VIEWS: 15 PAGES: 28 POSTED ON: 7/28/2011 Public Domain
Loop Quantum Gravity carlo rovelli 6 sf 3 8 !f q 7 5 3 6 1 s1 8 p 5 si 3 !i 7 Carlo Rovelli Loop Quantum Gravity Granada 2010 The theory The theory is deﬁned by the triple (H, W, A) H is Hilbert space W:H→C is a map that deﬁnes the dynamics A is an algebra of operators Carlo Rovelli Loop Quantum Gravity Granada 2010 The theory 1 H Hilbert space: ˜ H= HΓ Γ : Abstract graph : t(l) l Γ s(l) Graph Hilbert space: HΓ = L2 [SU (2)L /SU (2)N ] −1 Gauge transformations ψ(Ul ) → ψ(Vs(l) Ul Vt(l) ), Vn ∈ SU (2)N ˜ H = H/ ∼ where ∼ deﬁned identifying states on subgraph. The space HΓ admits a basis |Γ, jl , in labelled by a spin for each link and an intertwiner for each node. These states are called “spin network states”. Carlo Rovelli Loop Quantum Gravity Granada 2010 The theory II A Operator algebra: Ll = {Li }, i = 1, 2, 3 left-invariant vector ﬁeld for each link l : l “gravitational ﬁeld operator (tetrad)” Ul : “Holonomy of the Ashtekar-Barbero connection along the link”. Composite operators: Area: AΣ = Li Li . l l l∈Σ 2 Volume: VR = Vn , 2 Vn = | Li Lj Lk |. ijk l l l” 9 n∈R Angle: Li Li . l l The spin network basis |Γ, jl , in diagonalizes the area and volume operators. Carlo Rovelli Loop Quantum Gravity Granada 2010 The theory III W Dynamics: |ψ = |Γ, jl , in W(ψ) = djf Wv Wv ∂σ=ψ f v σ : spinfoam σ: two-complex ∆ with faces f and edges e colored with spins jf and intertwiners e ,i bounded by Γ, jl , in . σ = (∆, jf , ie ) : “spinfoam”. dj = 2j + 1 I) Wv = (PSL(2,C) ◦ Yγ ψv )(1 Yγ : Hj −→ Hj ⊂ H(p=γ(j+1), k=j) . Wv SU (2) rep SL(2, C) rep spinfoam vertex SU (2) ⊂ SL(2, C) Carlo Rovelli Loop Quantum Gravity Granada 2010 Main conjecture: (H, W, A) deﬁnes a (background independent) quantum ﬁeld theory whose classical limit is general relativity Carlo Rovelli Loop Quantum Gravity Granada 2010 In which sense this is a QFT ? In which sense it is background independent ? How do we compute transition amplitudes ? How do we extract physics ? What evidence do we have that its classical limit is General Relativity ? Carlo Rovelli Loop Quantum Gravity Granada 2010 In which sense this is a background independent QFT ? 1. Notice the following structure in H : There is a natural tensor map: T :H⊗H→H deﬁned by T (HΓ ⊗ HΓ ) = HΓ∪Γ ˆ Therefore W deﬁnes also maps W2 : H → H (cfr: W(ψin ⊗ ψ ﬁn ) = ψﬁn |e −iHt |ψin ) 6 sf ➜ A two-complex is cobordism of graphs. 3 8 !f A spinfoam is cobordism of spin networks. q 7 5 3 1 s1 ➜ A Hilbert space is associated to any connected graph 6 8 and an amplitude (a state) is associated to any two-complex p with the graphs as boundaries. 3 5 si !i 7 ➜ Compare: Atiyah Topological Quantum Field Theory Carlo Rovelli Loop Quantum Gravity Granada 2010 In which sense this is a background independent QFT ? Key differences from Atiyah TQFT: Boundary manifolds → graphs Cobordism manifolds → Two complexes 6 sf 3 8 !f q → 7 5 3 6 1 s1 8 p 5 si 3 !i 7 Finite → Inﬁnite dimensional boundary Hilbert spaces Carlo Rovelli Loop Quantum Gravity Granada 2010 Physical interpretation of HΓ : the spin network states A canonical quantization of general relativity [See Ashtekar’s talk] leads to a space of states which is (up to technical details) precisely H . Metric space, with a triad ﬁeld a Ashtekar- Barbero connection. Ul Ll This leads to HΓ with the operators Ul and Ll . Diffeomorphism invariance: Imbedded graph → Abstract graph Carlo Rovelli Loop Quantum Gravity Granada 2010 Physical interpretation of HΓ : the spin network states Basis |Γ, jl , vn in HΓ that diagonalizes area and volume: spin network basis Nodes: discrete quanta of volume (chunks of space, atoms of space) with quantum number vn . Links: discrete quanta of area, with quantum number jl . jl → vn A spin network state on a graph is a quantum state of geometry: These are not states in space. These are states of space. Carlo Rovelli Loop Quantum Gravity Granada 2010 Physical interpretation of HΓ : the spin network states jl vn → → Spin network diagonalize