Group Averaging of massless scalar fields in 1+1 de Sitter
aa r X i v : . [ g r- q c ] O c t Preprint typeset in JHEP style - HYPER VERSION
Group Averaging of massless scalar fields in deSitter
Donald Marolf and Ian A Morrison
Department of Physics, University of California at Santa Barbara, Santa Barbara, CA93106, USAE-mail: [email protected] , ian [email protected] Abstract:
Perturbative gravity in global de Sitter space is subject to so-called lineariza-tion stability constraints: If they are to couple consistently to the gravitational field, quan-tum states must be invariant under the de Sitter isometries. While standard Fock spacescontain no de Sitter-invariant states apart from (possibly) the vacuum, a full Hilbert spaceof de Sitter-invariant quantum states can be constructed via group averaging techiniques.We re-examine the simple toy model of de Sitter group averaging given by the free 1+1scalar field, expanding on an earlier analysis by Higuchi. Our purpose is twofold: to includethe scalar zero-mode, and to explicitly count the number of de Sitter-invariant states as afunction of an appropriately defined energy.
Keywords: de Sitter, curved space quantum field theory, linearization stability, groupaveraging. ontents
1. Introduction 12. Free scalar field in de Sitter 3
3. Group Averaging and the physical Hilbert space 64. Physical Entropy 85. Discussion 10
1. Introduction
Understanding quantum gravity in de Sitter space remains an important problem. A majormotivation is the relevance of de Sitter to cosmology: measurements of the CMB [1] areconsistent with a period of inflation in which the universe underwent a de Sitter-like phaseof rapid expansion, and observations of type Ia supernovae suggest [2] that our universemay have a small positive cosmological constant and may approach de Sitter space in thefar future. Thus any theory of quantum gravity should include a description of de Sitterspace, at least in some approximate form. Unfortunately, the study of de Sitter quantumgravity has been fraught with conceptual difficulties (see, e.g., [3, 4, 5]). In this paperwe examine one particular hurdle that arises in perturbative gravity about a de Sitterbackground.To summarize this hurdle, recall that field theories on spacetimes with Killing symme-tries have conserved charges. We wish to regard such a theory (together with linearizedgravitational waves) as the zero-order perturbative approximation to a theory of matterplus gravity. This context is particularly interesting when the background also has compactCauchy surfaces. Then the gravitational equivalent of Gauss’ law implies that the abovecharges must vanish in order for a solution to this zero-order theory to consistently coupleto dynamical gravity [6, 7, 8, 9, 10, 11, 12]. Since these constraints are not encoded in thelinearized field equations, they are known as linearization-stability constraints.In de Sitter space, the linearization-stability constraints require linearized quantumstates to be invariant under the de Sitter group SO ( D,
1) where D is the spacetimedimension [13, 14, 15]. Because the de Sitter group is non-compact, the standard Fockspace contains no de Sitter-invariant states except for a possible vacuum [14]. This meagerset of states is clearly insufficient to reproduce the rich physics of the corresponding classical– 1 –heory. Fortunately, however, one may use the standard Fock space (which we call the‘auxiliary’ Hilbert space H aux ) to build a new ‘physical’ Hilbert space H phys of de Sitter-invariant states via group averaging [14] . This technique considers linear superpositionsof auxiliary states [] ψ i of the form | Ψ i := Z g ∈ G dg U ( g )[] ψ i , (1.1)where G is the de Sitter group, dg is the unique (unimodular, left- and right-invariant) Haarmeasure of G , and U ( g ) gives the unitary representation of G on H aux . Such superpositionsare formally invariant under the de Sitter group. For compact groups the analogue of (1.1)converges and gives the projection of [] ψ i onto the trivial representation. However, sinceour G is non-compact, the state (1.1) is not normalizable in H aux . Nevertheless it canbe understood (see e.g. [18]) as a “generalized state” in a sense similar to that used fornon-normalizable eigenstates of operators with continuous spectrum (e.g., plane waves ininfinite