Path integral measure as determined by canonical gravity

Transcription

1 Path integral measure as determined by canonical gravity Atousa Ch. Shirazi FAUST Seminar Spring 2013

2 Motivation Dynamics of current spin foam approach is independent from canonical theory Need to use Liouville measure on the reduced phase space which preserves the phase space volume Dq exp is! DqDp exp is Plebanski-Holst formulation: has desired variables (! IJ µ,x IJ µ ) need to integrate out the connection

3 Outline Reduced phase space path integral approach for a general Hamiltonian system with constraints Applying that to Plebanski Holst formulation Integrating out the connection ADM path integral Conclusion

4 Reduced phase space path integral (Quantization of Gauge Systems, Henneaux and Teitelboim) First class constraints: Second class constraints: {F i,f j } = f k ijf k 0 {S i,s j } 6 0 Liouville measure on the whole phase space Liouville measure on the reduced phase space dqdp dq? dp? Liouville measure for noncanonical coordinates in phase space: [det{x µ,x }] 1 2 dx µ Change of variables: (q, p)! (q?,p?,s) dq? dp? 1 ds[det{s, S}] 2 = dqdp dq? dp? = (S)[det{S, S}] 1 2 dqdp

5 Reduced phase space path integral (Quantization of Gauge Systems, Henneaux and Teitelboim) Gauge fixing: dq? dp? = (F, )! second class (S)[det{S, S}] 1 2 dqdp {F, F} {F, } {,F} {, } [det{f, }] 2 Faddeev Popov term: Path integral: Expectation value: (O) := [det{s, S}] 1 2! FP = det({f, }) (S)! (F ) ( ) := DqDp exp is := DqDp p det({s, S}) det({f, }) [S] [F ] [ ]expis DqDp p det({s, S}) det({f, }) O [S] [F ] [ ]expis

6 Plebanski Holst formulation P-H action: ( ) S PH = X IJ ^ F IJ IJ ( ) X =(X 1? X) IJ F IJ = d! IJ +! I K ^! KJ Conjugate variables: (! IJ µ,x IJ µ ) Simplicity constraint: C µ X IJ = (1) Reduced phase space integral : = (II+) s 4! µ IJKL X IJ X KL 0 ± 1 apple ei ^ e J (I±) (II±) := IJKL Xµ IJ X KL ± 1 2apple IJ KLe K ^ e L D! IJ µ DX IJ µ 20 (C)V 9 V s exp i ( ) X IJ ^ F IJ V s = h 1 2 V = g 1 2 1) Canonical path integral measures for Holst and Plebanski gravity: I. Reduced phase space derivation, J. Engle, M. Han and T. Thiemann, 2010 Class. Quantum Grav

7 Integrating out the connection S[X,!] =(!, Â!)+(b,!) (, ):= d 4 x µ IJ µ IJ Gaussian integral: i I(X) = D! exp i h(!, Â!)+(b,!) I(X)ˆ=(detÂ) 1 2 exp i 4 (b,  1 b) Determinant of A: det  = V12 = (II+) D! IJ µ DX IJ µ 20 (C)V 9 V s exp is (1) ˆ= DX IJ µ 20 (C)V 3 V s exp is 1) Pure momentum path integral for spin foam, A. Ch. Shirazi & J. Engle (In Preparation)

8 ADM Formulation Canonical variables: (h ab, ab ) First class constraints: Hamiltonian and 3-diff constraints H 0 = h [h ach bd + h ad h bc h ab h cd ] ab cd (3) R(h)h 1/2 H a = h 1 2 hab D c bc H 0 No second class constraints := DqDp p det({s, S}) FP [S] [F ] [ ]expis H a Using: ADM = R Dh ab D ab DNDN a exp i R d 4 x( ab ḣ ab H G (h ab, ab )) { DN exp inh = (H), DN a exp in a H a = (H a )} N and N a are Lagrange multipliers No gauge fixing: No Faddeev-popov term

9 Final ADM path integral ADM = Dh ab D ab DNDN a exp i dt(âab,cd ab cd + ˆB ab ab + Ĉ)  ab,cd := Nh 1 2 [h ac h bd + h ad h bc h ab h cd ] 2 ˆB ab := ḣab 2D (a N b) Ĉ := Nh 1 2 (3) R Integrating out ADM = The exponential: ab ( (det Â) 1 2 Dhab DNDN a exp i 1 4 ˆB ab  1ab,cd ˆBcd + Ĉ)!L G d 4 x( 1 4 ˆB ab  1ab,cd ˆBcd + Ĉ) Determinant: (det Â) 1 2 = N 3 h 1 2 (1) Final ADM path integral ADM ˆ= N 3 h 1 2 Dhab DNDN a exp i d 4 xl G. 1) Pure momentum path integral for spin foam, A. Ch. Shirazi & J. Engle (In Preparation)

10 (1) (2) ADM ˆ= ˆ= Change of variables Comparing to our results (h ab,n,n a )! (g µ ) (1) Dg µ = Dg ab Dg 00 Dg 0a =detjdh ab DNDN a g ab = h ab g 00 = N 2 + h ab N a N b J ab,g 00,g 0a ) g 0a = h ab N ab,n,n a ) = det J = Dg µ = N 3 h 1 2 Dhab DNDN a exp is G DX IJ µ 20 (C)V 3 V s exp is G 2hN 2hNDh ab DNDN ab,g 00,g 0a ab,n,n a ) g ab g 00 g 0a N 0 2N 0 N a 0 2N a h ab h ab 1 0 N a ADM ˆ= R N 4 h 3 2 Dg µ exp is G Other change of variables: X IJ µ! (e I µ,c) (2) (e I µ)! (g µ, I J ) (3) DX IJ µ = V 6 De I µd 20 C Dg µ D I J = p gde I µ 1) Pure momentum path integral for spin foam, A. Ch. Shirazi & J. Engle (In Preparation) 2) Canonical path integral measures for Holst and Plebanski gravity: I. Reduced phase space derivation, J. Engle, M. Han and T. Thiemann, 2010 Class. Quantum Grav ) Path integral measure for first-order and metric gravities, R. Aros, M. Contreras and J. anelli, 2003 Class. Quantum Grav

11 Conclusion Breaking the manifest general covariance because of the appearance of 3-volume. Reason: Foliation of space-time! Price to pay for consistency with the canonical theory. The path integral is invariant under Bergman-Komar group but not Diff(M)! (Canonical Path-Integral Measures for Holst and Plebanski Gravity. II. Gauge Invariance and Physical Inner Product, M. Han, Class.Quant.Grav. 27 (2010) ) Continuum path integral Next step: Discretizing and quantizing the measure and implying to spin foam models.