Higher spin de Sitter Hilbert space

We propose a complete microscopic definition of the Hilbert space of minimal higher spin de Sitter quantum gravity and its Hartle-Hawking vacuum state. The funda- mental degrees of freedom are 2N bosonic fields living on the future conformal boundary, where N is proportional to the de Sitter horizon entropy. The vacuum state is normalizable. The model agrees in perturbation theory with expectations from a previously proposed dS- CFT description in terms of a fermionic Sp(N) model, but it goes beyond this, both in its conceptual scope and in its computational power. In particular it resolves the apparent pathologies affecting the Sp(N) model, and it provides an exact formula for late time vac- uum correlation functions. We illustrate this by computing probabilities for arbitrarily large field excursions, and by giving fully explicit examples of vacuum 3- and 4-point functions. We discuss bulk reconstruction and show the perturbative bulk QFT canonical commuta- tions relations can be reproduced from the fundamental operator algebra, but only up to a minimal error term ∼ e−O(N ), and only if the operators are coarse grained in such a way that the number of accessible “pixels” is less than O(N ). Independent of this, we show that upon gauging the higher spin symmetry group, one is left with 2N physical degrees of freedom, and that all gauge invariant quantities can be computed by a 2N × 2N matrix model. This suggests a concrete realization of the idea of cosmological complementarity.


Introduction and summary
Finding a precise and complete theory of quantum gravity has been a longstanding problem. For certain systems living in an infinitely deep gravitational potential well, shaped by a negative vacuum energy density, this problem has been solved: the Hilbert space and operator algebra of these theories are those of a conformal field theory living on the boundary of the well [1]. This discovery, known as the AdS-CFT correspondence, has had a profound impact on theoretical research in quantum gravity and quantum field theory over the past twenty years.
However, the universe we find ourselves in does not remotely resemble such spacetimes.
Rather than being trapped together in an infinitely deep gravitational well, galaxies surrounding us recede at ever increasing speeds, pushed apart by a small but positive vacuum energy density. Extrapolated to the far future, the geometry of our spacetime is neither asymptotically flat nor asymptotically anti de Sitter, but asymptotically de Sitter. Furthermore, the primordial universe that spawned us all also appears to be well-approximated by a dS-like geometry, albeit one with a much larger vacuum energy density.
Despite its evident importance, to this date, no precise, complete definition exists of any theory of quantum gravity in a four-dimensional universe with positive vacuum energy density, even when disregarding all other observational constraints such as the properties of particles beyond the graviton. Although low energy effective field theory approaches to this problem are perfectly adequate for many purposes, they also lead to many deep problems and conceptual paradoxes [2,3,4,5,6,7,8,9] as well as formidable technical challenges [10,11,12,13,14], and to longstanding disagreements on how to resolve them [15,16]. It is unlikely that definitive progress will be made on these issues in the absence of a theoretical framework on par with AdS- CFT. There have been several efforts to go beyond four-dimensional low energy effective field theory, towards a fundamental theory of quantum gravity in universes with a positive vacuum energy density. These include, but are certainly not limited to, string theory constructions of metastable de Sitter vacua [17,18,19,20], holographic considerations of the de Sitter observer's static region [21,22,23,24,25,26,27,28,29,30], more general holographic considerations of the landscape [31,32,33,34,35], and the dS-CFT correspondence [36,37,38]. For an overview, see e.g. [39,40,41]. However, these efforts fall short of providing concrete models with a precise, microscopic description of the fundamental degrees of freedom, Hilbert space and operator content, capable in principle of answering all physically sensible questions to any desired precision.
In this paper, we propose such a model. The perturbative low energy bulk field content of this theory includes a scalar, the graviton and an infinite tower of interacting massless higher spin fields, whose classical dynamics is governed by Vasiliev's minimal higher spin gravity equations of motion [43,44]. More specifically, we propose a precise microscopic definition of the Hilbert space of this theory, its operator algebra and its Hartle-Hawking vacuum state. We show that our construction is consistent with perturbative bulk field theory expectations within the realm of their applicability, including cosmological vacuum correlation functions. We demonstrate that the theory is furthermore capable of reaching deep into the nonperturbative regime, by computing the probability of arbitrarily large field excursions. We show that the perturbative bulk QFT Heisenberg algebra, a prerequisite for any attempt at reconstructing standard perturbative bulk quantum field theory, can be reproduced from the fundamental operator algebra, but only up to a minimal error term ∼ e −cN where N ∼ 2 dS /G Newt ∼ S dS , and only if the QFT is coarse grained and limited to access a maximal number of "pixels" of order N . In the same spirit, we argue that the computation of any gauge invariant observable in the theory can be reduced to a finite dimensional 2N × 2N matrix integral. Consistent with this, we show that the physical Hilbert space of gauge-invariant n-particle states is finite-dimensional for any given n.
Although much work remains to be done, there seem to be no insuperable obstacles to a precise identification and microscopic derivation of the de Sitter entropy S dS within this framework. In what follows we will give an overview of the basic formal elements of our construction, leaving the finer points and applications to the bulk of the text. The subdivision in sections of this summary follows the subdivision in sections of the remainder of the paper.

Preliminaries and review
Our construction provides a complete Hilbert space framework for the dS-CFT idea as envisioned in [36,37,38], and more specifically for the concrete proposal of [42]. As we review in section 2, the latter can be phrased roughly as the statement that the Hartle-Hawking wave function of the minimal higher spin dS 3+1 universe is given (up  Here D is minus the Laplacian, B xy represents a general source [45,46] coupling to bilinears χ x χ y , and the trace term on the right hand side appears due to the normal ordering : χχ : of the bilinears (defined in the usual way by subtracting a c-number such that the 1- model known to be dual to minimal Vasiliev gravity in AdS 3+1 [47,48], in effect flipping the sign of the cosmological constant and continuing AdS → dS [42].
In slightly more detail, a suitable decomposition of the source B into local differential operators,

2)
decomposes the source terms accordingly into the standard traceless conserved local (even) spin-s currents O i 1 ···is (z) = χ∂ i 1 · · · ∂ is χ + · · · : Here we introduced a convenient shorthand index notation I = (z, i 1 · · · i s ) labeling both spatial points z and tensor indices (i 1 · · · i s ). Contracted indices are understood to be summed and integrated over. For example for s = 0, we have D xy z = δ x z δ y z and O z =: χ z χ z :. The source components b I are interpreted in the bulk as boundary values of bulk higher spin fields; for example b z,ij is the conformal boundary value of the spatial metric fluctuation field h ij (z). Taking derivatives with respect to the sources b I generates the correlation functions of the O I . In particular the 2-point function is

Problems
The problem with this story is that it at most half a story. We explain this in some detail in section 3. The main issue is that just specifying a wave function without specifying the Hilbert space to which it belongs is physically meaningless. In particular, without an inner product, probabilities, expectation values and vacuum correlation functions cannot be computed. Naively postulating an inner product with a flat measure dB, or equivalently dH with H = D + B, turns out to be inconsistent for a number of reasons, and indeed ignoring this immediately leads to fatal normalizability problems in the nonperturbative regime [49,50].

Proposal
The answer turns out to be staggeringly simple. We argue in section 4 that the proper fundamental degrees of freedom are 2N bosonic fields Q α x , α = 1, . . . , 2N , x ∈ R 3 . We define a Hilbert space H consisting of O(2N)-invariant wave functions ψ(Q) with inner product ψ 1 |ψ 2 = dQ ψ 1 (Q) * ψ 2 (Q) , (1.5) where dQ is the flat measure. The vacuum wave function is ψ 0 (Q) = c e − 1 2 QDQ , ( 1.6) with c chosen such that ψ 0 |ψ 0 = 1. The true physical Hilbert space H phys is obtained as the subspace of H invariant under global higher spin transformations. We will have more to say about this below in section 1.6; for now we stick to H . The relation to the original source fields B appearing in (1.1) is given by

7)
Thus, given the bulk interpretation of the sources B xy , we may read off the bulk interpretation of the QQ bilinears B xy . In terms of the expansion ( 1.2), this means that if we define the analog of the bilinear currents (1. 3 b I 1 · · · b In = G I 1 J 1 · · · G InJn ψ 0 |B J 1 · · · B Jn |ψ 0 = c 2 N n G I 1 J 1 · · · G InJn dQ e − QDQ : (QD J 1 Q) : · · · : (QD Jn Q) : . Wishart matrix ensembles. The change of variables becomes singular when K > 2N , and the H-space description becomes singular in this case, because the matrix H has reduced rank 2N . However the Q-space description remains well-defined.

Probabilities and correlation functions
To illustrate how to use this framework in practice, we provide a number of sample computations in section 5. We begin by computing the probability P (b 0 ) of constant scalar mode excursions in global de Sitter with all other field modes integrated out. One of the most striking apparent pathologies of the Sp(N) wave functionψ HH defined in (1.1), interpreted (incorrectly) as a wave function on a Hilbert space with a flat inner product measure dB, is its exponential divergence for large negative b 0 [49]. This pathology is completely eliminated in our framework. We explicitly compute the probability density P (b 0 ), for any value of N , and find it is normalizable. In the large N limit it satisfies, up to O(1) constants in the exponential, P (b 0 ) ∼ e −N b 2 0 for small b 0 , P (b 0 ) ∼ e −N b 0 for large positive b 0 , and P (b 0 ) ∼ e −N |b 0 | 3 for large negative b 0 .
We proceed by explicitly computing the scalar-scalar-graviton three-point function and the scalar three-and four-point functions in the Hartle-Hawking vacuum. The scalar fourpoint function reduces to the computation of a three-dimensional four-mass box integral previously considered in the scattering amplitude literature [53]. The final result presented in [53] spans half a dozen of pages of complicated Mathematica output. We solve this integral in a different way, using the methods of [92], and obtain a remarkably simple result ( 5.38) that fits in a single line.