space).More concretely, one defines a new inner product on the group-averaged states (1.1): h Ψ | Ψ i := h ψ [] ·| Ψ i = Z g ∈ G dg h ψ [] U ( g ) [] ψ i . (1.2)The linear superposition (1.1) is meaningful when this “group-averaging inner product”converges.When the sense of the convergence is sufficiently strong, a theorem of [19] statesthat the group-averaging inner product is the unique inner product consistent with the ⋆ -algebra of bounded gauge-invariant observables in H aux . More formal discussions of groupaveraging can be found in [20, 21]. Other studies of de Sitter group averaging include[22, 23].The purpose of this paper is to analyze group averaging for a massless scalar fieldin 1 + 1 de Sitter, completing the analysis begun by Higuchi [14]. Higuchi was primarilyconcerned with de Sitter group averaging for 3+1 gravitons and used the 1 + 1 masslessscalar as a toy model. For simplicity, he omitted the scalar zero mode (which has noanalogue for gravitons). However, the physical massless scalar has a zero mode that shouldbe included in a more complete analysis. We do so below. We also compute the numberof de Sitter-invariant states as a function of energy flux through the de Sitter neck. Forenergies much greater than the de Sitter scale, a straightforward calculation shows thatthis entropy agrees with that of the naive auxiliary Hilbert space. This provides an explicitcheck of the argument presented in [22] that such a result should hold for generic fieldtheories.This paper is organized as follows. Section 2 briefly reviews the quantization of themassless scalar in de Sitter. Section 3 then follows Higuchi in using group averaging toconstruct an orthonormal basis of physical states from special auxiliary “seed states.” Thephysical entropy is computed in 4 and section 5 presents some final discussion. See [16, 17] for independent introductions of similar techniques in related contexts. – 2 – . Free scalar field in de Sitter
We begin with a brief overview of massless scalar fields in 1 + 1 de Sitter [24, 25, 26]. Itis useful to adopt conventions of conformal field theory [27, 28], and to write the 1+1 deSitter metric in the form ds = ℓ cos τ ( − dτ + dθ ) , (2.1)which is just a conformal factor times the metric on the cylinder. Here the conformal time τ has range − π/ < τ < π/ θ periodic θ ∼ = θ + 2 π , and ℓ is the de Sitter length scale.We adopt lightcone coordinates x ± = τ ± θ . The action of a free scalar field is S = − Z d x √− gg ab ∇ a φ ∇ b φ = Z d x ∂ + φ∂ − φ, (2.2)where g ab is the de Sitter metric and ∇ a the covariant derivative associated with g ab .In the second equality we note that the conformal factor from (2.1) cancels out of theaction, making the theory conformally invariant. The equation of motion for φ is thus ∂ + ∂ − φ ( x ) = 0, and the solutions are familiar left- and right-moving modes ∂ + φ ( x + ) = 12 √ π X m α m exp (cid:2) − imx + (cid:3) , ∂ − φ ( x − ) = 12 √ π X m e α m exp (cid:2) − imx − (cid:3) . (2.3)Upon integrating one finds φ ( x ) = φ π + α x + + e α x − + i √ π X m =0 (cid:20) α m m e − imx + + e α m m e − imx − (cid:21) = φ π + ( α + e α ) τ + ( α − e α ) θ + i √ π X m =0 m e − imτ h α m e − imθ + e α m e + imθ i . (2.4)We identify the term linear in τ as the momentum p ∝ ( α + e α ). The fact that φ ( x ) mustbe single-valued places further constraints on the mode expansion, depending on the targetspace of φ ( x ). We consider two cases:i) The target space of φ ( x ) is the real line. Single-valuedness of φ ( x ) requires φ ( τ, θ +2 π ) = φ ( τ, θ ); thus α = e α and the term linear in θ in (2.4) vanishes.ii) The target space of φ ( x ) is the circle S with radius R . Single-valuedness requires φ ( τ, θ + 2 π ) = φ ( τ, θ ) + 2 πRw , where w ∈ Z is the winding number of the field.From (2.4) we see that Rw is given by Rw = ( α − e α ). Furthermore, because φ ( x )is periodic, p is quantized: p = k/R, k ∈ Z . Periodic scalars in de Sitter havepreviously been considered in, e.g., [29].– 3 –or the remainder of this section we will keep p explicit so that our expressions apply toeither case; later we will specialize to case (ii) and write expressions in terms of k . Ourmode expansion is now φ ( x ) = φ π + 2 pτ + Rwθ + i √ π X m =0 m e − imτ h α m e − imθ + e α m e imθ i . (2.5)We now quantize our scalar field using canonical techniques, the end result of whichis the auxiliary Hilbert space H aux . The quantities φ , p , w , α m , and e α m are promoted tooperators. Imposing the canonical commutation relation [ φ ( τ, θ ) , φ ( τ, θ )] = iδ ( θ − θ )we find [ φ , p ] = i, [ α m , α n ] = [ e α m , e α n ] = mδ m, − n , (2.6)with all other commutators vanishing. In the usual fashion, α m and e α m are interpreted asleft- and right- moving creation operators ( m <