metric and have quantum spread extrinsic geometry Coherent states : peaked in a given (discrete) intrinsic and extrinsic geometry [Thiemann, Speziale CR, Livine, Bianchi Magliaro Perini] Triangulation interpretation: Regge or “twisted” [Dittrich, Bonzom, Speziale Freidel, Livine] Holomorphic representation: Basis of coherent states [Ashtekar Lewandowski Marolf Mourao, Bianchi Magliaro Perini] Carlo Rovelli Loop Quantum Gravity Granada 2010 Dynamics: W Amplitude associated to a state ψ of a boundary of a 4d region Probability amplitude P (ψ) = | W |ψ |2 4d region Wv 3d boundary boundary graph a spin network history σ : spinfoam Superposition principle W |ψ = W (σ) σ Locality: vertex amplitude W (σ) ∼ Wv . v Lorentz invariance I) Wv = (PSL(2,C) ◦ Yγ ψv )(1 Wv spinfoam vertex Carlo Rovelli Loop Quantum Gravity Granada 2010 Dynamics: W ψ SU (2) SL(2,C) Natural immersion HΓ ⊂ HΓ : SU (2) ⊂ SL(2, C) Yγ : Hj −→ Hj ⊂ H(p=γ(j+1), k=j) . [Engle Pereira CR, Livine, Speziale, Freidel Krasnov, Lewandowski Kaminski Kisielowski, 07-10] Wv i) If we replace Yγ with the identity, we obtain a TQFT which is well known: it is the Ooguri quantization of the theory S[B, A] = B∧F 1 ii) General relativity can be written as S[e, A] = ((e ∧ e) + e ∧ e) ∧ F ∗ γ 1 and B = (e ∧ e) + e ∧ e iff (in a ﬁxed gauge): B ij − γ ij k B 0k = 0 ∗ γ iii) Theorem [Ding CR 09]: on the image of Yγ : ψ|B ij − γ ij k B 0k |φ = 0 Carlo Rovelli Loop Quantum Gravity Granada 2010 Dynamics: W Asymptotic analysis iSRegge iSEinstein−Hilbert Wv ∼ e ∼ e [Barrett Dowdall Fairbairn Gomes Hellmann, Pereira] In the spin network basis W yields the cos of the action. In the holomorphic representation, only one of the two terms of the cos survives [Bianchi Magliaro Perini] Carlo Rovelli Loop Quantum Gravity Granada 2010 Dynamics: W I) Wv = (PSL(2,C) ◦ Yγ ψv )(1 This natural vertex amplitude appear to yield the Einstein equations in the large distance classical limit: A natural group structure based on SU (2) ⊂ SL(2, C) appears to code the Einstein equations. cfr : = e γµ δ(p1 + p2 − k) AB Carlo Rovelli Loop Quantum Gravity Granada 2010 Dynamics: W 6 sf 3 W |ψ = Wv (ψv (σ)). 8 !f q (2jf +1) 5 7 σ f v 3 6 1 s1 8 p I) Wv = (PSL(2,C) ◦ Yγ ψv )(1 5 si 3 !i 7 σ : spinfoam “Sum over histories” form of LQG: Dual interpretation: 1. Discrete version of: W (q) = Dg eiSEH [g] ∂g=q 11. Sum over Feynman graphs: Carlo Rovelli Loop Quantum Gravity Granada 2010 Dynamics: W 6 sf 3 W |ψ = Wv (ψv (σ)). 8 !f q (2jf +1) 5 7 σ f v 3 6 1 s1 8 p I) Wv = (PSL(2,C) ◦ Yγ ψv )(1 5 si 3 !i 7 σ : spinfoam “Sum over histories” form of LQG: Dual interpretation: 1. Discrete version of: W (q) = Dg eiSEH [g] ∂g=q 11. Sum over Feynman graphs: Carlo Rovelli Loop Quantum Gravity Granada 2010 Dynamics: W 6 sf 3 W |ψ = Wv (ψv (σ)). 8 !f q (2jf +1) 5 7 σ f v 3 6 1 s1 8 p I) Wv = (PSL(2,C) ◦ Yγ ψv )(1 5 si 3 !i 7 σ : spinfoam “Sum over histories” form of LQG: Dual interpretation: 1. Discrete version of: W (q) = Dg eiSEH [g] ∂g=q 11. Sum over Feynman graphs: Carlo Rovelli Loop Quantum Gravity Granada 2010 Quantization methods: Canonical quantization à la Dirac of the ADM constraints of General Relativity in Ashtekar-Barbero variables. Holonomies as main variables Diff invariance → abstract graphs Discretization and lattice quantization (à la QCD). GR = BF+constraints → constrained are implemented weakly on the image of Yγ Lattice spacing independence Quantum geometry methods All these converge to the structure (H, W, A) Carlo Rovelli Loop Quantum Gravity Granada 2010 Physical assumptions: General Relativity (with standard matter couplings, in Ashtekar formulation) Standard quantum mechanics (modiﬁed to be general covariant) Diffeomorphism invariance fully implemented Carlo Rovelli Loop Quantum Gravity Granada 2010 Extracting physics from the theory There is no physics without approximations. Graph expansion: Restricting the theory to HΓ is a truncation of wavelengths short with respect to the total size of the region considered (cfr: cosmology) (cfr: lattice