LQG Dynamics: Insights from PFT

Transcription

1 LQG Dynamics: Insights from PFT Madhavan Varadarajan Raman Research Institute, Bangalore, India (work done in collaboration with Alok Laddha) LQGDynamics:InsightsfromPFT p.1

2 Papers: Talk is based on arxiv: (al, MV). Formalism developed in CQG.27:175010,2010, PRD78:044008,2008. Thiemann: arxiv: LQGDynamics:InsightsfromPFT p.2

3 LQG Dynamics:Status Restrict attention to canonical theory. Constructionof Ĉhaminvolvesinfinitelymanyad-hocchoices so defn of quantum dynamics far from unique. - C ham (x)islocal.localfieldopertrsnotdefineable.basic opertrs nonlocal. -Fixtriangulation T.Construct C ham,t s.t. lim T C ham,t = C ham.replace C ham,t by Ĉham,T. -Problem: lim T Ĉ ham,t dependsonfinite Tchoicesdueto discont polymer repn. Consistency of theory reqires anomaly free repn of constraint algebra: {C ham [N], C ham [M]} = C diff [ β(n, M)] Usethistofixambiguitiesindefnof Ĉham? Problem: Constr algebra trivialises for all choices of lim T Ĉ ham,t LQGDynamics:InsightsfromPFT p.3

4 Can PFT help? PFT does help: - Detailed structure uncannily similar to LQG - Know the right answers! More in Detail: Constraints H +, H formliealgebra.cansolvethemin polymer repn unambiguously by Grp Averaging. C ham := H + H q, C diff = H + + H formdiracalgebra isomorphic to that of gravity. DoasinLQG:firstdiffavgthenconstruct Ĉham. Analysis yields suggestions for LQG. -oneshouldtailorrepnofregulatingholonomiestothatof states being acted upon -look beyond finite smeared density weight one operators LQGDynamics:InsightsfromPFT p.4

5 Plan Review of Classical and Quantum(polymer) PFT Constructionof ĈhamfollowingThiemanninLQG Quantum Constraint Algebra -TrivialisesasforLQGw.r.toThiemannURStopologyaswellas on LM habitat - Trivialization suggests use of higher density Ham constr. - Nontrivial repn of higher density constraint algebra on new habitat(time permitting!) Discussion LQGDynamics:InsightsfromPFT p.5

6 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. LQGDynamics:InsightsfromPFT p.6

7 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 ] 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 ± generateevolutionof ± fields. Evolution 2 independent diffeomorphisms! Gaugefix: X ± = t ± x deparameterize. get back standard flat spacetime free scalar field action. LQGDynamics:InsightsfromPFT p.7

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10 Caution: WesetSpacetimeTopology=S 1 R Not much known re:polymer repn when space is noncompact. Hence choose space= circle. There are complications coming from using single angular coordinate chart x on embedded circle. Also from using single spatial angular inertial coordinate X on the flat spacetime. Identifications of x and X mod 2π are needed. Thesecanbetakencareof.Willmentionsubtelitiesasand when dictated by pedagogy. LQGDynamics:InsightsfromPFT p.10

11 Quantum Kinematics: Embedding Sector Holonomies: Graph : set of edges which cover the circle. Spins :alabel 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 + LQGDynamics:InsightsfromPFT p.11

12 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. LQGDynamics:InsightsfromPFT p.12

13 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. LQGDynamics:InsightsfromPFT p.13

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15 Physical Hilbert Space by Group Averaging 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. Thereisanicegeometricalinterpretationforgrpavgeofa chrge net. LQGDynamics:InsightsfromPFT p.15

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17 PFTdynamicsalaLQG We follow Thiemann s seminal work: -Constructsolutionsto C diff := H + + H bygroupaveraging w.r.to spatial diffeos. - C ham := H + H X + X,define Ĉhamatfinitetriangulation. - Find its continuum limit on diffeomorphism invariant states - Evaluate its commutator and check for anomalies. Diffeo Averaging: - C diff = H + + H. - Finite diffeo φ corresponds to the gauge transformations φ = φ + = φ.( Physicalstatesarediffeoinvariant!) -Diffavgofachargenetisthesumoverallitsdistinctdiffeo images: γ + φ, k + φ, l + φ γ φ, k φ, l φ - Diff related states define same discrete slice, data. Diff avg single discrete slice. ( PhysicalstatesNOTnormalizablein H diff!) LQGDynamics:InsightsfromPFT p.17

