Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field
aa r X i v : . [ g r- q c ] A ug Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field
Christoph Charles ∗ Univ Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, F-69622, Villeurbanne, France (Dated: August 29, 2018)In order to study 3d loop quantum gravity coupled to matter, we consider a simplified modelof abelian quantum gravity, the so-called
U(1) model. Abelian gravity coupled to a scalar fieldshares a lot of commonalities with parameterized field theories. We use this to develop an exactquantization of the model. This is used to discuss solutions to various problems that plague eventhe 4d theory, namely the definition of an inverse metric and the role of the choice of representationfor the holonomy-flux algebra. Contents
I. Introduction II. Overview III. Classical model
IV. Quantization
V. Discussion & Future work VI. Conclusion Acknowledgements A. Details of the Hamiltonian analysis
B. Brackets between the ladder operators and the constraints References I. INTRODUCTION
In order to tackle the problem of quantum gravity, instead of studying the full theory of general relativity, it ispossible to study simpler models. One such model is pure 3d gravity, which describes a simplified universe with only2 spatial dimensions and 1 dimension of time and without matter. Since classical 3d gravity is a topological theory(it does not have local degrees of freedom), its quantum theory is much more tractable as was originally noticed byWitten [1]. Since then, the model has been studied in various other manners, including using Loop Quantum Gravitytechniques [2, 3]. Several directions can be considered from there. One could use the techniques developed to considera four-dimensional theory and therefore follow the LQG developments. Or it is possible to try and couple 3d gravityto matter, in order to get a more complete model. ∗ Electronic address: [email protected]
This last direction is however rather difficult since the main property of 3d gravity, namely its topological nature,is generically lost when coupling to matter. In the context of Loop Quantum Gravity, no complete model of 3dgravity coupled to matter, even a simple scalar field, is known [4] . This is partially due to difficulties in quantizingscalar fields in LQG [7–10], partially due to difficulties in constructing Dirac observables [11] but also simply to thedifficulties in writing the Hamiltonian constraints involving an inverse metric [12, 13].It does not mean that no reasonable conjecture is known. A surprising number of elements, at least from anLQG perspective [14–16], converge towards the idea that spacetime in 3d quantum gravity is best described by anon-commutative manifold when coupled to matter. In this regard, non-commutative field theory (see for instance[17]) would be the right effective field theory to describe quantum gravity phenomena, at least in three dimensions.This new non-commutative structure is particularly interesting because it seems to be specific to quantum gravityphenomena and as such, it does provide potential insights for studying the full 4d theory. Our goal in this paper istherefore to work towards the goal of developing a rigorous, non-perturbative theory of 3d quantum gravity coupledto matter (most probably just a scalar field) in the context of LQG. If such a theory can be developed, we will finallybe able to test the conjectures regarding the non-commutative structure of spacetime, at least in 3d.In this paper and as a first step in this project, we will study the quatum theory of matter coupled to 3d linear gravity. The linear term here refers to the fact that we will consider a simplification on the gravity side, by consideringan abelian gauge group (rather than the usual local Lorentz invariance). This model is inspired by Smolin’s remarkon the G → limit of gravity (where G is Newton’s constant) [18]. This model, called the U(1) model, correspondsto the usual linearized gravity theory but expressed in a diffeomorphism invariant manner. This simplification mightseem quite drastic, especially in 3d for which linearized gravity is quite trivial. Still, it does serve two purposes.First, pure 3d gravity, which has been studied so far, can be considered a simplification on the matter side. Here,we are trying to keep matter but rather simplify the gravity side in order to get new insights. Second, as we willsee, and perhaps unsurprisingly, this linear theory is exactly solvable and exactly quantizable (at least with a fewassumptions on the topology). The way it is solved however is interesting. Indeed, by writing every expressions in adiffeomorphism invariant manner, we will get formulas that are starting points for the full theory, either by deformingthem accordingly, or as initial point for a perturbative study. On top of these expected benefits, we will also getinteresting results and insights on how quantum matter and quantum spacetime interacts. In particular, our workreveals more precisely the role of the BF representation [19, 20] of the holonomy-flux algebra with respect to thesolutions of the theory but also the role of unconventional representations (inspired from [21–23]) in the constructionof the field operators.The main result of this paper is that, in this simplified setting of a scalar field coupled to 3d linear gravity, twosectors entirely decouple. One of the sector correspond to the matter sector. Its structure is exactly equivalent tothe free scalar field though expressed in a diffeomorphism invariant way. The second sector roughly corresponds togravity and is governed by equations similar to BF theory. This separation is possible because we can write theequivalent of creation and annihilation operators of the free field theory, with the additional property of commutingwith all the constraints. The first sector correspond to the states explored by the ladder operators while the secondsector correspond to the part on which the constraints act. This separation allows the definition of an explicit exact(though trivial) quantum theory. It is noteworthy however that the scalar field operators (the field operator and itscanonically conjugated momentum) cannot be expressed in the natural representations of the algebra we found, eventhough the ladder operators can. The problem is linked to the definition of the inverse of the determinant of thetriad, a problem widely encountered in LQG [12, 13]. It is possible to solve this problem in this simplified context byappealing to representations that are peaked on classical solutions of the Gauß constraints. This result might indicatea possible route for solving similar problems in non-linear or 4d theories.The paper is organized as follows. The first section gives a bird eye view on the ideas of the paper, staying quitegeneral but still giving more technical details than this introduction. The second section is devoted to the classicalstudy of the theory, in particular the decoupling of the two sectors classically. The third section is concerned withthe quantization of the theory. Two approaches are provided: the naive approach that correspond to the previousstudy and a second approach that allows the development of all the fundamental operators. Finally, the last sectiondiscusses various implications of the results with regard to future work. There is however substantial work trying to use matter as a clock [5, 6]. In that case, the scalar field is used to fix the gauge and theresulting theory is formulated as a diffeomorphism invariant theory. This actually evades the problem of Dirac observable we mentiona bit later.
II. OVERVIEW
The model we intend to study in the end is 3d quantum gravity coupled to matter. More specifically here, we wantto couple a scalar field to gravity in a quantum theory. For this, we can start from the standard action: S [ e, A, φ ] = Z S (cid:18) αǫ IJK e I ∧ F JK [ A ] + Λ6 ǫ IJK e I ∧ e J ∧ e K + 12 ⋆ d φ ∧ d φ + m ⋆ φ ∧ φ (cid:19) . (1)Here, S is the spacetime manifold. e is the triad. It is an R -valued -form that can be interpreted as an su (1 , -valued one using the Levi-Civita symbol. A is the spin connection. It is naturally an su (1 , -valued one form. F [ A ] isthen its curvature. φ is the scalar field. α , Λ and m are coupling constants. α contains the gravity coupling constant G and is, up to numerical factors G . Λ is the cosmological constant and m is the mass of the field. Finally, ⋆ is theHodge dual associated to the metric constructed out of the triad. We will choose the signature ( − + + +) , whichgoes with the sign in front of the mass term. There is a slight subtlety here. Normally, if g is the metric and ω is a p -form, then: ( ⋆ω ) µ ...µ n − p = 1 p ! p | det g | ω ν ...ν p ǫ ν ...ν p ρ ...ρ n − p g µ ρ ...g µ n − p ρ n − p . (2) ǫ µνρ is not a tensor here and is simply the Levi-Civita symbol (it is a tensor multiplied by a density). Namely, ǫ = 1 and all the other terms can be deduced by full anti-symmetry. But we have used the first order expression for theaction which uses det e and not the square-root of the determinant of the metric, which are equal only up to a sign.Here, we will rather use the following expression, which also solves the sign problem: ( ⋆ω ) µ ...µ n − p = 1 p !(det e ) ω ν ...ν p ǫ ν ...ν p ρ ...ρ n − p g µ ρ ...g µ n − p ρ n − p . (3)As we discussed, one can hope that this theory is exactly quantizable (or at least in some special cases like m = 0 ).It is however rather difficult because of a few road-blocks: • The gauge group is non-abelian. This leads to various difficulties when constructing well-defined version ofoperators. • The classical theory is not always solvable. For instance, a simple homogeneous scalar field coupled to 3dquantum gravity does not have an exact solution linking the volume of the universe to the value of the field.Though this is not an argument against the existence of a quantum version of the model exists, it is a noteworthydifficulty. • Even in the classical case of point particles coupled to 3d gravity, the exact solution is rather difficult toimplement and involves a lot of book-keeping. [33]The main idea of this paper is then to study a simpler model. We will study a scalar field coupled to linear gravity.This model is taken from Lee Smolin work [18]. It can be understood as a limit G → (that is α → ∞ ) of usualgravity with the additional constraint that AG (or αA ) is constant. This leads to the following (detailed) action: S [ e, A, φ ] = Z S h α ǫ IJK ǫ µνρ e Iµ ( ∂ ν A JKρ − ∂ ρ A JKν ) + Λ6 ǫ IJK ǫ µνρ e Iµ e Jν e Kρ − ǫ IJK ǫ µνρ e Iµ e Jν e Kρ (cid:0) e σM e τN η MN (cid:1) ∂ σ φ∂ τ φ − m ǫ IJK ǫ µνρ e Iµ e Jν e Kρ φ i d x. (4)In this writing, ǫ IJK is the standard Levi-Civita symbol. ǫ µνρ is not a tensor though, and follows the same conventionas the one we used for defining the Hodge star. Also, we have used the standard notation of e µI to write the inverseof the triad.In practice, we see that this amounts to removing the non-abelian term from the curvature of A . Everything else isleft untouched. This theory is particularly interesting because, while still diffeomorphism invariant, with some naturalconstraints, it is equivalent to the free scalar field. Indeed, assuming that S ≃ R , that the various fields behaveproperly at infinity (vanish quickly at infinity with their derivatives or converge at infinity for the triad), and thatthe triad is invertible everywhere (which we have more or less assumed when writing its inverse), then we can solvethe equations of motion. They are: • d e I = 0 for all I . This means that, since S is simply connected, there is a collection of fields Ψ I such that e I = dΨ I . • The usual equation of motion for the scalar field on a curved background: ⋆ d( ⋆ d φ ) − m φ = 0 . • For A , we get: α ǫ IJK ǫ µνρ F JKνρ [ A ] + (det e )Λ e µI = (det e ) (cid:20) e µI (cid:0) g στ ∂ σ φ∂ τ φ + m φ (cid:1) − e σI ∂ σ φ∂ µ φ (cid:21) . (5)This equation always has a solution as long as the right term has a vanishing divergence, which is just theconservation of energy.We see then, that A is completely fixed by the rest of the fields, that the equation on φ are correct as soon as we canshow that the space is flat. This is actually not always true. Indeed, all we have is: e I = dΨ I and e is invertible. Thistranslates to ǫ IJK dΨ I ∧ dΨ J ∧ dΨ K = 0 which means that the transformation from S to R encoded by Ψ is locally invertible. This sadly does not imply global invertibility. It should be noted however that this is part of the space ofsolutions. And when it is globally invertible, then is true that space is flat and we get the standard free field theory.So we still get something interesting: the free scalar field is an entire sector of our theory. At this stage, it isquite unclear if this sector can be quantized independently from the