Perturbative bulk QFT reconstruction
We discuss the reconstruction of perturbative bulk quantum fields from the fundamental operator algebra of H in section 6. A prerequisite for this is the reconstruction of the bulk QFT Heisenberg algebra. On conformal boundary fields α I , β J this is realized as The β I were identified with the QQ bilinears b I = G IK B K in ( 1.9). For several reasons, it is however manifestly impossible to find selfadjoint operators a J in the operator algebra of H exactly realizing these commutation relations. One reason is that if there were such operators, we could exponentiate them to unitary operators acting as translations b I → b I +c I , for arbitrary constants c I , inconsistent with the positivity of H = D + b I D I = DQ(DQ) T . Another reason is that in discretized models with K > 2N (so in particular in the continuum limit K = ∞), the b I are not independent.
When K, N → ∞ with κ ≡ K 2N fixed and κ 0.17, it is nevertheless possible to define self-adjoint operators a I such that the Heisenberg algebra is satisfied up to exponentially small corrections e −g(κ)N , where g(κ) is some O(1) function derived from the asymptotics of the Tracy-Widom distribution. We relate this exponential error directly to the probability of occurrence of nonperturbatively large fluctuations, defined in a precise sense.
When K > 2N , including in the continuum limit K = ∞, the naive construction fails, but it is still possible to define "coarse grained" self-adjoint operatorsb I ,ā I , reducing the number of effectively accessed spatial "pixels" to a finite number K eff , allowing again to reconstruct the perturbative bulk Heisenberg algebra up to exponentially small errors.
This indicates however that perturbative quantum field theory in higher spin de Sitter space breaks down beyond a K eff of more than O(N ) pixels, a number of the order of the de Sitter entropy S dS ∼ 2 dS /G Newton . We approach the problem of explicitly constructing H phys by first considering the discretized finite K models, where this construction is straightforward because the residual gauge symmetry group G is isomorphic to O(K), hence compact, and then taking the limit K → ∞. This allows us to explicitly construct a basis of gauge invariant n-particle states, and to interpret them as "group-averages" of ordinary n-particle states on H , generalizing the analogous SO(1,4) group-averaging procedure of perturbative gravity in dS [56,61,62].

Physical Hilbert space
We find that for each n, there is a finite number of such states.

Preliminaries and review
With the goal of making this work accessible to readers who aren't higher spin gravity experts, and to introduce some notation that will be useful in the remainder of the paper, we review in this section some basic properties of free massless higher spin fields in a fixed de Sitter background, as well as their Bunch-Davies/Hartle-Hawking vacuum wave function.
We emphasize the use of conformal boundary fields and their canonical Heisenberg algebra in this setting, as such fields will form the natural interface with operators defined in the boundary theory we propose in subsequent sections. We then review the dS-CFT idea relating the Hartle-Hawking wavefunction of an interacting bulk gravity theory to the partition function of a boundary conformal field theory, and its explicit realization in higher spin gravity as proposed in [40].

Free higher spin fields in de Sitter space
We work in planar coordinates with dS ≡ 1, so the background de Sitter metric in (d + 1)dimensions takes the form with η < 0 and x ∈ R d . Future infinity corresponds to η = 0, while η = −∞ is the past horizon of the planar patch.
We will frequently work in momentum space. Our conventions for the d-dimensional Fourier transform are To avoid dragging along factors (2π) d , we will use the notations Vasiliev gravity in dS 3+1 has a scalar field of mass m 2 = 2. The action of a free scalar with this mass is Canonical normalization corresponds to γ = 1, but for some purposes it is useful to consider a different normalization (for example "gravity" normalization, in which γ is proportional the the Newton constant), so we keep it arbitrary here. The mode expansion is

5)
where the coefficients satisfy the canonical commutation relations The free Bunch-Davies vacuum |0 is the state annihilated by all of the a k . The vacuum By decomposing e ikη = cos(kη) + i sin(kη), we can alternatively write the mode expansion in the form where the signs are picked for convention consistency with (2.25) below and Asymptotically for η → 0, we have, up to (1 + O(η 2 )) corrections, Note that α(k) and β(k) are the Fourier coefficients of local boundary fields α(x) and β(x), i.e. boundary fields that can be obtained locally from the bulk field φ(η, x) in the limit η → 0. These boundary fields transform under the SO(1,4) de Sitter isometry group as d = 3 conformal primary fields of dimension ∆ = 1 and ∆ = 2, respectively. In contrast, the boundary fieldβ(x) with Fourier coefficientsβ(k) = 1 k β(k) is non-locally related to β and thus φ:β where we used the Fourier transform formula (A.13). The relation between β andβ is known in the context of conformal field theories as the "shadow" transform [51]. We review this in appendix A; the general shadow transform for scalar operators is given in In the case at hand this maps the dimension∆ = 2 conformal primary field β(x) to the dimension ∆ = 3 −∆ = 1 conformal primary fieldβ(x). So althoughβ is non-locally related to the bulk field φ, it nevertheless transforms locally under the de Sitter isometry group.
Expressed in terms of the local boundary operators α and β, the canonical commutation relations (2.6) take the conventional local form for canonically conjugate fields: 12) with all other commutators vanishing. In terms of the the ∆ = 1 boundary operators α andβ on the other hand, we get (2.14) Note that these take the form of a ∆ = 1 3d CFT 2-point function, consistent with the SO(1,4) symmetry constraints. The vacuum 2-point functions of α andβ are similarly 15) which should be read for example as 0|α(x)α(x )|0 = γ 2 G(x, x ) with G as in (2.14). Again all of these take the form of ∆ = 1 CFT 2-point functions, as required by symmetry. However these correlation functions cannot possibly be reproduced by correlation functions of a conventional CFT, i.e. by a Euclidean path integral with insertions of α andβ, since such insertions would necessarily commute. In particular this would imply α(x)β(x ) = β (x )α(x) , inconsistent with the above. Indeed as we will discuss in more detail below, in contrast to AdS-CFT, a complete framework for dS-CFT, capable in particular of reproducing quantum mechanics in the bulk, requires more ingredients than just a boundary CFT.

Free higher spin fields in dS d+1
A free massless spin s field in dS d+1 is a totally symmetric tensor φ µ 1 ···µs satisfying the double-tracelessness conditions 16) and the equations of motion [63] where ∇ µ is the covariant derivative and the symmetrization is over the (µ 1 · · · µ s ) indices only. The equations of motion are invariant under the gauge transformations In what follows we will assume d is odd (and we will mostly have d = 3 in mind). In a suitable gauge, a basis of solutions is provided by the following set of canonically normalized positive frequency mode functions: with all field components involving one or more time-indices equal to zero. The functions are Hankel H functions of the first kind, k ∈ R d labels the momentum of the mode, and σ labels an orthonormal basis of polarization tensors e i 1 ···is (k) satisfying the tracelessness and transversality conditions e j ji 3 ···is = 0, k j e j i 2 ···is = 0, where indices are raised with the flat Euclidean metric δ ij , and orthonormality means (e σ , e τ ) ≡ e σ * i 1 ···is e τ i 1 ···is = δ στ . In the d = 3 case of interest, σ takes on two values, corresponding to the two helicity states of a massless spinning particle. In the far past η → −∞ we have, up to an overall phase which we recognize as the canonically normalized positive frequency mode associated with the standard Bunch-Davies free vacuum.
The mode expansion of the free spin s field takes the form 20) satisfying canonical commutation relations The Bunch-Davies vacuum |0 is the state annihilated by all of the a kσ . The vacuum 2-point function in momentum space is

22)
where Π i 1 ···is,i 1 ···i s is the projector onto spin s transverse traceless polarizations, For example for spin s = 1 and s = 2, we have, respectively, Decomposing H (1) ν (z) = J ν (z) + iY ν (z), we can alternatively write the mode expansion in terms of boundary fields analogous to (2.8): Note that for s > 2− d 2 , the mode with coefficientβ is the dominant one at late times, so we are departing here from the customary notation in the AdS-CFT literature, where α usually refers to the dominant mode.
In terms of the boundary fields α,β, the nonzero canonical commutators become The vacuum correlation functions are proportional to this same G; suppressing indices, Consistent with the transformation properties of α andβ under the SO(1,d+1) de Sitter isometry group, these correlators take the form of CFT 2-point functions of spin-s, ∆ = d − 2 + s primary fields, that is to say higher spin conserved currents (for s > 0). However, just like for the scalar case discussed earlier, they cannot possibly all be reproduced from a single conventional Euclidean CFT, because the operators insertions do not commute in the above.
The Fourier transform α i 1 ···is (x) of α i 1 ···is (k) is locally related to the η → 0 asymptotic bulk field φ i 1 ···is (η, x), but the Fourier transformβ i 1 ···is (x) ofβ i 1 ···is (k) is not. As can be seen from (2.29), the boundary field β locally obtained from φ is related toβ by the following transformation in momentum space: For the second equality we used the fact that β i 1 ···is (k) is transverse and traceless. Since The commutator of β with α takes the canonical δ-function form, 34) and the β-β 2-point function is Note that due to the power of k growing large and negative in the momentum space 2-point function, in position space,G(x, x ) acquires IR divergences for s ≥ 2 (logarithmic for the graviton, positive powers for the higher spin fields). Physically this can be thought of as being due to "drift" of the higher spin gauge fields due to accumulation of frozen-out,

Free theory
The wave function of the free Bunch-Davies vacuum state |0 in an eigenbasis |β of the boundary field operators β i 1 ···is (x) appearing in (2.33) is Odd spin fields appear in non-minimal higher spin gravity, but for simplicity we will only consider the minimal case here. In the free (Gaussian) approximation, the wave function of this theory is simply the product of the free wave functions given in (2.36) for all even spins: .

(2.39)
Of course, Vasiliev gravity is an interacting theory, so the full, interacting vacuum wave function will not be Gaussian. We turn to this next.

Interacting theory and dS-CFT
In interacting gravitational theories, a natural definition of a preferred cosmological vacuum state is given by the Hartle-Hawking wave function ψ HH [β], semiclassically obtained as a path integral with asymptotic boundary conditions specified by β in the future, and by "no boundary" Euclidean boundary conditions in the past [71,72]. This generalizes the representation of the ground state wave function in time translation invariant quantum mechanical systems as a Euclidean path integral. Perturbatively, the Hartle-Hawking wave function takes the general form

40)
where we collectively denoted all asymptotic field degrees of freedom by β I , with I labeling both spatial points and field species. If the perturbation theory is around a de Sitter background and the β I are asymptotic boundary fields defined along the lines of our definitions above, the β I transform as conformal primary fields under the background de A simple (but by no means unique) way of realizing these constraints more generally is to identify ψ HH [β] with the partition function of an actual CFT, where the fields β appear as sources: where the CFT operators O I are conformal primary fields of dimension The coefficients g I 1 ···In are then proportional to the connected correlation functions of the O I . This is the basic idea of the dS-CFT correspondence [36,37,38]. In this form, it is roughly speaking an analytic continuation of the AdS-CFT correspondence: in Euclidean AdS, ψ HH [β] becomes a bulk path integral with spatial boundary conditions parametrized by β, and the above equality becomes just the standard GKPW prescription [73,74].
Unfortunately, this is only morally speaking an analytic continuation, because the unitarity constraints in AdS and those in dS are not analytic continuations of each other: to obtain a unitary theory in Lorentzian AdS d+1 perturbation theory, the CFT must furnish unitary representations of SO(2,d), whereas to obtain a unitary theory in dS d+1 perturbation theory, the CFT must furnish unitary representations of SO(1,d+1) [75,76,77,78].
In the context of dS-CFT, this point was emphasized in [79,80]. In particular this gives rise to rather different allowed spectra of primary conformal dimensions ∆. This is man- A striking exception to these obstacles in relating AdS duals to dS duals, pointed out in [42], is the free bosonic O(N) vector model dual to parity-even minimal AdS 4 Vasiliev gravity [47,48], and cousins thereof [81,82]. This theory has N real bosonic fundamental scalar fields χ a (x), a = 1, . . . , N , with Euclidean action on R 3 given by  g IJK in (2.40) reveals that one gets the wrong sign for these coefficients in this way [42].
One does get the correct sign if, as in [83], instead one identifies . (2.44) In [42] it was furthermore shown that if one grants conditions on α rather than β for the scalar), in which case the bosonic and fermionic partition functions are no longer each others inverse [42]. We will review the free Grassmann model in more detail next.

