(14)-a1311 [and kinematical states of lqg];
> s.a. entanglement entropy.
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