Propagation in Polymer PFT

Transcription

1 Propagation in Polymer PFT Madhavan Varadarajan Raman Research Institute, Bangalore, India PropagationinPolymerPFT p.1

2 Non-uniqueness of Hamiltonian constraint in LQG: Classical Ham constr depends on local connection field A. But in LQG, no local connection operator, only holonomy operators.cantconstructhamconstroprtrby A Âin classical expression. Follow Thiemann Strategy: -Introducetriangulation T δ ofcauchyslice Σwith T δ 0 = Σ. -Defineapproximantsto (A, E)intermsofholonomy-fluxes ofsmallloops,surfacesassociatedwith T δ. -Substitute in expression for Ham constr to get approximant H δ sothathamconstraint = H δ 0. -Replaceclassicalholonomy-fluxin H δ byquantumoprtrs, get Ĥδ. -DefineHamconstroprtras continuumlimit Ĥδ 0. Remarkably, limit operator can be defined. Problem is it dependsonchoiceofhol-fluxapproximantsatfinite T δ. Infinitely Non-Unique!. PropagationinPolymerPFT p.2

3 Cut down on non-uniqueness by imposing further requirements E.g. Require operator be defined on Hilbert space(jurek,hanno,assanioussi...).require non-trivial anomaly free repn of constraint algebra(laddha,tomlin,..). In this talk: try to address Smolin s propagation requirement. Smolin: Requires that the quantum dynamics be such that it propagates perturbations from one part of quantum geometry to another(this propagation thought of as nonpert seed of graviton propagation in semiclassical LQG). Argues that requirement not met in LQG due to ultralocality of Ham constr action. PropagationinPolymerPFT p.3

4 Ultralocality of LQG Dynamics: Hamconstraintactsonverticesofaspinnet.Actiononly depends on vertex structure in infinitesmal nbrhood of vertex. Repeated actions of Ham constr builds a nest of structure localised at each vertex. So vertices far apart in graph nevertalktoeachotherthruhamconstr. Smolin argued that it was unlikely that such quantization could describe propagating degrees of freedom between macroscopically seperated regions of quantum geometry. Arguments intuitive since not enough known about physical interpretation of states. But compelling. PropagationinPolymerPFT p.4

5 In this talk.. I shall examine this issue in LQG type polymer quantization of 2d Parameterised Field Theory. PFTisgencovreformultnoffreescalarfieldpropagationon fixed flat spacetime. All steps of LQG program including ambiguity free defn of quantum dynamics can be completed(laddha,mv). Ultralocality problem exists. Can see clearly that repeated action of Ham constr does not give long range propagation. But nevertheless physical states describe scalar field propagation. Howcanthisbe? PropagationinPolymerPFT p.5

6 Plan Classical PFT Quantum Kinematics Unitary implementation of Finite Gauge transformations Physical States via Grp Avging Action of Quantum Hamiltonian Constraint Quantum states and discrete spacetime The ultralocality problem Propagation PropagationinPolymerPFT p.6

7 Classical PFT FreeScalarFieldAction: S 0 [f] = 1 2 d 2 Xη AB A f B f Parametrize X A = (T, X) X A (x α ) = (T(x, t), X(x, t)). S 0 [f] = 1 2 d 2 x ηη αβ α f β f, η αβ = η AB α X A β X B. Varythisactionw.r.to fand2newscalarfields X A : S PFT [f, X A ] = 1 2 d 2 x η(x)η αβ (X) α f β f δf: α ( ηη αβ β f) = 0 η AB A B f = 0 δx A :nonewequations, X A areundeterminedfunctions of x, t,so2functionsworthofgauge! x, t arbitrary general covariance So Hamiltonian theory has 2 constraints. Remark:Freesclarfieldsolnsare f = f + (T + X) + f (T X) Use split into left movers + right movers in Hamiltonian theory. PropagationinPolymerPFT p.7

8 Hamiltonian description T(x), X(x) become canonical variables. Use light cone variables T(x) ± X(x) := X ± (x). Phasespace: (f, π f ), (X +, Π + ), (X, Π ) Constraints:H ± (x) = [ Π ± (x)x ± (x) ± 1 4 (π f ± f ) 2 ] Gaugefix: X ± = t ± x deparameterize. get back standard flat spacetime free scalar field action. Define: Y ± = π f ± f {Y +, Y } = 0, {Y ± (x), Y ± (y)}=derivativeofdeltafunction {H +, H } = 0.P.B.algebrabetweensmeared H + s isomorphic to Lie algebra of diffeomorphisms of Cauchy slice.samefor H algebra. H ± generatespatialdiffeomorphismsof ±fields. FiniteEvoltn 2independentdiffeomorphisms (Φ +, Φ )! Φ + movesonly + fields, Φ movesonly - fields. PropagationinPolymerPFT p.8