The Sp(N) model
To establish notation which will be useful further on, we spell out some details here about the free fermionic Grassmann model. Consider Grassmann-valued scalar fields χ a x , x parametrizing spatial points, a = 1, . . . , N , with action where D equals minus the Laplacian on flat R 3 , minus the conformal Laplacian on the round S 3 , or any other similar operator defining a free CFT. In what follows we will assume for simplicity we are considering flat R 3 . Note that the index contractions of the χ a can no longer be performed with the O(N)-invariant metric δ ab , as this gives zero action for anticommuting fields. Instead of the symmetric δ ab , we must contract by a constant antisymmetric ab , which we can take to be of the following standard symplectic form:

(2.46)
This is invariant under the group Sp(N) of linear symplectic transformations preserving ab . We are assuming N is even here. Thus the group Sp(N) takes over the role of O(N) in this model. In particular, we view this group as gauged, in the sense that physical operators are restricted to be Sp(N)-invariant combinations of the χ a .
The two-point function is where ab = − ab , and (on R 3 ) Normal ordering is defined such that : The single-trace primary fields O i 1 ···is (x) consist of even spin s traceless conserved current bilinears, as in (2.43), i.e. O = c 0 : , and so on. The partition function Z Sp(N) [β] is the generating function for correlation functions of these operators: Here contracted x, y indices are integrated over, D xy = −∂ 2 δ 3 (x − y), and the trace term arises due to the normal ordering subtraction (2.49). In (2.50), the source deformation B takes the form

52)
The round brackets on the indices x, y denote symmetrization: . Collectively labeling the spin indices and spatial points by I = (i 1 · · · i s , z), we can succinctly rewrite the above expressions as (2.54) In this notation, the primary 2-point functions are For example for s = 0 we have G  Expressions similar to (2.55) can be written for all connected n-point functions, as can be seen most easily by rewriting (2.51) as

57)
where Again this object is to be understood in a renormalized sense. Since log Z is the generating function of connected correlation functions, the coefficients G I 1 ···In are proportional to the coincides with the free higher spin wave function (2.39), ψ 0,free (β), provided we identify This is essentially the expected relation between CFT central charge and the Newton constant, N ∝ 2 dS /G Newton . The conjecture of [42] can now be phrased more precisely as CFTs as well [84,85]. Roughly speaking this works because analytic continuation from AdS to dS entails a continuation of the curvature radius → i , so which is effectively realized by replacing commuting scalars by anticommuting scalars. In this work, we will only consider the free case.

Problems
Pleasing as it is, the above concrete realization of the idea (2.41) is by no means a complete answer of what higher spin quantum gravity is in de Sitter space. It is at most half the answer: 1. The main problem is that just specifying a wave function without specifying the Hilbert space to which it belongs is physically meaningless. This problem is a practical one: without a Hilbert space inner product, it is impossible to compute prob- theorem, and even if it did, its states would live on two-dimensional spatial slices, whereas the wave function ψ HH (β) lives on a three-dimensional slice. Related to this, time evolution in AdS corresponds to translations along the boundary, hence coincides with time evolution in the CFT, whereas is dS time evolution must emerge holographically. Another related issue is that in dS, the CFT sources β are themselves dynamical, again in contrast to AdS. 2. To specify a Hilbert space, one needs to define an inner product. In principle this could be as straightforward as specifying a domain for B and an integration measure Usually when we quantize classical systems, the integration measure is determined by the symplectic structure on the classical phase space: picking canonical Darboux coordinates, the measure is flat. However in the case at hand, we aren't directly quantizing classical Vasiliev gravity, and moreover the structure of the phase space of Vasiliev gravity is unknown. 3. One could proceed naively and assume the sources β I to be half of a set of Darboux coordinates (β I , α I ), as is the case in the free higher spin field theory discussed in section 2. 1.2. In the quantum theory these are then promoted to self-adjoint operators where dH is the flat measure. In this case, the integration domain is naturally restricted to positive definite matrices H, since this domain is GL(K) invariant, and since we do not require translations of H to be a symmetry of the theory. Evidently though, the proper K = ∞ continuum counterpart of this is tricky to define, and its existence is far from clear. 5. There are other technical issues with the definition and interpretation of ψ HH (β) = Z Sp(N) (β) in the continuum, once we go beyond perturbation theory, and want to make sense of it as a finite functional for general continuous sources β I of all spins. Indeed this requires adding infinitely many local counterterms to cancel off UV divergent contact terms. Even at the quadratic level in the sources, determining those is already a formidable task [86], and simple dimensional analysis makes it clear that the problem becomes in particular intractable for the higher spin sources β i 1 ···is , since these have negative dimension ∆ s = 2 − s, allowing in principle infinitely many local counterterms of arbitrarily high order.
We will take a different point of view here, and start instead with a description of a Hilbert space and a vacuum state |ψ 0 , to be identified with the Hartle-Hawking state, which is manifestly well-defined, and then show that at least in a formal sense it implements the invariant measure suggested above, that it satisfies the required symmetry properties, and that it reproduces the predictions of the bulk ψ HH (β) in perturbation theory in the large N limit.

Proposal
In this section, guided by an explicit construction of the generating function of cosmological correlation functions to leading order in the large N limit, we formulate our proposal for the

Large N argument
To get a crucial hint as to what the proper, well-defined description of the Hilbert space might be, consider again the wave function (2.60): The generating function for correlation functions of the (shadow) boundary fieldsb I = where we used (2.56), and we leave the measure unspecified at this point. Substituting To leading order in the N → ∞ limit, we can evaluate this in saddle point approximation, without knowledge of the measure, just by extremizing the exponent with respect to variations of B. The saddle point equations are Thus to leading order in the large N limit, we find We now recognize the right hand side of (4.5) as the generating function for correlation functions of bilinears in a bosonic O(2N) vector model: where the integration variables are 2N bosonic fields Q α x , α = 1, . . . , 2N . This can now be interpreted as the generating function for correlation functions of normal-ordered QQ bilinears in a very simple Gaussian vacuum state in a well-defined Hilbert space, as follows: 1. Define a Hilbert space H 0 with standard (flat measure) inner product (4.7)

Define a vacuum wave function
with c a normalization constant such that ψ 0 |ψ 0 = 1.

3.
Define normal ordering of QQ bilinears in the usual way by subtracting their vacuum expectation value: 4. Define single trace primary operators (analogous to (2.54))

11)
Note that the large-N limit considered above does not correspond to the free limit in the bulk, but to the (interacting) tree level approximation. Thus, in view of the results of [42], the left hand side of (4.11) should correspond to the generating function for vacuum correlation functions computed at tree level in the dS 4 higher spin Vasiliev theory. We can't really do any better than that on the left hand side, because we have not defined the measure, and because Vasiliev gravity at this point is only defined as a low-energy classical effective field theory, not as a complete perturbative quantum field theory. (There is no known action, no known phase space to quantize, and the theory breaks down as an effective theory at de Sitter curvature scale.) The right hand side of (4.11) on the other hand is perfectly well-defined. The Hilbert in AdS, they correspond to the boundary fields α I corresponding to normalizable modes in AdS, whereas here, they correspond to the shadowsβ I = G IJ β J of the boundary fields β I , as defined more explicitly in (2.33). In AdS, these correspond to non-dynamical sources, but in dS, they these modes are dynamical, dominating in fact the late-time structure of the universe.

Some simple illustrations
Before continuing to further refine this proposal and to provide more arguments in favor of it, let us pause here and give some simple examples to more concretely illustrate the perhaps somewhat abstract considerations given above.
According to our prescription, the exact generating function for vacuum correlation functions of the boundary fields is In particular the vacuum 2-point functions are   Going one order beyond the Gaussian approximation, we can read off from (4.12) that the vacuum 3-point functions are given by In position space this is a pure contact term ∝ δ 3 (

Equivalence of descriptions for K degrees of freedom
In this section we consider toy models in which the boundary field B xy in ψ HH (B) is replaced by a finite-dimensional K × K matrix, i.e. the continuous spatial indices x, y of the original model are replaced by discrete indices x, y = 1, . . . , K. This can be thought of as a lattice regularization of space with K lattice points. We will show that in this case, the proposed description in terms of ψ 0 (Q) is exactly equivalent to the original description in terms of ψ HH (B), for a well-defined and natural choice of measure [dB], provided K ≤ 2N . In section 6 we will see that this bound has significance in the continuum model as well: it is roughly speaking the maximal number of degrees of freedom that can be adequately described by a local field theory in the bulk.

K = 1
We begin by considering the simplest case, K = 1. In this case we can drop the x, y indices altogether, and ψ 0 in (4.8) reduces to the ground state wave function of a 2N -dimensional isotropic harmonic oscillator with coordinates q α , α = 1, . . . , 2N : where we have put D ≡ 1 for simplicity. The inner product on the Hilbert space H 0 is the We take this O(2N) symmetry to be gauged, which means we take the Hilbert space H to The corresponding canonically conjugate quantum operators take the same form, with the appropriate symmetrization of v to ensure hermiticity: Denote the basis of delta-function orthonormal eigenstates ofû by |u , i.e.
We can alternatively parametrize these canonical kets by In terms of this coordinate, (4.21) becomeŝ Note that h > 0, so, unlikeû = logĥ, the operatorĥ itself cannot have a well-defined hermitian canonical conjugate operator on the Hilbert space. Equivalently, unlike transla- To express the ground state |ψ 0 in terms of the |h basis, we cannot simply substitute the change of variables q 2 = N h into (4.19). Rather we have to computẽ The matrix element q|h is the wave function of |h interpreted as a state on the original

.27)
Expanding h = 1 + b around its expectation value ψ 0 |h|ψ 0 = 1, we may writẽ which we recognize as the K = 1 analog of the wave function (4.1). It can be represented as an Sp(N)-invariant Grassmann integral in the obvious way. What the present analysis tells us is that this form of the wave function will give exactly the same results as the q-space form, provided we use the natural measure [dh] = dh h , integrated over h > 0. As a check, let us compare the generating function e N ab ≡ ψ 0 |e N ab |ψ 0 for moments of the fluctuation variableb, computed using ψ 0 (q) versus usingψ 0 (h). First we use ψ 0 (q): Next we useψ 0 (h) to compute the same: (4.30) in agreement with (4.29).
It should be kept in mind that O(2N)-invariant operators o, for example the self-adjoint , in the following way: For the above examples this yields Note that these operator are self-adjoint with respect to the measure dh h . In particularṽ generates scale transformations h → e 2λ h. As a check one can verify that the operatorsh, v andw close under commutation, satisfying the same algebra as h, v, w.