18 Hamiltonian constraint Question:Canweconstruct Ĉhams.t.itkillsphysicalstateswe have just constructed? Define Ĉham,T(δ) s + s.(s ± γ ±, k ±, l ± ). Definestatedep Ts.t.everyvertexofstateisvertexof T.Let coordinatelengthofeveryedge(i.e.1cell)of Tbe δ. Ĉ ham,t = (H + + H ) T 1 X + X T 1 fromp.b.ofholonomieswithspatialvolume X + X function V (R) = R dx X + X.Canconstruct ˆV (R)and replacepbby [, ]. The resulting operator acts nontrivially only at vertices of the state: 1 s + s = δλ(v) s + s X + (v)x (v) T Overall factor of δ similar to LQG undensitized triad operator. Care needed to ensure that operator doesnt kill zero volume stateselsekernelof Ĉhamtoolarge! LQGDynamics:InsightsfromPFT p.18

19 H + H T H := H + H = Π + X + Π X +matter. Hgeneratesggetransfwith φ + = φ 1.Unitaryrepnoffinite ggetransfnotweaklycont.howcanwedefine Ĥ? KeyIdea:Trytowrite ĤTproportionalto ( finite gge transf. 1). Illustratewith Π + X + term. ˆX + (x) T s + s = xk + δ s + s. LQGDynamics:InsightsfromPFT p.19

20 ˆΠ + T Π + T = Edge holonomy 1 δ What does a small, finite, gauge tranformation do? Wedefine: ˆΠ + T (v) s + s = ˆΠ + ˆX+ T T s + s e i vk = e i( vk) iδ 2 Π + 1 i( v k)δ s + s. Π + 1 s + s = Û+ E (φ v,δ) 1 iδ 2 s + s LQGDynamics:InsightsfromPFT p.20

21 Ĉ ham Recallthat H = Π + X + Π X +matter, C ham = H( X + X ) 1 Get: Ĥ T (v) s + s = 1 (v) X + X T = δλ(v) Û + φ v,δ Û φ v, δ 1 iδ s + s 2 dx T δ Ĉ ham,t(δ) [N] s + s = v N(v)λ(v)(Û+ φ v,δ Û φ v, δ 1) s + s FiniteoperatorasinLQG.Killscorrectsolutionsatall T(δ)! Continuum limit ala Thiemann: Let Ψ be spatially diffeo inv state. Can show lim δ 0 Ψ(Ĉham,T(δ)[N] s + s )existsexactlyasinlqg. (Technically:Contlimitexistson H kin inurstopology.) Can also show that commutator between 2 Ham constraint vanishes in this continuum limit. For RHS, need to construct spatial diffeo operator. LQGDynamics:InsightsfromPFT p.21

22 Ĉ diff C diff = Π + X + +Π X +matter.similartechniquesyield Ĉ diff T (v) s + s Û + Û 1 φ = v,δ φ v,δ iδ s + s Û diff 1 φ = v,δ 2 iδ s + s 2 Ĉ diff [ N] T (v) s + s = Û diff 1 v Nx φ (v) v,δ iδ s + s NOT A FINITE OPERATOR. RHShasshift β x = q xx (NM MN ), q xx = ( X + X ) 1 T (v) = ( 1 (v) X + T) 2 δ 2 (λ(v)) 2 X ˆq xx Ĉ diff [ β] T (v) s + s = iδ v λ2 (NM MN )(v)(ûdiff φ v,δ 1) s + s Thiemann Cont Limit of this operator vanishes doubly! -duetooveralfactorof δ -becausediffeoinvstateskilledby Ûdiff φ v,δ 1 exactly same structure in LQG!. LQGDynamics:InsightsfromPFT p.22

23 LM Habitat C ham [N]isnotdiffinvduetolapse. Ĉ ham [N]doesnotmap diffinvstatestodiffinvstateshenceneedtointerpretcont limit thru URST. Notethatbutfor N(v)factors, Ĉham[N] almost mapsdiffinv statestodiffinvstates.l-mconstructaspaceof almost diff invdistributionsonwhichcontlimitof Ĉham,T(δ)canbetaken directlysothatthespaceismappedintoitselfby Ĉham. LMHabitat V LM : -Letnontrivialverticesof s +,s be v 1,..., v n. -Let f : (S 1 ) n C.Letdiffeoclassofstatebe [s +,s ]. -Let Ψ f,[s+,s ] := s + φ,s φ f(φv 1,.., φv n ). V LM isfinitespanofdistrbtnsofform Ψ f,[s+,s ]. f are called vertex smooth functions.(note:if f = const, get diffeo inv states.) LQGDynamics:InsightsfromPFT p.23

