Conference on Path Integrals (Praha) June Path integral quantization of the exact string black hole. speaker: Daniel Grumiller (DG)

Transcription

1 Conference on Path Integrals (Praha) June 2005 Path integral quantization of the exact string black hole speaker: Daniel Grumiller (DG) affiliation: Institute for Theoretical Physics, University of Leipzig, Augustusplatz 10-11, D Leipzig, Germany supported by an Erwin-Schrödinger fellowship, project J-2330-N08 of the Austrian Science Foundation (FWF) Based upon: DG, hep-th/

2 Outline 1. Motivation 2. Actions (2D dilaton gravity) 3. The exact string BH 4. Path integral quantization Talk by Wolfgang Kummer 5. Discussion 1

3 1. Motivation Why 2D? 2

4 1. Motivation Why 2D? dimensionally reduced models (spherical symmetry) strings (2D target space) integrable models (PSM) models for BH physics (information loss) study important conceptual problems without encountering insurmountable technical ones 2D gravity: useful toy model(s) for classical and quantum gravity most prominent member not just a toy model: Schwarzschild Black Hole ( Hydrogen atom of General Relativity ) 2-a

5 2. Actions (2D dilaton gravity) EH: S = d D x gr + surface R: Ricci scalar g: determinant of metric g µν Physical degrees of freedom: D(D 3)/2 JBD/scalar-tensor theories/low energy strings: S (2) = d D x g ( XR U(X)( X) 2 + 2V (X) ) X: dilaton field U, V : (arbitrary) potentials dilaton gravity in D dimensions Often exponential representation: X = e 2φ Extremely useful: First order action! S (1) [ = X a T a +XR+ɛ (X a X a U(X) + V (X)) Review: DG, W. Kummer, D. Vassilevich, hep-th/ ],

6 3. The exact string BH NLSM (target space metric: g µν, coordinates: x µ ) S σ d 2 ξ h ( g µν h ij i x µ j x ν + α φr + tachyon ) set B-field zero. neglect tachyon for the time being. conformal invariance: 2πT i i = βφ R + β g µνh ij i x µ j x ν! = 0 thus, β-functions must vanish. LO: 16π 2 β φ = 4b 2 4( φ) φ + R α β g µν = R µν + 2 µ ν φ conditions β φ = 0 = β g µν follow from S = d D x ge 2φ ( R + 4( φ) 2 4b 2) for D = 2: Witten BH and CGHS model: 2D dilaton gravity (U = 1/X, V = 2b 2 X) Note: b 2 = (26 D)/(6α ) In 2D: non-perturbative generalization to all orders in α but no action, just geometry: R. Dijkgraaf, H. Verlinde, and E. Verlinde, Nucl. Phys. B371 (1992)

7 X=0 B I + I _ i 0 X=0 I + I _ i 0 i 0, I +, I and B denote spatial infinity, future light-like infinity, past lightlike infinity and the bifurcation point, respectively; the Killing horizon is denoted by the dashed line 5

8 Why an action? Needed for mass definition ( Gibbons-Hawking term ) clarify which, if any, of previous mass definitions is correct Needed for entropy get insight into thermodynamics of a non-perturbative BH solution of string theory Not needed for surface gravity and Hawking temperature (still, various papers get different results here) Supersymmetrization Quantization Get a new theory in this way which is interesting on its own and which generalizes the CGHS model may couple geometric action to some matter action, study critical collapse, etc. 6

9 Action for the ESBH extensive review on 2D dilaton gravity: DG, W. Kummer, D. Vassilevich, hep-th/ nogo result DG, D. Vassilevich, hep-th/ R 7

10 Action for the ESBH extensive review on 2D dilaton gravity: DG, W. Kummer, D. Vassilevich, hep-th/ nogo result DG, D. Vassilevich, hep-th/ circumvent it by allowing matter but: don t want propagating physical degrees of freedom! R 7-a

11 Action for the ESBH extensive review on 2D dilaton gravity: DG, W. Kummer, D. Vassilevich, hep-th/ nogo result DG, D. Vassilevich, hep-th/ circumvent it by allowing matter but: don t want propagating physical degrees of freedom! suggestive: consider (abelian) gauge field R 7-b

12 Action for the ESBH extensive review on 2D dilaton gravity: DG, W. Kummer, D. Vassilevich, hep-th/ nogo result DG, D. Vassilevich, hep-th/ circumvent it by allowing matter but: don t want propagating physical degrees of freedom! suggestive: consider (abelian) gauge field result: it works! DG, hep-th/ [ S ESBH = X a T a + ΦR + BF M 2 + ɛ (X a X a U(Φ) + V (Φ)) with Φ ± = γ ± arcsinh γ and γ = X/B +: ESBH, : ESNS The potentials read V = 2b 2 γ, U ± = 1 γn ± (γ), with an irrelevant scale parameter b R + and N ± (γ) = ( ) 1 1 γ γ ± + 1γ2. Note that N + N = 1. 7-c ],