K ≤ 2N
We will now do the same exercise starting from a Gaussian wave function depending on    The matrix H xy is symmetric and non-negative. The normalization is chosen such that the expectation value of H xy in the state |ψ 0 is There are two qualitatively distinct cases to consider: In what follows we will assume we are in the first case, K ≤ 2N . A discussion of the case K > 2N and the continuum limit is postponed to section 4. 3. 3.
In analogy with (4.23) we define O(2N)-invariant eigenkets of the hermitian operator H xy , satisfyingĤ The normalization of the eigenkets is chosen such that the corresponding decomposition of unity on the physical Hilbert space H has a scale invariant measure, as in (4.23): Here the integral is over real positive symmetric matrices H xy . To see that this is scale invariant, observe that real symmetric matrices have K(K+1) Thus the wave function of the state |ψ 0 expressed in the basis |H is 42) normalized such that 1 = ψ 0 |ψ 0 = [dH] ψ 0 |H H|ψ 0 , that is to say where we used the GL(K)-invariance of the measure. Although we won't need it in this section, the normalization constant can be computed explicitly. We do this in appendix B, the final result being (B.11): Note that the restriction K ≤ 2N we made is necessary for this to be finite.
Instead of H xy , we can also consider the closely related matrices H xy obtained from H xy by raising indices using the "metric" D xy : with the same normalization constant c as in (4.42). Expanding this in the fluctuation we getψ which we recognize as the finite-dimensional analog of the wave function (4.1). Again this can be written as an Sp(N)-invariant partition function in the obvious way. What the present analysis tells us is that this form of the wave function will give exactly the same results as the Q-space form, provided we use the natural defined in (4.39), integrated over H > 0.
As a check, we can compare the generating functions for vacuum correlation functions of normal ordered QQ bilinears B I defined as in (4.10) and (4.9), but now with the D xy I a set of K ×K matrices. This reproduces all the results of the saddle point analysis of section 4.1, except that now the equality of generating functions is exact. In both descriptions it is given by (4.12), for any N ≥ K/2, not just in the N → ∞ limit, provided we use the natural GL(K)-invariant measure [dH] defined in (4.39), integrated over H > 0.

K > 2N and continuum limit
The case of actual interest has K > 2N -in the continuum limit, K = ∞. When K > 2N , the Q-space description of the Hilbert space H remains nondegenerate, but the H-space description as we defined it becomes singular. This can be seen explicitly for example from (4.44), which diverges for K > 2N . The reason for the breakdown at K > 2N is that the  .12), we can at the end take → 0.
Then we arrive again at the conclusion that the (analytically continued) generating function of correlation functions in the H-space description coincides with the one computed in the Q-space description, which remains given by (4.12) irrespective of whether K is smaller or larger than 2N .
For non-perturbative questions the H-space description will likely be inadequate. The point of view we take here is that the Q-space description is the more fundamental one.
Happily, it is also the simpler one. For K > 2N the Q-space description remains well-defined, while the description in terms of ψ HH (b) becomes singular. However the equivalence persists in perturbation theory defined by analytic continuation. Thus we will use the Q-space description as the fundamental definition of the Hilbert space H .

Conclusion and comments on
The Hilbert space H is not yet the physical Hilbert space H phys of higher spin de Sitter quantum gravity. To define H phys , we need to take into account additional constraints related to bulk higher spin gauge invariance. We will discuss these gauge symmetries and the construction of H phys in section 7. We will also explain there that the choice of D appearing in the definition of ψ 0 (Q) and the Sp(N) model can be thought of as a partial gauge fixing choice: for example, taking D to be the flat Laplacian on R 3 corresponds to picking planar coordinates in de Sitter, while taking it to be the conformal Laplacian on while H phys should be viewed as a nonperturbative completion of H Fock, phys . The following two sections will exclusively pertain to H rather than H phys , and the results we obtain should therefore be compared to analogous computations on H Fock rather than H Fock, phys . , is its apparent non-normalizability, as pointed out in [49]. This pathology is completely eliminated in our setup. In what follows, we will compute more specifically the probability distribution for the constant scalar mode on S 3 , and find that in contrast to the results of [49], it is normalizable.

Probabilities and correlation functions
We will switch from planar to the global de Sitter gauge here, for which the operator D appearing in the wave function ψ 0 (Q) ∝ e − 1 2 QDQ becomes minus the conformal Laplacian on S 3 . The flat Laplacian on R 3 is mapped to the conformal Laplacian on the round sphere of radius L in stereographic coordinates by a Weyl transformation A further spatial diffeomorphism x(u) maps this to the conformal Laplacian in any desired coordinate system u with metric ds 2 = h ij du i du j on the round sphere of radius L. In more detail, under these field redefinitions, the planar dS wave func- and R(h) = 6 L 2 is the Ricci scalar.

Sp(N) model
In [49] it was pointed out that the Sp(N) wave functionψ 0 (B) on S 3 diverges exponentially at large negative constant scalar deformation B xy = b 0 δ xy . This is easy to see. The spectrum of the minus the conformal Laplacian D on the round sphere with unit radius is λ = ( + 2) + 3 4 with = 0, 1, 2, . . . and degeneracy d = ( + 1) 2 . Equivalently, putting = k − 1, we have λ k = k 2 − 1 4 with k = 1, 2, . . . and degeneracy d k = k 2 . Deforming the conformal Laplacian by a constant b 0 just shifts this spectrum to λ k = k 2 − 1 4 + b 0 , so the corresponding wave function becomes This can be evaluated by first computing ∂ b F (b), which sums to an elementary function, and then integrating this back up to F (b 0 ) with integration constant fixed by F (0) = 0.
The result is  It should be kept in mind that this is just the wave function for a single deformation, with all other deformations kept zero. As such, |ψ 0 (b 0 )| 2 does not have an immediate interpretation as the probability distribution for measuring b 0 , tracing over everything else. To obtain this probability distribution, we would have to integrate over all possible other deformations as well. At first sight this seems like an impossible task, since this involves integrating over an infinite number of higher spin degrees of freedom, all coupled to each other. Remarkably, in the Q-model description, this actually becomes entirely straightforward. We turn to this next.

6)
and G 00 is the ∆ = 1 scalar 2-point function in the zero angular momentum sector on S 3 . Explicitly we can compute G 00 directly from its definition either in position space, , or in angular momentum space, 4 . We conclude that The probability distribution for the bilinear B 0 in the Q-model is computed by This is useful in particular to get the large-N probability distribution in various limits:

Probabilities of general field profiles
The above computation is readily generalized to general scalar field profiles. For example the probability distribution for a general shadow scalar profileβ(x) on R 3 is given by

16)
This can be solved numerically or analytically in suitable limits by standard methods, e.g.
in long wavelength, large field regimes, or perturbatively at small B. As a check, note that to lowest order in small B perturbation theory, this equation becomes d 3 y G(x, y) u(y) = |x−y| 2 . In momentum space this becomes u(k) = kβ(k) = β(k), that is to say the saddle point value of u(x) equals the local boundary field β(x) of the bulk theory to this order, and the probability density for β becomes P (β) ∝ e −N k 1 k β(k) β(−k) to this order, as expected from the free bulk theory. Non-Gaussian corrections to this are obtained by going to higher orders in perturbation theory. In principle we can evaluate non-perturbatively as well, by evaluating the functional determinant non-perturbatively, as we did for the constant scalar mode on S 3 in the previous section. Of course it is more efficient to compute such n-point functions directly in the Q model rather than to first compute the full P (β). We turn to this next.

Cosmological correlation functions
In this section we compute the vacuum scalar-scalar-graviton 3-point function as well as the scalar 4-point function. Although we won't consider spins higher than two, let us begin by briefly outlining how in principle these computations can be systematically generalized to arbitrary spins. To do so, we need an efficient way of generating higher spin currents.
Here we will consider the generating function used in [89]. Given 2N real fields Q α x , one employs the equivalence between traceless symmetric tensors and functions of a complex null vector z [90] to construct where the coefficients a k are given in terms of Gegenbauer polynomials C The definition is somewhat degenerate at d = 3, and we should rescale the currents by some function that is singular at d = 3. In order to get the correct normalization for the two-point functions, we choose where T s (u) is Chebyshev polynomial of order s, and (M ) n denotes the Pochhammer symbol. With this choice, the spin-two B I operator is We separately choose the normalization of the scalar operator B 0 as follows The two point function of higher spin fields B I with these normalizations are then [89] For scalar and spin two operators, this gives two-point functions consistent with the normalization of section 2.1.2, (5.23)

Three-point functions in momentum space
Three-point functions of higher spin conserved currents in three dimensions are extensively discussed in [91]. Conformal symmetries fix much, but not all, of their structure. In appendix A it is shown how the shadow transform simplifies in momentum space. Given the relevance of shadow transforms to our discussion, we are particularly As a simple example, we consider the three point function of two scalars and a graviton.