9 Spacetime interpretation of canonical data: PropagationinPolymerPFT p.9

10 PropagationinPolymerPFT p.10

11 Note: Not much known re:polymer repn on non-compact space. Henceset Σ=circlesothatSpacetimeTopology=S 1 R. There are complications coming from using single angular coordinate chart x on Σ and single spatial angular inertial coordinate X on the flat spacetime. Identifications of x and X mod 2π areneeded.thesecanbetakencareof.will mention subtelities when necessary. PropagationinPolymerPFT p.11

12 Quantum Kinematics: Embedding Sector Focus on + sector. Canonical coordinates: (Π + (x), X + (x)) (A, E) LQG Holonomiesof Π + : Graph setofedgeswhichcoverthecircle. Spins anintegerlabel k e foreachedge e. Holonomies e i k e Π + e e ElectricField X + (x) PoissonBrkts: {X + (x), e i k e e (for xinside e) e Π + } = ik e e i e k e e Π + PropagationinPolymerPFT p.12

13 ChargeNetworks: γ, k + ˆX + (x) γ, k + = hk e + γ, k +, xinside e. e i e k + e Π + e γ, k + = γ, k + + k + InnerProduct: γ, k + γ, k + = δ γ,γδ k+, k + Rangeof k e : hk e Za, aisabarbero-immirziparameterwith dimensions of length. Similarly for sector. Propagation in Polymer PFT p. 13

14 Quantum Kinematics: Matter Sector and H kin Can define holonomies and repn on matter charge networks. Chargenetwork: γ, l. H kin obtainedasproductof +, embeddingandmatter charge nets. PropagationinPolymerPFT p.14

15 PropagationinPolymerPFT p.15

16 Physical Hilbert Space by Grp Avging Can construct physical states via grp avging w.r.to gge transformations just as we do for spatial diffeos in LQG. GrpAvgofachrgenet γ +, k +, l + γ, k, l isthe distribution, γ + φ +, k + φ +, l + φ + γ φ, k φ, l φ,wherethesumis over all distinct gge related chrge nets. There is a nice geometrical interpretation for a chrge net and for physical states obtained by grp avging. PropagationinPolymerPFT p.16

17 Geometrical picture: First,gotofineenoughgraphsoastowrite γ +, k +, l + γ, k, l γ, k +, k, l +, l sothateachedgeof γhaspairofembeddingchrgelabels k +, k andpairofmatterlabels l +, l. k ± areeigenvaluesof ˆX ±.So (k +, k )defines (X +, X ) coordinate of point in flat sptime! Associate matter chrge pair (l +, l )withthispoint. Doingthisforalledgesofchargenet,wegetadiscrete Cauchy slice with quantum matter. Gggetransf Φ ± move +, - edgesindependently.gge transformed chrgnet has different pairs of +- embeddning and matter chrges. Defines new slice with new matter data. Thus quantum matter data propagates from 1 slice to another. Turns out that a(grp avged) physical state then defines discrete sptime with matter propagating on it. So quantization by grp avging encodes propagation. PropagationinPolymerPFT p.17

18 PropagationinPolymerPFT p.18

19 Classical Diffeo, Ham constraints Recall: H ± (x) = [ Π ± (x)x ± (x) ± 1 4 (π f ± f ) 2 ] C diff := H + + H = Π + (x)x + (x) + Π (x)x (x) + π f f. Generates spatial diffeo i.e. evoltn along slice. Intermsof Φ ± :Sptldiffeo=transfinwhich Φ + = Φ. C ham = H + H (densityweight2). Generates motion along timelike normal to slice(normal defined wrto flat spacetime metric). Finitetransfcorrespondstochoice Φ + = (Φ ) 1. PropagationinPolymerPFT p.19

20 Quantum Constraints a la LQG: Sincefordiffeos, Φ + = Φ Φ Û diff (Φ) = Û+(Φ + = Φ)Û (Φ = Φ).Cansolvebygrpavging. PropagationinPolymerPFT p.20

