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].
@ 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].
@ 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: It can be written as

AS = \(1\over2\)lP2vS (–Ov,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 = \(1\over2\)lP2v [ 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 = lP2v [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 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].
@ 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"].
@ 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].
@ Length: Loll CQG(97)gq/96; Thiemann JMP(98)gq/96; Bianchi NPB(09)-a0806; Ma et al PRD(10)-a1004.
@ 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].
@ 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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