An Invitation to the New Variables with Possible Applications

An Invitation to the New Variables with Possible Applications

An Invitation to the New Variables with Possible Applications Norbert Bodendorfer and Andreas Thurn (work by NB, T. Thiemann, AT [arXiv:1106.1103]) FAU Erlangen-N¨urnberg ILQGS, 4 October 2011 N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 1 Plan of the talk 1 Why Higher Dimensional Loop Quantum (Super-)Gravity? 2 Review: Hamiltonian Formulations of General Relativity ADM Formulation Extended ADM I Ashtekar-Barbero Formulation Extended ADM II 3 The New Variables Hamiltonian Viewpoint Comparison with Ashtekar-Barbero Formulation Lagrangian Viewpoint Quantisation, Generalisations 4 Possible Applications of the New Variables Solutions to the Simplicity Constraint Canonical = Covariant Formulation? Supersymmetry Constraint Black Hole Entropy Cosmology AdS / CFT Correspondence 5 Conclusion N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 2 Plan of the talk 1 Why Higher Dimensional Loop Quantum (Super-)Gravity? 2 Review: Hamiltonian Formulations of General Relativity ADM Formulation Extended ADM I Ashtekar-Barbero Formulation Extended ADM II 3 The New Variables Hamiltonian Viewpoint Comparison with Ashtekar-Barbero Formulation Lagrangian Viewpoint Quantisation, Generalisations 4 Possible Applications of the New Variables Solutions to the Simplicity Constraint Canonical = Covariant Formulation? Supersymmetry Constraint Black Hole Entropy Cosmology AdS / CFT Correspondence 5 Conclusion N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 3 Why Higher Dimensional Loop Quantum (Super-)Gravity? Quantum Gravity: Perturbative: Superstring theory / M-theory (ST / MT), require I Additional particles I Supersymmetry I Higher dimensions Non-perturbative: Loop Quantum Gravity I Various matter couplings & SUSY possible I 3+1 dimensions (Ashtekar Barbero variables) [however, Melosch, Nicolai '97; Nieto '04, '05] I What if LHC finds evidence for higher dimensions? ! Make contact between them? [Thiemann '04; Fairbairn, Noui, Sardelli '09, '10] Compare results in 3+1 dimensions: Landscape problem: Dimensional reduction of ST / MT highly ambiguous Compare results in higher dimensions: Starting points: I Higher dimensional Supergravities F are considered as the low-energy limits of ST / MT F have action of the type SGR + more I Symmetry reduced models (higher dim. & SUSY black holes or cosmology) ! Extend loop quantisation programme to higher dimensions and Supergravities [Jacobson '88; F¨ul¨op'93; Armand-Ugon, Gambini, Obr´egon,Pullin '95; Ling, Smolin '99-; Sawaguchi '01; Smolin '05,...] N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 4 Plan of the talk 1 Why Higher Dimensional Loop Quantum (Super-)Gravity? 2 Review: Hamiltonian Formulations of General Relativity ADM Formulation Extended ADM I Ashtekar-Barbero Formulation Extended ADM II 3 The New Variables Hamiltonian Viewpoint Comparison with Ashtekar-Barbero Formulation Lagrangian Viewpoint Quantisation, Generalisations 4 Possible Applications of the New Variables Solutions to the Simplicity Constraint Canonical = Covariant Formulation? Supersymmetry Constraint Black Hole Entropy Cosmology AdS / CFT Correspondence 5 Conclusion N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 5 extrinsic curvature Kab ADM Formulation [Arnowitt, Deser, Misner '62] M D+1 split Σt4 Foliation of M: x Σt3 M top. R × σ,Σt = Xt (σ), Xt : σ !M μ Important fields on σ: T μ δt = X μ δt N n δt a Lapse, Shift: N; N Σt2 ∗ x μ Spatial metric qab = (X g)ab, N δt ∗ Σ Extrinsic curvature Kab = (X Lnq)ab t1 1 = N (_qab − (LN~ q)ab) R R D p (D) ab a 2 [a; b = 1; :::; D] ) SEH = dt σ d x N det q (R ±[KabK − (Ka ) ]) ADM phase space Γ ab Canonical variables: qab, P (∼ extrinsinc curvature Kab) cd c d (D) Poisson brackets: fqab(x); P (y)gADM = δ(aδb)δ (x; y) 1st class constraints: R D a Totally constrained Hamiltonian: H = σ d x(NH + N Ha) Spatial diffeomorphism constraint Hap(q; P) (D) p 1 ab 1 a 2 Hamiltonian constraint H(q; P) = ± det q R + det q [PabP − D−1 (Pa ) ] N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 6 Extended ADM I Extension of ADM phase space I Introduce SO(D)-valued vielbein: i j i ai p ai qab = ea eb δij Kab = Kai eb E = det q e i; j; ::: 2 f1; :::; Dg (1) ai a i Poisson bracket relations: fE ; Kbj g = δbδj Increased number of degrees of freedoms ) new constraint needed: i a K[ab] = 0 , K[a eb]i = 0 , Gij := Ka[i E j] = 0 (2) Valid extension? ADM Possion bracket relations