QCD). Vertex expansion: Similar to the vertex expansion in QED: number of “elementary processes considered in the transition amplitude” Large distance expansion: Large with respect to the Planck length (classical limit). jl 1 Carlo Rovelli Loop Quantum Gravity Granada 2010 Results: 1. graviton propagator q q q f q ¢¢f q f q ¢¢f Graph Γ5 = l ¢ f 5 gg l ¢ £ . Vertex l ¢ f 5 gg l ¢ £ 4d region 5 5 £ 5 £ g ¢ f £ 5 q &q g ¢ f £ ¢ g& f£ q ¢ g& &q f£ Boundary state: ψL coherent state determined by the (intrinsic and extrinsic) geometry of the boundary of a ﬂat 4-simplex. → Background info input dynamically via the boundary state. Amplitude: Wmn = W |Lna · Lnb Lmc · Lmd |ψL abcd c corresponds to the perturbative QFT’s graviton propagator W abcd (xm , xn ) = 0|g ab (xn )g cd (xm )|0 c Matches to ﬁrst order ! [Bianchi Magliaro Perini] Carlo Rovelli Loop Quantum Gravity Granada 2010 Results: 1. graviton propagator q q q q ¢¢f d m q f q ¢¢f f Graph Γ5 = l ¢ f 5 gg l ¢ £ . Vertex l ¢ f 5 gg l ¢ £ 4d region 5 £ 5 £ 5 5 q q b g ¢& f £ c g ¢ f £ n ¢ g& f£ q g& ¢ &q f£ a Boundary state: ψL coherent state determined by the (intrinsic and extrinsic) geometry of the boundary of a ﬂat 4-simplex. → Background info input dynamically via the boundary state. Amplitude: Wmn = W |Lna · Lnb Lmc · Lmd |ψL abcd c corresponds to the perturbative QFT’s graviton propagator W abcd (xm , xn ) = 0|g ab (xn )g cd (xm )|0 c Matches to ﬁrst order ! [Bianchi Magliaro Perini] Carlo Rovelli Loop Quantum Gravity Granada 2010 Results: II. cosmology Triangulate a 3-sphere with two tetrahedra : these capture the ﬁrst d.o.f.’s in a mode expansion of a cosmological metric r r p Dual graph: ∆∗ = 2 r r z Boundary state on ∆∗ ∪ ∆∗ coherent state ψz peaked on 2 2 u homogeneous isotropic geometries on the 3-sphere. Amplitude W (z, z ) = W |ψz ⊗ ψz r r z z 2 +z 2 W (z, z ) ∼ zz e − 2t [See Vidotto’s talk] Reproduces the Friedmann dynamics: - Is peaked on the classical solutions - Satisﬁes a quantum constraint which reduces to the (gravitational part of the) Friedmann hamiltonian for →0 Carlo Rovelli Loop Quantum Gravity Granada 2010 Open issues No UV divergences! Infrared divergences? [Speziale Perini CR, Bonzom, Smerlak, Rivasseau Gurau Oriti] Scaling by radiative corrections? Matter in spinfoam? Cosmological constant? Relation covariant canonical formalism’s dynamics? [Ashtekar Campigia Henderson, Wilson-Edwin Nelson,Vidotto CR] Observations are possible. Several suggestions, but: No empirical support yet No solid veriﬁable prediction yet Carlo Rovelli Loop Quantum Gravity Granada 2010 Summary The theory is deﬁned by the triple (H, W, A) . It is a generalization of a topological QFT in the sense of Atiyah Kinematics: quanta of space with quantized volume and area [see Ashtekar and Livine’s talks] Dynamics: transition amplitudes computed in expansions Indications supporting the conjecture that it is quantum GR: derivation from canonical quantization of GR derivation from discretization of GR and GR=BF+constraints. asymptotic of the vertex results on the low-energy limit : n-points functions, cosmology [see Vidotto’s talk] Main physical applications Loop Cosmology → Big bounce [see G. Mena-Marugan, Martin-Benito, Tanaka, Olmedo’s talks] Black hole entropy for real black holes [see Barbero, Diaz-Polo, Borja, talks] Loop Quantum Gravity provides a still incomplete, but clean and full-scale tentative quantum theory of space time and gravitation. Carlo Rovelli Loop Quantum Gravity Granada 2010 Good news Main open problems 15 years ago: To construct a mathematically well deﬁned background independent quantum ﬁeld theory The “problem of time” Fate of GR singularities (Cosmological and Black Holes) Deriving a ﬁnite black hole entropy from ﬁrst principle Curing ultraviolet divergences of standard ﬁeld theories Computing quantum gravitational transition amplitudes. Most of the open problems in quantum gravity of 15 years ago have a solution today in the context of loop gravity Carlo Rovelli Loop Quantum Gravity Granada 2010