Geometrical Operators in Quantum Gravity

In General > s.a. 3D quantum gravity; phenomenology.
@ Reviews: Ashtekar gq/99, gq/01-in; Barbero a0804-in.
@ General references: Bojowald PRD(01)gq [inverse scale factor operator]; Corichi & Zapata IJMPD(08) [loopy and fuzzy].
@ Discreteness issue: Dittrich & Thiemann a0708, comment Rovelli a0708; Kaminski et al CQG(08)-a0709 [dynamical sector].
@ Length: Loll CQG(97)gq/96; Thiemann JMP(98)gq/96.
@ Angles: Major CQG(99)gq; Seifert gq/01-ug; Major & Seifert CQG(02)gq/01 [atoms of geometry].
@ In 3D theory: Carlip CQG(91) [geometry from holonomies]; Carbone et al CQG(02)gq/01, Pierri gq/02 [volume].
> Related topics: see gravitational thermodynamics.

Area Operators > s.a. 3D quantum gravity; canonical quantum gravity [covariant lqg].
* Area operator: It can be written as

A_S = l_P² _{v in S} (–O_v,S)^1/2 ,   with   O_v,S = _{I,J I,J} X ⁱ_I X ⁱ_J = –[2(J ^{d, i}_v,S)² + 2(J ^{u, i}_v,S)² – (J ^{d+u, i}_v,S)²] .

* Area eigenvalues: For a general state in the kinematical Hilbert space,

a_S = l_P² _v [ 2j ^d_v (j ^d_v+1) + 2j ^u_v (j ^u_v+1) – j ^d+u_v (j ^d+u_v+1)]^1/2,

where all j 's are (consistent) half-integers; Thus, for a gauge-invariant state with no tangential edges to S,

a_S = l_P² _v [j(j+1)]^1/2 ;

A spin network edge contributes 8 l_P² [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 a0711 [effect of theta-parameter ambiguity]; Engle & Pereira a0710 [in new spin foam model].
@ And fermions: Montesinos & Rovelli CQG(98)gq; Ross GRG(01) [torsion and spin].
@ Spectrum: Helesfai & Bene gq/03 [numerical]; Corichi RMF(05)gq/04.
@ 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].

Volume Operators
* Idea: A suitably regularized version of

V(R) = _R |det E|^1/2 ,   det E = (1/3!) _abc ^ijk E^a_i E^b_j E^c_k .

* 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 (3^1/2/8)^1/2.
* Remark: The function V_Sigma is the generating functional of the co-triad; V_Sigma eⁱ_a 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 gq/06 [matrix elements in loop basis].
@ Spectrum: Thiemann JMP(98)gq/96; Brunnemann & Thiemann CQG(06)gq/04; Meissner CQG(06)gq/05; Brunnemann & Rideout gq/06-in, CQG(08)-a0706, CQG(08)-a0706.
@ 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].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 20 jun 2008