(5.25)
The three point function is There are eight types of contractions among the three groups of normal ordered Q bilinears.
They all have the same contribution, which is The integrals can be performed following the methods of [92]. One finds The three point function can now be expressed more compactly as B 0 (p 1 )B 0 (p 2 )B 2 (p 3 |z) = − 32 √ 2 N 2 δ p 1 +p 2 +p 3 (I 1 (p 1 , p 2 , p 3 ; z) + I 2 (p 1 , p 2 , p 3 ; z)) (5.28) Finally, we rewrite the three-point function in terms of the standard local boundary fields B 0 and B ij 2 (which appear as local scalar and spin-2 sources in the Sp(N) model). By our general prescription, they are related to B 0 and B ij by the shadow transform B I = G IJ B J : where Π ij,i j (p) is the projector (2.23). Picking a transverse traceless gauge for B 2 , this sim- Putting everything together we end up with the following simple expression for the scalar-scalar-graviton 3-point function:

Scalar four-point function
It is also straightforward to calculate higher point functions in our setup. These are far less constrained by conformal symmetries. As a final example, we give here the scalar fourpoint function. For inflationary theories these were considered, for example, in [107,108].
In our case we have This integral in can be calculated 1 by considering the inverse momenta as follows [109].
Let us define P i ≡ i j=1 p i and denote the inverse of a vector with a tilde, for examplẽ q i = q i /q 2 . We then have that Using this, we can rewrite the integral appearing in the four-point function in terms of three-point function integrals, We introduce the quantities and calculate the three-point function integrals using the methods of [92], This leads to the scalar four-point function As a simple check, the above result is permutation invariant under exchange of the momenta and has the correct scaling properties. We have also verified the result numerically.
In the scattering amplitudes literature, the integral (5.33) is referred to as a four-mass box integral. Explicit analytic results in the d = 3 case of interest to us were previously obtained in [53]. Perhaps the approach we have taken here, based on [92,109], may be useful more broadly in that context as well, given that our expression (5.38) is dramatically simpler than the one obtained in [53]. 6 Perturbative bulk QFT reconstruction 6

.1 Bulk reconstruction and the Heisenberg algebra
So far we have shown how to compute probability distributions and vacuum correlation functions of higher spin de Sitter boundary field modes β I , identified in our framework with bilinear operators B I = 1 N : Q x D xy I Q y acting on the Hilbert space H , related more precisely by the shadow transform B I =β I = G IJ β J . We found results consistent with general expectations from perturbative bulk QFT, and illustrated for the constant scalar mode β 0 on S 3 how to go beyond the perturbative bulk QFT regime.
However, to go beyond observables that can be described entirely in terms of the boundary fieldsβ I , and in particular to reconstruct local bulk quantum fields φ i 1 ···is (η, x), we also need to identify the operators canonically conjugate toβ I , that is to say the operators α I appearing in the perturbative bulk QFT Heisenberg algebra (2.30), which in our condensed notation reads where in our conventions, as in (2.59), γ = 2 N . Once these conjugate operators have been identified, we can define free local quantum fields φ i 1 ···is (η, x) in the bulk simply by copying the free field expression (2.25), that is, in d = 3, Transformed to position space, these definitions express φ i 1 ···is (η, x) as convolutions of α i 1 ···is (x ) andβ i 1 ···is (x ) with certain boundary-to-bulk kernels K α (η, x−x ) and K β (η, x−x ), providing a dS analog to the HKLL construction [93] in AdS (except that in dS, we have two dynamical modes rather than the single normalizable mode in AdS; see also [94,95]). The boundary fields α, β must satisfy the Heisenberg algebra in order for the free bulk fields thus constructed to be local and causal, i.e. in order for the fields commute at spacelike separations. Bulk interactions can in principle be reconstructed order by order by comparing vacuum correlation functions computed in the Q-model to vacuum correlation functions computed perturbatively in the bulk interaction picture using these free fields.
However, as we will see, the exact bulk perturbative Heisenberg algebra (6.1) cannot be realized on H . Indeed this is obvious for a number of reasons: 1. The algebra (6.1) implies an infinite number of independent degrees of freedom per spatial point: one for the scalar, and two for each of the infinite tower of higher spin fields. However, the Q-model has only 2N degrees of freedom per spatial point.
Hence H cannot possibly accommodate (6.1). Nevertheless, we will show below that at large N , in discretized models with K O(N ), it is possible to define self-adjoint operator α I such that (6.1) is satisfied up to nonperturbatively small corrections, in the sense that on states not too far from the vacuum, the algebra is satisfied up to an operator-valued error term of order δ Heis ∼ e −g(κ)N , (6.3) where κ = K 2N and g(κ) is some order 1 function which is positive for κ 0. 17. Moreover, for K > 2N , including in the continuum limit K = ∞, the Heisenberg algebra can be realized with similar exponential accuracy provided we consider "coarse grained" operatorsβ I ,ᾱ I and restrict to a finite volume of space, such that the effective number of resolvable spatial "pixels" K eff is less than some order N number.

Even in discretized models
This suggests we can reconstruct perturbative bulk quantum field theory from the fundamental boundary Hilbert space H , but only up to a resolution in which at most O(N ) spatial pixels are resolved. This may seem peculiar, but keeping in mind that the de Sitter horizon entropy S dS ∼ 2 dS /G Newton ∼ N in this model, it is in line with general expectations on limitations of bulk effective field theory in de Sitter, based on the holographic principle and related ideas (see e.g. [24,96,97,98]), although realized here in an unusual and perhaps surprising way.

More general comments
Independent of the construction of a Heisenberg algebra, there are other reasons to expect a breakdown of conventional perturbative bulk QFT in out setup. One is that perturbative single-particle kets such as Of course this does not mean we cannot define normalizable "single-particle" states more general than single-scalar states in H . For example the states |F ≡ xy F xy B xy |0 , for smooth functions F xy with compact support, are normalizable, and can formally be thought of as infinite superpositions of higher spin particle states by Taylor expanding in r = (x − y). But it does mean that within in the Q-model, we cannot truly define an orthonormal basis of single-particle states of definite s, and thus cannot truly reproduce the  In a similar spirit, we noted in section 4.3.3 that in discretized models with K lattice points, although it was possible to formulate perturbation theory purely in terms of the Sp(N) model wave functionψ 0 (H), this required renormalization through analytic continuation as soon as K > 2N . More generally, we noted various qualitative differences in behavior of such models depending on whether K ≤ 2N or K > 2N . The underlying reason for this was that for K ≤ 2N , the O(2N)-invariant bilinears H xy = 1 N Q α x Q α y are independent variables, with H generically of rank K, while for K > 2N , the H xy are not independent, as H generically has reduced rank 2N (as illustrated in fig. 4.1). This distinction between K ≤ 2N and K > 2N will also play a crucial role in our construction of an approximate Heisenberg algebra below.
Finally, an important thing to keep in mind is that all of these considerations pertain to the Hilbert space H (which is to be compared to H Fock , as we do in this section), not to the physical Hilbert space H phys (which is to be compared to H Fock,phys ). We discuss the construction of the physical Hilbert space in detail in section 7. As we will see, the difference between H and H phys is rather enormous in higher spin de Sitter, because the residual gauge group G on H , i.e. the higher spin group, is infinite dimensional.
This reduces the number of gauge-inequivalent degrees of freedom to such extent that the physical Hilbert space of gauge-inequivalent n-particle states becomes finite dimensional for any given n. In particular this effectively removes all UV divergences of norms of single-particle states mentioned above. Moreover we will show that all gauge invariant quantities can be computed by a 2N × 2N matrix integral. Thus, from the point of view of H phys , the inability to reconstruct local bulk QFT beyond a certain resolution set by N is not all that dramatic, and is in fact quite natural.

Reconstructing the Heisenberg algebra
The goal of this section is to investigate to what extent we can reproduce the perturbative higher spin bulk QFT Heisenberg algebra (2.30) in our framework. Of course, the Hilbert space H 0 of wave functions ψ(Q) with inner product (4.7) has a standard Heisenberg algebra, given by  Finding such operators will be our goal in what follows.

Free field theory approximation
It is easy to find operators A J ]|ψ 0 = 0, by defining with P z α = −i∂ Q α z as in (6.5), so [Q α x , P β y ] = i δ αβ (D −1 ) xy . Then we have Here we were allowed to add the normal-ordering signs because the bilinear operator inside the brackets has zero vacuum expectation value, as can be seen using P α x |ψ 0 = iQ α x |ψ 0 . Thus ψ 0 |[A To do better, we can try to correct the A (0) I order by order in the 1 N expansion. We turn to this next, first for the discretized toy models with K spatial points, and then in the continuum limit.

K=1
We first consider the K = 1 toy model introduced in section 4. 3. 1. The analog of (6.6) in this case is 10) It will be useful to first switch to the h-space description, which for K = 1 is equivalent to Hilbert space H consists of wave functionsψ(h) on the domain h > 0, normalizable with respect to the inner product with These manipulations are valid for wave functionsψ(h) vanishing sufficiently fast at h = 0, so integrating by parts does not pick up any boundary terms. The non-hermiticity is easily fixed, however, by defining where {A, B} = AB + BA denotes the anticommutator, and we recall thatṽ appeared in (4.33) as the h-space version of the manifestly self-adjoint q-space operator v =: q α p α :.
The operatorã is still canonically conjugate to h, as [h,ã] = 2i N . Althoughã is hermitian, in the sense that it maps to itself upon integrating by parts on wave functions vanishing sufficiently fast at the origin, is not self-adjoint, essentially for the same reason as the ordinary momentum operator in quantum mechanics of a particle on a half-line is not self-adjoint. Because of this, it cannot be exponentiated to a unitary operator, and it does not have a spectral decomposition on H, so it is not a valid canonical conjugate variable.
In the last expression forã in ( 6.12), the appearance of the singular operator h −1 is what causes the failure ofã to be self-adjoint. If instead of h −1 = 1 q 2 we had a self-adjoint operator regular at q = 0, the resulting operator would have been self-adjoint as well. This observation immediately suggests a definition for a self-adjoint operator approximatingã, approximately realizing the Heisenberg algebra in a perturbative sense. The basic idea is simply to truncate the formal power series h −1 = (1 + b) −1 = ∞ k=0 (−b) k to a finite number of terms, defining is regular at the origin q = 0, this operator is manifestly self-adjoint and regular at q = 0. Its counterpart in the q-space description is, written out explicitly, Notice a (0) = 1 N : qp :, which is the K = 1 analog of the free theory approximation (6.7). The higher order a (n) add corrections to this reducing the error in the Heisenberg algebra: That is to say, on the vacuum, or on states perturbatively close to it, the self-adjoint operator a pert ≡ a (N/2) satisfies the Heisenberg algebra up to non-perturbatively small corrections at large N : There is a very simple physical interpretation of this result: the power series h −1 = k has radius of convergence equal to 1. The probability for b to exit this region of perturbative convergence is P exit = N N Γ(N ) , z) is the incomplete Gamma-function. Asymptotically at large N , this becomes P exit = e −(1−log 2)N , which exactly equals the minimal error term e −Σ in the Heisenberg algebra. Thus, rather pleasingly, we conclude that the minimal error term in the perturbative Heisenberg algebra equals the exponentially small probability of a nonperturbatively large fluctuation. Although this is of course a toy model with just one spatial point, this is qualitatively in line with general physical expectations in de Sitter space: nonperturbatively large "uphill" fluctuations away from the classical vacuum (e.g. of a scalar φ with some potential V (φ)) are possible in de Sitter, but exponentially suppressed by a factor of e −S , where S is the de Sitter entropy. In Vasiliev dS gravity with Sp(N) dual, the dS entropy is proportional to N .

