Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity

Transcription

1 Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity Madhavan Varadarajan (Raman Research Institute) Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 1

2 Preview Arena: Hamiltonian GR which rewrites Einstein eqns for 4d sptime metric as evolution eqns for 3d geometry Evolution= +. maps Σ to itself spatial reparameterization 3d diffeo. carries nontrivial dynamical content. My interest: quantum gravity, spec. LQG. wellunderstood. iskeyopenissue.since issonontrivial, consructinghamopertrfor VeryHard. Easier if we could develop better intuition for. This talk: ClassicalPBidentitytoaidthis.PBrelatesaspectsof to aspects of. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 2

3 Main Aim of Talk: Motivate and display new identity. While identity is classical, motivation for its detailed form from considerations in loop quantum gravity. Second Aim: Describe these considerations and thereby provide a status update on quantum dynamics in LQG. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 3

4 Plan of Talk: 1. Review of the Hamiltonian formulation of General Relativity 2.Achangeofvariables 3. Loop Quantum Gravity 4. The New Identity Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 4

5 1. Hamiltonian GR q ab, q ab q ab, p ab. {q, p} δ. q intrinsic geometry of Σ. p extrinsicgeometryof Σin M. Useful to decompose time flow into components normal, tangentialtoslice. t a = Nn a + N a N Lapse N a Shift Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 5

6 Hamiltonian density which evolves (q, p) normal to slice: H(q, p) Scalar/Hamiltonian Constraint. Hamiltonianforevoltnalong N nis H(N) = Σ NH. Hamiltonian density for evolution tangential l to slice: D a (q, p). Vector/DiffeomorphismConstraint. Hamiltonianforevolnalong Nis D( N) = Σ Na D a Evoltnalong t = N n + N: q = {q, H(N) + D( N)} ṗ = {p, H(N) + D( N)} Einstein Eqns also constrain values of q, p. Constraints+Evolutionequations G ab = 0. Solns q(x, t), p(x, t) Soln g ab (x, t) Changetimeflowbychanging N, N a.getsamesptime geometry. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 6

7 Spacetime Covariance: Theory is sptime covariant if its solns describe spacetime geometries. GRismanifestlysptimecovariantsincesolnsto G ab = 0are sptime metrics. SinceunderlyingtheoryisGR,theHamformulationofGRis also sptime covariant. However the sptime metric is constructed out of the primary variables of the formulation which are evolving 3d objects. So sptime covariance is not as explicit in the Ham formulation. This raises the following general question: Does any structure in the Ham formulation of a sptime cov theory acquire a characteristic form deriving from sptime covariance? Answer:(H-K-T) Yes! The P.B. algebra between generators of evolution obtains a characteristic form deriving from spacetime covariance. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 7

8 Idea:Hamformofanytheoryofsptimegeometry, g ab, hassptime=σ R.Evolutionpushes Σalongtimeflowand generates sptime. - Consider infinitesmal normal and tangential displacements of Σin M. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 8

9 - Work out algebra of hypersurface deformations: [ Ta ( N1 ),Ta ( N 2 ) ] =Ta(L N2 N1 ), [ No(M),Ta( N) ] =No(L N M) [ No(M1 )No(M 2 ) ] =Ta( N M1,M 2,q ab ) - Can show this imples that PB between evolution generators must have this structure. Fore.g.inGR: {D( N 1 ), D( N 2 )} = D(L N2 N1 ) {H(M), D( N)} = H(L N M) {H(M 1 ), H(M 2 )} = D( N M1,M 2,q ab ). In the classical theory, Sptime Covariance implies a characteristic PB structure between the generators of evolution. Ifwewantsomesortofaspacetimestructuretoemergein quantum theory, impose condition that PB algebra be represented in quantum theory. Refer to this condition as Quantum Spacetime Covariance Condition Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 9

10 2. A Change of Variables: Instead of 3-metric use a triplet of orthonormal fields E a i, i = 1, 2, 3sothat 3 i=1 Ea i Eb i qab.rotatedtriadgives same metric. Under such internal rotations i transforms as S0(3) Lie algebra valued index(equivalently, SU(2) Lie algebra valued index). E a i isasu(2)yangmillselectricfield!conjugatevariableis an SU(2)connection A i a, {A, E} δ Due to triad rotation gauge, extra degrees of freedom in (E, A)relativeto (q, p).cantakecareofthisandthen rexpress: H(q, p), D a (q, p) H(E, A), D a (E, A). Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 10

11 3. Loop Quantum Gravity: Canonical Quantization Approach: - Choose set of functions on phase space. Represent each function fasoperator ˆfonHilbertspaceofwavefunctions suchthat {f, g} = [ ˆf,ĝ] i h.quantumkinematics - Construct quantum correspondents of generator(s) of evolution. Quantum Dynamics. LQG Kinematics- 3 Key Features: 1. No structure depends on any unphysical background metric. 2. ConnectionRepn : ˆf(A)ψ(A) = f(a)ψ(a). Êψ(A) = i h δ δa ψ(a). Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 11