0) and annihilation operators ( m > L , L ± [27, 28] L m = 12 ∞ X n = −∞ : α m − n α n : , (2.7)which obey the algebra [ L ± , L ] = ± L ± , [ L , L − ] = 2 L , (2.8)and likewise for e L , e L ± .We can define a vacuum state []0 i as the state for which α m []0 i = e α m []0 i = 0 ∀ m > . (2.9)Such a vacuum state is not in general annihilated by α or e α . Instead, there is a two-parameter family of vacua distinguished by their eigenvalues of α and e α , i.e. the mo-mentum and winding of each vacuum. It is equivalent to label independent vacua by theireigenvalues h and e h of the Virasoro generators L and e L : h = 12 (cid:18) p + Rw (cid:19) , e h = 12 (cid:18) p − Rw (cid:19) ; (2.10)we therefore denote a vacuum by []0; h, e h i . We shall see shortly that the only de Sitter-invariant vacuum is the p = w = 0 vacuum []0; 0 , i . Excited states are created by actingon a vacuum with creation operators α m ( e α m ) for m <
0, and will be labeled using thesomewhat degenerate notation [] n, e n ; h, e h i , where n, ˜ n are the eigenvalues of L − h, e L − ˜ h and we refer to N := n + e n as the level of a state. Each creation operator α m ( e α m ) increasesthe eigenvalue of L ( e L ), and thus the level, by m .Let us also introduce the operator H = L + e L , (2.11)– 4 –hich generates translations in τ ; i.e., it is the Hamiltonian for the conformally rescaledproblem on the cylinder S × R , up to a constant offset associated with the Casimir energy.Since de Sitter space does not have a global future-directed timelike Killing field, H is notnaturally thought of as a de Sitter Hamiltonian. However, it does agree with the flux ofde Sitter stress-energy through the sphere at τ = 0 (again up to a constant offset). In thislatter form, this operator was an important ingredient in the analysis of [22]. We shall thusrefer to H as an “energy.” This operator acts on a state [] n, e n ; h, e h i as H [] n, e n ; h, e h i = (cid:16) h + e h + n + e n (cid:17) [] n, e n ; h, e h i = (cid:18) p + R w n + e n (cid:19) [] n, e n ; h, e h i , (2.12)and so the energy of such a state is E := h + n + e h + e n . Let us quickly review the symmetries of 1 + 1 de Sitter spacetime. This space has threeindependent Killing vector fields which we may take to be ∂ θ = 12 ( ∂ + − ∂ − ) , ξ a ∂ a = 12 (cos( x + ) ∂ + +cos( x − ) ∂ − ) , ξ a ∂ a = 12 (sin( x + ) ∂ + +cos( x − ) ∂ − ) . (2.13)Such isometries can be understood by embedding 1+1 de Sitter in 2 + 1 Minkowski space:there ∂ θ generates rotations preserving the Cartesian coordinate X , while ξ a and ξ a gen-erate boosts along the Cartesian spatial directions. The Killing fields act on H aux viaoperators J , B , and B which satisfy the SO (2 ,
1) algebra[ B , B ] = iJ, [ B , J ] = iB , [ B , J ] = − iB . (2.14)On the scalar field φ ( x ), their action is[ B , φ ( x )] = i £ ξ φ ( x ) = iξ a ∂ a φ ( x ) , (2.15)and likewise for J and B . One may express the SO (2 ,
1) generators in terms of Virasorogenerators via J = L − e L , (2.16) B = 12 (cid:16) L + L − + e L + e L − (cid:17) , (2.17) B = − i (cid:16) L − L − − e L + e L − (cid:17) . (2.18)We see that the de Sitter group is a diagonal subgroup of the SL (2 , C ) × SL (2 , C ) generatedby L , L ± , e L , e L ± .Constructing de Sitter-invariant states is non-trivial, as can be seen from the expres-sions of the generators (2.16)-(2.18). Because the boost generators contain both raisingand lowering Virasoro generators, it is difficult to construct a non-trivial state that is boost– 5 –nvariant. Indeed, it is easy to show that the only de Sitter-invariant state in our basis isthe p = w = 0 vacuum []0; 0 , i (see also [30]). Furthermore, it can be shown that thereexist no linear combinations of our basis states that are both de Sitter-invariant and nor-malizable [13]. Thus []0; 0 , i is the only de Sitter-invariant state in the auxiliary Hilbertspace H aux .
