Polymer Parametrized field theory

Transcription

1 .... Polymer Parametrized field theory Alok Laddha Raman Research Institute, India Dec 1 Collaboration with : Madhavan Varadarajan (Raman Research Institute, India) Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 1 / 41

2 Motivation Major under construction issues in LQG are a satisfactory definition of quantum dynamics, extraction of gauge invariant physics and emergence of classical GR. Quantum geometry is discrete at kinematical level, but what is the precise sense in which the discreteness manifests itself once all the constraints are solved. If the underlying theory (in some suitable sense) is discrete, how does continuum classical GR emerge from it Rationale behind this work : General covariant Quantum field theories can elucidate some of these issues. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 2 / 41

3 Simplest possible field theory Consider a free massless scalar field theory on a Minkowskian cylinder (S 1 R, η) S[Φ] = 1 2 dx + dx η AB A Φ B Φ. (X +, X ) are the usual light-cone co-ordinates... Φ(X +, X ) = [Zero mode, Right mover = Φ (X + ), Left mover = Φ (X )] X + X Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 3 / 41

4 Only slide with too many equations.... canonical analysis of the theory involves choosing a 1-parameter family of spacelike embeddings (X + (x, t), X (x, t)) (X + (x), X (x)) define a space-like embedding of (S 1, x) into (S 1 R, η). [here x is the angular co-ordinate] Phase space minus zero mode : (Y + (x) = Φ (X + (x)), Y (x) = X + Φ (X (x))) X Hamiltonian : H(x) = (Y + ) 2 (x) + (Y ) 2 (x) {Y ±, Y ± } 0, {Y + (x), Y (x )} = 0. We will call these fields (Y +, Y ) right and left moving modes. They dont talk to each other once we forget about the zero-mode. This theory is not generally covariant Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 4 / 41

5 What is a PFT Extend the phase space : (X + (x), Π + (x), Y + (x)), (X (x), Π (x), Y (x)) The theory that we get from this extension is known as parametrized field theory (PFT) In ordinary field theory, we work on a fixed foliation, whereas in PFT all foliations are allowed. Physics(solution to equations of motion) is independent of choice of foliation, whence there is gauge in PFT Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 5 / 41

6 What is a PFT contd. This gauge is nothing but general covariance! = PFT has two first class constraints and no true Hamiltonian Note : we added two extra fields (X +, X ) but have two first-class constraints. Whence these fields are pure gauge, and the physical constent of the theory is just free mass-less scalar field propogating on a flat cylinder Note : (X +, X ) play a dual role in the theory. They are light-cone coordinates on (S 1 R, η) and are also dynamical fields which define spacelike embeddings of S 1 in (S 1 R, η). Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 6 / 41

7 What is a PFT contd. The two constraints are H + (x) = Π + X + (x) + (Y + ) 2 (x), H (x) = Π X (x) (Y ) 2 (x) {H + [N + ], H + [M + ]} = H + [[N +, M + ]] {H [N ], H [M ]} = H [[N, M ]], {H +, H } = 0 Here N +, N are vector fields on S 1. Note the Lie-algebraic nature of the constraint algebra. Main reason why the model is tractable. Tempting to think that the constraints generate two copies of diffeomorphisms on S 1. Not quite as all fields are periodic except, X ± (2π) X ± (0) = ±2π Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 7 / 41

8 What is a PFT contd. Two intepretation of gauge transformations (We only look at (+) constraint) (i) Familiar : Given a spacelike embedding (X + (x), X (x)), H + [N + ] evolves it to a new embedding (X + (x) + δx + (x), X (x)), and evolves Y + by the scalar field Hamiltonian, and keeps Y unchanged. (ii) New and important : N + (x) is a vector field on S 1, so we can think of it as a periodic vector field on R. Gauge transformations generated by H + [N + ] on (+)-fields can be interpreted as action of periodic diffeomorphisms on R generated by N +. Periodic diff. : If φ(x) = y then, φ(x + 2π) = y + 2π Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 8 / 41

9 Observables in the theory Gauge transformations are spatial diffeos (on R) = any integrated scalar density is an observable. x Y + (x)f (X + )(x), x Y (x)g(x )(x) are observables. Familiar examples are Fourier modes (a n, a m) As the reduced phase space is the covariant phase space of scalar field theory, (a n, a m) form a complete set So we have a generally covariant field theory where the constraint algebra is a true Lie algebra(feature of two dimensions) and a complete set of observables is available. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 9 / 41

