Symmetry reductions in loop quantum gravity. based on classical gauge fixings

Transcription

1 Symmetry reductions in loop quantum gravity based on classical gauge fixings Norbert Bodendorfer University of Warsaw based on work in collaboration with J. Lewandowski, J. Świeżewski, and A. Zipfel International Loop Quantum Gravity Seminar December 8, 2015

2 2 Talk in a nutshell Aim: I I Symmetry reduce at quantum level Extract dynamics from full theory Results: I I Reduction to LQC Bianchi I [NB 14] k =0FRW[NB 15] Reduction to spherical symmetry SU(2) variables [NB, Lewandowski, Świeżewski 14-] Commutators in SU(2) vars. [NB, Zipfel 15] Abelian connections [NB 15] What else? I Simplified coarse graining & dynamics

3 3 Plan of the talk 1 Strategy 2 Example of the formalism 3 µ scheme in the full theory 4 Spherical symmetry and SU(2) 5 Conclusion

4 4 Outline 1 Strategy 2 Example of the formalism 3 µ scheme in the full theory 4 Spherical symmetry and SU(2) 5 Conclusion

5 5 Strategy 1. Suitable classical starting point I Gauge fix 2. Identify reduction constraints I Symmetry ) f i (p, q) =0 3. Quantise à la LQG 4. Impose reduction constraints I ˆf i i sym =0, [Ô sym, ˆf i ]=0 5. Extract dynamics

6 6 Outline 1 Strategy 2 Example of the formalism 3 µ scheme in the full theory 4 Spherical symmetry and SU(2) 5 Conclusion

7 7 Phase space & gauge fixing ADM {q ab, P cd } = c (a # d b) q a6=b =0 Diagonal metric gauge C a =0 q ab = 0 q xx q yy q zz 1 C A Gauge fixes spatial di eo constraint C a =0 C a =0 ) P a6=b (q aa, P bb ) # Gauge fixed phase space: {q aa, P bb } = b a

8 8 Adapted variables Full theory Relation to LQC := p q xxq yy q zz P := 2 3 P xx q xx + P yy q yy + P zz q zz p qxxq yy q zz Z / v, P / b := P xx q xx P yy q yy P := 1 2 := P xx q xx P zz q zz P := 1 2 log qyy q xx = P =0 log qzz q xx = P =0 {, P } = {,P } = {,P } = (3) {b, v} /1

9 9 Consequences of symmetry T 3 FRW model 0= = P = = P & P a6=b =0 + = =0 First class subset Spatial di eomorphisms: R d 3 P L ~N + P L ~N =0

10 10 Quantum kinematics and reduction Scalar fields [Thiemann, QSD5] Point holonomies h := e i P ( E Dh h 0 0 ), 2 R kin =, 0, 0 [ (R) h i := dr R h i = h i 8 2 R Reduction: # [ (R) = [ (R) =0 + di. invariance h, di = P Single vertex states 2 e i P ( ) e E, i ( ) h, di h 0, 0 di =, 0, 0 di

11 Reduced operators Diagonal operators [ ( ) h, di = h, di \P ( ) h, di = h, di Polymerised shift operators \ 1 (sin( P ) )( ) h, di = 2i, hdi E +, hdi E $ cuto for matter energy density / P 2 11

12 12 Quantum dynamics I 1. FRW part of Hamiltonian 2. Other terms vanish Dynamics \P ( ) 2 h, di 3 = (sin( P\ ) )( ) h, di + rescaling of variables LQC di erence equation in (v, b) variables [Ashtekar, Corichi, Singh 2 v, i= 3 G 4 v v +2 v +4, i + v 2 v 4, i ( v +2 + v 2 ) v, i 2

13 13 Quantum dynamics II 1. FRW part of Hamiltonian 2. Other terms vanish / P ab,butnot/ P /, =0 Spatial derivatives finite di erences ) vanish on single vertex states

14 14 Outline 1 Strategy 2 Example of the formalism 3 µ scheme in the full theory 4 Spherical symmetry and SU(2) 5 Conclusion

15 15 T 3 Bianchi I: classical preparations ADM {q ab, P cd } = c (a # Diagonal metric gauge # d b) Gauge fixed phase space: {q aa, P bb } = # New variables: K a, E b = e ae a = q aa, E a = p det qe a, K a = K ab e b, b a b a q a6=b =0 C a =0

