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Loop Quantum Gravity and Planck Stars Loop Quantum Gravity and Planck Stars Carlo Rovelli jl t vn → r = rin r =2m r Saclay, march 2014 Loop Quantum Gravity and Planck Starw jl i. Loop quantum gravity: the theory! vn → • Loop cosmology" " " ii. Planck Stars [F Vidotto, CR, 2014]! t • Observations r = rin r =2m r Carlo Rovelli Loop Quantum Gravity and Planck Stars no uncontrollable infinities existence, ! not uniqueness Is there a consistent quantum theory ! whose classical limit is general relativity," in 4 lorentzian dimensions, with its standard matter couplings? unification ! not addressed! What are its physical consequences?! Can we observe some specific effects? Carlo Rovelli Loop Quantum Gravity and Planck Stars Newton: Particles Space Time Faraday-Maxwell: Particles Fields Space Time Special relativity: Particles Fields Spacetime Quantum mechanics: Quantum-Fields Spacetime General relativity: General-covariant fields Quantum gravity: General-covariant quantum fields Carlo Rovelli Loop Quantum Gravity and Planck Stars Loop Quantum Gravity n0 2 L N = L [SU(2) /SU(2) ] l State space HΓ n Γ Operators: where Kinematics Graph (modes, links) Transition amplitudes W (hl)= dhvf δ(hf ) A(hvf ) C ZSU(2) f v Γ Y Y e v 1 hvf A(hf )= dge (2jf + 1) Trj[hf Yγ†geg− Yγ ] Vertex amplitude e0 f j SL(2,C) Xf Z Yf 2-complex (vertices, edges,C faces) Dynamics Simplicity map Y : γ Hj !Hj,γj j; m j, γ(j + 1); j, m | i 7! | i hf = hvf v Y With a cosmological constant Λ > 0: ! Amplitude: SL(2,C) ➞ SL(2,C)_q network evaluation. Carlo Rovelli Loop Quantum Gravity and Planck Stars Main results q 1. The amplitudes with positive cosmological constant are UV and IR finite: W < (Han, Fairbairn-Meusburger, 2011). ! C 1 2. The classical limit of the vertex amplitude converges (appropriately) to the Regge Hamilton function (with cosmological constant). " " (Barrett et al, Conrady-Freidel, Bianchi-Perini-Magliaro, Engle, Han..., 2009-2012).! 3. The boundary states represent classical 3d geometries. " " (Canonical LQG 1990’, Penrose spin-geometry theorem 1971). ! 4. Boundary geometry operators have discrete spectra. " " (Canonical LQG main results, 1990’). Carlo Rovelli Loop Quantum Gravity and Planck Stars General relativity : Spacetime is a field (like electric field) Quantum gravity : Space is formed by “quanta”: it is formed by elementary “bricks” Discreteness of Geometry. Fundamental length (like strings). Quantum theory : All fields are formed by “quanta” (like photons). Carlo Rovelli Loop Quantum Gravity and Planck Stars Geometry + quantum theory Geometry: - 6 lengths or, 4 normals (up to rotations) L~ 1 L~ a,a=1...4 i, j =1, 2, 3 Closure constraint: ~ a La =0 Volume: V 2 = 9 L~ P(L~ L~ ) 2 1 · 2 ⇥ 3 A = L~ a | a| Quantum geometry: i j 2 ij k [La,Lb]=lP δab ✏ k La Carlo Rovelli Loop Quantum Gravity and Planck Stars Complete set of commuting operators Aa,V Spectrum is discrete A =8⇥~G jl(jl + 1) p N Basis that diagonalises A ,V : ja,v ,ja a | i 2 2 A a ,V are only five quantities: geometry is never sharp Minimal volume Carlo Rovelli Loop Quantum Gravity and Planck Stars jl → vn → Spin network states j ,v | l ni Basic states of Loop Quantum Gravity Carlo Rovelli Loop Quantum Gravity and Planck Stars Boundary geometry: spinfoams 2 L N State space Γ = L [SU(2) /SU(2) ] H Al Operator: triad (metric) Polyhedron l l Area and volume form a complete set of commuting observables and have discrete spectra Nodes: discrete quanta of volume (“quanta of space”) Links: discrete quanta of area. Geometry is quantized: (i) eigenvalues are discrete (ii) operators do not commute → ! coherent states theory → semiclassical spacetime States describe quantum geometries: not quantum states in spacetime but rather quantum states of spacetime Carlo Rovelli Loop Quantum Gravity and Planck Stars Quantum field theory → The quanta of a field are particles (Dirac) (Discreteness of the spectrum of the energy of each mode) p ...p F 3 | 1 ni ( , ,W) a(k),a†(k) F A A 3 W F eynman rules ! Carlo Rovelli Loop Quantum Gravity and Planck Stars Quantum gravity → Quantum granularity of spacetime (1995) (Discreteness of the spectrum of geometrical operators such as volume) Γ,j ,v F 3 | l ni ( , ,W) L~ F A A 3 l W T ransition amplitudes ! Carlo Rovelli Loop Quantum Gravity and Planck Stars Large distance limit A “spinfoam”: a two-complex colored with spins v on faces and intertwiners on edges. e jf iSRegge iSRegge Theorem : For a 5-valent vertex A(jf ,ve) e + e− j 1 [Barrett, Pereira, Hellmann, ⇠ Gomes, Dowdall, Fairbairn 2010] [Freidel Conrady 2008, W eiS∆ iS[g] Bianchi, Satz 2006, Z Dg e C −−j !1 C −−C − ! Magliaro Perini, 2011] !1 Z ⇤ ⇤ Theorem : For a 5-valent vertex q iSRegge iSRegge ⇤~G A (jf ,ve) e + e− q = e [Han 2012] j 1,q 1 ⇠ ⇠ Carlo Rovelli Loop Quantum Gravity and Planck Stars 14 14 FromFrom General General Relativity Relativity to Quantum to Quantum Gravity Gravity 3. The3. The third third point point refers refers to the to the classical classical and and continuum continuum limit limit of the of the theory. theory. TheThe two two limits limits should should not not be confused.be confused. The The first first is the is the standard standard~ ~ ! ! 