others, but it is surely a fair assumption. Wehave a theory, therefore, that is diffeomorphism invariant and still contains the free scalar field. We should noticehere similarities with parametrized field theory (PFT) [24–26]. And indeed, working with PFT really correspondsto directly working with Ψ I . Compared to PFT, in addition to using directly the triad, we will also develop newdirections for quantizing such a theory.As the goal at this point is to write the corresponding quantum theory, we should be able to find quantities more orless equivalent to the creation and annihilation operators in standard quantum field theory. Indeed, if the free scalarfield is an entire sector of the theory, this sector should be in correspondence with the usual solutions. We expect inparticular corresponding ladder operators acting in this sector, though these quantities should probably be amendedto accommodate the new symmetries.What do we expect? A nice way to look at this is to consider an even simpler theory. Let’s study a simple harmonicoscillator, that we can describe by the following action: S = Z (cid:18) m ˙ x − kx (cid:19) d t. (6)Let’s write this in a Hamiltonian manner. The momentum is: p = m ˙ x. (7)This leads to the following Hamiltonian: H = p m + kx . (8)If we define ω = q km , we can now write: H = p m + m ω x . (9)Now let’s define the complex quantity: a = r mω x + i p √ mω (10)And we finally have: H = ωaa. (11)It is now well-known that a and a becomes creation and annihilation operators in the quantum theory.Let’s now turn to a diffeomorphism invariant version of this problem, starting with: S = Z (cid:18) m ˙ x ˙ t − kx ˙ t (cid:19) d s, (12)where now t is a variable depending on the parameter s and all derivatives are taken with respect to s . A reparametriza-tion will leave the action invariant which is therefore promoted to a diffeomorphism invariant one. We now have twomomenta p x and p t . And a complete Hamiltonian analysis will reveal that they must now satisfy a (first class)constraint which is: p t + p x m + kx , (13)which is quite unsurprisingly the Shcrödinger equation (in its classical form). The interesting question though is canwe adapt the a quantity so that it commutes with this constraint?Yes we can. The commutator of the current a and our constraint is nearly zero already. In fact, the commutatorwith p t is zero but there is a constant (which is just the quanta of energy) for the second part. We must thereforeadd a term that does not commute with p t . There are various ways to do that. The most interesting to us, is to justconsider the time dependent expression for a . Indeed, a follows the following equation of motion: d a d t = − i ωa. (14)As a consequence: a ( t ) = (cid:18)r mω x + i p √ mω (cid:19) e − i ωt . (15)Taken without modification, and by interpreting the t as the conjugate to p t , this quantity directly commutes withthe constraint. This observation is what motivates our construction for the full system.Our goal will be to reexpress the usual creation and annihilation operators in standard quantum field theory, sothat the quantities linked to position and time can be reinterpreted in function of our new variables (the triad andthe connection). If such a quantity can be constructed, it is by definition equal to the creation and annihilationoperators when the gauge is fixed. But if it also commutes with the constraints, as our small study suggests, thenit is a gauge-unfixed version of these operators and are really the natural operators in the diffeomorphism invariantworld.What we need to do then, is to get the Hamiltonian version of our problem. Then we will need to extract all theinteresting operators as we just illustrated. This is what we do in the next section. III. CLASSICAL MODELA. Hamiltonian analysis
Ok, we now have the action we want to study. Let’s start the Hamiltonian analysis proper. There are variousmathematical difficulties we will just ignore for now. Namely, there are questions surrounding the behaviour of thefields at infinity or the various possible topologies for S the spacetime manifold. We will concentrate on the simplestpossibility. All the other possibilities will just create a richer theory for which we will have neglected various sectors.We will assume that S is homeomorphic to R . We will also assume that all the matter fields vanish at infinity.Granted all this, we choose some decomposition of S as R × Σ with corresponding coordinates ( t, σ ) . t will be ourtime variable and σ will be the coordinates on the spatial slice Σ . We do assume that Σ is homeomorphic (and evendiffeomorphic) to R but not necessarily a flat slice though. We also make the strong assumption that Σ is spacelikewith respect to the metric and nowhere degenerate. This last assumption is reasonable though as, in a hamiltoniananalysis, we are interested in parametrizing the space of solutions which should correspond to the variables on aCauchy slice of spacetime.This allows the following writing: S [ e, A, φ ] = Z R L d t, (16)with: L = Z Σ h α ǫ IJK ǫ µνρ e Iµ ( ∂ ν A JKρ − ∂ ρ A JKν ) + Λ6 ǫ IJK ǫ µνρ e Iµ e Jν e Kρ − ǫ IJK ǫ µνρ e Iµ e Jν e Kρ (cid:0) e σM e τN η MN (cid:1) ∂ σ φ∂ τ φ − m ǫ IJK ǫ µνρ e Iµ e Jν e Kρ φ i d σ. (17)From there, we proceed as usual: define the momenta, reverse the expressions that can be, keep the rest as primaryconstraints. The details of the computation can be found in appendix A. Once all this is done, we can write theLegendre transform of the Lagrangian which is the Hamiltonian.After some computations (detailed in the appendix), we finally get: H = Z Σ h ∂ A IJ B IJ + 12 ∂ A IJa (cid:0) B aIJ − αǫ IJK ǫ ab e Kb (cid:1) + X µI ∂ e Iµ − A JK (cid:0) − αǫ IJK ǫ ab ∂ b e Ia (cid:1) − e I (cid:16) αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I h cd ∂ c φ∂ d φ − m n I φ − n I h Π − n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ (cid:17)i d σ, (18)with the following primary constraints: X I = 0 ,B IJ = 0 ,X aI = 0 ,B aIJ = 2 αǫ IJK ǫ ab e Kb . (19)Here, summations on small latin indices cover only spatial coordinates. Capital latin indices do cover the dimensions. X is the natural conjugate with respect to e , B the conjugate with respect to A and Π the conjugate of φ . We havealso used the following notations in the Hamiltonian: • h ab is the induced metric on Σ and can be written as h ab = e Ia e Jb η IJ . Due to our assumptions, it is spacelike. h ab is the corresponding inverse metric. • n I is the natural normal to Σ . It is a vector valued density and reads: n I = ǫ IJK ǫ ab e Ja e Kb .From there, we can pursue the constraint analysis. After some lengthy, but straightforward, computations (seeappendix A), we get the following system of constraints: X I , B IJ , X aI , B aIJ − αǫ IJK ǫ ab e Kb , − αǫ IJK ǫ ab F JKab [ A ] − Λ n I + n I h cd ∂ c φ∂ d φ + m n I φ + n I h Π + n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ, αǫ IJK ǫ ab ∂ b e Ia . (20)It can then be separated into first and second class constraints. We get two sets of second class constraints which arethe equivalent of the simplicity constraints in 3d [27]: (cid:26) X aI , B aIJ − αǫ IJK ǫ ab e Kb . (21)And we get a system of first class constraints: X I , B IJ , ∂ b B bIJ , αǫ IJK ǫ ab F JKab [ A ] + Λ˜ n I − ˜ n I ˜ h cd ∂ c φ∂ d φ − m ˜ n I φ − ˜ n I h Π − ˜ n J η JK ǫ cd ǫ IKL ˜ e Ld det ˜ h Π ∂ c φ. (22)where the tilded quantitites are constructed out of B rather than e .This allows the computation of the Dirac brackets: { e I ( x ) , X J ( y ) } D = − δ IJ δ ( x − y ) , { A IJ ( x ) , B KL ( y ) } D = − ( δ IK δ JL − δ IL δ JK ) δ ( x − y ) , { A IJa ( x ) , e Kb ( y ) } D = α det h ǫ ab ǫ IJK δ ( x − y ) , { A IJa ( x ) , B bKL ( y ) } D = − δ ba ( δ IK δ JL − δ IL δ JK ) δ ( x − y ) , { φ ( x ) , Π( y ) } D = − δ ( x − y ) , (23)all other (non-fundamental) brackets being zero (including brackets dealing with X aI ). With these brackets, it israther obvious that the second class constraints commute with all the other constraints. Interestingly, they can besolved, and the system can finally be rewritten as: ( αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I h cd ∂ c φ∂ d φ − m n I φ − n I h Π − n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ, ǫ ab ∂ b e Ia , (24)with the following brackets: (cid:26) { A IJa ( x ) , e Kb ( y ) } = α det h ǫ ab ǫ IJK δ ( x − y ) , { φ ( x ) , Π( y ) } = − δ ( x − y ) . (25)The B variables have been removed thanks to the second class constraints and the time component variables havebeen removed as they decouple from the rest and can be trivially solved. We now have the Hamiltonian formulationof our problem.How is this theory supposed to be linked to the free field theory? It is quite obvious that the constraint on the triadreally carries the information that space is flat. There are a few subtleties linked to the problem of global invertibilitywe mentionned earlier but appart from this, it should be interpreted as the fact that the integral of e is a vectorthat embed of surface Σ into R . The second constraint is familiar in its form (it is really the Einstein equation) butonly set the value of the spin connection A . Apart from topological obstructions (which we avoided by choosing thesimplest case), this equation always has a solution. So, where is the dynamics of the field encoded?The point we have to remember is that the dynamics do not impose anything on a given Cauchy surface. As aconsequence, φ and Π are completely free. The only constraint will come from the evolution in time which should beencoded here as an action of the diffeomorphism constraints (they can be constructed out of the Einstein equation byprojecting using e and n ). Therefore, the dynamics is not encoded in a constraint per se but rather in their action.The constraint must be contained in the brackets with the curvature constraints. Because the equivalence has beenestablished using the equations of motion earlier, we won’t dwell into the equivalence here, which would require acareful analysis of possible gauge fixation. Rather, we will admit that this Hamiltonian theory should at least containthe free field theory and try from there to construct interesting quantities. In particular, we will study in the nextsection if it is possible to construct the equivalent of the creation and annihilation operators. B. Creation and annihilation operators
So we are looking for operators that should reduce in the correct gauge fixing to the standard creation and anni-hilation operator for the scalar field. In the diffeomorphism invariant context though, we expect them to commutewith the constraints but still preserve a nice algebra among them, as was suggested on our simple harmonic oscillatorstudy.The difficulty resides in that the space manifold Σ is not necessarily flat. The expression must therefore be adapted.We can go about two methods of construction. A first method would be to take advantage of the fact that Σ , thoughnot flat, is supposed to be a Cauchy surface. This means that the field in the entire spacetime can be reconstructedfrom Π and φ on the surface. The creation and annihilation operators could then be deduced as coefficient of theFourier transform. This method would actually work (and it will be explored in section III C to prove a couple ofinteresting properties) but is more complicated than necessary for now. A second idea is just to make a simple ansatzand check that the resulting operators have the correct algebra, among themselves but also with the constraints.Let’s go back to the standard free field theory for a moment. We have the following action: S = − Z (cid:0) η µν ∂ µ φ∂ ν φ + m φ (cid:1) d x d t. (26)This action leads to the following Hamiltonian: H = 12 Z (cid:16) Π + ( ~ ∇ φ ) + m φ (cid:17) d x, (27)where, once again Π is conjugate to φ . Normally, we define: a ~k = 1 p πω ~k Z (cid:0) ω ~k φ + iΠ (cid:1) exp (cid:16) − i ~k · ~x (cid:17) d x, (28)where ω ~k = p ~k + m . This allows the simple expression: H = Z ω ~k a ~k a ~k d k. (29)And of course, we have the well-known algebra: { a k , a k ′ } = i δ ( k − k ′ ) , { H, a k } = − i ω k a k , { H, a k } = i ω k a k . (30)Can we have a similar algebra with the coupling to linear gravity? The problem comes from the Hamiltonian whichno longer exists but is replaced by a collection of constraints. The curvature constraints (which contain the Einsteinequation projected on Σ ) are however local. We can show the problem with this in the non-gravitational case, bylooking at the commutator not with the Hamiltonian H but rather with H ( x ) = (cid:16) Π + ( ~ ∇ φ ) + m φ (cid:17) which isthe integrand. We get: { H ( x ) , a k } = 1 √ πω k (cid:16) − ~ ∇ φ · ~k − i m φ + ω k Π (cid:17) exp (cid:16) − i ~k · ~x (cid:17) . (31)The resulting expression is not integrated over space, depends on the derivatives of φ and cannot simply be expressedin