10 Role of Poincare symmetries Free field theory on a flat cylinder is Poincare invariant. On cylinder there are no boosts, only translations. They translate (X +, X ), so in PFT they translate the embedding fields. Consider (X + (x), X (x)) (X + (x) + +, X (x) + ) These canonical transformations commute with the flow generated by constraints = These are the global symmetries of the theory (They map solutions to solutions.) Infact as the scalar field is massless, the full symmetry group is the conformal group. But as we will see, only the Poincare symmetries will survive in the quantum theory Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 10 / 41

11 Loop quantization We will focus on the right-moving(+) sector, and only mention the left-moving sector at crucial points. Note (in d=1) : scalar density = one-form, densitized vector field = scalar Π +, Y + are one-forms on S 1, (analogs of connection in LQG) X + is a scalar (densitized vector field), (analog of densitized triad) N + is a vector field. (analog of Shift vector) Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 11 / 41

12 Elementary variables In LQG our elementary variables are the cylindrical functions, f γ [A] and fluxes E i [S]. So in our case, we choose analog of cylindrical function for Π + and Y + and analog of flux for X +. Elementary variables By a graph γ we mean finite collection of closed intervals on S 1. Π + f γ, k [Π + ] := exp[i e k e Π e +] Y + g γ, l [Y + ] := exp[i e l e Y + ] e... X + X + (x) Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 12 / 41

13 Two Immirizi parameters Our elementary variables should be such that they separate points in phase space, for this k e, l e do not have to be reals, it suffices to take them as integers. Choose k e Za, l e Zɛ a, ɛ are analogs of the Immirizi parameter. These are dimensionful parameters which have been put in the definition of *-algebra that we will quantize. Some technicality : a := 2π N where N Z, N >> 1 From now on we will refer to k e as embedding charges, and l e as matter charges. (We will see why soon) Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 13 / 41

14 Action of finite gauge transformations Constructive Principle underlying Loop Quantization: In a generally covariant Hamiltonian system, consider a sub-algebra of the constraint algebra which is a true Lie-algebra. The elementary variables we want to quantize should be such that they behave covariantly under this sub-algebra. (Think of spatial diffeomorphisms in LQG) In our case the entire constraint algebra is a true Lie-algebra, and so our elementary variables should be covariant w.r.t all the constraints This is almost the case Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 14 / 41

15 Contd. Let (φ +, φ ) be finite transformations generated by (H + [N + ], H [N ]) then, (Recall : φ ± are periodic diffeomorphisms on R) φ ± diffeomorphism on S 1 for all fields except X ±. Let us denote the action of these gauge transformations on our elementary variables by α α φ +(f γ [Π + ]) = f φ + (γ)[π + ] α φ +(g γ [Y + ]) = g φ + (γ)[y + ] (2πm is explained on next slide) α φ +(X + )(x) = X + (φ + (x)) + 2πm φ acts trivially on right-moving(+) fields. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 15 / 41

16 Picture for φ + Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 16 / 41

17 The representation H kin Basic idea : We want to find a faithful rep. of the above algebra which carries unitary representation of the Gauge group. In LQG the canonical basis is the spin-network basis s = γ, j, c v Corresponding basis for PFT is given by s + = γ, {k e }, {l e } Similar basis for the left-moving sector Kinematical Hilbert space is a tensor product H + kin H kin As our states implicitly depend on two parameters ɛ and a, we infact have two parameter family of Hilbert spaces. We will have to use all of them when we do semi-classical physics. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 17 / 41

18 The rep. contd Recap : Elementary variables : (f γ,k [Π + ], g γ,l [Y + ], X + (x)) Let γ be some graph with an edge e.. ˆf e,k γ, (..., k e,...){l e } = γ, (..., k e + k,...), {l e } Action of ˆf e,k is exactly analogous to the action of holonomy operator in LQG ĝ e,l γ, {k e }, (..., l e,...) + = (phase factor) γ, {k e }, (..., l e + l,...) + Action of ĝ e,l is analogous to action of exponentiated flux times holonomy. Let x o e, ˆX + (x 0 ) s + = k e s + Action. of X + (x) is analogous to action of the flux operator... Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 18 / 41

19 The rep. contd. Y +, Π + are not well-defined operators but their exponentials are. The states s + are eigenstates of the Embedding operators ˆX + (x). We call them charge-network states. The data {k e } is quantum counter-part of classical embedding field X + (x) There is a nice spacetime picture for γ, {k + e }, {l + e } γ, {k e }, {l e }. We will see how this plays a crucial role in understanding the model later on. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 19 / 41

20 Spacetime interpretation of network states Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 20 / 41

21 The rep. contd One consequence : Classical spatial volume functional is dx X + (x)x (x) Corresponding operator is a self-adjoint operator with a pure point spectrum There is even a volume gap in the quantum theory : ( V ) min a (Discreteness of quantum geometry at kinematical level) Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 21 / 41