13 Plot of U as a function of γ Red: ESBH, Blue: ESNS, Black: Witten BH Asymptotic ( weak coupling ) limit (γ ): Witten BH: U = 1/Φ, V = 2b 2 Φ valid for both branches (ESBH, ESNS) Strong coupling limit (γ 0): ESBH branch: JT model (U = 0, V = b 2 Φ) ESNS branch: 5D Schwarzschild! (U = 2/(3Φ), V = 2b 2 (6Φ) 1/3 ) 8

14 4. Path Integral Quantization ESBH special case of 2D dilaton gravity apply general results! talk by Wolfgang Kummer Basic ideas: W. Kummer, H. Liebl, D. Vassilevich, hep-th/ Matter provides propagating d.o.f. Integrate out geometry exactly! Possible in first order formulation with EF gauge fixing fermion: gauge fields appear only linearly! Treat matter perturbatively Reconstruct geometry (Virtual BHs) Calculate S-matrix or corrections to classical quantities (like specific heat) Note: Generalization to SUGRA: L. Bergamin, DG, W. Kummer, hep-th/

15 Constraint algebra, gauge fixing Poisson brackets: { q i, p j} = δ j i q i = (ω 1, e 1, e+ 1 ), pi = (X, X +, X ) Algebra of secondary (first class) constraints: { G i (x), G j (x ) } = G k Ck ij δ(x x ) with the same structure functions C k ij crucial: q i linear in G i, absent in C k ij no ordering problems, BRST charge nilpotent without higher order ghost terms Relation to Virasoro algebra: G = G 1 ; H 0 = q 1 G 1 + ε a bq a G b ; H 1 = q i G i {G, G } = 0, {G, H 0 } = Gδ, {G, H 1 } = Gδ, {H 0, H 0 } = (H 1 + H 1 )δ, {H 0, H 1 } = (H 0 + H 0 )δ, {H 1, H 1 } = (H 1 + H 1 )δ Gauge fixing fermion chosen s.t.: ω 0 = 0, e 0 = 1, e+ 0 = 0 10

16 Matter Without matter: pointless (0 ppdof) interesting scattering: need matter focus on massless Klein-Gordon: L (m) = F (X)dφ ( dφ) if F (X) = const. minimal else nonminimal e.g. SRG: g (4) = X g (2) F X { G +, G } new ln F (X)L (m) G 1 but: BRST charge essentially unchanged! integrate out geometry exactly; ) W = Dfδ (f + iδ/δj e +1 W [f] W = D φ exp i d 2 x [kinetic + j Ê + ] e vertices DG, Kummer, Vassilevich, hep-th/ relevant: boundary values of X I (ADM mass); measure for 1-loop (Polyakov): φ = φf 1/2 ; vertices: nonpolynomial, nonlocal; backreactions included to each matter order; geometry reconstructed from matter! 11

17 Details of path integral For simplicity minimal coupling (F = const.): W = D φ exp i (L k + L s + L v )d 2 x L k = 0 φ 1 φ E 1 ( 0φ) 2 L s = σφ + j e + Ê irrelevant 1 L v = w ( ˆX), w = X I(y)V (y)dy φ = φf 1/2, Ê + 1 = I( ˆX) ˆX = a } + {{ bx 0 } ( 0φ) 2 + irrelevant X a = 0, b = 1, E 1 = K (m ) Ê + 1 D φ exp i = I(X) + I (X) }{{} 0 2 ( 0φ) E 1 + L k = exp ( i/96π xy R y x }{{} il Pol y fr x 1 ) DG, W. Kummer, D. Vassilevich, hep-th/

18 Effective line element temporal gauge : (A I ) 0 = a 0 = const. line element in outgoing Sachs-Bondi form: (ds) 2 = 2drdu + K(r, u)(du) 2 Killing-norm (SRG, asymp. Minkowski, LO): ( ( ) ) 2m K(r, u) = 1 r + ar θ(r 0 r)δ(u u 0 ), zeros of Killing-norm approximately at r = 2m i + (Schwarzschild horizon) and r = 1/a (Rindler horizon) I + non-vacuum-geometry concentrated on light-like cut; y curvature scalar as well; i 0 interpretation: virtual black hole (VBH) induced by effective matter interaction; I - S-matrix: sum over all VBH i - DG, W. Kummer, D. Vassilevich, gr-qc/ , review: hep-th/ DG,

19 Scattering amplitude P. Fischer, DG, W. Kummer, D. Vassilevich, gr-qc/ , DG, gr-qc/ , gr-qc/ , hep-th/ S-matrix for s-wave scattering: ingoing modes q, q ; outgoing ones k, k ; E := q + q T T δ(k + k q q )E 3 / kk qq 3/2 interesting part: scale independent T, momentum transfer Π := (k + k )(k q)(k q) T (q, q ; k, k ) := 1 Π E 3 ln Π2 E Π ln p2 E 2 3kk qq 1 2 r p s r,p ( r 2 s 2 ), p 2 p {k,k,q,q } k = αe 200 sigma alpha beta k = (1 α)e q = βe q = (1 β)e 14

20 5. Discussion Basic results: Action for the exact string black hole DG, hep-th/ Path integral quantization of generic 2D dilaton gravity with scalar matter DG, W. Kummer, D. Vassilevich, hep-th/ To-do list: Thermodynamical considerations Supersymmetrization (hep-th/ ) S-matrix calculations Application to 2D type 0A/0B strings 15