terms of the creation and annihilation operators. How can we solve these problems?What must happen is similar to what we have seen in the case of the harmonic oscillator: the curvature of A in thecurvature constraint will not commute with the operators and will exactly compensate. This is possible if some partof the creation-annihilation operators uses the triad. The natural way to do this, is to use the integral of the triad asa position operator.So, let’s start from this kind of expressions: a k = Z ( f ( k, σ, e, A ) φ + g ( k, σ, e, A )Π) d σ. (32)This is just the most generic linear expression. Can we go further? Well somewhat yes. We want two additionnalproperties:1. The expression should be covariant with respect to local gauge transforms.2. The expression should be covariant (or even invariant) with respect to diffeomorphism transforms.Concerning the first point, we do expect some covariance. Basically, k should be expressed in some local referenceframe and when it is changed, k should change meaning some covariance for a k . In the linear gravity scenario though,the reference frames cannot change by gauge transform (an interpretation of this is that only infinitesimal changeshave been kept). We therefore expect full invariance. This leads to the simple condition that a k should commutewith the Gauß constraint ( d e = 0 ). As e is invariant under Gauß transforms, then this means that a k can depend on A only through its curvature.Something similar can be said for diffeomorphism invariance. In principle, in the full theory, we only expect somekind of covariance. One problem for instance is that the integral of (parallel transported) e depends on the path andso the annihilation operator could be linked to some integration path choice. In that case, diffeomorphism transformmight lead to some transformation of the operators. We are in the linear gravity case though. And in that case, it isway easier to solve. The integral of e does not depend on the choice of path (thanks to the Gauß constraint). So wecan make similarly the reasonnable assumption that a k should be invariant under diffeomorphism transforms.This leads to the following expression: a k = Z (cid:16) ˜ f ( k, σ, e, F [ A ]) φ + ˜ g ( k, σ, e, F [ A ])Π (cid:17) d σ. (33)with the additional constraint that a k commutes with the curvature constraints. We can make one additional as-sumption: that a k does not depend on A at all. This seems reasonable enough since we don’t really see how thiswould enter the equation anyway and the standard creation operator doesn’t have any dependence on curvature (atleast for scalars).So, we have the following working hypothesis. The annihilation operator has the following form: a k = Z ( h ( k, σ, e ) φ + h ( k, σ, e )Π) d σ. (34)And: { D I , a k } D = 0 . (35)A nice addition is to use our guess about the depency in the triad for the position operators.. We offer the followingansatz: a k = 1 √ π Z (cid:0) A ( k, e, σ ) k I n I φ + i B ( k, e, σ )Π (cid:1) e − i ~k · R σ ~e d σ. (36)This expression is directly inspired from the standard expression for the annihilation operator. Let’s explain a fewbits: • The factor k I n I is a density. This way A is a scalar. It might not be the right density to put (for instance p n I n I would work too) but this doesn’t matter since it can be corrected with the right expression for A (whichwould then be the ratio between two densities). It is a natural density to consider though since it very muchlooks like the energy component of k . • The integral term R σ ~e is a bit weird to say the least. First, ~e is simply the triad taken to be a vector-valuedone-form. Now the integral only has an end point of coordinates σ . But the fact that there is no start pointis actually important: we cannot take a specific point as reference. Indeed, the exponential of the triad createscurvature at one point and destroys it at the other. Here, we need an operator that only create curvatures at aspecific point.This operator really corresponds to the Ψ we encountered earlier such that dΨ I = e I . Because of this relationship with the triad, there is still a sense in which the difference of is an integral of the triad. By extension,we use this notation with only one end-point to the integral.There is a way to make this more rigorous for a non-compact spatial slice. Because, all the information iscontained in a Dirac bracket, we can consider the action of the integral as the start points goes to infinity.Though the integral is not well-defined, its Dirac bracket still exists and correspond exactly to what we need.It turns out that the correct values are: (cid:26) A ( k, e, σ ) = 1 ,B ( k, e, σ ) = 1 . (37)This leads to the following, and in fact quite familiar, expression: a k = 1 √ π Z (cid:0) k I n I φ + iΠ (cid:1) e − i ~k · R σ ~e d σ. (38)A lengthy - but not difficult - computation shows that indeed (see appendix B): { D I , a k } D = 0 . (39)More interestingly, the algebra of these operators can be computed explicitly. It requires some technology we willdevelop in the next section. C. Fourier transform and full algebra
A point must be underlined here: in usual free field theory, the creation and annihilation operators have a niceinterpretation as Fourier coefficients of the 3d field solution of the equation of motion. A similar property holds truehere, granted a few assumptions.Our spacetime is R (this was one of our simplifying assumptions). We also assumed that Σ (the space manifold)is homeomorphic to R . We will go a bit further here and assume that the embedding of Σ into R given by the There are other possibilities that reflect this though: for instance k I n I ( Q ) p n I n I where Q is some fixed reference point on the manifold.But once more, this can be done by adjusting A , thought this might be taken as some explicit dependancy on σ . So let’s not forget thispossibility later on. R ~e is a Cauchy surface for the free field theory. This assumption is reasonable: when we choosea slice Σ of spacetime, our goal is not to break diffeomorphism invariance but to parameterize the space of solutionsfor the problem. It is natural therefore to choose a Cauchy surface to do so. It is even natural to think that if wedon’t choose a Cauchy surface, the Hamiltonian analysis will not be well-defined. We will leave this question openhowever and just assume a correct choice of Σ .What we mean by this assumption is the following. Let φ : R R be a field that satisfies the standard free scalarfield equation: − ∂ t φ + ∆ φ − m φ = 0 . (40)Let’s now interpret Σ as a submanifold of R with embedding given by ~ Ψ = R ~e . We assume that knowing φ and itsderivative along the normal on this embedding is sufficient (and also necessary) to know φ on the whole R . Thismeans that we can now extend naturally some fields on Σ to the whole R spacetime.On the Σ slice, we have two fields we are interested in φ and Π . Π can naturally be connected to a derivative of φ in the time-direction (see appendix A): Π = − (det e ) g τ ∂ τ φ = − ~n · ~ ∇ φ. (41)Here, ~n is the normal density on Σ induced by the triad and ~ ∇ φ is the gradient of φ (as a spacetime field) expressedin the coordinates we used for the embedding. This means that φ and Π on Σ can naturally be extended to a fieldon the whole spacetime R . Now, we can use the Fourier transform as usual on R and get coefficients that will turnout to be the a k we defined earlier (up to some Dirac deltas factor). But of course, the formula will be more generaland apply to any couple of fields we might define on Σ .Now, let’s turn back to our expression for a k : a k = 1 √ π Z (cid:0) k I n I φ + iΠ (cid:1) e − i ~k · R σ ~e d σ. (42)Our claim is that, this is (up to a factor we will make explicit shortly) the Fourier coefficients for the extension of φ in R according to the previous rules. There is a rather simple way to check this thanks to linearity. We just have toconsider the case of: ( φ ( σ ) = A δ ( σ − σ ) √ det h , Π( σ ) = Bδ ( σ − σ ) . (43)We have put the determinant for φ , because φ is a scalar and we want A not to depend on the choice of coordinates. Π however is a density, and so to have B coordinate independent, the determinant factor should be avoided. In thatcase: a k = 1 √ π k I n I ( σ ) p det h ( σ ) A + i B ! e − i ~k · R σ ~e . (44)Let’s now consider a field Φ( x, t ) solution of the equation of motion in R . We can write it in a general form as follows: Φ( ~x ) = Z δ ( k + m ) b k e i ~k · ~x d k. (45)The b k are therefore the Fourier coefficients (up to a Dirac delta factor) of Φ . Let’s now consider the plane P goingthrough R σ ~e and tangent to Σ (or more precisely tangent to its embedding) at this point. This plane is spacelike andas such can be used as a Cauchy surface for the field Φ .There is always a Lorentz transformation sending (1 , , to the normalized normal of the plane P , granted thechosen orientation is the same (there is an infinite amount of such transformation but anyone will do, we can forinstance take a boost). Let’s note such a Lorentz transformation L . We can now write a parametrisation of the pointsof P as follows: ~x P ( ˜ X ) = −−−−−−→ L ⊲ (0 , ˜ X ) + Z σ ~e. (46)Here we chose the following notation: to a vector ~z can be associated a 2d spatial vector ˜ z and a time component z t .By extension, any 2d vector will be written ˜ w as we used for the coordinates on the plane denoted ˜ X . Also, ⊲ is usedto indicate the action of the Lorentz group onto 3d vectors. We can now write initial conditions on the plane P for Φ : ∀ ˜ X ∈ R , ( Φ( ~x P ( ˜ X )) = Aδ ( ˜ X ) , −−−−−−−−→ L ⊲ (1 , , · ~ ∇ Φ( ~x P ( ˜ X )) = Bδ ( ˜ X ) . (47)1These initial conditions correspond to the values of equation 43. Indeed, thanks to the Minkowski structure ofspacetime, nothing can propagate faster than light. With the conditions of equation 43, this translates to Φ( x ) = 0 for any point outside of the lightcone of the point at σ . Now, the transformations laws under diffeomorphism arecompletely local which guarantees that Φ is a Dirac delta on any Cauchy surface passing through σ . The fact that Φ is a scalar even gives the coefficient of transformation which is . We must however be careful, as the Dirac deltais a density, which is why the determinant is eaten up. A similar result holds for the derivative: it is zero nearlyeverywhere and locally can be expressed with respect to the gradient on Σ and Π . Because, we chose a surface tangentto Σ , the gradient does not appear and we can conclude.We can now use the standard derivation of b k in terms of A and B . Let ~k be a 3d vector with k + m = 0 and k t > . Then, we get: Z Φ( ~x P ( ˜ X ))e − i ~k · ~x P ( ˜ X ) d ˜ X = A e − i ~k · ~x P (0) . (48)We can also compute: Z Φ( ~x P ( ˜ X ))e − i ~k · ~x P ( ˜ X ) d ˜ X = Z Z δ (( k ′ ) + m ) b k ′ e i ~k ′ · ~x P ( ˜ X ) d k ′ e − i ~k · ~x P ( ˜ X ) d ˜ X = Z Z δ (( k ′ ) + m ) b k ′ e i ( L − ⊲ ( ~k ′ − ~k ) ) · ( L − ⊲~x P ( ˜ X ) )d k ′ d ˜ X = Z Z δ (( L ⊲ k ′ ) + m ) b L⊲k ′ e i( ~k ′ − L − ⊲~k ) · ( L − ⊲~x P ( ˜ X ) )d k ′ d ˜ X = Z Z δ (( k ′ ) + m ) b L⊲k ′ e i( ˜ k ′ − ( ˜ L − ⊲~k )) · ˜ X e i( ~k ′ − L − ⊲~k ) · ( L − ⊲~x P (˜0) )d k ′ d ˜ X = (2 π ) Z δ (( k ′ ) + m ) b L⊲k ′ δ ( ˜ k ′ − ( ˜ L − ⊲ ~k ))e i( ~k ′ − L − ⊲~k ) · ( L − ⊲~x P (˜0) )d k ′ = (2 π ) Z δ (cid:18) k ′ t − q ~k ′ + m (cid:19) + δ (cid:18) k ′ t + q ~k ′ + m (cid:19) | k ′ t | b L⊲k ′ δ ( ˜ k ′ − ( ˜ L − ⊲ ~k ))e i( ~k ′ − L − ⊲~k ) · ( L − ⊲~x P (˜0) )d k ′ . (49)This last line splits into two terms. For the first line, the main observation is that: δ k ′ t − q ~k ′ + m ! δ ( ˜ k ′ − ( ˜ L − ⊲ ~k )) = δ ( ~k ′ − L − ⊲ ~k ) (50)as there is a unique vector of square norm − m with given spatial support and with positive time component. Thesecond term is more involved. We get: δ (cid:18) k ′ t + q ( − ~k ′ ) + m (cid:19) δ ( ˜ k ′ − ( ˜ L − ⊲ ~k )) = δ ( ~k ′ − L − ⊲ ~k ) , (51)where ~x is the vector deduced from ~x by inverting its time component, namely ( − x t , ˜ x ) . This leads to: Z Φ( ~x P ( ˜ X ))e − i ~k · ~x P ( ˜ X ) d ˜ X = (2 π ) Z | k ′ t | δ ( ~k ′ − L − ⊲ ~k ) b L⊲k ′ e i( ~k ′ − L − ⊲~k ) · ( L − ⊲~x P (˜0) )d k ′ + (2 π ) Z | k ′ t | δ ( ~k ′ − L − ⊲ ~k ) b L⊲k ′ e i( ~k ′ − L − ⊲~k ) · ( L − ⊲~x P (˜0) )d k ′ = 2 π L ⊲ k ) t (cid:16) b k + b k e − ( L − ⊲k ) t ( L − ⊲~x P (˜0) ) t (cid:17) (52)Similarly, we can compute: Z −−−−−−−−→ L ⊲ (1 , , · ~ ∇ Φ( ~x P ( ˜ X ))e − i ~k · ~x P ( ˜ X ) d ˜ X = B e − i ~k · ~x P (0) , (53)2and also: − Z −−−−−−−→ L ⊲ (1 , , · ~ ∇ Φ( ~x P ( ˜ X ))e − i ~k · ~x P ( ˜ X ) d ˜ X = − π ( b k − b k e − ( L − ⊲k ) t ( L − ⊲~x P (˜0) ) t ) . (54)We can conclude: ( b k = π (cid:0) ( L − ⊲ k ) t A + iB (cid:1) e − i ~k · ~x P (0) ,b k = π (cid:0) ( L − ⊲ k ) t A − iB (cid:1) e − i ~k · ~x P (0) . (55)Now: ( L − ⊲ k ) t = ( L − ⊲ ~k ) · −−−−−→ (1 , , , )= ~k · ( L − ⊲ −−−−−→ (1 , , , ))= k I n