22 Unitary representation of the Gauge group Recall that in LQG spatial diffeomorphisms act unitarily by basically moving the graphs around Û(φ) γ, {j e }, {c v } = φ 1 (γ), {j φ 1 (e)}, {c φ 1 (v)} Situation (for all the constraints!) is similar in PFT Consider a finite canonical transformation (φ +, φ ) generated by the constraints. We will denote the corresponding unitary operators by (Û(φ+ ), Û (φ )). Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 22 / 41

23 Comparison with spatial diffeos in LQG Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 23 / 41

24 Space-time picture Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 24 / 41

25 Quantum observables So we see that there is no (triangulation or otherwise) ambiguity in the quantization of gauge transformations, but what about observables Recall : In the right-moving sector O F are observables. := dxy + (x)f (X + )(x) As Y + is not quantized on H kin = naive quantization of observables will involve choice of triangulation That in a weak-star topology, triangulation can be taken to infinity for the Hamiltonian constraint is a gift of nature Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 25 / 41

26 Quantum observables contd.. New type of observables... Try to quantize e io F. Intermediate steps do require a choice of triangulation However, in the end triangulation can be taken to infinity and a well-defined unambiguous operator exists! Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 26 / 41

27 Quantum observables contd. ê io F γ, {ke }, {l e } + = (phase factor) γ, {k e }, {l e + F (k e )} + So upto a (highly relevant) phase factor, these observables donot embedding charges k e but change matter charges. These observables are unitary operators, and strongly commute with Û(φ+ ) φ + : Û(φ + )e io F Û(φ + ) 1 = e io F These observables are independent of any choice of triangulation! Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 27 / 41

28 Represenation of the Poincare group Recall, in classical theory, (X + (x), X (x)) = (X + + +, X (x) + ) ˆV [ + ] γ, {k e }, {l e } = γ, {k e + + }, {l e } (1) and we have a similar action on ˆV [ ] on H kin. Unlike in classical theory the parameters ± Za, with ± min = a. This breaking of Poincare symmetries is rather similar to what happens in lattice field theories. (We dont have a lattice field theory, yet). Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 28 / 41

29 Recap We started with a massless scalar field theory on a cylinder cast in a form such that it has general co-variance In the hamiltonian formalism the constraint algebra is a true Lie algebra We saw that Loop quantization yields a (two parameter family of) (non-seperable) Hilbert space which carries a unitary rep. of classical gauge group There exists an algebra of classical observables {e io F } which can be quantized without any dependence on some triangulation. A discrete subgroup of the Poincare group is represented unitarily Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 29 / 41

30 Solving the quantum constraints In LQG solution to the diffeomorphism constraint is via a Rigging map η : S η [S] S [S] S Here [S] is the orbit of S under diffeomorphism group and η [S] represent a (huge) amount of ambiguity in the solution of the constraint. One can use the same technique to solve all the constraints in our model η + ( s + ) η ( s ) = η [s + ]η [s ] s + [s + ] s + s [s ] s [s ± ] denotes the set of all charge-networks which are gauge related to s ±. η [s + ] denotes the enormous amount of ambiguity in the definition of the Rigging map. We will see how to reduce it soon. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 30 / 41

31 Super-selection in physical Hilbert space H phy is non-separable If a subspace of (physical) Hilbert space remains invariant under action of all the observables, then we say that this subspace is super-selected and it suffices to restrict attention to this space. Recall : a = 2π = there is a finest sequence of embedding charges N (0,..., (N 1)a). Consider a kinematical subspace spanned by { γ, {k e } = (0,..., (N 1)a), {l e } }(i.e. fix a graph and the embedding data once and for all) Apply rigging map : η + ( γ, {k e } = (0,..., (N 1)a), {l e } ) This is a subspace of H phy. It can be shown that this subspace is invariant under the action of all the observables and the discrete Poincare group. We will call it H ss. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 31 / 41

32 Some pleasant technicalities H ss is separable. Once we demand that all the observables and the (discrete) Poincare symmetry should be well represented on H ss, the Rigging map ambiguities η [s] reduce to just one-parameter worth of ambiguity! η [s] = λ. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 32 / 41

33 Physics in super-selected space We will now argue that by working in H ss, we are working with a lattice field theory. Heuristically, as a charge-network corresponds is a discrete analog of spatial slice. Corresponding physical state (sum of all gauge-related states) will be a discrete approximant to the underlying spacetime, but there is a more precise way to see it. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 33 / 41

