Fock representation of gravitational boundary modes and the discreteness of the area spectrum
aa r X i v : . [ g r- q c ] J un Fock representation of gravitational boundary modes andthe discreteness of the area spectrum
Wolfgang Wieland
Perimeter Institute for Theoretical Physics31 Caroline Street NorthWaterloo, ON N2L 2Y5, Canada
May 2017
Abstract
In this article, we study the quantum theory of gravitational boundary modeson a null surface. These boundary modes are given by a spinor and a spinor-valued two-form, which enter the gravitational boundary term for self-dualgravity. Using a Fock representation, we quantise the boundary fields, andshow that the area of a two-dimensional cross section turns into the differenceof two number operators. The spectrum is discrete, and it agrees with theone known from loop quantum gravity with the correct dependence on theBarbero – Immirzi parameter. No discrete structures (such as spin networkfunctions, or triangulations of space) are ever required — the entire derivationhappens at the level of the continuum theory. In addition, the area spectrumis manifestly Lorentz invariant.
Contents1 Introduction 22 Boundary and corner terms for self-dual variables 43 Landau quantisation of area 84 Topological quantisation 175 Summary and conclusion 23 . Introduction In loop gravity, the quantum states of the gravitational field are builtfrom superpositions of spin network functions, which consist of gravitationalWilson lines for an SU (2) (respectively SL (2 , C ) ) spin connection A ABa .Wherever the Wilson lines meet, all free indices of the parallel transport [Pexp( − R γ A )] AB must be saturated and contracted with an invariant tensor(an intertwiner). Otherwise gauge invariance is violated.In the presence of inner boundaries the situation is different. The Wil-son lines can now have open ends at the boundary, where they create a sur-face charge, namely a spinor-valued surface operator ˆ π A . Gauge invarianceis restored when both the connection and the boundary spinors transformaccordingly. Suppose now that there are N such punctures that carry N spinors ˆ π A , . . . , ˆ π NA (see figure 1), such that we can introduce the followingspinor-valued surface density ˆ π A ( z ) = N X i =1 ˆ π iA δ (2) ( z i , z ) , (1)where δ (2) ( · , · ) is the two-dimensional Dirac distribution at the boundary. Inthe N → ∞ continuum and ~ → semi-classical limit this surface densitywill define a classical field π A ( z ) . What is the geometric significance of thissurface density in general relativity?The answer becomes most obvious when considering self-dual (complex)gravity [6]. The action in the bulk is given by the BF topological actionplus a constraint, namely S M [Σ , A, Ψ] = i8 πG Z M Σ AB ∧ F AB [ A ] − Ψ ABCD Σ AB ∧ Σ CD , (2)where Σ AB is the self-dual Plebański two-form, F AB is the curvature ofthe self-dual connection and Ψ ABCD = Ψ ( ABCD ) is a spin (2 , Lagrangemultiplier (the Weyl spinor) imposing Σ ( AB ∧ Σ CD ) = 0 (the simplicity con-straint). If we want to consider a manifold with boundaries (at, say, largebut finite distance) boundary terms have to be added, otherwise the varia-tional problem is ill-posed. The defining feature of the self-dual action (2)is that all fields carry only unprimed (left-handed) indices A, B, C, . . . . Is The coupling of spin networks to boundaries was first studied in the context of nullsurfaces that satisfy the isolated horizon boundary conditions [1–3], in our case no suchrestrictions are required [4, 5]. SL (2 , C ) gauge invariant boundary term that has this featureas well (i.e. contains only left-handed fields)? In the case of null boundaries,such a boundary term exists [4, 5], and its existence relies on the followingobservation: The pull-back Σ ABab ←− of the self-dual two-form to a null bound-ary can be written always as a symmetrised tensor product of a spinor ℓ A and a spinor-valued two-form η Aab , which are both intrinsic to the bound-ary. An SL (2 , C ) gauge invariant boundary term can be then introducedquite immediately, and it is simply given by the three-dimensional boundaryintegral S ∂ M [ η , ℓ | A ] = i8 πG Z ∂ M η A ∧ Dℓ A , (3)where Dℓ A = d ℓ A + A AB ℓ B is the exterior covariant derivative of ℓ A . Theorigin of this boundary term is further explained in section 2, see also [4, 5]for further references.Notice then that the boundary term (3) is essentially an integral over asymplectic potential. Performing a decomposition along the null bound-ary, we can identify canonical variables at the boundary. The configurationvariable is given by the spinor ℓ A (a null flag), its canonically conjugatemomentum is a spinor-valued surface density π A = i16 πG ˆ ǫ ab η Aab , (4)where ˆ ǫ ab is the Levi-Civita density (a tensor-valued density) on a cross-section of the boundary.The purpose of this paper is to give a more thorough analysis of theseboundary variables, in both classical and quantum gravity. First of all(section 2), we explain the geometric origin of the boundary fields ℓ A and η Aab for a four-dimensional causal diamond, whose boundary is null (seefigure 2 for an illustration). Next, we introduce the appropriate boundaryand corner terms. We then show that the boundary action (3) contributesa corner term to the pre-symplectic potential on a space-like three-surface,which intersects the boundary transversally. Section 3 deals with the quan-tum theory. Starting from the boundary fields, we construct four pairs ofharmonic oscillators and define the corresponding Fock vacuum. Upon in-troducing the Barbero – Immirzi parameter, we can then write the oriented area of the two-dimensional corner as the difference of two number oper-ators. The spectrum is discrete and matches (up to ordering ambiguities)the loop quantum gravity area spectrum [7, 8]. The result is obtained with-out ever introducing discrete structures, such as spin network functions or3riangulations of space. In addition, the derivation is manifestly Lorentz in-variant. Finally (section 4), we explain the compatibility of the result withloop gravity in the spin network representation.The paper is part of a wider effort [4, 5, 9] to understand null surfaces,causal structures and internal boundaries in non-perturbative and canonicalquantum gravity in terms of the spinorial representation of loop quantumgravity [10–12]. A similar formalism using metric variables (rather thanspinors) is being developed by Freidel and collaborators [13, 14], see also[15] for gravity in three dimensions. In addition, our results are probablyrelevant for the so-called BF representation [16, 17] as well. PSfrag replacements π A π A π A C Figure 1: In loop quantum gravity the quantum states of the gravitational field are builtfrom gravitational Wilson lines (lying in a three-dimensional spatial hypersurface). TheseWilson lines can hit a two-dimensional boundary C , where they create a surface charge,namely a spinor-valued surface density π A .