3. Group Averaging and the physical Hilbert space
We now construct de Sitter invariant states via group averaging. We study the resultingphysical Hilbert space H phys and provide an orthonormal basis. We follow closely in thesteps of [14] and, in particular, define the space H seed = { [] ψ i seed } of “Higuchi seed states”which are:i) SO (2)-invariant, i.e. J [] ψ i seed = 0 , (3.1)ii) annihilated by the lowering operators L and e L , L [] ψ i seed = e L [] ψ i seed = 0 , (3.2)iii) in the subspace corresponding to eigenvalues E > H (recall 2.12). We note that H preserves the conditions (3.1), (3.2) and so can be diagonalized in H seed .Furthermore, we will confine attention to a basis of such states which are eigenstates of E with inner products seed h ψ [] ψ i seed = ( E − δ ψ ,ψ . (3.3)Here δ ψ ,ψ denotes the complete set of Kronecker deltas needed to specify that [] ψ i seed and [] ψ i seed represent the same state in our basis .The group averaging of such states is easy to control. Criterion (ii.) has the effectthat B [] ψ i seed = ( L − + e L − )[] ψ i seed , seed h ψ [] B = seed h ψ []( L + e L ) . (3.4)As a result seed h ψ [] B [] ψ i seed = 0 for all seed states. Recalling the commutation relations[ L ± , L ] = ± L ± , [ L , L − ] = 2 L , (3.5)we may compute seed h ψ []( B ) [] ψ i seed by commuting creation operators to the left (an-nihilation operators to the right), with the result seed h ψ []( B ) [] ψ i seed = seed h ψ []( L + e L )[] ψ i seed = E seed h ψ [] ψ i seed , where E is the energy of either state. Continuing in thismanner, one can readily see that seed h ψ []( B ) m [] ψ i seed = 0 m odd , (3.6) seed h ψ []( B ) m [] ψ i seed = f ( m, E ) seed h ψ [] ψ i seed m even , (3.7)– 6 –here f ( m, E ) is a function of m and E . In particular, [14] showed that seed h ψ [] e iλB [] ψ i seed = (cid:18) cosh λ (cid:19) − E seed h ψ [] ψ i seed (3.8)if the zero-mode is ignored, i.e. for the Fock space over the p = w = 0 vacuum. However,since dependence on p, w enters only through E , we see that (3.8) holds in general. Workingin terms of the the energy flux E turns out to make the inclusion of several effects of thezero mode quite straightforward.We wish to compute the group averaging inner product of two seed states. We beginby specializing expression (1.2) to the case of SO (2 , SO (2 ,
1) we can write any group element as a product of two SO (2) rotations and aboost [31, 32]: U ( g ) = e iαJ e iλB e iγJ . (3.9)Here e iαJ is the SO (2) rotation through angle α (0 ≤ α ≤ π ) and e iλB is the boost along ξ a with rapidity λ (0 ≤ λ ≤ ∞ ). In a similar fashion, the Haar measure can be decomposedas dg = 14 π dα dγ dλ sinh λ, (3.10)where π dα and π dγ are both Haar measures on SO (2). The group averaged inner productis then h Ψ | Ψ i = 14 π Z dα dγ dλ sinh λ h ψ [] e iαJ e iλB e iγJ [] ψ i = Z ∞ dλ sinh λ h ψ [] P e iλB P [] ψ i . (3.11)In the second line we have identified the projector P onto SO (2)-invariant states P = 12 π Z π dα e iαJ . (3.12)Now consider the inner product of two physical states built from seed states | Ψ , i := R dgU ( g )[] ψ , i seed . From (3.11) we have h Ψ | Ψ i = Z ∞ dλ sinh λ seed h ψ [] P e iλB P [] ψ i seed = Z ∞ dλ sinh λ seed h ψ [] e iλB [] ψ i seed = seed h ψ [] ψ i seed Z ∞ dλ sinh λ (cid:18) cosh λ (cid:19) − E = seed h ψ [] ψ i seed E − δ ψ ,ψ . (3.13)In the second line we used the fact that seed states are SO (2)-invariant; in the third line weused (3.8). Evaluating the integral and inserting our normalization (3.3) leads to the final– 7 –esult. This is essentially the same calculation as was performed previously in [14] withoutthe zero mode. We emphasize again that our formalism allows a quick generalization tothe case of non-vanishing zero-mode.We see from (3.13) that the set of de Sitter-invariant states {| Ψ i i} built from the seedstates { [] ψ i i} forms an orthonormal set. One can also show that this set spans the spaceof de Sitter-invariant states constructed from linear combinations of auxiliary states with E >
1. The proof is exactly as in [14]. Thus we have an orthonormal basis of states asdesired.We conclude this section with a discussion of auxiliary states with E ≤
1. It is naturalto ask whether such states contribute to the physical Hilbert space and, if so, how we canincorporate them into our formalism. Fortunately, for any vacuum there are only a fewsuch states (and there are none for h + ˜ h > The case p = w = 0 : The only states with E ≤ , i and thesingle-particle states α − []0; 0 , i , e α − []0; 0 , i . The vacuum is de Sitter-invariant andcan be a state in the physical Hilbert space. However, this state must be treatedseparately since for []0; 0 , i group averaging does not converge. This separate treat-ment may be justified via the observation (see [18, 33]) that []0; 0 , i is superselectedfrom states where group averaging does converge. Turning now to the single-particlestates, one notes that they each have angular momentum ±
1. As a result, the groupaverage of such states over the rotation group SO(2) ⊂ SO(2,1) already vanishes andwe do not expect these states to contribute the physical Hilbert space.