K ≤ 2N
The above construction for K = 1 is readily generalized to the case K ≤ 2N introduced in section 4. 3.2. In this section we won't notationally distinguish operatorsÕ acting oñ ψ(H) from operators O acting on ψ(Q) like we did in earlier, trusting the context will make clear which one is meant. As in the K = 1 case discussed above, it is possible to formally construct hermitian operators A I satisfying the Heisenberg algebra (6.6). Explicitly, they are given by where, when acting on wave functionsψ(H),  To write A I in a form analogous to the last expression for a in ( 6.12), first note that and B I = Tr(D I B) are represented on ψ(Q) as It can be checked explicitly from these expressions that the Heisenberg algebra (6.6) is For n = 0, this reproduces (6.8). For n = ∞, it formally reproduces the exact algebra However a more careful analysis shows again that the limit n → ∞ does not converge, but rather reaches an optimal approximation point at some value of n of order N . To see this, note that substitution of (6.24) in ( 6.25) gives Since B ∝ 1 √ N at large N in the Gaussian approximation, we might naively expect the error term δ (n) IJ to become arbitrarily small when n increases, but just as in ( 6.16), this simpleminded estimate fails when n becomes of order N . Consider first the vacuum expectation value of δ (1) Therefore we find for δ (1) IJ , and by a similar but somewhat longer computation for δ (2) IJ : More generally we have δ (n) 29) where Tr(BD) n can be computed by a matrix integral independent of D: The first matrix integral can be written in terms of the eigenvalues λ x > 0 of H as where we used the eigenvalue permutation symmetry to rewrite the trace expectation value in terms of a single eigenvalue λ K : Tr(H − 1) n = K x=1 (λ x − 1) n = K (λ K − 1) n . To find the minimal error in large N saddle point approximation, we have to extremize the integrand with respect to the eigenvalues λ x and with respect to n. Extremization with respect to n simply gives λ K = 2. Thus, the minimal value of δ (n) in a large N saddle point approximation is obtained as δ (n) | min ≈ P (λ K = 2), where P (λ K = 2) is the probability density for the eigenvalue λ K to equal 2. (The optimal value of n can in principle be found from the extremization equation with respect to variations of λ K .) Notice that the perturbative expansion (6.23), i.e. H −1 = (1 + B ) −1 = k (−B ) k also breaks down exactly when one of the eigenvalues λ of H exceeds λ ≥ 2. When N K, all eigenvalues clump near λ = 1. When K grows, their spread increases due to the eigenvalue repulsion induced by the Vandermonde measure factor. Thus when K gets too large, the eigenvalues will reach λ ∼ 2 with probability essentially equal to 1. Then the minimal error will be of order 1 and the Heisenberg algebra cannot be approximately realized following this construction.
Before we enter this regime though, when K is still sufficiently small, the probability of λ K making it has high up as λ = 2 is exponentially small at large N . In fact it will be essentially equal to the probability of the largest eigenvalue exceeding 2, which in the limit is given by the Tracy-Widom distribution [110]. Applied to the specific (Wishart) matrix ensemble (6.30) of interest the probability density for the largest eigenvalue λ is given in this limit by [111] where f 1 (x) is the β = 1 Tracy-Wishart probability density function and The relevant asymptotics of f 1 (x) are the x → ∞ asymptotics, which can be found e.g. in [112]: f 1 (x) ∼ e − 2 3 x 3/2 . In the limit (6.33), we thus get for the minimal Heisenberg error A plot of the coefficient g(κ) is given in fig. 6.1. Note that g(κ) is positive only on the interval (0, 3 − 2 √ 2) ≈ (0, 0.171573). Beyond this range, P (λ > 2) becomes of order 1 in this asymptotic limit, due to eigenvalue repulsion effects, and the perturbative Heisenberg algebra error cannot be made small. Physically, the reason is that in this regime, some fluctuation takes us out of the perturbative regime with probability essentially 1. In two dimensions, this corresponds to an effective number of pixels equal to K eff = V 4π 2 ≈ 10 6 , 10 4 , 10 2 . Realizing the perturbative QFT Heisenberg algebra at a resolution K eff requires N 3K eff

K > 2N and continuum limit
It is clear already from the low order error terms (6.28) that the above construction of an approximate Heisenberg algebra breaks down when K > 2N , so most definitely also in the continuum limit K = ∞. Our asymptotic analysis confirms this, and in fact indicated a breakdown well before this point, namely for K 2N > 3 − 2 √ 2 ≈ 0.171573. The basic reason for this was that when K gets large, matrix eigenvalue repulsion effects push B outside the radius of convergence of the perturbative expansion (6.23): the probability that H = 1 N QQ T has an eigenvalue larger than 2 becomes essentially equal to 1 in this regime.
When K > 2N , the impossibility of realizing the Heisenberg algebra also follows from a much simpler observation, namely the fact that Π xy , the formal canonical conjugate to H xy defined in ( 6.20), no longer exists, because H = (DQ)(DQ) T becomes non-invertible (as it has reduced rank 2N < K). Similarly, in the continuum limit, one cannot expect to be able to construct the perturbative QFT Heisenberg algebra, because the full collection of perturbative higher spin fields yields an infinite number of degrees of freedom per spatial point, whereas the fundamental variables Q only constitute 2N degrees of freedom per spatial point. This means in particular that the full collection of QQ-bilinears B I is not an independent set of variables, making it manifestly impossible to construct a Heisenberg algebra of the form (6.6).
The above observations suggest that we can at most hope to construct an approximate Heisenberg algebra for a much smaller set of degrees of freedom than naively suggested by the perturbative bulk QFT. Here we will consider an idea realizing this intuition.
We will define a "coarse grained" version of the operators A I , B I , in such way that the effective number of coarse grained spatial "pixels" K eff becomes less than the order N bound suggested by the finite K models, so we can effectively use the construction of section 6.3. 3. The fundamental theory itself remains the same; only the operators we consider get coarse grained. To this end, we define coarse grained fundamental fieldsQ as where W (u) is a "window" function satisfying d 3 u W (u) = 1 (so d 3 u W (u) = 1 as well).

A convenient choice for practical computations is a Gaussian
Moreover we restrict the range of x to some region of volume V 3 . The coarse graining has the effect of reducing the effective spatial resolution to pixels of size ∼ 3 , so we may expect to get an effective number of degrees of freedom of order K eff ∼ V / 3 . The vacuum

2-point function ofQ is
For the Gaussian window function (6.38) this can be explicitly computed by first writing then performing all Gaussian integrals, and finally doing the integral over t. The result is where Erf(z) is the error function, Erf(z) = 2 √ π z 0 dx e −x 2 , which satisfies Erf(z) ≈ 1 − e −z 2 √ πz for z 1 and Erf(z) ≈ 2z √ π for z 1. Thus, as was to be expected, theQQ 2-point function is essentially indistinguishable from the original QQ 2-point function when |x − y| , while it gets regularized for |x − y| , capping off smoothly in the x = y coincidence limit at a finite value 1 4π 3/2 . This is illustrated in fig. 6.3 on the left.   43) which is finite as anticipated. We next define coarse-grained versions of B xy and B I simply by replacing Q everywhere byQ, but keeping everything else unchanged: This renders formerly UV-divergent quantities finite. Recall for example that in the discretized, finite K model, we had Tr(DB) 2 = K 2N (K + 1), which diverges in the continuum limit K = ∞. In the coarse grained case we get instead where we used the fact that V 3 , so K eff 1.
The vacuum 2-point functions of the bilinearsB I are .
Since we necessarily have |x − y| L in the finite volume V = L 3 under consideration, we need √ s L for the above condition to be realizable, which puts a restriction on the range of spins for which the coarse grained setup can in any way be a good approximation to the original setup, to wit With this in mind, we are ready to repeat the construction of section 6.3.3 to define approximate canonical conjugates A I to the B I , at least within this range of spins. To this end, we define a coarse grained version of P α x defined in (6.7), J ], we now get a somewhat more complicated expression than (6.26), because we no longer get exact cancelations in the sum over k: The first term in (6.51) reproduces the continuum Heisenberg algebra ( 6.6) to the extent that (6.47) is satisfied. Up to subleading terms in the large K eff expansion, the vacuum expectation values of the error terms are The first expression is just the coarse grained version of the analogous error term appearing in the finite K model. The terms in the second expression are essentially zero in the regime (6.47). To see why, consider for example the s = 0 case D uv When |x − y| , we can put g ,xy ≈ 1 4π|x−y| , g ,yz ≈ 1 4π|y−z| , up to corrections of order e −|x−y| 2 /4 2 . But then the integral over z equals the electrostatic potential at the point y of a spherically symmetric charge density ρ(z) = (δ − δ 0 ) z x centered at x, essentially concentrated within a region |z − x| O( ), with zero total charge. In other words it is the electrostatic potential of a neutral, spherically symmetric atom, at distances much larger than the radius of the atom. This vanishes, so (6.56) vanishes, up to terms of order e −|x−y| 2 /4 2 . Similar considerations hold for higher spins s, as long as (6.47) is satisfied, as well as for the 1/K eff corrections. We conclude that, at least when n is not too large, the error term ∆ IJ is negligible in the regime of interest. Since the remaining δ IJ error term is similar to the δ IJ error term in the finite K model, we may expect minimal error estimates similar to (6.36) to hold in the present case as well.
Thus we conclude that in the regime (6.47), as long as This is adequate for physical observables that can be computed accurately with these coarse-grained operators, such as suitable cosmological correlation functions at a coordinate distance scale r such that r L. The fundamental limitation in this construction is that the UV and IR cutoffs must be chosen such that the number of distinguishable spatial pixels K eff ∼ L 3 3 is smaller than N . Moreover, by (6.48), there is a bound on the spin s that can be resolved in this way, requiring s N 2/3 . Although other constructions might be possible circumventing these limitations, this does hint at a breakdown of perturbative bulk quantum field theory when trying to go beyond these bounds. Since the horizon entropy of Vasiliev de Sitter is of order N , this would be in line with expectation from the holographic principle (see [24,96,97,98] for related discussions), although realized in a novel way. We leave a more thorough investigation of these observations to future work.

Physical Hilbert space
As we already briefly discussed in section 4.4 In this section we will take a closer look at these gauge symmetries and the construction of H phys . We first point out that the choice of operator D appearing in ψ 0 (Q) and in the Sp(N) model can be thought of as a partial gauge choice, and that this choice leaves a residual gauge group G which can be thought of as the higher spin symmetry group, or a generalization thereof. This means that H phys can be defined, formally at least, as the G-invariant subspace of H . We discuss some of the subtleties which quite generally arise when carrying out such a program in practice, and how they are resolved in our framework.