24 Ĉ ham [N]anditscommutatoron V LM Canshowthat s + s lim δ 0 Ψ f,[s+,s ](Ĉham,T(δ)[N] s + s ) = i Ψ g i,[s + i,s i ]( s+ s ), i.e.continuumlimitofconstraintmaps V LM intoitself. Can show that commutator of 2 smeared ham constraints vanishes. Can show RHS also vansishes i.e. lim δ 0 Ψ f,[s+,s ]( C diff,t(δ) ( β) s + s ) = 0.Again, doubly : -duetooverallfactorof δ -dueto Ûdiff φ v,δ 1,oneobtainsdifferenceofevaluationsof fatpointswhichareseperatedby δ. CouldonegetnontrivialactionofRHSwithextrafactorof δ 2? -one δ 1 tocanceloverall δ -one δ 1 toconvertdifferenceoffunctionsintoderivative. AnswerisYES!Toseewhathappens,consider Ĉdiff[N]. LQGDynamics:InsightsfromPFT p.24

25 Ĉ diff [ N]on V LM Recall Ĉ diff,t(δ) [ N] = v lim δ 0 Ĉ diff,t(δ) [ N] Ψ f,[s+,s ] = Ψ g,[s+,s ]with g = L N f := i Nx (v i ) f(v 1,..,v n ) v i. Ûdiff 1 φ v,δ Nx iδ. Thisactionyieldsafaithfulrepnon V LM ofpbalgebraof diffeo constraints. Lesson for LQG: Should be open to consideration of operatorswhichdonothavefiniteactionon H kin. Howcanwegetthe2extrafactorsof δ 1 incommutatorof 2 Ham constraints in PFT? Answer:Byusingdensity2Hamconstraint H = qc ham insteadofdensity1c ham. {H(N), H(M)} = C diff (L N M), N, M N, M. Duetoextra δ 1,contlimof ĤT(δ)on V LM blowsup.its commutator is also ill defined. Anomaly on LM habitat. LQGDynamics:InsightsfromPFT p.25

26 Ĥ, ĈdiffonNewHabitat V + (Slick, quick argument for anomaly free density 2 constr algebra on V +.) Recall H ± generateactionof φ ± onphasespace. φ ± are basically diffeos. Define V + aslmtypehabitatfor + sector s + φ f(φ+ v + 1,.., φ+ v + n ) (v + i areverticesof s + ). Ĥ + [N + ]actsas Liederivative wrto N +. Getrepnof {H + [N + ], H + [M + ]}on V +.Similarlyfor - Define V + = V + V.Since+,-sectorscommuteand H = H + H, C diff = H + + H,getanomalyfreerepnof their constr algebra as well. LQGDynamics:InsightsfromPFT p.26

27 Discussion:PFT ideas to consider for LQG Consider pbility of allowing repns of holonomies at finite T to be state dependent. Thelackofweakcontinuityofoperatorson H kin arenot necessarily a hindrance to the defn of their generators on an appropriate space of distributions thru mechanisms of finite T and continuum limit of dual action. The choice of density one constraints hides the underlying non- triviality of the constraint algebra; choice of more singular operators of higher density weight may be necessary to probe the constraint algebra. Keyissue:Canwehandlealgebraofhigherdensityconstrin LQG? Call the commutator of 2 Ham constraints the LHS. Call the oprtr correspondent of PB brkt between them, the RHS. LQGDynamics:InsightsfromPFT p.27

28 RHS InPFT,fordensity1case,RHSvanishesinThiemannURSTas wellasonlmhabitat.inlqgdensityoneresultsaresameas for PFT(T-L-M-G-P). In PFT, density 2 constraints yield diffeo smeared by c- number vector field; not kinematically finite butwelldefinedonlm. Question: Can we define the diffeo constraint smeared with c-numbershiftinlqg?cancheckopertratfinite TisNOT finite(exactlybyonepowerof δ 1 asinpft).isitdefinableon LMhabitat?Moreprecisely,doesthereexistadefnof ˆF i ab T s.t.incontinuumlimit, Ĉdiff[N]isdefinedonLMhabitatand its commutator algbera isomorphic to Lie algebra of v.f.s? Looks like answer may be YES!(Work in progress, AL-MV). LQGDynamics:InsightsfromPFT p.28

29 LHS Question:ForRHStohavenetfactorof δ 1 asin Ĉdiff,T(δ)[N] how should we rescale Ham constraint in LQG? Answer:Cancheckthatweneedtorescaleitby q 1 6 δ 1. But:Rescaled constraints not well defined on LM habitat. Howdowelookfornewhabitat?IsthereanalogofZero Volume habitat of PFT? GLMP rescale ham constraints by hand and show that cant get diffeo from commutator unless Ham constr moves vertices around. AL-MV diffeo constraint work should feed into this. LQGDynamics:InsightsfromPFT p.29