2. Boundary and corner terms for self-dual variables
Consider then general relativity in the self-dual formulation [6]. Theconfiguration variables in the bulk are the self-dual Plebański area two- Bahr, Dittrich and Geiller [16, 17] have proposed recently a radical reformulationof loop quantum gravity in the continuum, which is built over a distributional vacuumpeaked at flat or constantly curved three-geometries. In this new representation, onefinds a more complicated area spectrum [17]. In our continuous Fock representation, theoriginal loop gravity area spectrum is recovered (up to quantisation ambiguities). Thetwo representations are therefore likely unitarily inequivalent, such that normalised statesin one representation may only reappear as distributions in the other. Σ AB = Σ ( AB ) and the sl (2 , C ) connection A AB with curvature F AB =d A AB + A AC ∧ A C B . Physical motions are given by those field configurationsthat extremise the topological BF action S M [Σ , A ] = (cid:20) i8 πG β + i β Z M Σ AB ∧ F AB (cid:21) + cc . (5)in the class of all fields that satisfy the simplicity constraints Σ ( AB ∧ Σ CD ) ! = 0 , (6a) Σ AB ∧ ¯Σ A ′ B ′ ! = 0 , (6b) Σ AB ∧ Σ AB + ¯Σ A ′ B ′ ∧ ¯Σ A ′ B ′ ! = 0 . (6c)The simplicity constraints guarantee that the area two-form Σ AB is compat-ible with the existence of a Lorentzian metric g ab = ǫ AB ¯ ǫ A ′ B ′ e AA ′ a e BB ′ b fora tetrad e AA ′ a = − ¯ e A ′ Aa , such that either Σ AB = ∓ e AC ′ ∧ e BC ′ , or Σ AB = ∓ i2 e AC ′ ∧ e BC ′ . (7)The equations of motion for any one of these solutions are then the torsionlesscondition, ∇ Σ AB = 0 ⇔ ∇ [ a Σ ABab ] = 0 (8)and the Einstein equations, which demand that the curvature be Ricci flat F AB = Ψ ABCD Σ CD , (9)where ∇ = d + [ A, · ] is the exterior covariant derivative and Ψ ABCD =Ψ ( ABCD ) is the spin (2 , Weyl spinor.The action (5) contains two coupling constants, namely Newton’s con-stant G , which is a mere conversion factor between units of action and unitsof area (for ~ = c = 1 ), and the Barbero – Immirzi parameter β , which isa pure number ( β > ). The addition of the Barbero – Immrizi parameteris actually necessary: Had we not introduced β , and worked with the ac-tion (5) for β → ∞ (or β → ) instead, the equations of motion would be We will later restrict ourselves to only one of these four solution sectors, namely thefirst Σ AB = − e AC ′ ∧ e BC ′ , which corresponds to Σ AA ′ BB ′ = − ¯ ǫ A ′ B ′ Σ AB − ǫ AB ¯Σ A ′ B ′ = e AA ′ ∧ e BB ′ and a signature ( − +++) metric g ab . We are then considering the gravitational field in a compact four-dimen-sional causal diamond M as drawn in figure 2. The boundary ∂ M consists offour components: Three-dimensional null surfaces N + and N − , and spacelikehypersurfaces Σ + and Σ − at the top and bottom cutting off the diamondbefore the null surfaces recollapse into a point or caustic.Next, we have to add boundary and corner terms such that the varia-tional problem is well-posed. For self-dual variables on a null surfaces, theseboundary terms have been studied in the earlier papers [4, 5] of the series.The boundary action is built from certain boundary fields, which are in-trinsic to a null surface: On a null surface N there always exists, in fact,a spinor-valued two-form η Aab ∈ Ω ( N : C ) and a two-component Weylspinor (a spinor-valued -form) ℓ A ∈ Ω ( N : C ) such that the pull-back ϕ ∗ N : T ∗ M → T ∗ N of the self-dual two-form Σ ABab to the null boundaryturns into the symmetrised spin (1 , tensor product (cid:2) ϕ ∗ N Σ AB (cid:3) ab ≡ Σ AB ab ←− = η ( Aab ℓ B ) . (10)It can be then shown (see again [4, 5] for the details) that the boundaryspinors ( η Aab , ¯ ℓ A ′ ) determine the entire intrinsic geometry of the null sur-face. For instance, there is the spin ( , ) vector component ℓ α ≡ i ℓ A ¯ ℓ A ′ , (11)and it determines the internal null surface generators ℓ α = e αa ℓ a . The spin This can be seen as follows: If Σ AB is a solution of the simplicity constraints (6),we can build a new such solution simply by replacing Σ AB by i × Σ AB . At the level ofthe self-dual variables, multiplication by the imaginary unit amounts to take the Hodgedual in the internal indices α, β, γ, . . . . If the simplicity constraints are satisfied, thereexists then a tetrad e α such that either Σ AB (case i) or i × Σ AB (case ii) is given by theself-dual part of ± e α ∧ e β , see [6]. If we now insert any such Σ AB for e.g. β → ∞ backinto the action, we are left with the Einstein – Hilbert action in the first case, but in thesecond case we only get a topological term, namely / πG R M e α ∧ e β ∧ F αβ (equally for β → and βG = const . ). The resulting equations of motion would be the torsionlesscondition ∇ e α = 0 alone, and there would be no Einstein equations, since the variation of e α would only yield the Bianchi identity F αβ ∧ e β = 0 , which is already a consequence ofthe vanishing of torsion T α = ∇ e α = 0 . The intrinsic geometry is determined completely by a degenerate signature (0++) metric q ab , whose degenerate direction determines the direction [ ℓ a ] ∋ ℓ a : q ab ℓ b = 0 of thenull generators. ε ab = − i η Aab ℓ A , (12)on the other hand, defines the area two-form, which is intrinsic to the bound-ary. In fact, the two-dimensional and oriented area of any two-dimensionalcross-section of the boundary is given by the integral Ar[ C ] = − i Z C η A ℓ A . (13)For the area to be real constraints must be satisfied, namely the realityconditions η Aab ℓ A + cc . = 0 . (14)The action for the entire region consists then of the action (5) in the bulkplus a boundary and corner term, namely S [ A, Σ , η , ℓ, α ] == i8 πG β + i β (cid:20) Z M Σ AB ∧ F AB + Z N + η + A ∧ (cid:0) ( D − ω ) ℓ A + − ψ A + (cid:1) ++ Z N − η − A ∧ (cid:0) ( D + ω ) ℓ A − + ψ A − (cid:1) + Z C o α ( ℓ − A ℓ A + − (cid:21) + cc . (15)where D a = ∇ ← a is the pull-back of the exterior covariant derivative from thebulk to the boundary. The action is to be extremised in the class of all fieldsthat satisfy the simplicity constraints (6) for given