16 16 T 3 Bianchi I: reduction constraints T 3 Bianchi I ae b =0=@ ak b & P a6=b =0 + Spatial di eos: Abelian Gauß law: R First class subset d 3 E a L ~N K a + P L ~N =0 R d 3!@ ae a =0

17 17 Quantum kinematics & reduction Standard LQG quantisation for U(1): [Corichi, Krasnov 97] 1. Holonomies h =exp i R K ads a, fluxes E(S) = R E a S abc dx b ^ dx c 2. Reduction ) gauge / spatial di eo invariance # Single vertex states x, y, zi $ p 1, p 2, p 3i LQC [Ashtekar, Wilson-Ewing 09] h z z h y y Reduced operators h x x Areas A(T 2 x), A(T 2 y ), A(T 2 z) Reduced Wilson loops T 3

18 18 Quantum dynamics Polymerisation R K ads a sin( R K ads a )/ U(1)! =1 ) old LQC dynamics [Ashtekar, Bojowald, Lewandowski 03, has been formulated using R Bohr ] R Bohr! 2 R ( new LQC dynamics 1/ x =sizeofuniverseinx-direction [Ashtekar, Pawlowski, Singh 06; Ashtekar, Wilson-Ewing 09] Full theory lessons LQG on fixed graph [Giesel, Thiemann 06] $ U(1) I Problems for coarse states (?) R I FRW: Kads a / p distance see also [Charles, Livine 15] µ dynamics from coarse graining? [Gielen, Oriti, Sindoni 13; Alesci, Cianfrani 14]

19 19 Outline 1 Strategy 2 Example of the formalism 3 µ scheme in the full theory 4 Spherical symmetry and SU(2) 5 Conclusion

20 20 Spherical symmetry: classical preparations S r 2 σ 0 r θ,ϕ σ 1. Radial gauge q ra = ra [Duch, Kamiński, Lewandowski, Świeżewski 14] [NB, Lewandowski, Świeżewski 14, 15] q ab = q AB 1 C A Σ 2. SU(2) connection variables A i A, Ej B 3. C a =0 ) P ra (A i A, E B j ) Reduction constraints P ra =0, spatial di eomorphisms preserving S 2 r

21 21 Quantum kinematics & reduction [NB, Lewandowski, Świeżewski 14] Standard LQG quantisation 1. Kinematics ) spin networks S 2 r 1 [...[ S 2 r n 2. Reduction ) di invariance on S 2 r 2 S r n Σ=[0, ) S 2... Quantisation... r Symmetric operators 1. Areas of the S 2 r! R(r) 2 := Averaged trace of momenta! P R (r) := 2 R(r) R R S 2 r S 2 r d 2 p det q AB d 2 P AB q AB

22 22 Regularising [ ˆR, ˆP r ] [NB, Zipfel 15] Z R(r) 2 / S 2 r Poisson bracket tricks [Thiemann: QSD1, QSD4] p V k V k P R (r) / 1 Z d 2 {H, V } R(r) Sr 2 V k / ijk Ei A Ej B AB H := FAB i n i AB n i = ijk Ej A Ek B AB k ijk Ej A Ek B ABk Simplest non-trivial spin network: Operators non-trivial at kink e 1 e 2 Graph-preserving regularisation Graphical calculus [Alesci, Liegener, Zipfel 13] j Kink state

23 23 Results of [ ˆR, ˆP r ] [NB, Zipfel 15] i P R P R ei P R e $ F i 2i AB h AB h 1 AB Classical reduction D ± h ˆR, ˆP R i E =0.5i Quantum reduction j ± 1 2 h ˆR, ˆPR i j 0.1 i + O(j 1 ) Several problems Strong regularisation dependence Kink state degenerate Problems absent for trivalent vertex! future work

24 24 Outline 1 Strategy 2 Example of the formalism 3 µ scheme in the full theory 4 Spherical symmetry and SU(2) 5 Conclusion

25 25 Conclusion Strategy I Gauge fixing I ˆf i i sym =0, [Ô sym, ˆf i ]=0! Loop quantum cosmology I µ scheme in full theory I Single-vertex truncation This work was supported by the Polish National Science Centre grant No. 2012/05/E/ST2/03308.! Spherical symmetry I Partial results in SU(2) variables Lessons / open questions I µ-scheme for coarse states I Coarse graining Thank you for your attention!