0 limit.0 limit. The The second second derives derives from from the the expansion expansion that that define define the the theory. theory. ThisThis expansion expansion is a is truncation a truncation of the of the degrees degrees of freedom of freedom very very similar similar to thatto that of lattice of lattice gauge gauge theory. theory. (But (But the the pattern pattern of invariance of invariance and and the the scalescale structure structure is di is↵erent di↵erent in Yang in Yang Mills Mills theory theory and and general general relativity, relativity, withwith heavy heavy consequences, consequences, as we as wesee see below.) below.) The The classical classical limit limit of lattice of lattice QCDQCD on a on fixed a fixed triangulation triangulation is of is course of course a classical a classical lattice lattice theory. theory. The The classicalclassical limit limit of covariant of covariant loop loop quantum quantum gravity gravity is Regge is Regge calculus calculus on aon a finitefinite triangulation. triangulation. As well As well known, known, the the continuum continuum limit limit of Regge of Regge calculus calculus is fullis full general general relativity. relativity. The The structure structure of the of the theory theory is therefore is therefore as in as the in the TableTable 1.1; 1.1; where wherej, j,andand∆ are∆ are defined defined later later on. on. C C Loop-gravityLoop-gravity j j !1!1 ReggeRegge theory theory transitiontransition amplitudes amplitudes −−!−−! curvature curvature !1 !1 !1 !1 L L C C ∆ ∆ − −− − −− − −− − −− ⌧ ⌧ Continuum limit Continuum limit L L ExactExact j j !1!1 GeneralGeneral relativity relativity transitiontransition amplitudes amplitudes −−!−−! − − − − − − − − − − − − − − − − − − − − − − − − −− − − − − − − −− ClassicalClassical limit limit −−−−−−−−−−−−−−−−−−−−!−−−−−−−−−−−−−−−−−−−−!L L Planck TableTable 1.1 1.1RelationRelation between between continuum continuum limit limit and and classical classical limit limit of the of the transitiontransition amplitudes. amplitudes. - Physical QFT’s are constructed via a truncation of the d.o.f. (cfr: QED: particles, QCD Lattice).! LetLet us- Physical now us now calculation give give the are the performed full full definition within definition a truncation. of theory. of theory. The The kinematics, kinematics, which which is is essentiallyessentially the the same same as in as the in the canonical canonical theory, theory, after after solving solving the the di↵eomor- di↵eomor- phismphism constraint. constraint. The The Hilbert Hilbert state stateof theof the theory theory admits admits a family a family of sub- of sub- H H spacesspaces labelledlabelled by anby (oriented)an (oriented) abstract abstract graph graphΓ formedΓ formed by N by nodesN nodes HΓ HΓ n andn andL linksL linksCarlol. Thel .Rovelli The Hilbert Hilbert space space is the Loopis the same Quantum same as Gravity the as the Hilbert and Hilbert Planck space Stars space of of HΓ HΓ a SUa (2)SU(2) lattice lattice Yang Yang Mills Mills theory. theory. It is It formed is formed by square-integrable by square-integrable (in (in the the HallHall measure) measure) functions functions (U l)(U ofl)LSU of LSU(2)(2) group group elements elementsUl,U invariantl, invariant un- un- derder the the gauge gauge transformations transformations (U )(U ) (V U(VVU),V where), whereV V SU(2)SU(2) and and l !l ! s l s t l t n 2n 2 s ands andt aret are the the source source and and target target node node of the of the link linkl. Thel. The “electric “electric field” field” operatorsoperators on thison this state state space space are are the the quantum quantum operators operators corresponding corresponding to to thethe triad triad field, field, and and the the area area and and volume volume operators operators are are constructed constructed in terms in terms of theseof these as in as the in the previous previous section. section. Results: I. Theory 1. The amplitudes (with positive cosmological constant) are UV and IR finite: (Han, Fairbairn-Moesburger, 2011). W q < ! C 1 3. The classical limit of the vertex amplitude converges to the Regge Hamilton function. " (Barrett et al, Conrady-Freidel, Bianchi-Perini-Magliaro, Engle, Han..., 2009-2013)." " " " " " " ⇤ ⇤ " iSRegge iSRegge ⇤~G Av e + e− ,q= e j 1,q 1 ⇠ ⇠ Carlo Rovelli Loop Quantum Gravity and Planck Stars Results: II. Applications 1. Scattering - n-point functions (CR, Alesci, Bianchi, Perini, Magliaro, Ding, Zhang, Han) 2. Black hole thermodynamics - microscopic derivation of Bekenstein-Hawking entropy. (Krasnov, CR, Ashtekar, et al, Bianchi 2012).
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