34 Physics in... Look at the algebra of observables e io F which should be well represented on this space. We require two things : (i) êio F should be well defined on H ss (ii) If we assume that there is a semi-classical state in this Hilbert space, then [êio F, ê io G ] semi = {e io F, e io G } This will happen iff F (k e)+f (k e+1 ) 2 F ( (ke+k e+1) 2 ) This condition tells us that the set of functions F for which Dirac observables are well defined on H ss is tied to the choice of embedding charges. F s are piece-wise constant over spacing ( X + ) = a. Hence test functions which generate the algebra cannot resolve spacetime at scales finer then a. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 34 / 41

35 On Poincare invariance Recall + Za Once we are in super-selected Hilbert space, nice things happen with global translations. ˆV [a]η( γ, (0,...(N 1)a), {l 1,..., l N } ) = η( γ, (0,...(N 1)a), {l N, l 1..., l N 1 } ) i.e. Action of ˆV on a physical state is same as a physical state with cyclically permuted matter charges. So if a state ψ is invariant under cyclic permutations of the matter charges then η( ψ ) is Poincare invariant There exist an infinite number of Poincare invariant states in the theory! Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 35 / 41

36 Recovering Continuum theory We have a background independent(all constraints moded out) lattice field theory. But classical theory is on a continuum Free scalar field theory on a flat spacetime can be quantized on a Fock space, and the Fock vacuum is a canonical semi-classical state So question we want to ask is, can we define a suitable approximant to Fock vacuum in H phy Only information we have is, (1) Approximation will be better as ɛ, a 0. (2) Our state should be Poincare invariant Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 36 / 41

37 Posing the problem Here s how we frame the question Fock vacuum requires a notion of creation and annihilation operators, these are not well defined operators in our theory, so define suitable approximants to them Look for a (two parameter family of) states Ψ = Ψ + Ψ which satisfy â n Ψ = 0 + error-terms n << 2π a By error-terms we mean, these terms should vanish in the limit ɛ, a 0. If we can find such a state, then we say that it approximates Fock vacuum for long wavelength modes of the scalar field Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 37 / 41

38 The solution Suitable approximants to â n are obtained by discretizing the classical a n = Y + (x)e inx + (x) on the spacetime lattice that we already have and quantizing the resulting expression. (This is the only place where we need a choice of triangulation) Ψ + = l exp[ ɛ2 a 2 f ({l e })] γ, (0,..., (N 1)a), l Where f is a Positive definite function of (l 1,..., l n ). Continuum limit : Fix an ɛ 0, a 0 once and for all. Let ɛ = ɛ 0 ( a a 0 ) with > 3 Then Ψ approximates Fock vacuum better and better in the limit a 0. Unfortunately I know of no intuitive way to understand this limit (I will come back to this point on the next slide) Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 38 / 41

39 In the Continuum limit Some interesting features worth noting are, The state η(ψ ) is (discrete) Poincare invariant.(as it should be) As ɛ, a 0, non-trivial contributions to â n Ψ comes from charges which are such that ɛ a (l e) = O(1). And there are some highly delicate cancellations between these contributions. The expectation value of the commutator < [â n, â m] > = nδ n,m Ψ 2 + error-terms n, m << A. (This is not too surprising as The weyl algebra of scalar field is faithfully represented on the kinematical Hilbert space) The reason why there is no nice way to understand this computation is that the physical interpretation of matter charges l e is missing. These are not eigen-values of any commuting set of operators. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 39 / 41

40 Summarise Two dimensional PFT is a perfect field theoretic model where the program of loop quantization has been completed All our constructions, quantization of gauge transformations, quantization of an algebra of obs. is free of any choice of triangulation We have a physical Hilbert space with a very small ambiguity in the definition of physical inner product Due to superselection, the final quantum theory we get is a theory of scalar field operators living on a regular lattice. There exist a state Ψ which in a precise sense approximates the Fock vacuum for long wavelength modes of the scalar field. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 40 / 41

41 Things to try Do PFT on R 2. Embeddings satisfy non-trivial boundary conditions X ± (x) x as x This will be a nice toy model for LQG on non-compact spaces. (Ei a (x) Ei,0 a (x) as x ) This model is also canonically related to the CGHS model A continuum path integral for PFT already exists (Madhavan 2004) The measure factor is an analog of FV measure for QG. Discretize this Path integral and get a spin-foam rep. for PFT Compare it with the physical inner product that we obtained in this work We quantized the flow generated by constraints. One could also do Dirac quantization with constraints or a Master constraint and see how physical Hilbert spaces compare. Alok Laddha (RRI) Polymer Parametrized field theory Dec 1 41 / 41