Gauge invariance
For most of our concrete computations in the continuum limit, we took the operator D gravity even in perturbation theory [55,56]; that is to say, physical states must be defined to be invariant under SO (1,4 Depending on restrictions one may wish to place on the set of admissible field redefinitions R, there may be subtleties in making the above discussions more precise. In conventional definitions of the higher spin symmetry group, one works at the level of the Lie algebra rather than at the level of finite group transformations, or equivalently at the level of infinitesimal group transformations R = 1 + L. The conventional (real) higher spin Lie algebra is then roughly speaking defined as the space of finite-order differential operators L satisfying L T D + DL = 0 [57]. For example with D = −∂ 2 , this is satisfied for translations L = ∂ i as well as higher order L = ∂ i 1 · · · ∂ in with n odd, taking into account that ∂ T i = −∂ i . It is however a subtle matter whether these generators can be exponentiated to finite group transformations in a given representation of the algebra [120].
On the other hand, we can implement the residual gauge invariance constraints already at the level of the Lie algebra, by requiring physical states to be annihilated by the Lie algebra generators, so this does not affect our ability to define H phys . Alternatively, one could drop the restriction to transformations R generated by finite differential operators

Physical Hilbert space at finite K
The construction of gauge invariant operators and states in H phys is most straightforward in the finite-K discretized models, for which ψ 0 (Q) ∝ e − 1 2 Tr(Q T DQ) , with D xy a positive definite symmetric K × K matrix. The full gauge group is GL(K), acting as Q → R −1 Q, D → R T DR, with the analog of the residual "higher spin" symmetry group being the subgroup G satisfying R T DR = D. The group G is isomorphic to O(K).

The operator algebra of H consists of linear combinations of O(2N)-invariant products
of the operators Q α x and P x α = −i∂ Q α x , such as Q α x Q α y , P x α P y α , Q α x P y α + P α y Q α x and products thereof. The states of H are generated by acting with these operators on the vacuum state It is convenient to consider normal-ordered operators such as : H x 1 y 1 H x 2 y 2 H x 3 y 3 :, which are defined as usual by removing all vacuum self-contractions, or equivalently by expressing the operators Q α x in terms of creation and annihilation operators and moving all annihilation operators to the right. (Here the relevant annihilation operator is a α x ≡ (D −1 ) xy P y α , since this satisfies a α x |0 = 0.) Then we may define "n-particle" states in H as |H x 1 x 1 · · · H xnx n ≡ : H x 1 x 1 · · · H xnx n : |0 . (7. 2) The normal ordering ensures that states with different n are automatically orthogonal to each other. With these definitions, we have for example, denoting g xy ≡ (D −1 ) xy , H xx |H yy = 1 2 g xy g x y + g xy g x y . 3) The normalization in (7.1) was chosen such that there is no N -dependence in this n = 1 expression, and such that for general n, the leading term at large N is of order N 0 .
The operator algebra of H phys consists of combinations of Q α x and P x α invariant under both O(2N) and G. For the construction of states in H phys , it suffices to consider Ginvariant operators f (H) acting on |0 . Such operators are generated by traces: denoting Q x α ≡ D xy Q α y , let us define Tr(HD) n = 1 (N K) G-invariant states are then produced by acting with normal-ordered products of such operators on the vacuum |0 : |T n 1 · · · T n k ≡ : T n 1 · · · T n k : |0 . (7.5) This state can be interpreted as a gauge-invariant n-particle state, where n = n 1 + · · · + n k .
For low n, it is easy to construct a basis of invariant n-particle states in H phys : Sectors with different n are orthogonal due to the normal ordering in (7.5), and the inner products for a given n are readily computed: The normalization in (7.4)  The inner product between two such projected states is (ψ 1 |ψ 2 ) = ψ 1 |1 phys |ψ 2 = rs (C −1 ) rs ψ 1 |r s|ψ 2 . (7.12) One can alternatively think of the projection operator 1 phys as being the result of a "group averaging" procedure over the residual gauge group G = O(K): where U (g) denotes the unitary representation of G on H , and [dg] is the Haar measure normalized such that G [dg] = 1. Thus the projected inner product (7.12) is the finite-K analog of the "group averaged inner product" for perturbative quantum gravity in de Sitter space as defined in [56] (see also [61,62] and references therein). We avoid the need for any explicit integrations here due to the fact that we know in advance the complete list of invariant states at each level n, and we avoid the complications of having a noncompact group (at finite K) because G = O(K) is compact. We will discuss the continuum limit in section 7.4 To compute the H phys -projected/group-averaged inner product (7.12) explicitly for the basis elements of H defined in (7.2), we need to compute overlaps of these with the invariant basis of H phys . This is straightforward but somewhat tedious (though easily automated in Mathematica). For example, H xx H yy |T 2 1 = 1 K 2g xx g yy + 1 N g xx g yy + g xy g x y H xx H yy |T 2 = 1 K g xy g x y + g xy g yx + 1 2N g xy g yx + g xy g x y + 2g xx g yy . (7.14) Thus, 16) where the second expression is schematic and meant to convey the nature of the K-and N -dependence. Note that the structure of these H phys -projected inner products are very different from the original inner products on H such as (7.3).
Using these results, we can construct group-averaged states of unit norm; for example Then we have more generally also (xx |yy ) = 1, for any choice of x, x , y, y , expressing the fact that all single-particle states in H are gauge equivalent to each other in this theory.

Reduction to 2N × 2N matrix model when K ≥ 2N
From the above it is clear that the computation of any gauge invariant quantity, e.g.
overlaps of invariant states such as (7.5) or vacuum expectation values of gauge invariant operators, can be reduced to the computation of the vacuum expectation value of products of trace operators T n ∝ Tr(QDQ T ) n as defined in (7.4). Note that since the final result will be independent of D, we might as well set D ≡ 1 from the start. Thus, all gauge invariant quantities can be reduced to the computation of Gaussian K × 2N matrix model integrals of the form Here and below the normalization of the integration measure is chosen such that I 0,0,0,... = 1.
When K ≤ 2N , we can follow the reasoning of section 4. 3.2 and change variables to the K × K symmetric positive definite matrix H ≡ QQ T to rewrite this in terms of a Wishart matrix model,  The point to note here is that H = QQ T is a K × K matrix, while M = Q T Q is a (2N ) × (2N ) matrix. Thus, when K ≥ 2N we can instead change variables to M and repeat the same reasoning to rewrite (7.18) in terms of a 2N × 2N Wishart matrix model: where dm is the flat measure on the space 2N × 2N real symmetric matrices, normalized such that the above integral equals 1 for k = 0. As a check, note that according to this formula, (: Tr m 2 +λ Tr m | λ=0 = 1 2N ∂ 2 λ e N λ 2 | λ=0 = 1 in agreement with (7.7).

Physical Hilbert space in continuum limit
In this section we will consider the construction of H phys in the continuum limit. We will define the continuum limit here as the limit K → ∞ with D limiting to, say, the Laplacian on flat R 3 , or the conformal Laplacian on the round S 3 . That is to say, we think of the finite K model as a lattice version of the continuum theory with K lattice points, and D the discrete Laplacian on the lattice. The key question then is whether we obtain finite quantities in the continuum limit K → ∞. In QFT, this typically involves delicate renormalization prescriptions. As we will see, the theory under consideration behaves in a much simpler way: once we impose standard normalization of states, all UV divergences disappear. This is essentially due to the fact that there exist only a finite number of n-particle states for each n.
As we have seen above, it is certainly possible to define a basis of H phys of invariant states such as (7.6) whose inner products remain finite in the limit K → ∞. For example from (7.7)-(7.8) we get in this limit The physical interpretation of these quantities is obscure, however. The situation is somewhat better for invariant states |ψ) = 1 phys |ψ obtained by group averaging of non-invariant states |ψ , or equivalently projection onto H phys , as defined in (7.11). For example if we take |ψ to correspond to some ordinary n-particle state in H , then we can think of as |ψ) as the state symmetrized with respect to the residual gauge symmetry group G. The group G includes for example rotations of the S 3 , so the group-average includes averaging over the center of mass position of the n particles on the sphere as well as their orientation, expressing there is no invariant physical meaning to the absolute position and orientation of n particles on an S 3 spatial slice of de Sitter space.
If we use properly normalized group-averaged states, i.e. unit normalized states such as (7.17), factors of K cancel out and inner products between such states are finite even in the limit K → ∞. To illustrate this, let us work out the inner products of group averaged 2-particle states to leading order at large N . In this limit, we have T 2 |T 2 = 1, T 2 1 |T 2 1 = 2, T 2 |T 2 1 = 0, hence, using (7.14) also to leading order at large N , where g xy = 1 4π|x−y| in the continuum limit. The corresponding unit-normalized states are |xx , yy ) ≡ K 4(g x(y g y )x ) 2 + 2(g xx g yy ) 2 |H xx H yy ) . (7.26) The inner product between two unit-normalized states is (xx , yy |vv , ww ) = 2 g x(y g y )x g v(w g w )v + g xx g yy g vv g ww 2(g x(y g y )x ) 2 + (g xx g yy ) 2 2(g v(w g w )v ) 2 + (g vv g ww ) 2 , ( 7.27) which is independent of K and thus remains finite in the continuum limit. To compute more specifically the inner product of unit-normalized group-averaged states corresponding to scalar field insertions |β(x)β(y) and |β(v)β(w) whereβ(x) is the shadow transform of the scalar field mode β(x), we need to consider the coincident point limit x = x, y = y, v = v, w = w: (xx, yy|vv, ww) = 2g 2 xy g 2 vw + g xx g yy g vv g ww 2g 4 xy + g 2 xx g 2 yy 2g 4 vw + g 2 vv g 2 ww . (7.28) Taking into account that lim x →x g xx = lim x →x 1 4π|x −x| diverges in the continuum limit, this actually collapses to (xx, yy|vv, ww) = 1 , (7.29) indicating that all 2-particle scalar states are gauge equivalent. It can be checked that this final result remains the same at finite N .
Defining H I ≡ D xy I H xy , using the notation originally introduced in (2.54), we can similarly construct 2-particle states |H I H J with particles of arbitrary even spin, their groupaveraged counterparts |H I H J ) and their unit-normalized versions which we denote by |IJ).
To leading order at large N , the overlap between such states is

31)
where s I is the spin of the single-particle state labeled by I. If both |IJ) and |KL) have at least one higher-spin particle, then (7.30) reduces to (IJ|KL) = G IJ G KL |G IJ ||G KL | = ±1, while if |IJ) has at least one higher spin particle while |KL) has two spin-0 particles, it reduces to (IJ|KL) = 2G IJ G KL √ 2|G IJ |Λ 2 = 0 in the continuum limit. Thus, consistent with the fact that the physical Hilbert space of two-particle states is two-dimensional, all inner products of this form collapse to either 0 or ±1, depending on the spin content of the states: if s I + s J = 0 and s K + s L = 0 ±1 if s I + s J > 0 and s K + s L > 0 0 otherwise A similar but longer computation shows that at finite N , (7.30) becomes At finite N there is some mixing between the two sectors which we identified above as orthogonal in the N → ∞ limit: if |IJ) has two spin-0 particles and |KL) has at least one higher spin particle, we now get with 1 phys again the projector (7.10), and Z an appropriate operator renormalization factor.
Vacuum probability distributions P (A phys ) for the physical observable described by this operator can be inferred e.g. from its moments 0|A n phys |0 = Z n 0|A 1 phys A 1 phys A 1 phys · · · A|0 .

