Developments on the radial gauge

Transcription

1 Developments on the radial gauge Jędrzej Świeżewski Faculty of Physics, University of Warsaw in collaboration with Norbert Bodendorfer, Paweł Duch, Wojciech Kamiński and Jerzy Lewandowski ILQGS, November 10, 2015

2 Plan of the talk Introduction What is the radial gauge all about? Spacetime radial gauge Who is interested? What can we tell them? What can we learn from that? Conclusion Jędrzej Świeżewski 2

3 Introduction observer s observables: coordinates 1. introduce the observer 2. define coordinates adapted to the spatial metric spatial slice 3. define relational observables being pull-backs of canonical fields observer 4. their Poisson algebra is under control Duch, Kamiński, Lewandowski, JŚ JHEP05(2014)077, JHEP04(2015)075 radial gauge: 1. metric has the form q ab = q AB 0 2. Dirac bracket algebra is {q AB, P CD } = {, } = {, P AB } =0etc. {, P ra } = nontrivial 3. solve the vector constraint for P ra and plug it into the Hamiltonian Bodendorfer, Lewandowski, JŚ Phys.Rev. D92 (2015) 8, Jędrzej Świeżewski 3

4 A quantum reduction to spherical symmetry q AB, P AB fix radial gauge loop quantize general relativity loop quantize loop quantum gravity symmetry reduce symmetry reduce symmetry reduce midisuperspace models loop quantize quantum spherically symmetric models Bodendorfer, Lewandowski, JŚ Phys.Lett. B747 (2015) Jędrzej Świeżewski 4

5 Spacetime radial gauge radial gauge spacetime radial gauge radial lines are spatial geodesics iff: (3) arr =0 radial lines are spacetime geodesics iff: (4) µrr =0 (3) arr = 1 2 qab (2q rb,r q rr,b ) so the gauge conditions are (4) trr = 1 N K rr (4) arr = N a N K rr qab (2q rb,r q rr,b ) so the gauge conditions are q ra = ra q ra = ra K rr =0 Jędrzej Świeżewski 5

6 Spacetime radial gauge * * also known as holographic, axial or Fefferman-Graham gauge AdS/CFT correspondence boundary perspective: construct operators in CFT which mimic observables in the bulk AdS to deal with the gauge redundancy in the bulk a suitable gauge is introduced C F T a desired feature is that at least matter fields are commuting at spacelike separation are they commuting in spacetime radial gauge? yes no 1 2 Kabat, Lifschytz, Decoding the hologram: Scalar fields interacting with gravity, Physical Review D 89 (2014) Donnelly, Giddings, Diffeomorphism-invariant observables and their nonlocal algebra arxiv: [hep-th] Jędrzej Świeżewski 6

7 Spacetime radial gauge H[N] = C a [N a ]= Z Z d 3 xn d 3 xn a 2apple p P ab 1 P ab q 2 P 2 2r b P b a + a p q 2apple R(3) + h matt 1. the constraints are C =(H, C r,c A,K rr,q rr 1,q ra ) 2. Dirac bracket is given by {O 1, O 2 } D = {O 1, O 2 } 8X, =1 {O 1, C }(M 1 ) {C, O 2 } 3. where M (r, ; r, ) ={C (r, ), C ( r, )} = R (3) 3 rr 2K Ar K Ar + t r 2 0 2K ra @ B = 0 2@ r K rb r q AB apple 6 0 p det q (r, ; r, ) 4. and t matt (r, ) (r, ; r, ) = apple p q h matt (r, ) q rr ( r, ) q AB h matt (r, ) q AB ( r, ) 1 2 hmatt (r, ) (r, ; r, ) Jędrzej Świeżewski Bodendorfer, Duch, Lewandowski, JŚ arxiv:

8 Spacetime radial gauge C =(H, C r,c A,K rr,q rr 1,q ra ) R (3) 3 rr 2K Ar K Ar + t r 2 0 2K ra @ B 0 2@ r K rb r q AB apple 6 0 p det q = apple 0 F F T G 2. inverse is given by apple 0 F F T G 1 = apple (F 1 ) T GF 1 (F 1 ) T F to find F 1 we need to solve 4X Z =1 d rd 2 F (r, ; r, )N ( r, ) =M (r, ) 4. that 2 r N +( R (3) rr +2K Ar K Ar t matt )N +2K ra N A = M 2@ r N r B N B = M r N A +2@ r (K ra N)=M A Jędrzej Świeżewski Bodendorfer, Duch, Lewandowski, JŚ arxiv:

9 Spacetime radial 2 r N +( R (3) rr +2K Ar K Ar t matt )N +2K ra N A = M 2@ r N r B N B = M r N A +2@ r (K ra N)=M A N A (r, )+2K ra N(r, ) =... N [M ] N A [M ] N r [M ] N [M ] =(F 1 ) M Jędrzej Świeżewski Bodendorfer, Duch, Lewandowski, JŚ arxiv:

10 Spacetime radial gauge C =(H, C r,c A,K rr,q rr 1,q ra ) 1. {O 1, O 2 } D = {O 1, O 2 } 8X apple (F {O 1, C } 1 ) T GF 1 (F 1 ) T F 1 0, =1 {C, O 2 } 2. { (r 1, 1 ), (r 2, 2 )} D = Z drd 2 N[{ r apple (r 1, 1 ), C }](r, ) p det q(r, ) N [{C, (r 2, 2 )}](r, ) Z drd 2 N [{ (r1, 1 ), C }](r, ) apple p N[{C r det q(r, ), (r 2, 2 )}](r, ) 3. it leads to the following conclusions: a) in spacetime radial gauge matter fields do not commute b) in spacetime radial gauge matter fields have non-local brackets c) the non-localities vanish in weak-gravity limit since apple G Newton d) to guarantee simple matter sector, the gravitational gauge fixing conditions should be mutually commuting Jędrzej Świeżewski Bodendorfer, Duch, Lewandowski, JŚ arxiv:

11 Conclusion Radial gauge is a useful tool for dealing with spatial diffeomorphisms in canonical GR leads to a novel definition of spherical symmetry on the quantum level Spacetime radial gauge is of interest for AdS/CFT correspondence has non-local Dirac brackets (also in the matter sector) commutativity of gravitational gauge fixings is crucial for commutation properties of matter sector Jędrzej Świeżewski 11