reproduced on extended phase space cd cd c d (D) fqab(E); P (E; K)gjG=0 = fqab; P gADM = δ(aδb)δ (x; y) (3) New constraints close amongst themselves: fG; Gg ∼ G qab(E); Pcd (E; K) (and in particular H; Ha) are Dirac observables w.r.t. new constraint Gij st )H ; Ha and Gij constitute 1 class constraint algebra by construction N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 7 Ashtekar-Barbero Formulation Canonical transformation to Ashtekar-Barbero variables [Sen; Ashtekar; Immirzi; Barbero] SPIN c SPIN j Introduce spin connection Γaij [e] s.t. @aebi − Γabeci + Γaij [e] eb = 0 Crucial: Defining and adjoint representation of SU(2) equivalent! Only in D = 3: Canonical transformation ai 1 ai kl SPIN fE ; K g −! f E ; A := 1=2 Γ [e] + γK g γ 2 =f0g: Immirzi Parameter (4) bj γ bj j bkl bj R ) Simple Poisson algebra fA; Eg ∼ 1 and 1st class constraint algebra Canonicity of the above transformation non-trivial a New constraint Gij = Ka[i E j] ) SU(2) Gauß law constraint: 1 1 G = γK E a + k (@ E a + ΓSPIN [e]E al ) ij a[i γ j] 2γ ij a k akl 1 = k (@ E a + lmA E a ) (5) 2γ ij a k k al m Higher dimensions? SPIN bj No obvious way of combining Kai and Γaij [e] to a connection conjugate to E in a mathematically sensible way! N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 8 Extended ADM II Extension of ADM phase space II Introduce SO(D + 1) or SO(1; D) \hybrid" vielbein: I J J aJ p aJ qab = ea eb ηIJ Kab = KaJ eb E = det q e I ; J; ::: 2 f0; 1; :::; Dg (6) Motivation: 2nd order Palatini formulation of General Relativity aI a I Poisson bracket relations: fE ; KbJ g = δbδJ I New constraints: K[ab] = 0 , K[a eb]I = 0 insufficient! Use IJ [I aJ] ( G := Ka E (7) Proof of validity of extension II analogous to extension I case Connection formulation? HYB c HYB J \Hybrid" spin connection [Peldan '94] ΓaIJ [e] s.t. @aebI − ΓabecI + ΓaIJ [e] eb = 0 HYB BUT: No obvious way of combining KaJ and ΓaIJ [e] to a connection conjugate to E bJ in a mathematically sensible way (if D 6= 2)! N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 9 Plan of the talk 1 Why Higher Dimensional Loop Quantum (Super-)Gravity? 2 Review: Hamiltonian Formulations of General Relativity ADM Formulation Extended ADM I Ashtekar-Barbero Formulation Extended ADM II 3 The New Variables Hamiltonian Viewpoint Comparison with Ashtekar-Barbero Formulation Lagrangian Viewpoint Quantisation, Generalisations 4 Possible Applications of the New Variables Solutions to the Simplicity Constraint Canonical = Covariant Formulation? Supersymmetry Constraint Black Hole Entropy Cosmology AdS / CFT Correspondence 5 Conclusion N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 10 The New Variables - Hamiltonian Viewpoint Extension of ADM phase space III Introduce \generalised" vielbein, transforming in the adjoint representation of SO(D + 1) or SO(1; D): IJ IJ aIJ p aIJ qab = eaIJ eb Kab = KaIJ eb π = det q e I ; J; ::: 2 f0; 1; :::; Dg (8) Motivation: 1st order Palatini formulation of General Relativity (cf. next to next slide) aIJ a I J Poisson bracket relations: fπ ; KbKLg = δb δ[K δL] New constraints: Gauß and simplicity constraint IJ [I jK a J] aIJ bKL a[IJj bjKL] G := Ka π K and S := π π (9) Proof of validity of extension analogous to extension I and II case Canonical transformation to new connection formulation HYB HYB ΓaIJ [π] : Extension of ΓaIJ [e] off the simplicity constraint surface S = 0 , πaIJ = n[I E ajJ] [Freidel, Krasnov, Puzio '99] (10) Canonical transformation (non-trivial): aIJ 1 aIJ HYB fπ ; K g −! f π ; A := Γ [π] + βK g β 2 =f0g; 6= γ! (11) bKL β bKL bKL bKL R G IJ becomes SO(D + 1) or SO(1; D) Gauß law constraint: IJ aIJ [I j aKjJ] G = @aπ + Aa K π (12) Formulation works with SO(D + 1) and SO(1; D) independent of spacetime signature! N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 11 Comparison with Ashtekar-Barbero Formulation Ashtekar-Barbero formulation LQG bk Canonical variables Aaj , E are real Simple Poisson algebra fALQG ; Eg ∼ 1 Compact gauge group SU(2) i First class constraints H; Ha and G Physical information: LQG SPIN k Aaij − Γaij [e] = γ ij Kak (13) G=0 Relation to other formulations: AB −! ADM New formulation, D = 3 NEW bKL Canonical variables AaIJ , π are real Simple Poisson algebra fANEW ; πg ∼ 1 Compact gauge group SO(4) IJ aIJ bKL First class constraints H; Ha; G and S Physical information: NEW HYB NEW HYB Aaij − Γaij [π] ≈ S − gauge; Aa0j − Γa0j [π] ≈ βKaj (14) S=0 time gauge G=0 NEW −! Ex.

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