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An Invitation to the New Variables with Possible Applications

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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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