I ( σ ) p det h ( σ ) , (56)where we used n divided by its norm as an expression for the normal to P . Thus: b k = π (cid:18) k I n I ( σ ) √ det h ( σ ) A + iB (cid:19) e − i ~k · ~x P (0) ,b k = π (cid:18) k I n I ( σ ) √ det h ( σ ) A − iB (cid:19) e − i ~k · ~x P (0) . (57)We can finally rewrite this in the more traditional manner: b k = π (cid:18) k I n I ( σ ) √ det h ( σ ) A + iB (cid:19) e − i ~k · ~x P (0) ,b − k = π (cid:18) − k I n I ( σ ) √ det h ( σ ) A − iB (cid:19) e i ~k · ~x P (0) . (58)And then: b k = sgn( k t ) √ π a k , (59)and this is true for any k such that k + m = 0 .All this means that, up to a numerical factor, the sign of k t and a Dirac delta, the a k coefficients really are theFourier coefficients of the field we get by specifying the initial conditions of Φ and Π on Σ embedded into R . This isespecially useful to compute the brackets between the a k coefficients. Let’s compute the following bracket: { δ ( k + m ) a k , δ ( k ′ + m ) a k ′ } = δ ( k + m ) δ ( k ′ + m ) { a k , a k ′ } = δ ( k + m ) δ ( k ′ + m ) 12 π Z Z { k I n I ( x ) φ ( x ) + iΠ( x ) , k ′ J n J ( y ) φ ( y ) + iΠ( y ) } e − i ~k · R x ~e − i ~k ′ · R y ~e d x d y = δ ( k + m ) δ ( k ′ + m ) i2 π Z Z (cid:0) − k I n I ( x ) δ ( x − y ) + k ′ J n J ( y ) δ ( x − y ) (cid:1) e − i ~k · R x ~e − i ~k ′ · R y ~e d x d y = δ ( k + m ) δ ( k ′ + m ) i2 π Z ( k ′ − k ) I n I e − i( ~k + ~k ′ ) · R x ~e d x. (60)Though this last form is pretty compact, it is better to expend it back a bit as follows: { δ ( k + m ) a k , δ ( k ′ + m ) a k ′ } = δ ( k ′ + m ) h δ ( k + m ) √ π R (cid:16) ( − i √ π e − i ~k ′ · R x ~e ) k I n I + i( √ π k ′ I n I e − i ~k ′ · R x ~e ) (cid:17) e − i ~k · R x ~e d x i (61)From what we just saw, the term in large square brackets is (up to a numerical factor and a sign) the Fourier coefficientof a field with initial values on Σ given by: ( φ = − i √ π e − i ~k ′ · R x ~e , Π = √ π k ′ I n I e − i ~k ′ · R x ~e . (62)3But we know such a field: it is simply the field Φ( x ) = − i √ π e − i k ′ · x on the whole R spacetime. And its Fouriertransform is proportional to a Dirac delta δ ( k + k ′ ) . From that, we conclude (with the factors correctly computed): { δ ( k + m ) a k , δ ( k ′ + m ) a k ′ } = − isgn( k t ) δ ( k ′ + m ) δ ( k + k ′ ) . (63)This is exactly the kind of algebra we wanted for creation-annihilation operators. It is correctly adapted to thediffeomorphism invariant case as no frame of reference can be preferred. Let’s note here that the sign is the reversefrom the usual since we have: a k = − a − k (64)with the extra sign coming from the fact that we put the sgn( k t ) factor out of a k . IV. QUANTIZATIONA. First approach
We can now turn to the quantization of the system. In principle, we should start with some natural construction ofthe algebra of observables, starting with canonical variables. This is however notoriously difficult for matter coupled togravity [8–10]. As a first approach, let’s avoid the usual difficulties by choosing another set of fundamental variables.The first point to note is that we have the creation and annihilation operators which are quite natural. They are forinstance used in the construction of the Fock space and it does make sense to keep them as fundamental. The secondpoint to note is that the creation and annihiliation operators, by construction, commute with the triad operators andwith the curvature constraints. They commute with the triad because they do not depend on the connection, and wedevoted a large part of this paper (see appendix B) to prove it commutes with the curvature constraints. Conversely,the triad operators and the curvature constraints are particularly interesting as fundamental variables since they areconjugate to each other. Finally, we have proven previously that the a k can be interpreted as Fourier coefficients(section III C), which means we can reconstruct (at least classically) the field phi and its momentum Π . This alsomeans that, classically, if we now the triad and the curvature constraints, we can reconstruct the curvature of theconnection everywhere. This is enough to reconstruct the spin connection up to a gauge. Therefore, the followingcollection: • a k for all k ∈ R such that k + m = 0 (which contains both creation and annihilation operators based on thesign of k ), • D I ( x ) for all I and x , • and e Ia ( x ) for all I , a and x gives a complete description of the gauge invariant phase-space. This collection divides into two sectors that commutewith each other and that, remarkably, we know how to quantize separately. The creation-annihilation algebra leadsto the well-known Fock quantization (with a few caveats). And the algebra of the curvature and triad operators canlead to a quantization around a state similar to the BF vacuum [19, 20] as we will shortly show.There is one important point to underline here: all this works only when restricting to the gauge-invariant subspaceof the phase space. It is not always possible to solve for this subspace explicitly, and it is not possible for the non-abelian case. In the abelian case however, not only is it possible, it greatly simplifies a number of expressions. Indeed,the algebra between the D I is only simple if the Gauß constraints is checked. The same thing holds for the bracketsbetween D I and a k which in all generality is linear in the Gauß constraints. In general then, we would have todeal with partial gauge-fixing, the choice of path and other niceties. And such a treatment will be necessary for thenon-abelian case. However, as a first approach, and when considering our simple linear theory, it is possible to avoidsuch consideration. And this is what will do in all the constructions from now on.Let’s start with the Fock quantization. We have shown that the creation-annihilation operators respect an algebrasimilar to the standard one. There is a caveat though, as this algebra is labeled by vectors in R (rather than R )but with the additional constraint of being on the mass shell. This corresponds to functions living on the two-sheethyperboloid, with the condition that reflection with respect to the origin gives rise to a complex conjugation.If we want to map this algebra onto the usual one, we have to project these functions over the hyperboloid ontothe plane R . This can be done quite easily (though not in a covariant way) by considering only one sheet of4the hyperboloid (the other one can be recovered by conjugation) and forgetting about the time component of themomentum k . For instance, let’s restrict to the k t > sheet. We can define: c ˜ k = a ( √ ˜ k + m , ˜ k ) . (65)The c operators now check an algebra that is even more familiar: { c ˜ k , c ˜ k ′ } = 0 , { c ˜ k , c ˜ k ′ } = 0 , { c ˜ k , c ˜ k ′ } = 2i p ˜ k + m δ (˜ k − ˜ k ′ ) . (66)We notice here an energy factor. This is due to the unusual convention used for the a as we did not divide by thesquare root of the energy. Though this was natural to preserve a covariant expression, this means that the square of a operators (that is N k = a † k a k ) does not count particles but rather directly counts energy quantas. From there, theusual Fock quantization is known. It is useful however, for the sake of completeness, to develop it in a language closerto our originally found algebra, that is with: { δ ( k + m ) a k , δ ( k ′ + m ) a k ′ } = − isgn( k t ) δ ( k ′ + m ) δ ( k + k ′ ) . (67)This will lead to a more covariant expression more suited to the quantum gravity problem.We must start with the one particle Hilbert space H . First let H be the two-sheet hyperboloid embedded in R defined by: t − x − y = m (68)where ( t, x, y ) are the coordinates in R . Now, H will be the space of functions from H into C equipped with thefollowing scalar product: h ψ | φ i = Z δ ( k + m ) ψ ( k ) φ ( k )d k. (69)This is the momentum representation for our one-particle. Because, we are interested in real valued fields, we willadd the following constraint: ∀ k ∈ R , ∀ φ ∈ H , φ ( k ) = − φ ( − k ) . (70)Note the minus sign corresponding to the fact that a k = − a − k . With this definition H is trivially a pre-Hilbertianspace. By choosing a plane in R to parametrize H , we get however that: h ψ | φ i = Z p ~k + m ψ ( k ) φ ( k )d k. (71)This shows that H is isomorphic to L ( R ) with the caveat that the wave-functions must be divided √ E in themapping. This factor is actually quite important as it appeared in our algebra for the a k and this will allow a simplerrepresentation of the creation-annihilation operators.Now, we define the following sequence of Hilbert spaces:1. H = C , the -particle Hilbert space, also called the vacuum Hilbert space,2. H = H , the -particle Hilbert space as previously explained.3. H n = Sym( H ⊗ n ) , for n ≥ , the symmetric part of the tensor product of n copies of H and represents the n -particle Hilbert space for bosonic particles..The Fock space H φ is defined by: H φ = M n ∈ N H n . (72)Now, we can define the creation and annihilation operators a k . There are two cases. First, let’s consider k suchthat k + m = 0 and k t < . We define ˆ a k by its restriction ˆ a k,n on H n . For n ≥ , we define ˆ b k,n : ˆ b k,n : ( H ⊗ n → H ⊗ ( n − | v i ⊗ | v i ⊗ · · · ⊗ | v n i 7→ √ n P ni =1 v i ( k ) | v i ⊗ | v i ⊗ · · · ⊗ d | v i i ⊗ · · · ⊗ | v n i (73)5As standard, d | v i i means that | v i i is omitted from the list. ˆ a k,n is the restriction of ˆ b k,n to H n . For n = 0 , we have: ˆ a k, : (cid:26) H → H v (74)which corresponds to the fact that the vacuum is annihilated by all annihilation operators.Similarly, we can define a k for k such as k + m = 0 and k t > . This will act in the (algebraic) dual spaces. Let’sdefine ˆ b k,n : ˆ b k,n : ( ( H ⋆ ) ⊗ n → ( H ⋆ ) ⊗ ( n +1) h v | ⊗ h v | ⊗ · · · ⊗ h v n | 7→ √ n +1 P n +1 i =1 h v | ⊗ h v | ⊗ · · · ⊗ h u | ⊗ h v i | ⊗ · · · ⊗ h v n | , (75)with: ∀| v i ∈ H , h u | v i = v ( k ) . (76) ˆ a k,n is the restriction of ˆ b k,n to H n . This concludes the matter sector.For the gravity sector, we have two sets of observables. We have the curvature constraints which, as long as wedon’t restrict to the constraint surface, are legitimate observables. We will write D I ( x ) from now on and rememberthat they are densities. And we have the triad e Ia ( x ) . They are not exactly conjugate. The conjugate arise when weintegrate them along a line (possibily starting from infinity as mentioned in section III B). Then R P ( σ ) e I is conjugateto D I ( x ) and commutes with the a operators. When we integrate, we loose some information. But it is remarkablethat we don’t loose gauge-invariant information: thanks to gauge-invariance, the integral of e only depends on theend-point of the integral. That means we completely characterize the subspace defined by de I = 0 . This is thissubspace that we will quantize.The curvature constraints D I ( x ) are densities while, the integral of the triad acts as a scalar function. This setup issimilar to Loop Quantum Gravity where conjugate quantities are carried by dual geometrical constructs. It is in factexactly equivalent to the usual Loop Quantum Gravity setup except that here, because we have used gauge-invariantquantities, the support is on surfaces and points rather than lines. As a first approach however, we will not quantizein the standard fashion - that is using the Ashtekar-Lewandowski representation or its equivalent - but will ratherconsider the equivalent of the BF representation [19, 20]. Indeed, we have two choices: either we start from a vacuumstate where e = 0 everywhere or we start with a vacuum state that has D I ( x ) = 0 everywhere. The second case is akinto the BF vacuum and is very relevant to our problem: this vacuum state is precisely the solution to the constraints.So let’s quickly sum up the construction in the abelian case.Let’s define the Hilbert space H G . Let R be the space of functions over Σ valued in R that are zero everywhereexcept for a finite number of points. Now H G is the space of square integrable functions over R equipped with thefollowing scalar product: h Ψ | Ψ i = X ~f ∈R Ψ ( ~f )Ψ ( ~f ) . (77)The sum is well-defined (though possibly infinite) thanks to the square integrable condition. Note that this spacecan be constructed by a projective limit (as it is standard in Loop Quantum Gravity). In that case, we would havefunctions depending on R labels for a finite number of points. Two functions with support on a different set ofpoints would be equivalent (regarding cylindrical consistency) if they do not depend on the labels of the points thatare no shared and if the dependency is the same for shared points. This is however not needed here thanks to thecombination of two properties. First, because we look at the gauge-invariant subspace, the support is points ratherthan graph, things are greatly simplified. And because the gauge group is