boundary conditionson N ± : δω a = 0 , δ [ ψ A ± ] a = 0 , (16a)on Σ ± : δA AB ← a = 0 , (16b)on C ± : δℓ A ± = 0 . (16c)The resulting equations of motion are the Einstein equations (9) and the tor-sionless condition (8) in the bulk. At the null boundary, additional boundaryequations of motion appear: The variation of the boundary spinors deter-mines the exterior covariant derivatives D a ℓ A and D [ a η Abc ] in terms of theexternal potentials ω a and ψ Aa , which are held fixed in the variational It is here that we restrict ourselves to only the first solution sector (7) of the simplicityconstraints (6). See also footnote 2. The paper [5] explains the geometric significance of ω a and ψ Aa as a measure for theextrinsic curvature of the null boundary. (cid:2) ϕ ∗ N ± Σ AB (cid:3) ab = ℓ ± ( A η ± B ) ab , (17a) (cid:2) ϕ ∗ C o η ± A (cid:3) ab = ℓ ± A α ab , (17b)where e.g. ϕ ∗ C is the pull-back ϕ ∗ C : T ∗ N → T ∗ C o . Equation (17a) is ob-tained from the variation of the connection along the null surface, whereas(17b) follows from the variation of ℓ A ± at the intersection C o = N + ∩ N − .Finally, there is also the variation with respect to the two-form α at thecorner C o = N + ∩ N − , and it simply fixes the normalisation ℓ − A ℓ A + = 1 of thespin dyad ( ℓ A − , ℓ A + ) at the corner.The variation of the action determines both the equations of motionand the covariant symplectic potential (at the pre-symplectic or kinematicallevel), namely δS = EOM · δ + Θ ∂ M ( δ ) . (18)For each one of the boundary components, there is then a term in the pre-symplectic potential, namely Θ Σ ± = i8 πG β + i β (cid:20)Z Σ ± Σ AB ∧ d A AB + Z C ± η ± A d ℓ A ± (cid:21) + cc ., (19a) Θ N ± = ∓ i8 πG β + i β Z N ± h η ± A ℓ A ± ∧ d ω + η ± A ∧ d ψ A ± i + cc . (19b)In the following, we will restrict ourselves to only one such component,namely Σ + ≡ Σ . The symplectic potential Θ Σ consist of a three-dimensionalintegral over the interior, and an additional two-dimensional integral over thecorner C ≡ C + . The goal of the remaining part of the paper is to studythe quantisation of the phase space at this two-dimensional corner alone.The canonical analysis of the entire phase space including the new boundaryvariables η ± Aab and ℓ A ± will be left to a forthcoming publication in this series.The approach so far is therefore incomplete: we will quantise the symplecticstructure at the corner, but we will leave the degrees of freedom in the bulkclassical.
3. Landau quantisation of area
In this section, we will develop our main result, namely a new repre-sentation of quantum geometry that reproduces the discrete loop quantum8Sfrag replacements Σ − Σ + N + N − C o C − C + Figure 2: We are considering the gravitational field in a four-dimensional causal region M ,whose boundary has four components, namely the three-dimensional null surfaces N + and N − , which have the topology of a cylinder [0 , × S , and the spacelike disks Σ − and Σ + at the top and bottom. The boundary has three corners, which appear as the boundaryof the boundary, namely ∂ N + = C + ∪ C − o and ∂ N − = C o ∪ C − − . All these manifoldscarry an orientation, which is induced from the bulk: ∂ M = Σ − − ∪ N − ∪ N + ∪ Σ + . gravity area spectrum in the continuum, without ever relying on a discreti-sation of space, lattice variables or a gauge fixing to a compact gauge group.In addition, our construction is manifestly Lorentz invariant.Our starting point is the classical phase space at the corner. The Poissonbrackets for the boundary variables are determined by the corner term i8 πG β + i β Z C (cid:0) η A d ℓ A − cc . (cid:1) (20)appearing in the symplectic potential (19a). The spinor ℓ A plays the role ofthe configuration variable. Its conjugate momentum is given by the spinor-valued surface density π A := i16 πG β + i β ˆ ǫ ab η Aab , (21)where β > denotes the Barbero – Immirzi parameter and ˆ ǫ ab is the two-dimensional and metric-independent Levi-Civita density at the corner. Thefundamental Poisson brackets are given by (cid:8) π A ( z ) , ℓ B ( z ′ ) (cid:9) C = δ BA δ (2) ( z, z ′ ) , (22a) (cid:8) ¯ π A ′ ( z ) , ¯ ℓ B ′ ( z ′ ) (cid:9) C = δ B ′ A ′ δ (2) ( z, z ′ ) , (22b) If { ϑ , ϑ } are coordinates on C , this density is defined by ˆ ǫ ab = d ϑ i ∧ d ϑ j ∂ a ∂ϑ i ∂ b ∂ϑ j . δ ( · , · ) is the two-dimensional Dirac distribution at the corner. Allother Poisson brackets among the canonical variables vanish identically.The spinors ℓ A and π A are not arbitrary. The reality conditions (14)constrain the spin (0 , singlet π A ℓ A to satisfy C = i β + i π A ℓ A + cc . = 0 . (23)The reality conditions are necessary for the spinors to be compatible witha real and Lorentzian metric in a neighbourhood of the corner. On the C = 0 constraint hypersurface in phase space, we can then find the followingidentities for the area in terms of the canonical variables, namely Ar[ C ] = − i Z C η A ℓ A ≈ Z C (cid:0) η A ℓ A − cc . (cid:1) ≈ π i βG Z C ( π A ℓ A − cc . ) . (24)where “ ≈ ” means equality up to terms that vanish for C = 0 . The righthand side is clearly real and well-defined on the entire phase space. Defining Ar[ C ] := 4 π i βG Z C ( π A ℓ A − cc . ) , (25)we can extend, therefore, the definition of the area away from the C = 0 constraint hypersurface, thus turning the area into a partial observable [18]on the entire kinematical phase space over the corner.To quantise the theory, we construct harmonic oscillators from π A and ℓ A .We will then define the canonical Fock space for these oscillators and com-pute the spectrum of the area operator (92) at the quantum level. To defineharmonic oscillators, we need, however, additional geometrical backgroundstructures at the corner. An example illustrates the situation: Consider aparticle in a complex plane, z = x + i y is the position, p = 1 / p x − i p y ) denotes the conjugate momentum. The Poisson brackets are { p z , z } = { ¯ p z , ¯ z } = 1 . The Landau operators a := p Ω / z + iΩ − p z ) and b := p Ω / z + iΩ − ¯ p z ) are then built by taking the sum of the