21 Theimann type Ham Constr: Can construct finite triangulation constraint which acts as infinitesmalversionof Φ + = (Φ ) 1. Let φ δ,v bediffeowhichisidentityoutsidesmallnbrhoodof v andwhich pulls point vtoright. Ĉ (δ) ham pulls + edgesat vtorightby φ δ,v, pulls - edgesat vtoleftby φ 1 Ĉ (δ) ham Ψ (Û+ v N(v) (φ δ,v )Û (φ 1 δ Û + (φ δ,v )Û (φ 1 δ,v ) Ûham,δ,v. δ,v ). δ,v ) 1 Ψ Equally possible to make choices in which oprtr pulls + left, - righti.e.inwhichwereplace Ûham,δ,vbyitsadjoint. We impose constraint and its kinematic adjoint in quantum theory. NOTE:Weareinterestedinactionof Ûham,δforsuffsmall δ. PropagationinPolymerPFT p.21

22 Actionof Ûham,δ,v PropagationinPolymerPFT p.22

23 Summary: Ĉ ham,δ actsnontriviallyonlyininfinitesmalvicinityofvertices dragging + one way and - the other. Including matter labels, charge net describes data on a lattice version of Cauchyslice.Repeatedactionof Ĉham,δcanonlygenerate singletimesteptopastandfutureandthenevolutiongets stuck. Only immediate null related lattice point is generated with matter data. No large scale propagation Thishappensduetoultralocalactionof Ûham,δ... PropagationinPolymerPFT p.23

24 Ultralocality problem Recall:Finiteggetransf U ± (Φ ± )candrag +, - edgespast eachotherandgetbeyondsingletimestepasfollows: Problem: Above, necessary intermediate step is disappearanceof (k + 1, k 1 )edgei.e. k 1 isdraggedpast k+ 1. PropagationinPolymerPFT p.24

25 Duetoultralocalactionof Ûham,δwealwayshavesome (k + 1, k 1 ) edge,cantdrag k 1 past k+ 1. PropagationinPolymerPFT p.25

26 Propagation: Change of Perspective Ratherthanaskingifrepeatedactionsof give propagation, we should ask if joint kernel of diffeo constraintand Ĉhamdescribespropagation. Ûham,δ(or Ĉham,δ) Element Ψ of kernel is(distribtnl) sum of chargenet(bra) states. Ψ describes propagation if following statement holds: If the bra corresponding to some discrete Cauchy slice with matterdataisinsumthenthebracorrespondingtoanyfinite evolutionofthissliceanddatamustalsobeinthesum. Byfiniteevolutionwemeanactionofanyfiniteggetransf Φ ± so this statement is equiv to showing elements of kernel same as physical states obtained by grp avging wrto finite transf generatedby H ±. The key to showing statement holds for joint kernel of diffeo constrandhamconstraint Ĉhamistoshowthatifbrawith (k + 1, k 1 )edgeisin Ψsoisbrawithoutthisedge(previous slide). Propagation in Polymer PFT p. 26

27 Proof: Weuselanguage c isin Ψ tomeanthat c isasummand in the sum representing distribution Ψ. Let Ψbeinkernelofdiffeo,Hamconstr.Thenitfollowsthat c isin Ψifandonlyif -allsptldiffeoimagesof c arein Ψ. - Ûham,δ c arein Ψforsuffsmall δ (sothat Ψ(Ûham,δ 1) c ) = 0in δ 0limit) -Similarly Û ham,δ c arein Ψforsuffsmall δ. We shall show that the desired chargenets(with and without the (k 1 +, k 1 )edge)arein Ψbyrelatingthemthrutheaction of Ûham,δ, Û ham,δ. PropagationinPolymerPFT p.27

28 Proof(Contd): PropagationinPolymerPFT p.28

29 Conclusions Ultralocal action of Ham constraint does not preclude propagation. Propagationistobeseenasapatternwrittenontheentire set of kinematic bras which comprise a given physical state ratherthanastherepeatedactionofthehamconstrainton a fixed kinematic ket. Seeninthiswaypropagationisaresultof: -Particular Û 1 structureofthe(finitetriangulation)ham constraint. -Theimpositionofthekinematicadjoint Û 1 alsoasa Ham constraint. General structural lessons are robust and should be used in LQG to find phys correct Ham constraint.seem to find v.interestingpreliminryresultsfor U(1) 3 model. PropagationinPolymerPFT p.29

30 PropagationinPolymerPFT p.30