abelian, much simpler expressions can begiven still. Nonetheless, the construction is similar in spirit: we have a normalized vacuum state which is: Ψ ( f ) = (cid:26) if f = ~ , otherwise. (78)Here ~ is understood to be the function that is constant over Σ and equal to the vector ~ . Then, excitations can beconstructed with the action of the exponential of the integrated triad (which we will construct shortly). The Hilbertspace is then the completion of the linear span of these excitations. This means that we have an Hilbertian basis givenby the indicator functions once more. A member Ψ f of the basis is given for each function f of R and is defined by: Ψ f ( g ) = (cid:26) if g = f, if g = f. (79)6The operator corresponding to D I ( x ) must be regularized. As D I ( x ) is a density, it is natural to consider thefollowing integrated quantities: R N ( x ) D I ( x )d σ where N is some test function. We will therefore define the operator ˆ D I [ N ] . It is defined by its action on the basis in the following manner: ˆ D I [ N ]Ψ f = X P ∈ Σ N ( P ) f ( P ) I ! Ψ f . (80)This action is not always well-defined but it is on a dense subset of the space (namely the span of states Ψ f withfunctions f that have finitely many non-zero points). We see here that the basis we constructed diagonalizes the ˆ D I [ N ] operator. Similarly, we can defined the exponentiated operator for the triad. We do not need to regularize thistime (except through the integral). Let ~k be in R and P on Σ . We define ˆ E ( ~k, P ) by its action of the basis: ˆ E ( ~k, P )Ψ f = Ψ ˜ f , (81)where: ˜ f ( Q ) = (cid:26) f ( Q ) if Q = P,f ( P ) + ~k if Q = P. (82)As such ˆ E ( ~k, P ) is the quantization of exp (cid:16) − i ~k · R P ~e (cid:17) .Note that the non-exponentiated version of the operator does not exist. In practice, this means we have used theBohr compactification of R for the values of the integrals. This can be seen by the fact that the dual (present ineigenvalues of the curvature constraints) is R equipped with a discrete topology. This trick is handy to circumvent theproblem of using non-compact groups. Sadly, the Bohr compactification is only injective for maximally almost periodicgroups which the gauge group of the non-abelian theory ( SU(1 , ) is not. This is what prevents the standard Ashtekar-Lewandowski construction for non-compact gauge group. It should be noted however that such an obstruction is notpresent for the BF vacuum [20]. It might very well be then, that the current construction generalizes to the non-abeliancase.Finally, the kinematical Hilbert space is simply H G ⊗ H φ with the operators naturally extended. The solution tothe constraints is simply: ( C Ψ ) ⊗ H φ ≃ H φ where Ψ is the vacuum for H G . It is trivial to see that this space isisomorphic to the standard Hilbert space for a free field theory. Though this construction is interesting to get a feelof how the theory works in the quantum realm, it is not satisfying on at least two accounts:1. First, it relies too much on a change of variable. Normally, to get a direct link with the classical theory, one wouldstart with canonical variables and represent them, and then try to express constraints and similar operators.Here, not only have we not done that, it is not even possible to express the original operators. For instance, it isincredibly difficult (if not outright impossible) to extract the curvature operator out of the constraints. Indeed,to do that, we require both the fields operators (which we don’t have) and the inverse of the metric (whichdoes not even exist as an operator). Similarly, the natural expression for the momentum operator for the fielddepends on the normal operator, which does not exist because of the Bohr compactification we used.2. Second, it relies heavily on the abelian structure of the theory. All this approach was only possible because wecan decouple completely two sectors that we might want to call the gravitational and the matter sector (thoughthe curvature cosntraint has a bit of matter in it). This is not something we can hope for in a non-abeliantheory. So the method is way too specific to our case.It does not mean it is not useful though: this acts as a guideline. We now know what the theory looks like and whatto expect from different constructions.The ideal construction however would start from the curvature operator, the triad and the field operators and thenget the constraints. At least, it should be possible to reconstruct all these operators. This is however not possible inour case. Indeed, the curvature operator (or the holonomy operator) appears only in the curvature operator for now.As a consequence, we will first need the scalar field operator and the momentum operator to be able to retrieve it.However, from the work done in section III C, we can use the Fourier transform in R to get expressions of φ and Π in terms of the creation and annihilation operators. We get: ( φ ( σ ) = R δ ( k + m ) sgn( k t ) √ π a k e i ~k · R σ ~e d k, Π( σ ) = R δ ( k + m )( ~k · ~n ) sgn( k t ) √ π a k e i ~k · R σ ~e d k. (83)7The expression of Π is particularly problematic as it relies on the existence of an operator for the normal n , whichdoes not exists in our representation.One might want to try and use the more standard Ashtekar-Lewandowski representation H AL . In that case, it ispossible to construct a normal operator n in a way similar to the area operator in LQG [28]. However, in that case,we face another problem: given a state of the form | i ⊗ | φ i ∈ H AL ⊗ H φ where | i is the AL vacuum and | φ i issome state in H φ , we have ˆΠ | i ⊗ | φ i = 0 irrespective of the state | φ i . This might be possible to cure, by forgettingabout classical expressions and rather concentrating on reproducing the algebra at the quantum level. This wouldbe however surprising since the expression for Π is quite regular involving only exponentials and polynomials in thetriad that commute among themselves and should not require regularization.We want to suggest another direction in this paper, that we will start exploring in the next section. Though, wedo not have a complete proof for a successful construction, the arguments we just laid out fail in this context. Thissolution, though it seems unnatural at first, has - in hindsight - geometrical justification. The idea is to use the workdone by Koslowski and Sahlmann [21–23] and to develop a representation peaked on a classical non-degenerate spatialmetric. Though perfect diffeomorphism invariance (for the vacuum) is lost, there is still a notion of diffeomorphismcovariance available and the geometrical interpretation we will offer justifies the choice of a particular background, atleast for abelian gravity. We develop this approach in the following section. B. Ashtekar-Lewandowski representation peaked on a classical vacuum
The difficulty we face is linked to the non-existence of non-exponentiated versions of the triad operators on theHilbert space. This is quite standard in Loop Quantum Gravity: the standard constructions only allow for oneoperator out of a conjugated pair to be defined, the other one is only defined through its exponentials. In the usualAshtekar-Lewandowski representation [16, 28, 29] for instance, the holonomy operators are well-defined but only theexponentiated versions are defined. In the BF representation defined by Dittrich et al. [19, 20], the triad is onlydefined through its exponentials, but some version of the logarithm of the holonomies are defined . In our case, wehave developed the equivalent of the BF representation, since the conjugate to the triad is defined. Moving to thestandard Ashtekar-Lewandowski representation will not help however. Indeed, our problem is not only linked withthe possibility of writing a simple triad operator but also the possibility of inverting it, at least to some extent aswe want to be able to write the inverse determinant of the spatial metric. And the usual Ashtekar-Lewandowskirepresentation does not allow for that (at least not in any known ways ) since the vacuum is degenerate everywhereand all the excited states are degenerate almost everywhere. If we want to write the inverse determinant, we willtherefore need a new representation of the holonomy-flux algebra (or of its equivalent in our case - since we consideredonly the gauge-invariant sector).It is noteworthy that some other representations have been discussed already in Loop Quantum Gravity, mostnotably [21–23]. This representation is very similar to the Ashtekar-Lewandowski representation, except the vacuumis not peaked on degenerate geometry but rather on a given classical metric. Of course, diffeomorphism invariance ofthe vacuum is lost, which explains how the LOST theorem [30] is evaded, and is replaced by a notion of diffeomorphismcovariance. This representation is however very interesting to us because the metric is everywhere non-degenerate forthe vacuum. Even for most of the excited states, the metric is non-degenerate and when it is not, it is only degenerateon a finite number of points. As long as we can reproduce the classical algebras correctly, this leads to very naturalexpressions for the inverse determinant of the metric. However, we have now traded another issue which is the choiceof the background metric, which seems a bit counter-productive with regard to the standard Loop Quantum Gravityapproach.Before tackling this problem however, let’s sum up Koslowski’s and Sahlmann’s approach in [21–23] and adaptit to our case. The construction uses the dual structure to the one we have done in section IV A. In the previousconstruction, the operators acting on surfaces (the constraints) were diagonal, and excitations were created by actingon points. Here, it is the reverse: the point operators are diagonal and the surface operators create the excitations.This means we need some projective techniques to deal with it correctly.We can define a Hilbert space H ∆ for a given triangulation ∆ of Σ . This Hilbert space is the completion of thespan of the basis given by R labels of the triangles that are non-zero for a only finite number of triangles. We canmake this precise in the following manner: let F ∆ be the space of functions for the triangles of ∆ into R such that There are in fact technical difficulties in this case because of the non-abelian nature of the gauge group. However, the limit for loopsgoing to zero is usually well-defined (though group-valued) and play the same role. Though Thiemann developed some ideas in this regard [12], there are severe questions on whether his approach is successful [13]. H ∆ are functions from F ∆ into C that are square integrable for: h ψ | φ i = X f ∈F ∆ ψ ( f ) φ ( f ) . (84)The full (continuous) Hilbert space is defined as: H KS = [ ∆ H ∆ ! . ∼ . (85)Here the union is a disjoint union over all possible triangulations of Σ . We must now define the equivalence relation ∼ .For this, we need the notion of a refinement of a triangulation. A triangulation ∆ ′ is a refinement of ∆ if for anytriangle in ∆ is the union of triangles in ∆ ′ . We can then map any function of F ∆ into F ∆ ′ . For f ∈ F ∆ , we define f ′ ∈ F ∆ ′ as: f ′ ( t ) = f ( T ) , with t ⊆ T. (86)Similarly, we can write extend a state ψ ∈ H ∆ into ψ ′ ∈ H ∆ ′ as follows: ψ ′ ( f ) = (cid:26) ψ ( g ) if g ′ = f, otherwise. (87)We can finally get to our equivalence relation necessary to define H KS . Two states ψ ∈ h ∆ and ψ ′ ∈ H ∆ ′ are equivalentif and only if there exists a refinement ∆ ′′ of both ∆ and ∆ ′ such that the extension of ψ and ψ ′ in H ∆ ′′ are identical.Note that if this is true, it is true for any refinement of both triangulations. Note also that there is always a refinementof both triangulations but there is no guarantee that the extension of ψ and ψ ′ will match.Up to this point, the definition actually follows the techniques of the BF vacuum in order to adapt the constructionto quantities carried by surfaces and points (rather than lines). But what will distinguish H KS from both the BFrepresentation and the standard AL representation is the construction of the operators.First, let’s start with the simplest operator: the integrated curvature constraint. Let ∆ be a triangulation of Σ and φ a function from the triangles into R non-zero only a finite number of triangles. If ∆ ′ is a refinement of ∆ , wedefine: [ e i D [ φ ] : (cid:26) H ∆ ′ → H ∆ ′ ψ ψ ′ (88)with: ψ ′ ( f ) = ψ ( f + φ ) . (89)The final sum is done by extending φ to ∆ ′ . This is standard action, completely equivalent, so far, to the one in theAL-representation. This action can be extended on coarser representation. It is compatible with the quotient andtherefore carries to whole space H KS .Second, we can consider the triad operator. This is done in two steps. As a first step, let ∆ be a triangulation.Let’s denote | ψ f i the state in H ∆ defined by: ψ f ( g ) = (cid:26) if f = g, otherwise, (90)with f ∈ F ∆ . These states form a (Hilbertian) basis of H ∆ . We can now define: d O [ φ ] | ψ f i = X σ ∈ Σ φ ( σ ) · f ( σ ) | ψ f i , (91)with φ is a function from Σ into R with finitely many non-zero values. Thus P σ ∈ Σ φ ( σ ) · f ( σ ) is understood asa sum over these finitely many values and f ( σ ) is the label for the triangle of ∆ that σ belongs to . We recognize In practice, this means that this sums is not well-defined if the point σ fulls on an edge or a vertex of the triangulation. This is notimportant for us as we can just reduce the domain of the operator. ˜ e : Σ → R . And consider the following operator: d e [ φ ] = Z φ · ˜ e + d O [ φ ] . (92)This operator trivially has the same algebra but is peaked on a classical configuration for the triad. This is the maindifference of the KS representation (compared to the usual AL one).Now, all this construction relies on a choice of background metric and even, to be more precise, a choice ofbackground triad. This choice seems arbitrary at first, but in our case there is a very natural way to select a class ofmetrics. Indeed, we have to remember that we are considering the gauge-invariant subspace which translates to thecondition: de I = 0 . (93)This condition entails that, if we restrict once more to a simply connected manifold, the triad derives from a potential Ψ I . This functions acts as an embedding of Σ into R (if the metric is invertible). But it also means that theintegrated triad is zero on any closed loops. And this is valid also on the vacuum state. This means that thebackground triad must satisfy all these conditions and in particular correspond to an embedding into R . Up totopological questions, that we have discarded as we are considering the simplest case, this means that the metric isfixed up to diffeomorphism. This entails in turn that the construction will indeed depend on the metric but once thediffeomorphism constraints will be enforced, diffeomorphism invariance will be restored in a way which is independentfrom the choice of the initial metric (as long as it is invertible). So, from now on, let’s just choose a backgroundembedding into R and use the triad that derives from it.Let’s turn back to the full representation, including the matter sector. Our goal was to able to write expressionslike: ( φ ( σ ) = R δ ( k + m ) sgn( k t ) √ π a k e i ~k · R σ ~e d k, Π( σ ) = R δ ( k + m )( ~k · ~n ) sgn( k t ) √ π a k e i ~k · R σ ~e d k. (94)This suggested that the gravity sector needed a new representation. The Fock space used for matter is howevercompletely equipped for such expressions. We will therefore rather keep it. This leads to the full Hilbert space: H Full = H KS ⊗ H φ . (95)Before moving to the next section, let’s make a final remark: though this representation gives natural inverse operators,in a sense, this does not matter. What matters is the algebra of the operators. In the end, we must find two naturalpairs of collections of operators, corresponding to the field and momentum operator on the one hand and to the triadand curvature operator on the other. Moreover, these operator should lead to expressions for the constraints thatmatch the previously found algebra. If the naive inversion fails, this will mean that this technique fails. This is whatin the end should guide such construction. And these tests are still to be done with the method we just suggested. V. DISCUSSION & FUTURE WORK
Granted the previous idea can be made to work, the natural question is whether this can be extended outside ofthe abelian theory. Indeed, the representation we chose depended on a background which, for the abelian case, canbe chosen naturally. This however depended on the resolution of the Gauß constraints. In the non-abelian case,such a procedure might not be that well-defined. A few points are encouraging though: this representation gives anatural understanding of how matter propagates on an (abelian) quantum spacetime. Indeed, as we mentioned earlyon in this paper, the theory we developed is, at least in some sector, equivalent to a free scalar field theory. Withsuch a theory, spacetime is completely classical. Our theory however is completely quantum mechanical, includingspacetime. On the constraint surface, the triad in particular is completely ill-defined (in a quantum mechanical sense)and only the curvature has a precise value. We might wonder how a field might propagate freely here. The answer,according to the construction we have just done, is simple: spacetime really is flat. The degeneracy of the triad doesnot come from a true quantum degeneracy but rather is caused by the superposition of all the states coming from theaction of the diffeomorphism constraints. The final state therefore is a superposition of classical flat space but seenfrom all possible coordinate systems. This is of course possible only because there are no local degrees of freedom in3d gravity. Though, it might be possible to extend these techniques to non-abelian 3d gravity, the implications are0not quite as clear for the 4d case. An interesting idea, that has been explored almost accidentally in the context ofcosmology (see for instance [31]) as a first approach, is that only local degrees of freedom (that is gravitational waves)are quantum in that sense.Let’s get back to the 3d problem. Even in that case, once we want to get to the full non-abelian theory, a fewroadblocks appear. One of the major problem is path-dependency. Indeed, we defined the following operator as acreation operator: a k = 1 √ π Z (cid:0) k I n I φ + iΠ (cid:1) e − i ~k · R σ ~e d σ. (96)There we used the integral R σ ~e which did not depend on the path chosen as long as the Gauß constraints weresatisfied. A natural extension to the non-abelian case would be: b k = 1 √ π Z (cid:0) k I n I φ + iΠ (cid:1) e − i ~k · R σ g⊲~e d σ, (97)where g is the holonomy of the connection along the integration path and acts as parallel transport. Though thisexpression is gauge-invariant, it depends on the path chosen for the integration, even when the Gauß constraintsare checked. This makes the correct generalization quite unclear. Two points should be underlined here however.First, similar problem have been dealt with in the construction of the BF representation and have been solved bya systematic choice of paths for gauge-fixing [32]. This is moreover close to book-keeping techniques needed forthe classical solution of the problem [33] which seems to support such an approach. Second, this problem can bepartially recovered in the abelian case, if one wants to define the theory more generally without imposing first theGauß constraints. This might be needed anyway to be able to check the brackets of all the quantum operators we areinterested in from the end of section IV B. This will therefore be an interesting intermediate step to consider.The abelian case also relied on the commutativity between the operators a k and the constraints D I . It would besurprising, to say the least, that such a setup could be possible in the non-abelian case, for the operators b k and thecorresponding constraints ˜ D I . Several scenarios can be envisioned, the most probable to our eyes though is that,though the b k will not commute with the constraints, it should still be possible to make them into the algebra ofcreation and annihilation operator for some non-commutative field theory. In that case, they would allow us to writea basis of states on which it is reasonable to to a perturbative study. Ideally of course, some exact cases could befound, like a m → limit, one-particle states or maybe some cosmological setup. In any case, the non-commutativityis not a problem as long as we can interpret it to be almost commutative in some limit. This, however, will only bepossible if we can develop the full set of operators φ , Π , e and A independently from the techniques we have employedin the commutative case. This means that one of the most important point moving forward is concluding the programopened by section IV B.Let’s mention one last point before wrapping up: the idea of studying the abelian theory as a starting point, possiblyfor perturbative expansion is not new and was originally introduced by Smolin [18]. In our case however, we wantedit in particular to be able to study the geometry of the quantum spacetime. According to Connes’work (for instance[34]), this is better encoded in the Dirac operator governing the propagation of fermions rather than just the metric.A similar approach would then start with fermions coupled to abelian gravity. This is however rather ill-defined atthe moment. Indeed, the gauge group does share the same topology as SU(1 , , making the distinctions betweenbosons and fermions less clear. Moreover, it is not completely straightforward how the abelian connection should becoupled to the fermions. This is therefore an interesting point to explore further in future work. VI. CONCLUSION
In this paper, we considered a simplified model for 3d quantum gravity coupled to a scalar field. The model wastaken from Smolin work [18], corresponds to a specific G → limit of standard 3d gravity, and can be formulated asstandard BF theory (coupled to a scalar field in our case) but with an abelian gauge group. In four dimensions, thiscorresponds to a linearization of gravity but still expressed in a diffeomorphism invariant way. In three dimensions,the theory is still topological, but the dynamics is simplified. We showed in this paper in particular that a full sector ofthe theory is completely equivalent to a free scalar field, the gravity field only being there to allow for a diffeomorphismcovariant formulation. This sector is actually fairly similar to what was already developed with parametrized fieldtheories [24–26], although in higher dimensions and with a different language.We showed furthermore that this equivalence with a free scalar field theory leads to the formulation of a creation-annihilation algebra of operators, even in a diffeomorphism invariant setting. This algebra can in principle be extended1to other sectors of the theory as long as the metric is everywhere invertible. Though the natural formulation is a bitdifferent due to diffeomorphism invariance, the algebra is completely equivalent to the standard one for the free scalarfield. The interesting point is that all these operators commute with the constraints for the abelian theory. Thismeans they allow the construction of a set of solutions of the constraints, and mirror the fact that the classical abeliangravity theory (coupled to a scalar field) is equivalent, at least in some sector, to the classical free scalar field theory.This also means that these expressions are a good starting point for studying the non-abelian theory, for instanceto try and quantize the theory perturbatively. This also allows the construction of a full quantization of the lineartheory based on these operators as new variables. The quantum theory splits into two sectors. One is the sector thatencodes the various excitation of the scalar field, and can be mapped one to one to the free scalar field theory. Thesecond can be understood as the gravity sectors that more or less decouples in this abelian theory. It can be mappedonto the BF theory and be solved exactly.The drawback of such an approach is that some natural operators do not exist or are extremely difficult to construct.In particular, the momentum operator for the scalar field, and the holonomy operator for the gravity field, requirethe definition of (non-exponentiated) triad operators and an inverse-metric operator. This implies in particular, thateven the canonical variables of the theory cannot be expressed simply or may be downright impossible to write. Thisis not really a specific problem of our approach: we used the equivalent of the BF representation in our constructionwhich has similar difficulties for constructing triad operators or inverse-triad operators. However, in our case, thesedifficulties become a problem when trying to write a correlation operator for the scalar field for instance, which isa quantity we will eventually want to be able to compute. Using the older and somewhat more standard Ashtekar-Lewandowski representation only partially solves the problem. If it is indeed possible to define a non-exponentiatedtriad operator, the fact that the metric is degenerate almost everywhere for almost all states create huge problemswith our approach which precisely requires the opposite. Moreover, natural expressions for the momentum operatorof the scalar field are pathological, even though they only require exponential and polynomial terms in the triads,which should not need any regularization for the quantum case.We offered a possible way out. Though the construction needs to be studied more thoroughly, the drawbacks