configurationvariable (which is ℓ A in our case) and the complex conjugate momentumvariable (which is ¯ π A ′ ). Notice now that the complex coordinates ( a, b ) de-pend on an additional length scale, namely Ω , which is required because p z and z have opposite dimensions of length. The same happens for ( π A , ¯ ℓ A ′ ) .The momentum variable is a density weight one spinor, to sum ¯ π A ′ with ℓ A we need to first divide by an appropriate surface density d Ω , and thenmap primed into unprimed indices before taking the sum of ℓ A and ¯ π A ′ .10e therefore need two additional structures: a two-dimensional fiducial vol-ume element d Ω and a complex structure (essentially a Hermitian metric)mapping primed into unprimed indices A ′ → A .Accordingly, we choose a fiducial and non-degenerate area two-form ◦ ε ab ∈ Ω ( C : R > ) , such that the surface density d Ω = 12 ˆ ǫ ab ◦ ε ab = Ω ( ϑ, ϕ ) dcos ϑ ∧ d ϕ, (26)is positive ( ϑ and ϕ are spherical coordinates with respect to some fiducialround background metric δ = d ϑ + sin ϑ d ϕ at the corner). We will thenalso need the inverse ◦ ε ab of d Ω (a section of the anti-symmetric tensorbundle T N ∧ T N over the corner), which is defined implicitly by ◦ ε ac ◦ ε bc = [id N ] ab , (27)where id N : T N → T N is the identity.Next, we choose an internal and future oriented normal n α : η αβ n α n β = − such that we have a Hermitian metric in the spin bundle over the corner,namely δ AA ′ = σ AA ′ α n α , (28)where σ AA ′ α are the internal and four-dimensional soldering forms. Wethen have a norm and can set k ℓ k = δ AA ′ ℓ A ¯ ℓ A ′ , k π k = δ AA ′ π A ¯ π A ′ . (29a)Notice that k ℓ k is a scalar, whereas k π k is a surface density of weight four.Having introduced both a fiducial volume element (the surface density d Ω ) and a complex structure (the Hermitian metric δ AA ′ = σ AA ′ α n α ), wecan introduce now the Landau operators a A [ d Ω , n α ] ≡ a A = d Ω √ (cid:18) δ AA ′ ¯ ℓ A ′ − i2 ◦ ε ab π Aab (cid:19) , (30a) b A [ d Ω , n α ] ≡ b A = d Ω √ (cid:18) ℓ A + i2 δ AA ′ ◦ ε ab ¯ π A ′ ab (cid:19) , (30b) The most natural choice is given by the surface normal n a of Σ itself, such that n α = e αa n a . Notice that this turns n α into a field-dependent and internal four-vector,which depends (as a functional) on the pull-back of the tetrad to Σ . A matrix representation is given by the four-dimensional Pauli matrices σ AA ′ α =( , σ , σ , σ ) . The relation between the tetrad is given by e AA ′ a = i √ σ AA ′ α e αa . n α (which is an internal, future oriented nor-malised four-vector) and d Ω (which is the half-density √ d Ω ). The funda-mental Poisson commutation relations (22) translate now into commutationrelations for two pairs of harmonic oscillators over the sphere, namely (cid:8) a A ( z ) , a ∗ B ( z ′ ) (cid:9) C = i δ AB δ (2) ( z, z ′ ) , (31a) (cid:8) b A ( z ) , b ∗ B ( z ′ ) (cid:9) C = i δ AB δ (2) ( z, z ′ ) , (31b)where we introduced the conjugate spinors a ∗ A = δ AA ′ ¯ a A ′ , b ∗ A = δ AA ′ ¯ b A ′ . (32)In quantum theory, the Fock vacuum | , { d Ω , n α }i is then given as thestate in the kernel of the annihilation operators, ∀ z ∈ C : ˆ a A ( z ) (cid:12)(cid:12) , { d Ω , n α } (cid:11) = ˆ b A ( z ) (cid:12)(cid:12) , { d Ω , n α } (cid:11) = 0 . (33)Next, we introduce the canonical number operators for the two oscillatorsover any point in C , namely N a = a ∗ A a A = 12 h d Ω k ℓ k + ( d Ω) − k π k + i( ℓ A π A − ¯ π A ′ ¯ ℓ A ′ ) i , (34a) N b = b ∗ A b A = 12 h d Ω k ℓ k + ( d Ω) − k π k − i( π A ℓ A − ¯ ℓ A ′ ¯ π A ′ ) i . (34b)We can then also introduce the squeeze operators a A b A = − h d Ω k ℓ k − ( d Ω) − k π k + i(¯ ℓ A ′ ¯ π A ′ + π A ℓ A ) i , (35a) ( a A b A ) ∗ = − h d Ω k ℓ k − ( d Ω) − k π k − i(¯ ℓ A ′ ¯ π A ′ + π A ℓ A ) i . (35b) For the purpose of this paper, the two most relevant operators are thearea operator (92) and the reality conditions (14). Choosing a normal or-dering, the area operator is nothing but the difference of the two numberoperators, namely : Ar[ C ] : = 2 π i βG Z C (ˆ π A ˆ ℓ A + ˆ ℓ A ˆ π A − h . c . ) == 4 πβG Z C (cid:0) ˆ a † A ˆ a A − ˆ b † A ˆ b A (cid:1) . (36)12he spectrum of this operator in the Fock space over the vacuum (32) isdiscrete. This becomes particularly obvious if we introduce the followingbasis: Consider spinor spherical harmonics Y AJMs ( ϑ, ϕ ) , such that we havea mode expansion ˆ a A ( ϑ, ϕ ) = d Ω ∞ X J =1 / J X M = − J X s = ± ˆ a JMs Y AJMs ( ϑ, ϕ ) , (37a) ˆ b A ( ϑ, ϕ ) = d Ω ∞ X J =1 / J X M = − J X s = ± ˆ b JMs Y AJMs ( ϑ, ϕ ) . (37b)The resulting Poisson brackets are given by an infinite tower of harmonicoscillators, (cid:2) ˆ a JMs , ˆ a † J ′ M ′ s ′ (cid:3) = (cid:2) ˆ b JMs , ˆ b † J ′ M ′ s ′ (cid:3) = δ JJ ′ δ MM ′ δ ss ′ , (38)where we chose the canonical normalisation Z C d Ω δ AA ′ ¯ Y A ′ J ′ M ′ s ′ Y AJMs = δ JJ ′ δ MM ′ δ ss ′ . (39)The area operator is now just the sum of the differences of the two numberoperators in each mode, namely : Ar[ C ] : = 4 πβ G ∞ X J =1 / J X M = − J X s = ± (cid:16) ˆ a † JMs ˆ a JMS − ˆ b † JMs ˆ b JMs (cid:17) . (40)Hence, there is a fundamental discreteness of area in quantum gravity. Thepossible eigenvalues { a n } of area are given by the multiplies a n = 4 πβ G n = a o n , n ∈ Z (41)of the fundamental loop gravity area gap a o = 8 πβ G = 8 πβ ℓ , (42)where β > is the Barbero – Immirzi parameter and ℓ P = p ~ G/c is thePlanck length. Notice that the area spectrum contains both positive and The spinor spherical harmonics are defined with respect to some fiducial two-dimensional round metric δ = d ϑ + sin ϑ d ϕ at the corner C . Ar g [ C ] = Z C d x d y s det (cid:18) g ( ∂ x , ∂ x ) g ( ∂ x , ∂ y ) g ( ∂ y , ∂ x ) g ( ∂ y , ∂ y ) (cid:19) (43)instead. In fact, in loop gravity, one finds [7, 8] the following main eigenvaluesfor the metrical area of a surface, namely a n ,n ,... = 8 πβ G X j ∈ N n j p j ( j + 1) , n i ∈ N . (44)At the classical level, both notions of area agree up to a local sign, whereasin quantum theory, the eigenvalues of the two operators (43) and (40) aredifferent, but they approach each other in the semi-classical limit, namelyfor j → ∞ , ~ j = const . Finally, there are also the reality conditions (14), which we now need toimpose at the quantum level as well. Separating real and imaginary parts of π A ℓ A , we get C = i β + i π A ℓ A + cc . == 1 β + 1 h ( π A ℓ A + ¯ ℓ A ′ ¯ π A ′ ) + i β ( π A ℓ A − ¯ ℓ A ′ ¯ π A ′ ) i . (45)Choosing a normal ordering, and going back to the definition of the num-ber and squeeze operators (35a, 35b), we can quantise this operator ratherimmediately, namely by saying : C : = 1 β + 1 h i (cid:0) ˆ a A ˆ b A − (ˆ a A ˆ b A ) † (cid:1) + β (cid:0) ˆ a † A ˆ a A − ˆ b † A ˆ b A (cid:1)i . (46)A short calculation confirms that the area operator (36) commutes with thequantum reality conditions (46), namely that ∀ z ∈ C : h : C ( z ) : , : Ar[ C ] : i = 0 . (47)14n the same way, one can also show that the area operator and the realityconditions (46) commute with the generators of local SL (2 , C ) gauge trans-formations, which are given by Π AB ( z ) = −
12 ˆ π ( A ( z )ˆ ℓ B ) ( z ) , (48a) ¯Π A ′ B ′ ( z ) = −
12 ˆ π † ( A ′ ( z )ˆ ℓ † B ′ ) ( z ) . (48b)Notice that there is no ordering ambiguity in here, because [ˆ π A ( z ) , ℓ B ( z ′ )] = − i ǫ AB δ (2) ( z, z ′ ) is anti-symmetric in A and B , whereas Π AB = Π BA is sym-metric.In summary, the area operator, the reality conditions (46) and the gener-ators of local SL (2 , C ) gauge transformations can be diagonalised simultane-ously. In (41) we gave the spectrum of the area operator at the kinematicallevel (i.e. prior to imposing the reality conditions). The area operator com-mutes with both the reality conditions (46) and the SL (2 , C ) generators,and the spectrum at the level of the physical (or gauge invariant) bound-ary Hilbert space can therefore only be a subset of the kinematical areaspectrum.To impose the reality conditions (46) at the quantum level, we first intro-duce the corresponding finite gauge transformations U [ λ ] := exp( − i R C λC ) for gauge parameters λ : C → R . Any such gauge transformation generatesa conformal transformation of the fiducial area element d Ω in addition to alocal U (1) phase rotation. This can be seen by writing the constraint (46)as a sum of a local squeeze operator K ( z ) = 12i h ˆ a A ( z )ˆ b A ( z ) − (cid:0) ˆ a A ( z )ˆ b A ( z ) (cid:1) † i , (49)which is responsible for the conformal transformation of the fiducial areaelement, and the U (1) generator L ( z ) = 12 h ˆ a † A ( z )ˆ a A ( z ) − ˆ b † A ( z )ˆ b A ( z ) i . (50)We then have : C ( z ) : = − β + 1 ( K ( z ) − βL ( z )) , (51)which is a Lorentz invariant version of the so-called linear simplicity con-straints [19, 20]. 15t is then straightforward to see that the reality conditions generate thefollowing gauge transformations, namely exp (cid:16) i Z C λ : C : (cid:17) ˆ a A h d Ω , n α i ( z ) exp (cid:16) − i Z C λ : C : (cid:17) == e − i ββ λ ( z ) ˆ a A h e λβ d Ω , n α i ( z ) , (52) exp (cid:16) i Z C λ : C : (cid:17) ˆ b A h d Ω , n α i ( z ) exp (cid:16) − i Z C λ : C : (cid:17) == e − i ββ λ ( z ) ˆ b A h e λβ d Ω , n α i ( z ) , (53)where we used the notation ˆ a A [ d Ω , n α ]( z ) to stress that the annihilationoperators depend (as a functional) on the fiducial background structures d Ω and n α , and (as an ordinary function) on the points z ∈ C (see also thedefinition of ˆ a A and ˆ b A in (30a) and (30b) above).Two states in the Fock space are then said to be gauge equivalent, if thereis a local gauge parameter λ : C → R that maps one state into the other, inother words Ψ ∼ Ψ ′ ⇔ ∃ λ : C → R : Ψ ′ = exp (cid:16) i Z C λ : C : (cid:17) Ψ . (54)In particular, any two Fock vacua that only differ by a choice for the areadensity d Ω are gauge equivalent, (cid:12)(cid:12)(cid:12) , (cid:8) d Ω , n α (cid:9)E ∼ (cid:12)(cid:12)(cid:12)(cid:12) , (cid:8) e λβ d Ω , n α (cid:9)(cid:29) . (55)By imposing the reality conditions at the quantum level, the dependence ofthe Fock vacuum on the fiducial background area density d Ω is thereforesimply washed away. Still, the boundary Fock vacuum depends on a choicefor a fiducial four-normal n α . This dependence remains, but it is a result ofhaving only quantised the boundary. Had we quantised also gravity in thebulk, we would have had to impose the glueing conditions (17a) as well. Atthe classical level, they are given by the constraint C AB = 4 π i βGβ + i ˆ ǫ ab Σ ABab − ℓ ( A π B ) = 0 , (56)which links the boundary spinors π A and ℓ A to the pull-back of the self-dual area two-form Σ ABab . Now, ˆ ℓ ( A ˆ π B ) is the self-dual generator of local16 L (2 , C ) frame rotations. For a local gauge element Λ AB : C → sl (2 , C ) , wefind, in fact exp (cid:16) Z C iΛ AB ˆ π A ˆ ℓ B − hc . (cid:17) ˆ a A h d Ω , n α i ( z ) × exp (cid:16) − Z C iΛ AB ˆ π A ˆ ℓ B − hc . (cid:17) = g AB (Λ) ˆ a B h d Ω , g αβ (Λ) n β i ( z ) , (57)equally for ˆ b A ( z ) , where g (Λ) denotes the SO (1 , respectively SL (2 , C ) gauge transformation g (Λ) = exp(Λ) . The Fock vacuum at the boundarydepends parametrically on a future oriented four-vector n α , and any twodifferent choices for n α are related, therefore, by a Lorentz transformationthat sends one vacuum into the other, exp (cid:16) Z C iΛ AB ˆ π A ˆ ℓ B − hc . (cid:17)(cid:12)(cid:12)(cid:12) , (cid:8) d Ω , n α (cid:9)E = (cid:12)(cid:12)(cid:12) , n d Ω , g αβ (Λ) n β oE , (58)which is a direct consequence of (57). The boundary Fock vacuum is there-fore only Lorentz covariant, but not Lorentz invariant. At the level of theHamiltonian theory, it can be shown (see [5] for references) that the glueingconditions are the generators of simultaneous SL (2 , C ) gauge transforma-tions in the bulk plus boundary. Hence we expect that local Lorentz in-variance of the boundary states can be restored only by the coupling to thebulk, such that the quantum states for the bulk plus boundary geometry areentangled, Ψ = R d n Ψ ∂Σn ⊗ Ψ Σn , and local SL (2 , C ) gauge invariance followsfrom the average over all possible directions of n α .
4. Topological quantisation
In the previous section, we developed the quantisation of the gravitationalboundary fields using a Landau representation for the boundary spinors ℓ A and π A . The goal of this section is to explain the compatibility with loopquantum gravity in the usual Ashtekar – Lewandowski representation.The Fock vacuum (33) at the boundary depends parametrically on achoice for a fiducial area density d Ω and a Hermitian metric δ AA ′ = σ AA ′ α n α .The dependence on d Ω is gauged away by imposing the reality conditions(46). Two different Fock vacua that differ only by a choice for d Ω are gaugeequivalent, and the Fock vacuum | , { d Ω , n α }i is therefore gauge equivalentto a totally squeezed state such as Ψ ∅ = lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12) , n e − λtβ d Ω , n α o(cid:29) , (59)17or a gauge parameter λ : C → R > as in (58) above. Such a totally squeezedstate does not exist as a vector in the Hilbert space (it does not definea Cauchy sequence). Formally, it yields an eigenstate of the momentumoperator ˆ π A with vanishing eigenvalue (this can be seen from the definitionof the annihilation operators (30) by sending ◦ ε ab ≡ d Ω → , hence ◦ ε ab →∞ ). Using a functional Schrödinger representation, we then formally have ∀ z ∈ C : − i δδℓ A ( z ) Ψ ∅ [ ℓ A ] = − i δδ ¯ ℓ A ′ ( z ) Ψ ∅ [ ℓ A ] = 0 , (60)hence Ψ ∅ [ ℓ A ] = const . In loop quantum gravity, such a state is very wellknown: it represents the spinorial analogue of the Ashtekar – Lewandowskivacuum [21–23] restricted to the corner. Yet in here, this state appears asjust one representative of an infinite family of gauge equivalent states (58).Let us now see how to build excited states over this vacuum and imposethe reality conditions (14) at the quantum level. The basic idea is to lookat topological excitations for which the vacuum (59) is excited only overa certain number of punctures z , z , . . . z N ∈ C , such that the quantumstate of the two-dimensional geometry can be described by an N body wavefunction Ψ f [ ℓ A ] = f (cid:0) ℓ A ( z ) , . . . , ℓ A ( z N ) (cid:1) , (61)in a yet unspecified N particle Hilbert space H N . The entire boundaryHilbert space will be then given as a direct sum H = C ⊕ H ⊕ H ⊕ . . . (62)of all N particle Hilbert spaces. For the moment, the statistics is left un-specified. In particular, all punctures are thought to be distinguishable.At the level of the spin bundle, the N body wave function f ( ℓ A , . . . , ℓ AN ) sends the C z i fibres over the punctures into the complex numbers, henceit defines a map f : C z × · · · × C z N → C . (63)The actual location of the punctures z = ( z , . . . , z N ) is gauged away by theaction of small diffeomorphisms: Consider a diffeomorphism ϕ = exp( ξ ) : C → C at the corner that admits a horizontal lift ϕ ↑ = exp( ξ ↑ ) : S → S into the spin bundle S ( C , C , π S ) (the base manifold is the corner itself, the We are considering the spin bundle S ( C , C , π S ) over the corner, where each fibre C z = π − S ( z ) over a point z ∈ C is homeomorphic to C . C ). Two states are then said to be gauge equivalent, if thebundle morphism ϕ ↑ sends one state into the other, that is Ψ f ∼ Ψ ϕ ∗↑ f , where: Ψ ϕ ∗↑ f [ ℓ A ] := Ψ f [ ϕ ↑ ◦ ℓ A ] . (64)For large gauge transformations, on the other hand, we expect that theyhave a non-trivial action in the quantum theory [24]. This observation couldbe used, in fact, to determine the statistics of the N particle wave function(the exchange of two punctures can always be made undone by performinga large diffeomorphism). For the time being, we content ourselves withconsidering only the simplest case, where all punctures are thought to bedistinguishable. What is then the inner product on the N particle Hilbert space? Firstof all, we require that the left translation along the fibres ( U g Ψ f )[ ℓ A ] = f (cid:16) [ g − ( z )] AB ℓ B ( z ) , . . . , [ g − ( z N )] AB ℓ B ( z N ) (cid:17) (65)for local SL (2 , C ) gauge transformations g be unitary. This restricts severelythe functional form of f . The principal series of the unitary and irreduciblerepresentations of SL (2 , C ) are labelled and uniquely characterised by twonumbers, namely by a spin j = Z / and an additional quantum number ρ ∈ R , and together they parametrise the two Casimir operators ~L − ~K and ~L · ~K of the Lorentz group, see [25] for a detailed account. A concreterealisation of these ( ρ, j ) -representations of SL (2 , C ) is given by homogenousfunctions, ∀ ζ ∈ C − { } : f ρ,j (cid:0) ζℓ A (cid:1) = ζ − i ρ + j − ¯ ζ − i ρ − j − f ρ,j (cid:0) ℓ A (cid:1) , (66)where SL (2 , C ) acts as in (65) above. The most general N body wave func-tion (61) can be then built from complex superpositions of such homogenousfunctions, which should satisfy for all ζ i ∈ C − { } and i = 1 , . . . , N that f ρ,j (cid:0) ℓ A ( z ) , . . . ,ζℓ A ( z i ) , . . . , ℓ A ( z N ) (cid:1) == ζ − i ρ i + j i − ¯ ζ − i ρ i − j i − f ρ,j (cid:0) ℓ A ( z ) , . . . , ℓ A ( z N ) (cid:1) . (67)What is then the measure with respect to which these states are nor-malised? The integration measure on C d ℓ = 116 d ℓ A ∧ d ℓ A ∧ d¯ ℓ A ′ ∧ d¯ ℓ A ′ , (68) The null boundary is a three-dimensional manifold, and it seems quite plausible there-fore that anyonic statistics will play an important role if the dynamics is taken into accountas well, see [24] for detailed thoughts about this idea.
