The Free Polymer Quantum Scalar Field and. its Classical Limit

Transcription

1 The Free Polymer Quantum Scalar Field and its Classical Limit Madhavan Varadarajan Raman Research Institute, Bangalore, India (work done in collaboration with Alok Laddha) TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.1

2 Plan Classical Parameterized Field Theory(PFT):Lagrangian, Hamiltonian Polymer Quantum Kinematics Quantum Dynamics Dirac Observables Physical States Spacetime Discreteness and Emergent Lattice Field Theory Fock from Polymer and the Continuum Limit Quantum Dynamics revisited TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.2

3 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. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.3

4 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 Gaugefix: X ± = t ± x deparameterize. get back standard flat spacetime free scalar field action. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.4

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7 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. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.7

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9 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 + TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.9

10 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. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.10

11 Quantum Kinematics: Matter Sector Holonomy: e i e l e e Y + Recallthat {Y + (x), Y + (y)}isnon-vanishing(=derivativeof delta function). WeylAlgebra : e i e l e Y + e e i l e Y + e e = exp[ i h 2 α( l, l)] e i (l e+l e e ) Y + e Chargenetwork: γ, l e i e l e Y + e γ, l = exp[ i h 2 α( l, l)] γ, l + l Rangeof l e :Modulosometechnicalities, l e Zǫ,ǫisanother Barbero-Immirizilikeparameterwithdimensions (ML) 1 2. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.11

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13 Dirac Observables Sinceggetransfactasdiffeos,theintegralover x [0, 2π]of any periodic scalar density constructed solely from the phase space variables, is an observable so that O ± f := S 1 dxy ± (x)f(x ± (x))isanobservableforrealperiodic f. Inpolymerrepn Y ± arenotgoodoperatorsonlytheir exponentials are. So cant construct a triangulation indep O ± f. But (expio ± f )canbeconstructed! exp dxy + (x)f(x + (x)) γ, k, l := e i h 2 α( f, l) γ, k,( l + f) where f e = f( hk e ). TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.13

14 Physical Hilbert Space by Group Averaging Ggeinvstatessatisfy Û ± (φ ± )Ψ = Ψ φ ±. Formally:Ψ = ψ,sumoveralldistinct ψ ggerelatedtp ψ. Sum not normalizable in kinematic Hilbert space. Better tothinkofsumofbras. γφ, k φ, l φ livesinspaceofdistributions: Distributions are linear maps from finite span of chrge nets intocomplexes.sumofinnerprodtofeachbrainsumona given chrge network ket is finite(most terms are zero). Grp averaging technique yields correct physical inner productonspaceofgrpaveragesofchargenets. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.14

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16 Finest Lattice TurnsoutthatGrpAverageofacertainfamilyofchargenets yields a superselected sector corresponding to a regular spacetime lattice a. Roughlyspeakingthefamilyisthatofallchargenetssuch that the difference of the embedding charges on successive edgesistheminimalpossiblei.e.on γ ±, hk ± e I+1 hk ± e I = ±a. ThespanofGrpAveragesofallchargenetsinthisfamilyis left invariant by the action of all the Dirac observables. Since the minimal possible increment, a, in the embedding chargesisusedtodefinethissector,thereisnostateoutside this sector corresponds to any finer discrete spacetime structure. Hence the name Finest Lattice. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.16

17 Emergence of Lattice Field Theory Restrict attention to Superselected Finest Lattice Sector True degrees of freedom encoded in Dirac Observables. Classical sympl structure is represented by commutators of basic kinematic operators. Dirac Observables are composite opertrs.sodonotexpect(donotget!)repnofclassical continuum sympl structure for Dirac Observables i.e. of true d.o.f. Recall:e io+ i f = e S 1 dxy + (x)f(x + (x)).donotgetrepnofclassical P.B.forallchoicesof f. However: Do get repn of classical sympl structure for those continuumfunctions f(x ± )whicharepiecewiseconstanti.e. constant on edges of dual lattice. Thus emergent sympl structure of true degrees of freedom that of lattice field theory. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.17

18 Canalsoshowthatdynamicsoftruedegreesoffreedomis that of lattice field theory. Remark:localisingsupportof ftooneduallatticeedgecan get lattice approximant to local field operator. Discrete (lattice) Fourier transform of this yields approximant to the creation-ann modes. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.18

19 Fock vacuum 2 point function Want polymer state which approximates behaviour of Fock 2 ptfunction.precisenatureofapproxisthrudefnof continuum limit. NotethattheB-Iparameters a, ǫdictatethesmallest increment of embedding,matter chrges. Also a is the lattice spacing.hencecontinuumlimitis a 0, ǫ 0. Accordingly we consider 2 parameter family of polymer quantizations with 2 parameter family of states and 2 parameter family of Dirac observables which are lattice approximant to the annihilation- creation modes. Require exp value of quadratic combinations of mode operators for wavelengths >> lattice spacing to approach Fock vacuum exp value in the continuum limit. We explicitly construct such a set of states. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.19

20 Summary We constructed Polymer quantization of PFT with following features: Absence of Ad-hoc Triangulations in oprtr actions Correct implementation of constraint algebra: Classical constraints ensure f oliation indep of the dynamics. Foliation indep implies a consistent spacetime dynamics. Correctly implemented, quantum constraints play similar role(kuchař). Quantum sptime covariance is tied to our faithful repn of grp of gge transformations. Continuum Limit: Crucial that limit is indep of h so seperation ofnotionsof quantumand continuum.limitnotof1 parameter set of ad- hoc triangulations in single quantiztn; rather, limit of 1(B-I) parameter family of unitarily inequivalent quantizations. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.20

21 Further research... Thiemann Quantzn and Sptime Covariance:We used density weight 2 constraints. Classically equiv to 1 sptl diffeo and 1 Hamiltonian constr. As in LQG solve sptl diffeo by averaging, imposehamconstrondiffinvdistributions(moreonthisif time...) Non- comp sptl toplgy: Related to CGHS model. Breaking of Local Lorentz Invariance: No dispersion due to ± seperation. Lattice breaks LLI. Eff theory? Lorentzian to Euclidean PFT?: etc.,etc... TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.21

22 Thiemann type quantization We use density wt 2 constraints: H ± (x) = [ Π ± (x)x ± (x) ± 1 4 (π f ± f ) 2 ]. Constraint algebra is Lie algebra Define C diff [ (x) = H + + H, ] C diff (x) = Π + (x)x + (x) + Π (x)x (x) + π f (x)f (x). Define C ham = 1 X + (x)x (x) (H+ H ) Constraint algebra is Dirac algebra, same algebra as for gravitywithsptlmetric=pullbackofflatsptimemetrictothe Cauchy slice. FirstimposesptldiffeoconstrthenHamconstronsptldiffinv distrbtns. TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.22

23 Results Can write 1 X + (x)x (x) asopertrthruthiemann-liketricks. Solns obtained here, trivially solve sptl diff constraint If straightfwd regulrztn done for Ham constr ala LQG, no soln ofoursissolntohamconstr! ThereisnontrivialregforwhichoursolnsaresolnsofHam constr! TheFreePolymerQuantumScalarFieldanditsClassicalLimit p.23