ofthe previous two approaches disappear. The idea is to construct a representation peaked on a given classical statefor the spatial metric. This idea was explored by Koslowski and Sahlmann [21–23] as an equivalent to condensedstate around a classical configuration. Though strict diffeomorphism invariance of the vacuum was lost, a sens ofdiffeomorphism covariance can still be retained. However, if this breaking was natural in their case, it seems moredubious when studying the theory from a more fundamental standpoint. We showed however that a specific vacuumcan be selected using the Gauß constraints in the linear case and corresponds to a flat space. Because the vacuumis nowhere degenerate, all the problems with the previously mentioned representations are lifted. Interestingly, theconstruction also allows a very nice interpretation of how the spacetime on which a free scalar field propagates isrecovered in a setup where the triad is supposed to be completely degenerate in a quantum sense. In fact (in theabelian case), the classical spacetime is there all along and the degeneracy only comes from the superposition of allthe diffeomorphism equivalent way of describing the system.Finally, we left several questions open for further inquiry. Most notably, as we just said, the new representation weoffered should be studied further. Indeed, even though the straightforward problems have been lifted, the study ofthe construction of the full operator set is still to be done. We left it ou however because a full and complete studywould include a more complete treatment of the Gauß constraints which we just assume to be satisfied. Lifting thiscondition requires dealing with gauge fixing, choice of path when integrating, etc. These points must be consideredat some point as they are needed for the non-abelian theory but were left out of this first investigation. Similarly,we have left out all questions regarding the various possible sectors of the theory, the role of topology, the possiblerestrictions when considering compact spaces, etc. Though this is certainly worth investigating on its own merit, ourgoal was to get a first grap on how to develop a non-abelian theory. In this regard, though all this is very important,it will most probably be quite different when changing the Lorentz gauge group. Acknowledgements
I would like to thank Stefan Hohenegger for the numerous discussions that helped and guided this project, andwithout whom none of this would have been possible. I would also like to thank John Barrett for the variousconversations that launched the initial idea for this paper.2
Appendix A: Details of the Hamiltonian analysis1. Primary constraints and Hamiltonian
We have the following action as a starting point: S [ e, A, φ ] = Z S h α ǫ IJK ǫ µνρ e Iµ ( ∂ ν A JKρ − ∂ ρ A JKν ) + Λ6 ǫ IJK ǫ µνρ e Iµ e Jν e Kρ − ǫ IJK ǫ µνρ e Iµ e Jν e Kρ (cid:0) e σM e τN η MN (cid:1) ∂ σ φ∂ τ φ − m ǫ IJK ǫ µνρ e Iµ e Jν e Kρ φ i d x. (A1)Let’s start the hamiltonian analysis by choosing a integration manifold. We will simply choose S = R to avoid someproblems on compact manifolds and with non-trivial topology. We will though also neglect boundary terms, assumingnice behaviour at infinity.Let choose some decomposition of S as R × Σ with corresponding coordinates ( t, σ ) . t will be our time variable and σ will be the coordinates on the spatial slice Σ . We only assume that Σ is diffeomorphic to R but not that it is aflat slice.This allows the following writing: S [ e, A, φ ] = Z R L d t, (A2)with: L = Z Σ h α ǫ IJK ǫ µνρ e Iµ ( ∂ ν A JKρ − ∂ ρ A JKν ) + Λ6 ǫ IJK ǫ µνρ e Iµ e Jν e Kρ − ǫ IJK ǫ µνρ e Iµ e Jν e Kρ (cid:0) e σM e τN η MN (cid:1) ∂ σ φ∂ τ φ − m ǫ IJK ǫ µνρ e Iµ e Jν e Kρ φ i d σ. (A3)We can now define the various momenta.Let’s note B the momentum conjugated to A , X the momentum conjugated to e and Π the momentum conjugatedto φ . The definitions are: B µIJ ( σ ) ≡ δLδ ( ∂ A IJµ ( σ )) ,X µI ( σ ) ≡ δLδ ( ∂ e Iµ ( σ )) , Π( σ ) ≡ δLδ ( ∂ φ ( σ )) . (A4)Here, it is understood that ∂ means derivative with respect to the time variable t .This leads to our primary constraints. Let’s start with the easy ones: X µI = 0 . (A5)This comes from the fact that the action does not depend at all on the derviatives of e .Let’s now turn to the variable B . We must distinguish two cases. First, B is easy to study as no time derivate of A appears in the action. Therefore: B IJ = 0 . (A6)The story is a bit different for B a ( a = 0 ). Here we rather get: B aIJ = 2 αǫ KIJ ǫ µ a e Kµ = 2 αǫ IJK ǫ ab e Kb . (A7)There is no constraint on Π as the relation we get is invertible in ∂ φ . More precisely, we get: Π = − ǫ IJK ǫ µνρ e Iµ e Jν e Kρ (cid:0) e M e τN η MN (cid:1) ∂ τ φ = − (det e ) g τ ∂ τ φ. (A8)This can be inverted into: ∂ φ = − e ) g (cid:0) Π + (det e ) g a ∂ a φ (cid:1) . (A9)3We have assumed here that the metric is invertible.We can, at last, write the Hamiltonian which is defined as: H ≡ Z Σ (cid:18) B µIJ ∂ A IJµ + X µI ∂ e Iµ + Π ∂ φ (cid:19) d σ − L. (A10)Thanks to the constraints, most of the first terms vanish. We will only get the Π term, as well as the B a terms. Atthe end of the day, we must also make sure that the final expression does not depend on ∂ φ . We must therefore takesome time to rewrite L so that any time component is made explicit and not bulked together with the spatial ones.So let’s try to declutter L a bit: L = Z Σ h α ǫ IJK ǫ µνρ e Iµ ( ∂ ν A JKρ − ∂ ρ A JKν ) + Λ6 ǫ IJK ǫ µνρ e Iµ e Jν e Kρ − ǫ IJK ǫ µνρ e Iµ e Jν e Kρ (cid:0) e σM e τN η MN (cid:1) ∂ σ φ∂ τ φ − m ǫ IJK ǫ µνρ e Iµ e Jν e Kρ φ i d σ = Z Σ h e I α ǫ IJK ǫ ab ( ∂ a A JKb − ∂ b A JKa ) + αǫ IJK ǫ ab e Ib ∂ A JKa + αǫ IJK ǫ ab e Ia ∂ b A JK + e I Λ2 ǫ IJK ǫ ab e Ja e Kb − e I ǫ IJK ǫ ab e Ja e Kb g ∂ φ∂ φ − e I ǫ IJK ǫ ab e Ja e Kb (cid:0) e M e cN η MN (cid:1) ∂ φ∂ c φ − e I ǫ IJK ǫ ab e Ja e Kb (cid:0) e cM e dN η MN (cid:1) ∂ c φ∂ d φ − e I m ǫ IJK ǫ ab e Ja e Kb φ i d σ. (A11)Now assuming we can neglect the condition at the boundary (for example by asking all the fields to vanish at infinity),we can rewrite this a bit: L = Z Σ h ∂ A IJa (cid:0) αǫ IJK ǫ ab e Kb (cid:1) + 12 A JK (cid:0) − αǫ IJK ǫ ab ∂ b e Ia (cid:1) + e I (cid:16) αǫ IJK ǫ ab F JKab [ A ] + Λ2 ǫ IJK ǫ ab e Ja e Kb − ǫ IJK ǫ ab e Ja e Kb g ∂ φ∂ φ − ǫ IJK ǫ ab e Ja e Kb (cid:0) e M e cN η MN (cid:1) ∂ φ∂ c φ − ǫ IJK ǫ ab e Ja e Kb (cid:0) e cM e dN η MN (cid:1) ∂ c φ∂ d φ − m ǫ IJK ǫ ab e Ja e Kb φ (cid:17)i d σ. (A12)Let’s now define : n I = 12 ǫ IJK ǫ ab e Ja e Kb . (A13)This will allow the following more compact expression: L = Z Σ h ∂ A IJa (cid:0) αǫ IJK ǫ ab e Kb (cid:1) + 12 A JK (cid:0) − αǫ IJK ǫ ab ∂ b e Ia (cid:1) + e I (cid:16) αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I g ∂ φ∂ φ − n I (cid:0) e M e cN η MN (cid:1) ∂ φ∂ c φ − n I (cid:0) e cM e dN η MN (cid:1) ∂ c φ∂ d φ − m n I φ (cid:17)i d σ. (A14)4Let’s go back to the Hamiltonian. We have: H = Z Σ h ∂ A IJ B IJ + 12 ∂ A IJa (cid:0) B aIJ − αǫ IJK ǫ ab e Kb (cid:1) + X µI ∂ e Iµ + Π ∂ φ − A JK (cid:0) − αǫ IJK ǫ ab ∂ b e Ia (cid:1) − e I (cid:16) αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I g ∂ φ∂ φ − n I (cid:0) e M e cN η MN (cid:1) ∂ φ∂ c φ − n I (cid:0) e cM e dN η MN (cid:1) ∂ c φ∂ d φ − m n I φ (cid:17)i d σ. (A15)In this expression, we must now write ∂ φ in terms of Π using: ∂ φ = − e ) g (cid:0) Π + (det e ) g a ∂ a φ (cid:1) . (A16)Lets concentrate only on the relevant terms T : T ≡ Π ∂ φ + 12 e I n I g ∂ φ∂ φ + e I n I (cid:0) e M e cN η MN (cid:1) ∂ φ∂ c φ = − e ) g Π (cid:0) Π + (det e ) g a ∂ a φ (cid:1) + 12 e I n I g (cid:20) e ) g (cid:0) Π + (det e ) g a ∂ a φ (cid:1)(cid:21) − e I n I (cid:0) e M e cN η MN (cid:1) e ) g (cid:0) Π + (det e ) g a ∂ a φ (cid:1) ∂ c φ = − e ) g Π − g (cid:0) e M e cN η MN (cid:1) Π ∂ c φ + det e g g a g b ∂ a φ∂ b φ − det eg (cid:0) e M e cN η MN (cid:1) g a ∂ a φ∂ c φ = − e ) g Π − g c g Π ∂ c φ − det e g (cid:0) g a ∂ a φ (cid:1) . (A17)Let’s put this in one single package: H = Z Σ h ∂ A IJ B IJ + 12 ∂ A IJa (cid:0) B aIJ − αǫ IJK ǫ ab e Kb (cid:1) + X µI ∂ e Iµ − A JK (cid:0) − αǫ IJK ǫ ab ∂ b e Ia (cid:1) − e I (cid:16) αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I g cd ∂ c φ∂ d φ − m n I φ + n I g ( g a ∂ a φ ) (cid:17) − e ) g Π − g c g Π ∂ c φ i d σ. (A18)Note that: g cd − g c g d g = h cd (A19)where h cd denotes the inverse of the induced metric on Σ . In particular, it does not depend on e I . Similarly: e ) g = 1(det e ) det h det g = − det e det h , (A20)which is linear in e I . We can see therefore that every single one of the last terms is linear in e I . We can sum up thisin the following formula: H = Z Σ h ∂ A IJ B IJ + 12 ∂ A IJa (cid:0) B aIJ − αǫ IJK ǫ ab e Kb (cid:1) + X µI ∂ e Iµ − A JK (cid:0) − αǫ IJK ǫ ab ∂ b e Ia (cid:1) − e I (cid:16) αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I h cd ∂ c φ∂ d φ − m n I φ − n I h Π − n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ (cid:17)i d σ. (A21)5
2. Constraint analysis
So let’s start the constraint analysis. First, we must list all the constraints. The first constraints are the primaryconstraints. Explicitely, they read: X I = 0 ,B IJ = 0 ,X aI = 0 ,B aIJ = 2 αǫ IJK ǫ ab e Kb . (A22)Their Poisson bracket with the Hamiltonian must be zero on shell so that the constraints are conserved. We will beusing the following sign convention: { q, p } = − (A23)where q represents a fundamental variable ( φ , e or A ) and p the corresponding conjugated momentum ( Π , X or B ).Let’s study this. First: { X I , H } = − αǫ IJK ǫ ab F JKab [ A ] − Λ n I + 12 n I h cd ∂ c φ∂ d φ + m n I φ + n I h Π + n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ. (A24)We will simply write this quantity D I . No Lagrange multiplier appears here, so necessarily, D I = 0 , which is indeedthe curvature constraint. Similarly: { B IJ , H } = 2 αǫ IJK ǫ ab ∂ b e Ia . (A25)Here, we can identify some version of the Gauß constraint, which we will write G IJ = 0 .It is easy to see that no other constraint arise as the other commutators all involve Lagrange multipliers and canbe inverted. Therefore the system of equations is now: X I , B IJ , X aI , B aIJ − αǫ IJK ǫ ab e Kb , − αǫ IJK ǫ ab F JKab [ A ] − Λ n I + n I h cd ∂ c φ∂ d φ + m n I φ + n I h Π + n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ, αǫ IJK ǫ ab ∂ b e Ia . (A26)The first two constraints are obviously first class. The last four are not, but that does not mean we have found allthe first class constraints.It can be checked that the following constraint is first class: ∂ b B bIJ = 0 . (A27)It obvisouly commutes with every constraint and it is a constraint as a linear combination of the Gauß constraintfound so far and the simplicity constraint. Finally, it is quite obvious that: αǫ IJK ǫ ab F JKab [ A ] + Λ˜ n I −
12 ˜ n I ˜ h cd ∂ c φ∂ d φ − m n I φ − ˜ n I h Π − ˜ n J η JK ǫ cd ǫ IKL ˜ e Ld det ˜ h Π ∂ c φ = 0 (A28)where the tilded quantitites are constructed out of B (rather than e ), is a first class constraint.Counting the number of degrees of freedom, we find that necessarily, the last constraints are second class. That is: (cid:26) X aI , B aIJ − αǫ IJK ǫ ab e Kb , (A29)are second class. This allows the computation of the Dirac brackets: { e I ( x ) , X J ( y ) } D = − δ IJ δ ( x − y ) , { A IJ ( x ) , B KL ( y ) } D = − ( δ IK δ JL − δ IL δ JK ) δ ( x − y ) , { A IJa ( x ) , e Kb ( y ) } D = α det h ǫ ab ǫ IJK δ ( x − y ) , { A IJa ( x ) , B bKL ( y ) } D = − δ ba ( δ IK δ JL − δ IL δ JK ) δ ( x − y ) , { φ ( x ) , Π( y ) } D = − δ ( x − y ) , (A30)6all other (non-fundamental) brackets being zero (including brackets dealing with X aI ). With these brackets, it israther obvious that the second class constraints commute with all the other constraints. Interestingly, they can besolved, and the system can finally be rewritten as: ( αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I h cd ∂ c φ∂ d φ − m n I φ − n I h Π − n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ, ǫ ab ∂ b e Ia , (A31)with the following brackets: (cid:26) { A IJa ( x ) , e Kb ( y ) } = α det h ǫ ab ǫ IJK δ ( x − y ) , { φ ( x ) , Π( y ) } = − δ ( x − y ) . (A32)The B variables have been removed thanks to the second class constraints and the time component variables havebeen removed as they decouple from the rest and can be trivially solved. We now have the Hamiltonian formulationof our problem. This concludes this appendix. Appendix B: Brackets between the ladder operators and the constraints