19s clearly SL (2 , C ) invariant, but the homogenous functions are not normal-isable with respect to the L ( C , d ℓ ) inner product. The divergence can beremoved, however, by dividing out the integration over the gauge orbits ofthe reality conditions (23). We introduce the vector field V C = i β + i ℓ A ∂∂ℓ A − i β − i ¯ ℓ A ′ ∂∂ ¯ ℓ A ′ . (69)and define the three-form d µ ( ℓ ) = V C y d ℓ = 18 i β + i ℓ A d ℓ A ∧ d¯ ℓ A ′ ∧ d¯ ℓ A ′ + cc ., (70)where “ y ” denotes the interior product. The inner product between two N particle states is then given by the integral (cid:10) Ψ f , Ψ f ′ (cid:11) N = Z C z /C d µ ( ℓ ) · · · Z C zN /C d µ ( ℓ N ) f ( ℓ A , . . . ) f ′ ( ℓ A , . . . ) , (71)where we integrate over a gauge fixing surface, such as G ( ℓ i ) = k ℓ i k = const . that intersects the gauge orbits ℓ Ai ∼ e i β +i λ ( z i ) ℓ Ai of the reality conditions (23)exactly once. The inner product is now invariant under small deformationsof the gauge fixing surface, if and only if the integrant satisfies for all i =1 , . . . , N the constraint h(cid:16) i β + i ℓ Ai ∂∂ℓ Ai + 2 (cid:17) + cc . i f ( ℓ A , . . . , ℓ AN ) f ′ ( ℓ A , . . . , ℓ AN ) = 0 . (72)This condition is found by deforming the gauge fixing surface (e.g. G ( ℓ i ) = k ℓ i k = const . ) and using Stokes’s theorem. Suppose now that this conditionis satisfied (we will see in a moment that physical states always satisfy thiscondition). It then follows that the inner product is SL (2 , C ) gauge invariant,such that equation (65) realises a unitary representation of SL (2 , C ) . Stateswith different homogeneity weights are then necessarily orthogonal.Next, we define operators acting on these states. Consider an open neigh-bourhood U z ⊂ C around a point z ∈ C and define the following smearedEuler homogeneity operators, namely E N [ U z ]Ψ[ ℓ A ] := Z U z ℓ A (cid:0) z ′ (cid:1) δδℓ A ( z ′ ) Ψ[ ℓ A ] , (73) ¯ E N [ U z ]Ψ[ ℓ A ] := Z U z ¯ ℓ A ′ (cid:0) z ′ (cid:1) δδ ¯ ℓ A ′ ( z ′ ) Ψ[ ℓ A ] . (74)20onsider then one of our basis states, which are built from homogenousfunctions f ρ,j as in equation (67) above, and define the corresponding wavefunctional Ψ f ρ,j [ ℓ A ] = f ρ,j (cid:0) ℓ A ( z ) , . . . , ℓ A ( z N ) (cid:1) . (75)A short moment of reflection reveals that any such state is an eigenvector ofthe Euler operators with eigenvalues given by E N [ U z ]Ψ f ρ,j = N X i =1 χ U z ( z i ) (cid:0) − i ρ i + j i − (cid:1) Ψ f ρ,j , (76) ¯ E N [ U z ]Ψ f ρ,j = N X i =1 χ U z ( z i ) (cid:0) − i ρ i − j i − f ρ,j , (77)where χ U z ( z ′ ) denotes the characteristic function of U z ⊂ C . If (72) is satis-fied, the inner product is SL (2 , C ) invariant. States of different homogeneityweights are then orthogonal and the adjoint operators must satisfy, therefore, E † N [ U z ]Ψ f ρ,j = N X i =1 χ U z ( z i ) (cid:0) i ρ i + j i − (cid:1) Ψ f ρ,j , (78) ¯ E † N [ U z ]Ψ f ρ,j = N X i =1 χ U z ( z i ) (cid:0) i ρ i − j i − f ρ,j . (79)Next we have to quantise the reality conditions (23) and find their kernelin the state space spanned by the homogenous wave functions (75). At theclassical level, the reality conditions imply that for any open neighbourhood U z ⊂ C around any point z ∈ C the constraint i β + i Z U z π A ℓ A + cc . = 0 (80)is satisfied. Choosing a symmetric ordering, we define the operator : Z U z π A ℓ A : = 12i (cid:0) E N [ U z ] − ¯ E † N [ U z ] (cid:1) . (81)Physical states Ψ phys are now given by those wave functionals that are an-nihilated by the reality conditions, hence h i β + i : Z U z π A ℓ A : + hc . i Ψ phys = 0 . (82)21t is now immediate to impose the reality conditions, and identify theirsolution space. The eigenvalues of the Euler homogeneity operators are givenin (76) and (78), such that the only allowed values for the quantum numbers ρ i and j i must satisfy the relation i β + i ( ρ i + i j i ) + cc . = 0 ⇔ ρ i = βj i . (83)In defining the inner product (71), we mentioned that the integrals areindependent of the gauge fixing only if the integrant satisfies the constraint(72). We can now verify this condition: any N body physical state can bewritten as a superposition of homogenous functions Ψ phys [ ℓ A ] = X j ...j N c j ...j N f ( βj ...βj N ) , ( j ...j N ) (cid:0) ℓ A ( z ) , . . . , ℓ A ( z N ) (cid:1) (84)that all satisfy for all i = 1 , . . . , N the reality conditions ρ i = βj i . Fromthere, it is easy to see that (72) is satisfied, such that the inner product (71)between physical states is indeed independent of the gauge fixing. The N particle Hilbert space is then given by the Cauchy completion (with respectto the norm induced by the inner product (71)) of the complex span of allsuch normalisable states Ψ phys .Having imposed the reality conditions at the quantum level, we can nowintroduce the area operator and compare its spectrum with what we foundin the last section, see (41). In the classical theory, the area (13) of aneighbourhood U z on C is given by the integral Ar[ U z ] = − i Z U z η A ℓ A . (85)Choosing a normal ordering, we can now quantise this operator simply bysaying : Ar[ U z ] : = − πG ββ + i E [ U z ] , (86)where E [ U z ] is the Euler homogeneity operator (76). Consider then a ho-mogenous function f j ≡ f ρ,j such that the reality conditions (83) are satis-fied: ρ i = βj i for all i = 1 , . . . , N . The corresponding wave functional Ψ f j isan eigenstate of the area operator, with eigenvalues given by : Ar[ U z ] : Ψ f j = 8 π βG N X i =1 χ U z ( z i ) j i Ψ f j . (87)22he possible eigenvalues of the area of the entire corner are then simplygiven by a n = 4 π βG n, n ∈ Z / , (88)which agrees with the bosonic quantisation that we introduced earlier (41).Hence, we arrive at the same conclusion as before. In quantum gravity areais quantised.Before concluding, one last remark. The interpretation of Ψ f as a wave function for N particles is actually quite appropriate. Suppose, for a mo-ment, that the classical metric in the neighbourhood of the corner is flat,and that the corner itself is a round two-sphere. The null vector ℓ α thatshines out of this sphere is the square