The case p ≤ , w = 0 , or p = 0 , R w / ≤ : Here the only states with E ≤ h, e h i . Such vacua are not de Sitter invariant, though group averagingagain fails to converge. In this case we expect an appropriately renormalized form ofgroup averaging to converge, though we leave this for future work. The resulting deSitter-invariant states will again be superselected from states for which no such renor-malization was needed. See [18, 33, 34, 35, 36] for further examples and discussionof this phenomenon. The case | k | = | w | = 1 and R = √ : Such states also have J = ± = 0 and againgroup average to zero under SO(2) ⊂ SO(2,1). We expect no physical states fromsuch seed states.
4. Physical Entropy
We now compute the density of physical states. One typically computes this density as afunction of energy. However, as previously remarked, there is no natural conserved notionof energy in de Sitter space. Moreover, those charges which are associated with de Sitterisometries must vanish for physical states. We will thus need to find some other notion ofenergy to use below.A natural approach is to follow [22] and to consider the energy E defined in section2, which measures the flux of stress energy through the surface τ = 0. Since the definition– 8 –f E as the eigenvalue of H is not de Sitter invariant, this quantity is not a priori definedon physical states. Nevertheless, a de Sitter-invariant notion of E was defined in [22]. Inour present language, the de Sitter invariant energy is the operator whose eigenstates areprecisely the physical states | Ψ i i obtained by group averaging Higuchi seed states [] ψ i i seed ,and such that the eigenvalue of | Ψ i i is just the eigenvalue of H for [] ψ i i seed . The one to onemap between physical states and seed states, and that fact that the | Ψ i i form an orthogonalbasis of H phys , imply that this de Sitter-invariant energy is a self-adjoint operator on H phys .Furthermore, because any two states related by the action of some U ( g ) yield the samestate under group averaging, we see that defining H in some other reference frame (i.e.,replacing H by U ( g ) HU ( g − )for some g in the definition of a Higuchi seed state) would leadto the same de Sitter-invariant notion of energy. In a very rough sense, this energy operatorconsiders the energy flux of a physical state through each possible de Sitter neck (associatedwith each possible choice of reference frame) and reports the smallest value obtained. Forsimplicity, we will again use E to denote the eigenvalue of this de Sitter-invariant energy.It is clear that counting the density of physical states is equivalent to counting thenumber of Higuchi seed states as a function of E . The density diverges when the scalartarget space is non-compact, so we focus on the case with S target space. As usual, weperform the calculation separately for the Fock space over each vacuum []0; h, e h i . Our taskis thus to compute ln N seed ( N ), the logarithm of the number of Higuchi seed states as afunction of the level N (recall N = E − h − e h ) above each vacuum []0; h, e h i . To do so, wemust first examine the seed state criteria in more detail. We begin with SO (2) invariancewhich requires J [] n, e n ; h, e h i = ( kw + n + e n )[] n, e n ; h, e h i = 0 , (4.1)so that an SO (2)-invariant state at level N must satisfy n = 12 ( N − kw ) , e n = 12 ( N + kw ) . (4.2)Since n and e n must be non-negative integers, SO (2)-invariant states are possible only atlevels N = kw + 2 m , where m is a non-negative integer.Next consider the property L [] ψ i seed = 0. Because the left- and right-moving sectorscommute, we can decompose the Fock space over a given vacuum into left- and right-movingHilbert spaces H aux = H L ⊗ H R ; an auxiliary state with levels ( n, e n ) is then in the productspace H n ⊗ H e n . Because L acts only on left-movers we can focus on H n . The annihilationoperator L lowers n by 1, i.e. L [] n, e n ; h, e h i = [] n − , e n ; h, e h i . In fact, L is a surjectivemap from H n to H n − : L : H n → H n − . (4.3)The number of states in H n annihilated by L is therefore given by the difference indimension H n annihilated by L ! = dim( H n ) − dim( H n − ) = P ( n ) − P ( n − , (4.4)where P ( x ) is the number of integer partitions in x [28]. For n = 1 we have P (1) − P (0) = 0,while for n > P ( n ) − P ( n − >