In this appendix, we consider the bracket (using the Dirac bracket found in the previous appendix A) between thecurvature constraints and the would-be creation and annihilation operators. Namely, we want to compute { D I , a k } D (for which we will now drop the D index from now on) where: ( D I = αǫ IJK ǫ ab F JKab [ A ] + Λ n I − n I h cd ∂ c φ∂ d φ − m n I φ − n I h Π − n J η JK ǫ cd ǫ IKL e Ld det h Π ∂ c φ,a k = √ π R (cid:0) k I n I φ + s iΠ (cid:1) e − i ~k · R σ ~e d σ, (B1)where s is a sign to be determined. To deal with this problem properly, we will need to integrate D I with a test field.We will therefore compute the following bracket: { Z N I ( τ ) D I ( τ )d τ, a k } (B2)where both terms now have regular dependency on the variables and N I is the test field we just mentioned.In this bracket, we can distinguish three kinds of terms, when expanding D I . First, the bracket involving thecosmological constant term is trivial. Indeed, this terms only depends on the triad and as a k does not depend on A atall, the bracket is zero. Second, we have the bracket involving the curvature of A . This part of D I does not depend onthe matter field. As a consequence, only the dependence on the triad in a k will be of importance. Third, and finally,we will have the part of D I which involves the matter fields but does not involve the connection. And there only, thedependence on the matter fields in a k will be important for computing the brackets. The hope is of course that theselast two terms compensate. It is quite intuitive that it is possible since this would correspond to a k creating energyon the matter field and compensating by giving the correct curvature to satisfy the Einstein equation.Let’s start by computing the following bracket: A = { Z N I ( τ ) αǫ IJK ǫ ab F JKab [ A ]( τ )d τ, a k } . (B3)We have: A = Z N I ( τ ) α ǫ IJK ǫ ab { ∂ a A JKb ( τ ) − ∂ b A JKa ( τ ) , a k } d τ = Z N I ( τ ) αǫ IJK ǫ ab { ∂ a A JKb ( τ ) , a k } d τ = Z Z N I ( τ ) α √ π ǫ IJK ǫ ab (cid:16) φ ( σ ) { ∂ a A JKb ( τ ) , k L n L ( σ )e − i ~k · R σ ~e } + s iΠ( σ ) { ∂ a A JKb ( τ ) , e − i ~k · R σ ~e } (cid:17) d σ d τ = Z Z N I ( τ ) α √ π ǫ IJK ǫ ab (cid:16) φ ( σ ) k L { ∂ a A JKb ( τ ) , n L ( σ ) } e − i ~k · R σ ~e + ( k L n L ( σ ) φ ( σ ) + s iΠ( σ )) { ∂ a A JKb ( τ ) , e − i ~k · R σ ~e } (cid:17) d σ d τ (B4)7Let’s compute the two intermediary brackets. First, we have: { ∂ a A JKb ( τ ) , n L ( σ ) } = ∂∂τ a { A JKb ( τ ) , ǫ LMN ǫ cd e Mc ( σ ) e Nd ( σ ) } = ǫ LMN ǫ cd e Mc ( σ ) ∂∂τ a { A JKb ( τ ) , e Nd ( σ ) } = ǫ LMN ǫ cd e Mc ( σ ) ∂∂τ a (cid:18) α det h ( σ ) ǫ bd ( σ ) ǫ JKN δ ( τ − σ ) (cid:19) = 12 α det h ( σ ) ǫ bd ( σ ) ǫ NJK ǫ NLM ǫ cd e Mc ( σ ) ∂∂τ a ( δ ( τ − σ ))= 12 α det h ( σ ) h bb ′ ( σ ) ǫ cd h dd ′ ( σ ) ǫ b ′ d ′ ( δ JM δ KL − δ JL δ KM ) e Mc ( σ ) ∂∂τ a ( δ ( τ − σ ))= 12 α det h ( σ ) h bb ′ ( σ )(det h ( σ )) h cb ′ ( σ )( δ JM δ KL − δ JL δ KM ) e Mc ( σ ) ∂∂τ a ( δ ( τ − σ ))= 12 α ( δ JM δ KL − δ JL δ KM ) e Mb ( σ ) ∂∂τ a ( δ ( τ − σ ))= 12 α ( δ JL δ KM − δ JM δ KL ) e Mb ( σ ) ∂∂σ a ( δ ( τ − σ )) . (B5)We used the equality between the two derivatives for δ (up to a sign) on the last line to avoid the appearance ofderivatives of N I in the full bracket.Now, we also have (we include the initial ǫ for simplifications): ǫ IJK ǫ ab { ∂ a A JKb ( τ ) , e − i ~k · R σ ~e } = ǫ IJK ǫ ab ∂∂τ a { A JKb ( τ ) , e − i ~k · R σ ~e } = ǫ IJK ǫ ab Z ξ (cid:18) − Z σ i k P η P Q δ ( ξ − ζ ( s )) d ζ c d s d s (cid:19) e − i ~k · R σ ~e ∂∂τ a { A JKb ( τ ) , e Qc ( ξ ) } d ξ = ǫ IJK ǫ ab Z ξ (cid:18) − Z σ i k P η P Q δ ( ξ − ζ ( s )) d ζ c d s d s (cid:19) e − i ~k · R σ ~e × ∂∂τ a (cid:18) α det h ( ξ ) ǫ bc ( ξ ) ǫ JKQ δ ( τ − ξ ) (cid:19) d ξ = − α Z ξ (cid:18)Z σ i k P η P I δ ( ξ − ζ ( s )) d ζ a d s d s (cid:19) e − i ~k · R σ ~e ∂∂τ a ( δ ( τ − ξ )) d ξ = 1 α Z ξ (cid:18)Z σ i k P η P I δ ( ξ − ζ ( s )) d ζ a d s d s (cid:19) e − i ~k · R σ ~e ∂∂ξ a ( δ ( τ − ξ )) d ξ = − α Z ξ (cid:18)Z σ i k P η P I ∂∂ξ a ( δ ( ξ − ζ ( s ))) d ζ a d s d s (cid:19) e − i ~k · R σ ~e δ ( τ − ξ )d ξ = 1 α Z ξ (cid:18)Z σ i k P η P I dd s ( δ ( ξ − ζ ( s )))d s (cid:19) e − i ~k · R σ ~e δ ( τ − ξ )d ξ = i k P η P I α (cid:18)Z σ dd s ( δ ( τ − ζ ( s )))d s (cid:19) e − i ~k · R σ ~e = i k P η P I α δ ( τ − σ )e − i ~k · R σ ~e (B6)The last line should also contain an opposite contribution from the start point of the integral. To make this omissionrigorous, we have to consider that N I has compact support. In that case, once the start point is sufficiently far, itscontribution will always be zero. This however means that we have some restrictions on the distribution spaces wemight consider.8Let’s put all these computations together. We get: A = Z Z N I ( τ ) α √ π ǫ IJK ǫ ab (cid:16) φ ( σ ) k L { ∂ a A JKb ( τ ) , n L ( σ ) } e − i ~k · R σ ~e + ( k L n L ( σ ) φ ( σ ) + s iΠ( σ )) { ∂ a A JKb ( τ ) , e − i ~k · R σ ~e } (cid:17) d σ d τ = Z Z N I ( τ ) α √ π (cid:18) ǫ IJK ǫ ab φ ( σ ) k L α ( δ JL δ KM − δ JM δ KL ) e Mb ( σ ) ∂∂σ a ( δ ( τ − σ )) e − i ~k · R σ ~e + ( k L n L ( σ ) φ ( σ ) + s iΠ( σ )) i k P η P I α δ ( τ − σ )e i ~k · R σ ~e (cid:19) d σ d τ = Z Z N I ( τ ) √ π (cid:18) ǫ IJK ǫ ab φ ( σ ) k J e Kb ( σ ) ∂∂σ a ( δ ( τ − σ )) e − i ~k · R σ ~e (cid:19) d σ d τ + Z N I √ π (cid:16) ( k L n L φ + s iΠ)i k P η P I e − i ~k · R σ ~e (cid:17) d σ ≈ Z N I √ π (cid:0) ǫ IJK ǫ ab ( − ∂ a φ ) k J e Kb + ǫ IJK ǫ ab φk J e Kb (i k P η P Q e Qa ) + ( k L n L φ + s iΠ)i k P η P I (cid:1) e − i ~k · R σ ~e d σ ≈ Z N I √ π (cid:0) − ( ǫ IJK ǫ ab k J e Kb ) ∂ a φ + ( ǫ IJK ǫ ab η P Q k J k P e Kb e Qa + η P I k L k P n L )i φ − s ( η P I k P )Π (cid:1) e − i ~k · R σ ~e d σ, (B7)where everything with the ≈ is only true on-shell and more precisely when the Gauß constraints are verified.Let’s now turn to the second half of the computation: B = { R N I ( τ ) (cid:16) − n I ( τ ) h cd ( τ ) ∂ c φ ( τ ) ∂ d φ ( τ ) − m n I ( τ ) φ ( τ ) + n I ( τ )2 det h ( τ ) Π( τ ) + n J ( τ ) η JK ǫ cd ǫ IKL e Ld ( τ )det h ( τ ) Π( τ ) ∂ c φ ( τ ) (cid:17) d τ, a k } . (B8)Once more, let’s split this expression into simpler components. We will have: B = { Z N I ( τ ) (cid:18) − n I ( τ ) h cd ( τ ) ∂ c φ ( τ ) ∂ d φ ( τ ) (cid:19) d τ, a k } , (B9) B = { Z N I ( τ ) (cid:18) − m n I ( τ ) φ ( τ ) (cid:19) d τ, a k } , (B10) B = { Z N I ( τ ) (cid:18) − n I ( τ )2 det h ( τ ) Π( τ ) (cid:19) d τ, a k } , (B11) B = { Z N I ( τ ) (cid:18) − n J ( τ ) η JK ǫ cd ǫ IKL e Ld ( τ )det h ( τ ) Π( τ ) ∂ c φ ( τ ) (cid:19) d τ, a k } . (B12)Let’s compute each one of them separately, starting with B : B = { Z N I ( τ ) (cid:18) − n I ( τ ) h cd ( τ ) ∂ c φ ( τ ) ∂ d φ ( τ ) (cid:19) d τ, a k } = − Z N I ( τ ) n I ( τ ) h cd ( τ ) ∂ c φ ( τ ) { ∂ d φ ( τ ) , a k } d τ = − Z N I ( τ ) n I ( τ ) h cd ( τ ) ∂ c φ ( τ ) ∂∂τ d { φ ( τ ) , a k } d τ (B13)This calls for the following computation: { φ ( τ ) , a k } = { φ ( τ ) , √ π Z (cid:0) k I n I ( σ ) φ ( σ ) + s iΠ( σ ) (cid:1) e − i ~k · R σ ~e d σ } = i s √ π Z { φ ( τ ) , Π( σ ) } e − i ~k · R σ ~e d σ = − i s √ π Z δ ( τ − σ )e − i ~k · R σ ~e d σ = − i s √ π e − i ~k · R τ ~e (B14)9Putting it back into B , we get: B = − Z N I ( τ ) n I ( τ ) h cd ( τ ) ∂ c φ ( τ ) ∂∂τ d { φ ( τ ) , a k } d τ = − Z N I ( τ ) n I ( τ ) h cd ( τ ) ∂ c φ ( τ ) ∂∂τ d (cid:18) − i s √ π e − i ~k · R τ ~e (cid:19) d τ = s √ π Z N I n I h cd ∂ c φk P η P Q e Qd e − i ~k · R σ ~e d σ = Z N I √ π ( sη P Q k P h ab n I e Qb ) ∂ a φ e − i ~k · R σ ~e d σ, (B15)where the last expression was written in a form similar to that of A .Let’s now consider B : B = { Z N I ( τ ) (cid:18) − m n I ( τ ) φ ( τ ) (cid:19) d τ, a k } = − Z N I ( τ ) m n I ( τ ) φ ( τ ) { φ ( τ ) , a k } d τ = − Z N I ( τ ) m n I ( τ ) φ ( τ ) (cid:18) − s i √ π e − i ~k · R τ ~e (cid:19) d τ = Z N I √ π ( sm n I )i φ e − i ~k · R σ ~e d σ. (B16)Let’s turn to B : B = { Z N I ( τ ) (cid:18) − n I ( τ )2 det h ( τ ) Π( τ ) (cid:19) d τ, a k } = − Z N I ( τ ) n I ( τ )det h ( τ ) Π( τ ) { Π( τ ) , a k } d τ. (B17)We must now compute: { Π( τ ) , a k } = { Π( τ ) , √ π Z (cid:0) k I n I ( σ ) φ ( σ ) + s iΠ( σ ) (cid:1) e − i ~k · R σ ~e d σ } = 1 √ π Z k I n I ( σ ) { Π( τ ) , φ ( σ ) } e − i ~k · R σ ~e d σ = 1 √ π Z k I n I ( σ ) δ ( τ − σ )e − i ~k · R σ ~e d σ = k L n L ( τ ) √ π e − i ~k · R τ ~e . (B18)This gives: B = − Z N I ( τ ) n I ( τ )det h ( τ ) Π( τ ) { Π( τ ) , a k } d τ = − Z N I ( τ ) n I ( τ )det h ( τ ) Π( τ ) (cid:18) k L n L ( τ ) √ π e − i ~k · R τ ~e (cid:19) d τ = Z N I √ π (cid:18) − k L n L n I det h (cid:19) Πe − i ~k · R σ ~e d σ. (B19)0Finally, let’s turn to B : B = { Z N I ( τ ) (cid:18) − n J ( τ ) η JK ǫ cd ǫ IKL e Ld ( τ )det h ( τ ) Π( τ ) ∂ c φ ( τ ) (cid:19) d τ, a k } = − Z N I ( τ ) n J ( τ ) η JK ǫ cd ǫ IKL e Ld ( τ )det h ( τ ) (cid:18) { Π( τ ) , a k } ∂ c φ ( τ ) + Π( τ ) ∂∂τ c { φ ( τ ) , a k } (cid:19) d τ = − Z N I ( τ ) n J ( τ ) η JK ǫ cd ǫ IKL e Ld ( τ )det h ( τ ) (cid:18)(cid:18) k M n M ( τ ) √ π e − i ~k · R τ ~e (cid:19) ∂ c φ ( τ ) + Π( τ ) ∂∂τ c (cid:18) − s i √ π e − i ~k · R τ ~e (cid:19)(cid:19) d τ = Z N I √ π (cid:18)(cid:20) − n J η JK ǫ ad ǫ IKL e Ld det h k M n M (cid:21) ∂ a φ + (cid:20) s n J η JK ǫ cd ǫ IKL e Ld det h k P η P Q e Qc (cid:21) Π (cid:19) e − i ~k · R σ ~e d σ (B20)Before moving to the full expression, let’s try and simplify the terms in ∂ a φ on one side and Π on the other. First,for ∂ a φ , we have: C = sη P Q k P h ab n I e Qb − n J η JK ǫ ad ǫ IKL e Ld det h k M n M . (B21)And for, Π , we have: C = − k L n L n I det h + s n J η JK ǫ cd ǫ IKL e Ld det h k P η P Q e Qc . (B22) C is slightly simpler, let’s start with it. Indeed, we now we’d like to find − sη P I k P so that it exactly compensatesthe term in A . So let’s compute: η P I k P = δ JI η P J k P . (B23)We will now try to find another way to write δ JI . For this, let’s consider the tetrad d defined by, for all spatialdirections a , d Ia = e Ia and for the time direction, d I = η IJ n J √− n (where n is the Minkowski square of n I ). If the triad isnon-degenerate (which we assumed), d is invertible by construction and det d = −√− n . Therefore, we can write: δ JI = d Jµ d µI = d J d I + d Ja d aI = η JR n R √− n ǫ IMN ǫ cd e Mc e Nd −√− n ) + e Ja ǫ ab ǫ IMN d Mb d N ( −√− n )= η JR n R n I n + ǫ ab ǫ IMN e Ja e Mb η NL n L n = − η JR n R n I det h − ǫ ab ǫ IMN e Ja e Mb η NL n L det h . (B24)The last line uses det h = − n . Therefore: η P I k P = η P J δ JI k P = − η P J k P (cid:18) η JR n R n I det h + ǫ ab ǫ IMN e Ja e Mb η NL n L det h (cid:19) = − k L n L n I det h − ǫ ILK e Ld η KJ n J ǫ cd e Qc det h η P Q k P = − k L n L n I det h + ǫ IKL e Ld η JK n J ǫ cd e Qc det h η P Q k P (B25)And so, we get (for s = 1 ): C = sη P I k P , (B26)which is exactly what we wanted.1Let’s turn to C . Once more, we know what we would like. We would like to compensate the term − ǫ IJK ǫ ab k J e Kb coming from A . So, we would like C to be equal to the opposite. Once more, let’s start from the desired expression: ǫ IJK ǫ ab k J e Kb = ǫ IJK ǫ ab δ JS k S e Kb = − ǫ IJK ǫ ab (cid:18) η JR n R n S det h + ǫ cd ǫ SMN e Jc e Md η NL n L det h (cid:19) k S e Kb = − ǫ IJK ǫ ab η JR n R n S det h k S e Kb − ǫ IJK ǫ ab ǫ cd ǫ SMN e Jc e Md η NL n L det h k S e Kb = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M − ǫ IJK ǫ ab ǫ cd ǫ SMN e Jc e Md η NL n L det h k S e Kb = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M − ǫ IJK ǫ ab ǫ cd ǫ SMN e Jc e Md η NL ǫ LP Q ǫ ij e Pi e Qj h k S e Kb = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M − ǫ IJK ǫ ab ǫ cd ( η SQ η MP − η SP η MQ ) e Jc e Md ǫ ij e Pi e Qj h k S e Kb = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M − ǫ IJK ǫ ab ǫ cd η SQ e Jc ǫ ij h id e Qj det h k S e Kb = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M + ǫ IJK ǫ ab η SQ e Jc h jc e Qj k S e Kb = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M + ǫ IJK e Jc ′ e Kb ′ ǫ b ′ c ′ ǫ bc h ǫ ab η SQ h jc e Qj k S = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M − ǫ dc det h n I ǫ ad η P Q h bc e Qb k P = − n J η JK ǫ ad ǫ IKL e Ld det h k M n M + n I η P Q h ba e Qb k P (B27)And so, we get (once more for s = 1 ): C = ǫ IJK ǫ ab k J e Kb (B28)which is once again what we wanted.The only thing that remains is the term in φ . This time it is more natural to look at the term in A and try to getthe necessary term to compensate in B , namely to compensate sm n I . We have: D = ǫ IJK ǫ ab η P Q k J k P e Kb e Qa + η P I k L k P n L = ǫ IJK η P Q k J k P ǫ ab e K ′ b e Q ′ a δ KK ′ δ QQ ′ − δ QK ′ δ KQ ′ η P I k L k P n L = ǫ IJK η P Q k J k P ǫ ab e K ′ b e Q ′ a ǫ LKQ ǫ LQ ′ K ′ η P I k L k P n L = ǫ IJK η P Q k J k P n L ǫ LKQ + η P I k L k P n L = ( δ LI δ QJ − δ LJ δ QI ) η P Q k J k P n L + η P I k L k P n L = η P J k J k P n I − η P I k J k P n J + η P I k L k P n L = k n I = − m n I (B29)which is indeed − sm n I for s = 1 . Putting all this together, we do get: A + B + B + B + B ≈ . (B30)Or to put it in the original question terms: { Z N I ( τ ) D I ( τ )d τ, a k } ≈ (B31)2if s = 1 . It is to be noted that this result holds on-shell, when the Gauß constraint is checked. Otherwise, the bracketis linear in the Gauß constraints. [1] E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. 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