Conclusions
In this section we constructed the physical Hilbert space H phys as the subspace of H invariant under residual gauge transformations G, and started exploring its structure. We defined the continuum limit by first considering finite-K models with residual gauge group G = O(K) and then taking the limit K → ∞, in which G becomes a version of the higher spin symmetry group. This allowed for a precise definition and computation of groupaveraged states and their inner products. The resulting H phys has a topological flavor: sectors with definite "particle number" n only have a finite number of physically distinct states, and all gauge invariant quantities are computed by a 2N × 2N matrix model. In particular this effectively eliminates the usual UV divergences appearing in the continuum limit of H . Although we did not make this precise, it also suggests that the need for coarse graining to construct a perturbative Heisenberg algebra on H , as discussed in section 6.3.4, actually does not amount to a real loss of resolution from the point of view of H phys , given the even greater coarseness of H phys at least at finite n. The full Hilbert space H phys of higher spin de Sitter space remains infinite-dimensional, because there is no bound on n. However it seems quite plausible to us that a suitable reduced density matrix obtained from the vacuum state, with a bulk interpretation of being the density matrix of a "local observer", will nevertheless have a finite entropy. Identifying this density matrix and computing its entropy would constitute a microscopic derivation of the de Sitter entropy in this framework.

Outlook
We have provided a precise microscopic definition of the Hilbert space H and H phys of minimal parity-even Vasiliev higher spin de Sitter gravity, its operator algebra and its vacuum state |0 . H transforms unitarily under the higher spin symmetry group G and H phys is its G-invariant subspace. While our construction answers many questions, it raises many more. We outline a few in what follows, and speculate on possible answers.

I. Local bulk physics?
How do we identify observables accessible to a local observer in our framework, how does time emerge, and how do we reconstruct local dynamics? There are several possible approaches to these problems. 1. The most straightforward approach would seem to be a direct reconstruction of perturbative bulk quantum field theory analogous to [93]. The appropriate setting for this is the Hilbert space H . We discussed some steps this program in section 6. The starting point is the reconstruction of the bulk QFT Heisenberg algebra of boundary fields. We have shown this is possible, but only up to operator-valued error terms whose vacuum expectation value is of order e −cN , and only in a coarse grained sense, with the Heisenberg algebra effectively accessing no more than order N ∼ S dS spatial pixels on the boundary at future infinity. The error term is due to the existence of large quantum fluctuations exiting the radius of convergence of perturbation theory: such fluctuations are exponentially unlikely when restricting to observables accessible to the coarse-grained theory, but occur with probability 1 in the original fine-grained theory. This suggests a breakdown of bulk low energy effective field theory on time and length scales that are either too short or too long. For length scales this is fairly obvious, and for time scales this is suggested by the fact, roughly speaking, that by the usual boundary-bulk UV-IR correspondence, the UV coarse-graining cutoff corresponds to a late time cutoff in the bulk, while the IR coarse-graining cutoff L corresponds to an early time cutoff. More coarsely, the necessity of coarse graining implies the nonexistence of a perturbative bulk QFT description carrying an exact representation of the full de Sitter isometry group. While all of these features are certainly reminiscent of limitations on effective field theories of inflationary universes inferred from consistency with holography and other considerations [3,24,96,97,98], clearly more work is needed to make this connection precise. 2. To detect the emergence of time, it may however not be necessary to first reconstruct the perturbative bulk quantum fields and their canonical Heisenberg algebra. One could for example consider n-point functions of the B I such as those computed in section 5.3, without any need to define canonically conjugate operators A I , and infer bulk dynamical features directly from their analytic structure, or from their reinterpretation as amplitudes obtained by integrating bulk vertices over an emergent spacetime, along the lines of [106]. Alternatively, as in [64,66], one may be able to detect the emergence of time from the ultrametric, tree-like organization intrinsic to the probability distributions of the local boundary fields B I , analogous to the intrinsic tree-like organization of species living at any given time. For species, this organization can be understood from evolutionary branching over time, due to accumulation of "frozen-out" fluctuations in DNA. In inflating spacetimes, this organization can be understood from branching of the wave function into an ensemble of effectively classical field profiles, due to accumulation of frozen-out quantum fluctuations. 3. To define local observables, it may likewise not be necessary to first reconstruct the Heisenberg algebra, as one may be able to build observables such as local charge densities more directly from the microscopic operators representing such charge densities in the Q-model. For example the operator : ∂ i Q P : can be thought of as a momentum density, in the sense that : ∂ i Q P : is the generator of spatial translations. 4. Finally, one could reasonably take the point of view that the only physical object is really H phys , and that this should form the starting point for any discussion of the physics of the model, including local bulk physics. This would render the limitations on the constructability of the Heisenberg algebra on H less immediately relevant, although an effective bound of order N on the number of accessible pixels reappears in a different guise in this context. As we have defined it in section 7, H phys is quasitopological, in the sense that the subspace of gauge-invariant n-particle states is finite-dimensional for any finite n, and in the sense that all gauge-invariant quantities are computable as correlation functions of O(2N)-invariant traces in a 2N ×2N matrix model. The question then arises how the familiar locally propagating field degrees of freedom are recovered in this framework. Conceivably this requires some form of spontaneous breaking of the residual higher spin gauge symmetry group G, perhaps by the branching of the wave function into frozen-out effectively classical field profiles, producing something akin to the background of "fixed stars" of Mach's principle. The gauge symmetry can also be (partially) broken explicitly by picking a local observer, or a coordinate patch with a choice of boundary conditions. Identifying the proper group of asymptotic (gauge vs physical) symmetries is quite subtle in general, and sensitive to the choice of boundary conditions, which in turn depend on the physics questions of interest [121,122,123,124].

II. Entropy?
The most obvious question left open in this work is probably the identification and microscopic computation of the de Sitter horizon entropy [21] in our framework. From the bulk point of view, the de Sitter entropy is naturally thought of as the entropy of the reduced density matrix of the static patch obtained from the global Hartle-Hawking vacuum state.
In view of the highly nonlocal nature of the relation between bulk and boundary fields, and the obstruction to reconstructing a bulk effective field theory probing more than N pixels, it is not immediately obvious how to translate this bulk definition to a natural quantity in the boundary Q-model. One could try to go the other way, and consider natural quantities in the Q-model with an entropic interpretation. The simplest one would be just the vacuum entanglement entropy of a region R on the future boundary, computed naively in the Gaussian state ψ 0 (Q) = e − 1 2 QDQ on H . The local nature of this wave function (as opposed to the nonlocal wave functions typically arising as ground states of free field theories) leads to an intriguing simplification, allowing to express the entanglement entropy purely in terms of a "wave function" Ψ Σ (q) = dQ R | q e − R QDQ living on the region's boundary Σ = ∂R.
Note that Ψ Σ (q) is the vacuum wave function of a radially quantized free CFT, living on a 2d surface, not to be confused with the Hartle-Hawking state ψ 0 (Q) living on the 3d spatial slice. This notion of entanglement entropy still leads to the usual UV-divergent area law, because H still has infinitely many short-distance degrees of freedom. A physically better motivated computation would impose the constraints arising from the infinite-dimensional residual gauge group, reducing H to a quasi-topological physical Hilbert space. A complicating factor is that picking a region explicitly breaks the global residual gauge group G to a subgroup G in × G out . It is conceivable that this reduces the computation of the physical entanglement entropy of R to a matrix model larger than (7.21), involving not just the matrix M = Q T Q of (7.20), but matrices M in,in = Q T in Q in , M out,out = Q T out Q out and M in,out = Q T in Q out = M T out,in . Quite plausibly this would lead to a finite result of order N . However, its interpretation as the dS entropy would still be far from clear. It would seem more akin to the dS entanglement entropy considered in [118], but even this interpretation is not obvious, in view of the fact that the map from the local QQ bilinears B I to the local boundary fields β I involves a nonlocal shadow transform, B I = G IJ β J .

III. Generalizations?
Although we motivated our constructions by comparison to results that were derived with a dS-CFT framework in mind, the structure we ended up with differs from this frame- This indicates it will not necessarily be straightforward to broadly generalize the model considered in this paper without significant new ingredients.
Certainly it should be possible though to generalize it to non-minimal or parity-odd Vasiliev gravity in four dimensions, presumably leading to a scalar bosonic U(2N) Q-model and a fermionic spinor Q-model, respectively. We could also consider the theory formulated on late-time slices S of more general topologies, as was done in [50] for the original Sp(N) model. In fact, although we usually had the examples S = R 3 or S = S 3 in mind, nothing prevents us from considering more general cases in our setup, such as S = S 1 × S 2 .
This will no longer have the full conformal group exactly realized as a subgroup of G acting on H in the continuum limit, but will still have the same conformal correlation functions of the B I at scales much smaller than the size of S. It will moreover lead to exactly the same invariant states and inner products on H phys , at least when the continuum limit is defined as a K → ∞ limit of discretized models as in section 7.4. Note that we never couple the theory to a Chern-Simons gauge field, thus avoiding the disastrous divergences for nontrivial topologies noted in [50]. For local CFTs with a gauged symmetry such as O(N), it is necessary to couple the theory to a gauge field to ensure the path integral on It may nevertheless be possible to do so, and this may lead to other interesting theories such as the family of parity-violating Vasiliev theories in dS 4 . Such theories might also admit non-trivial large N expansions.
Perhaps further study of the model presented in this work and generalizations thereof will suggest a set of physical consistency conditions akin to the unitarity and crossing symmetry constraints of the CFT bootstrap program. Solutions to these consistency conditions would correspond to consistent microscopic models of universes with a positive vacuum energy, including our own. Deriving all of known physics from a set of bootstrap equations would certainly be rather pleasing, but the obstacles to get there might still be quite insuperable.
we can construct a "conjugate" primary field Then we may define shadowsÕ I through the following relation: The integral measure consists of two parts. One depends on the eigenvalues only and the other one only involves the O(K) part. As long as the integrand is O(K) invariant, the O(K) dependent measure will only give an overall constant and it suffices to integrate over the eigenvalues (see for example [114]) where f (H) is an O(K)-invariant function, λ i 's are eigenvalues of H and N is the constant 'volume' of the O(K) group, which will be computed shortly.
After certain rescalings, (B.1) can be expressed as which is a special form of Selberg integral [113]  Given that the Selberg integral requires x > 0 our result holds for 2N ≥ K.
What remains is to find the constant N . Consider the following Gaussian integral over symmetric real matrices I = dG e −1/2 Tr G 2 .

(B.6)
On the one hand, I is the product of a collection of ordinary Gaussian integrals .

(B.7)
On the other hand, by noting than the integrand is O(K)-invariant, I can be reduced to an integral over eigenvalues which are now valued in the whole real line: Putting everything together leads to One can check the above formula explicitly for low values of K and N .