12 3. Basic connection dep fns associated with edges and are called holonomies, h e (A). - h e = P exp e A a - A i a SU(2)Liealgebra h e (A) = SU(2)matrix. -Eachcomponentof SU(2)matrix h e (A) B C ĥb e C To summarise: LQG Kinematics consists of a background indep connection repn with connection oprtrs labelled by edges. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 12

13 LQG Dynamics: -Dynamicsalong Σgeneratedby D( N).Classicaly,this generates diffeomorphisms of (A, E). QMly due to background indep, these spatial diffeos are represented by Unitary operators. Key development: diffeomorphism inv measure on space of connections(early 90 s Ashtekar-Isham,Lewandowski..) -Dynamicsnormalto ΣistheKEYOPENISSUE.Needto construct Ĥ(M). Problem: H(E, A) depends on local fields like connection. Basic connection operators nonlocal holonomies. Need to construct Âfrom ĥe. Classically: A h small edge 1 δ with δ 0. QMly: Limit does not exist on operators due to background indep! Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 13

14 Strategy: -Consideratriangulation T δ of Σi.e.divide Σintocoordinate cellsofsize δ 3 - Within each cell, approximate local quantities in H to order δ by holonomies and suitable triad fns. -Obtainapproximant H I.Approximate H(N)by H δ (N) cells δ3 N I H I. -Replacefnsin H I byoperatorsandget Ĥδ(N). -Take δ 0.Hopethatwhileindividualoperator approximants dont have limit, maybe conglomeration of suchoperatorswhichdefine Ĥδ(N)doeshavelimit. Thiemann was able to construct such continuum operator! Butoperatoractiondependsonchoicesmadeatfinite δ such as shape of edges which label holonomies. Infinite Ambiguity. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 14

15 Idea: Use Quantum Sptime Cov to restrict choices i.e. require PB algebra of evolution generators be appropriately represented in quantum theory. Recall: {H(M 1 ), H(M 2 )} = D( N M1,M 2,q ab ). RescaleHamdensity H ( q) α H.Definerescaled Hamiltonian: H (α) (N) := Σ N q α H. PBbecomes: {H (α) (M 1 ), H (α) (M 2 )} = D( q 2α NM1,M 2,q ab ). Duetopropertiesof ˆqoprtr,turnsouttobebettertoimpose theabove α 0condtninquantumtheory. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 15

16 α 0 condtn: [Ĥα(M 1 ), Ĥα(M 2 )] = i h ˆD( ˆq 2α NM1,M 2,ˆq ab ) TwooperatorsonLHS.2limits δ 1, δ 2 0. On RHS one Diffeo operator one limit. Need to choose finite triangulation operators so that continuum LHS= RHS. Easier to analyse if finite triangultn LHS, RHS structurally similar. Question: RHS written as commutator of 2 opertrs? RHS is diffeo, so commutator of 2 diffeos? Classical precursor to this question: Can we write classical RHSasPoissonBracketbetween2diffeos? LookforanewPBidentity: D( q 2α NM1,M 2,q ab ) = {D(? ), D(? )} What can these shifts be? Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 16

17 4. The New Identity Fromlapses M 1, M 2,defineElectricShiftsascombination: q α M 1 Ei a := (α) N1i a, q α M 2 Ei a := (α) N2i a New Identity(Tomlin, MV): 3 i=1 {D((α) N a 1i ), D((α) N a 2i )} = 2αD( q 2α NM1,M 2,q) N c M 1,M 2,q = q cd (M 1 d M 2 M 2 d M 1 ) This in turn implies: 3 i=1 {D((α) N a 1i ), D((α) N a 2i )} = 2α{H (α)(m 1 ), H (α) (M 2 )} Triads crucial- not pble with 3-metrics! Identity trivialises precisely for α = 0! RelatesPBofdiffeoswithPBofHamconstr. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 17

18 SinceIdentityrelatesPBof2diffeostoPBof2Ham constraints, suggests following question: Canwewriteevolutiongenby H(N)intermsofsuchdiffeos? Yes! Ashtekar: Ashtekar showed in 80 s that expressions for constraints simple polynomials if we use complex self dual connections instead of real SU(2) ones. Motivated by identityandsomeearlierwork,heshowed,intermsofself dual triad-connection variables, that Ham evolution itself can be viewed in terms of triad dependent spatial diffeos and internal gauge transformations. Using this we now have more geometrical interpretation of Ham constraint. Suggests we go back to quantum theory and try to use this to define operator action. First exploratory steps recently by Laddha in context of Euclidean gravity. Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 18

19 Summary Canonical Gravity:Sptime Cov + Quantization = New Identity? New Identity found by(tomlin, MV). New way of looking at classical Einstein eqns(ashtekar). Opens up new window on definition of quantum dynamics. First possible steps(laddha). Canonical Gravity: Spacetime Covariance, Quantization and a New Classical Identity p. 19