0. The same argument applies for the action– 9 –f e L on the right-moving sector H e n . Combining our observations (4.2) and (4.4), we findthat the number of seed states at level N is N seed ( N ) = P (cid:18) N + kw (cid:19) (cid:20) P (cid:18) N − kw (cid:19) − P (cid:18) N − kw − (cid:19)(cid:21) × P (cid:18) N − kw (cid:19) (cid:20) P (cid:18) N + kw (cid:19) − P (cid:18) N + kw − (cid:19)(cid:21) . (4.5)One can easily re-write this as a function of the energy flux E , though in practice this willnot be required to establish agreement with the density of states in H aux .Let us now compare the number of seed and auxiliary states for both small and large N . Note that the number of states in H aux with given ( n, e n, h, ˜ h ) is simply N aux ( n, e n ) = P ( n ) P ( e n ) and the total number of states at level N is N aux ( N ) = N X n ′ =0 P ( n ′ ) P ( N − n ′ ) . (4.6)Because P ( x ) ∼ x for x of order 1, for small N there are dramatically fewer seed statesthan auxiliary states: N seed ( N ∼ ∼ N, N aux ( N ∼ ∼ N . (4.7)However, it is more interesting to compare entropies in the thermodynamic limit of large N . For N → ∞ we may use the Hardy-Ramanujan formula for the asymptotic behavior of P ( x )[37]: P ( x ) ≈ x √ (cid:20) π r x (cid:21) as x → ∞ . (4.8)Inserting (4.8) into (4.5) and using Cardy’s formula [38] to compute (4.6) yields S seed ( N ) ≈ S aux ( N ) ≈ π √ √ N , (4.9)so that in this limit the seed state entropy agrees with the entropy of the auxiliary Hilbertspace as claimed. This provides an explicit confirmation of the general argument given inthe appendix of [22].
5. Discussion
We have studied the behavior of 1+1 massless scalars under de Sitter group averaging,building on earlier work by Higuchi [14]. The new element was to include the scalar zeromode. We constructed an orthonormal set of de Sitter-invariant states which forms abasis of the physical state space (up to the minor exceptions discussed in section 3). Wehave also computed the entropy of this physical space. As anticipated in [22], to leadingorder at large E this entropy agrees with the entropy of the auxiliary Hilbert space. Thisobservation supports the claim that group averaging will yield enough states to reproduceclassical physics in the ~ → B (here L + ˜ L and L − + ˜ L − ) and E which in our case followedfrom the Virasoro algebra. Similar relations do hold, however, for conformally-coupledscalar fields in arbitrary dimension, i.e. for scalars on dS d +1 which satisfy the equation ofmotion g ab ∇ a ∇ b φ ( x ) = (cid:18) d − (cid:19) φ ( x ) . (5.1)In such cases it is straightforward to use Higuchi’s algorithm to construct an orthonormalbasis of physical states. (The 3+1 case was studied in [39].) Additionally, in the rightchoice of gauge, both free gravitons and free gauge vector fields in 3 + 1 dimensions haveboost matrix elements identical to those of conformally coupled scalar fields, and one mayagain construct an orthonormal basis [14].The more general case remains open for future work. However, one expects the masslessscalar field in higher dimensions to be qualitatively similar to the case discussed here. I.e.,we expect that there is some analogue of our quantity E which in some sense measuresthe total excitation of the state, including contributions from both particles and the zeromode. One expects group averaging to fail when the quantity is very small, but to convergewhen it is sufficiently large. Acknowledgements
The authors are grateful to Atsuchi Higuchi for many discussions of group averaging andfor sharing his unpublished notes [39]. This work was supported in part by the US NationalScience Foundation under Grant No. PHY05-55669, and by funds from the University ofCalifornia.
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