ℓ α = − σ AA ′ α ℓ A ¯ ℓ A ′ of the spinor ℓ A ( ϑ, ϕ ) = (cid:18) ℓ ( ϑ, ϕ ) ℓ ( ϑ, ϕ ) (cid:19) = (cid:18) cos ϑ e i ϕ sin ϑ (cid:19) . (89)Using a stereographic projection, we can then use this spinor itself as acoordinate on the sphere, such that a point is marked by the ratio ℓ ( z ) ℓ ( z ) = z = e i ϕ tan ϑ (90)of the spin up and down components. The N body homogenous wave func-tions (75) can be then written as a product of a universal prefactor timesan N body wave function f ( z , . . . , z N ) , which only depends on the complexcoordinates (90) labelling the locations z , . . . , z N of the punctures on thesphere. The most general state in the N body Hilbert space can be thenwritten in the following form, Ψ f [ ℓ A ] = X j ,...,j N c j ...j N N Y i =1 (cid:0) ℓ ( z i ) (cid:1) − i j i ( β +i) − (cid:0) ¯ ℓ ′ ( z i ) (cid:1) − i j i ( β − i) − × f j ...j N ( z , . . . , z N ) , (91)for complex coefficients { c j ...j N } and N body wave functions f j ... ( z , . . . ) on the fiducial round sphere.
5. Summary and conclusion
Let us summarise. First of all (section 2), we studied the boundarysymplectic structure at a two-dimensional cross-section of a null surface (see23gure 2). The canonically conjugate variables at the boundary consist of aspinor ℓ A and a spinor-valued surface density π A . We then showed that thearea of the cross section can be written as a surface integral of the Lorentzinvariant contraction (i.e. the helicity) of the spinors, Ar[ C ] = 4 π i βG Z C (cid:0) π A ℓ A − cc . ) . (92)To be compatible with a real and Lorentzian metric, the spinors have tosatisfy certain constraints, namely the reality conditions (23).Next (section 3), we built two pairs of harmonic oscillators out of thespinors. This required additional fiducial background structures, namely atwo-dimensional surface density d Ω and a Hermitian metric δ AA ′ = σ AA ′ α n α .We then quantised the oscillators using a bosonic representation obtaininga Fock vacuum | , { d Ω , n α i , which depends parametrically on the fiducialbackground structures. The oriented area (92) of the cross section turnedinto the difference of two number operators, with no dependence on the fidu-cial background structures. In quantum theory, the area becomes quantised(the integral (92) is essentially the generator of global U (1) transformationof the spinors), and the spectrum of the cross sectional area is equidistant. The possible eigenvalues have infinite degeneracy and are all multiples of thefundamental loop gravity area gap a o = 8 πβ ~ G/c .Finally (section 4), we studied how the Fock representation for the bound-ary fields fits together with loop quantum gravity in the usual spin networkrepresentation. We imposed the reality conditions (23) at the quantum level,and saw that two different Fock vacua | , { d Ω , n α i and | , { d Ω , n α i thatdiffer only by a choice for the fiducial area element are gauge equivalent (thereality conditions are the sum of a squeeze operator (49) and an U (1) gener-ator (50), the squeeze operator generating conformal transformations of thefiducial area density d Ω ). Imposing the reality conditions amounts there-fore to consider only gauge equivalence classes of states that are related Such area spectra also appear in other approaches, such as the semi-classic quanti-sation of black holes due to Bekenstein – Mukhanov [26], and non-commutative geometry[27, 28]. There are no normalisable states in the Fock space that would lie in the kernel of thereality conditions (46). The Fock space that we introduced in section 3 is therefore onlykinematical, physical states represent distributions (elements of the algebraic dual of theFock space). Yet by duality, the area spectrum can already be inferred from the states inthe Fock space alone.
24y the gauge transformations (52, 52). Thus, we washed away the depen-dence of the Fock vacuum on the fiducial background area density d Ω . TheFock vacuum for any given choice of d Ω is therefore gauge equivalent toa totally squeezed state such as (59). Such a totally squeezed state is wellknown from loop quantum gravity, where it represents the analogue of theAshtekar – Lewandowski vacuum at a two-dimensional surface. Finally, weintroduced topological excitations over this vacuum, such that the boundaryspinors are excited only over a finite number of punctures z , . . . , z N ∈ C .Every such puncture carries a unitary representation of the Lorentz group,which are classified by quantum numbers ρ ∈ R and j ∈ Z / . We thenquantised the reality conditions and imposed them at the quantum level:Only those representations contribute for which ρ = βj , which is the samekind of constraint that appears in the definition of the loop gravity transitionamplitudes [19, 20]. Once the reality conditions are imposed, the spectrumof the area operator is therefore discrete, and it matches the one that wederived using the Fock representation in the continuum, see (41) and (87).In summary, our quantisation of the gravitational boundary modes an-swers and addresses some long standing and well-founded doubts and reser-vations against loop gravity, see [29]. It was often remarked that the deriva-tion of the area spectrum relies on using SU (2) gauge connection variables,whereas the geometrically relevant choice for gravity seems to be rather SL (2 , C ) . Our calculation is manifestly Lorentz invariant (this becomes par-ticularly clear in section 4), and no gauge fixing to compact gauge groupsis ever required. Another critique came from the usage of spin networkfunctions. It was argued that by working with spin network functions thediscreteness was introduced right from the onset. Yet, in our framework allfields are continuous, and no discretisation was ever introduced. In addition,we have quantised the boundary variables on a null surface, such that thediscreteness of area in quantum gravity seems to be indeed compatible withboth local Lorentz invariance and the universal causal structure of the lightcone. Acknowledgments.
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