Quasinormal Quantization in deSitter Spacetime

A scalar field in four-dimensional deSitter spacetime (dS_4) has quasinormal modes which are singular on the past horizon of the south pole and decay exponentially towards the future. These are found to lie in two complex highest-weight representations of the dS_4 isometry group SO(4,1). The Klein-Gordon norm cannot be used for quantization of these modes because it diverges. However a modified `R-norm', which involves reflection across the equator of a spatial S^3 slice, is nonsingular. The quasinormal modes are shown to provide a complete orthogonal basis with respect to the R-norm. Adopting the associated R-adjoint effectively transforms SO(4,1) to the symmetry group SO(3,2) of a 2+1-dimensional CFT. It is further shown that the conventional Euclidean vacuum may be defined as the state annihilated by half of the quasinormal modes, and the Euclidean Green function obtained from a simple mode sum. Quasinormal quantization contrasts with some conventional approaches in that it maintains manifest dS-invariance throughout. The results are expected to generalize to other dimensions and spins.


Introduction
In this paper we present a new and potentially useful approach to an old problem: the quantization of a scalar field in four-dimensional de Sitter spacetime (dS 4 ), which has an SO(4, 1) isometry group. One standard approach begins with the spherical harmonics of the S 3 spatial sections, and proceeds by solving the wave equation for the time-dependent modes.
Linear combinations of these modes that are nonsingular under a certain analytic continuation are then identified as the Euclidean modes and used to define the quantum Euclidean vacuum. The vacuum so constructed exhibits manifest SO(4)-invariance and can also be shown to possess the full SO(4, 1) symmetry of dS 4 . Another common approach singles out the southern causal diamond and relies on a special Killing vector field, denoted L 0 , which generates southern Killing time and whose corresponding eigenmodes have real frequency ω. This construction displays manifest SO(3) × SO(1, 1) symmetry and again leads to the dS-invariant Euclidean vacuum. The modes employed in these and similar constructions are not in SO(4, 1) multiplets and hence SO(4, 1)-invariance of the final expressions is nontrivial.
For example, the action of the dS 4 isometries on the southern diamond L 0 eigenmodes shifts the frequency by imaginary integer multiples of 2π/ℓ (where ℓ is the dS radius) while the usual southern diamond modes all have real frequencies.
It is natural to adopt scalar modes which lie in highest-weight representations of SO (4,1) and therefore boast manifest dS-invariance. These turn out to be nothing but the quasinormal modes of the southern diamond, which have complex L 0 eigenvalues and comprise four real or two complex highest-weight representations. * They are singular on the past horizon and decay exponentially towards the future, as opposed to the conventional southern diamond modes which oscillate everywhere. In order to quantize in a quasinormal mode basis, a norm is needed. The singularities on the past horizon render the Klein-Gordon norm singular, which is presumably why the quasinormal modes have not typically been used for quantization. However a variety of other equally suitable norms have been employed for various reasons in dS [5,6,7,8,9,10,11]. One of them -the so called R-norm [11] -differs from the Klein-Gordon norm by the insertion of a spatial reflection through the equator of the S 3 slice, thereby exchanging the north and south poles. We demonstrate that the R-norm is finite for quasinormal modes and hence suitable for quantization. We also show that the Euclidean vacuum has the simple and manifestly dS-invariant definition as the state annihilated by two of the four sets of quasinormal modes. Moreover the Euclidean Green function is shown, as anticipated in [12], to be obtainable from a simple sum over the quasinormal modes. We caution the reader that quasinormal modes have singularities on the past horizon which we regulate with an iǫ-prescription. Our statements about completeness and mode sums depend on taking the ǫ → 0 limit at the end of our calculations.
The real Killing vectors which generate the dS isometries have an SO(4, 1) Lie bracket algebra and are antihermitian with respect to the Klein-Gordon norm. However they have mixed hermiticity under the R-norm. Multiplication by appropriate factors of i produces complex Killing vector fields which are antihermitian under the R-norm. The Lie algebra of these R-antihermitian vector fields turns out to generate SO (3,2), which is precisely the symmetry group of a 2+1-dimensional CFT, and the transformed notion of hermiticity is exactly the one conventionally employed when studying CFT 3 on the Euclidean plane [9].
† For every bulk scalar Φ one expects a dual operator O in the boundary CFT 3 . The bulk state with one quantum in the lowest quasinormal mode is dual to the CFT 3 state associated to an O insertion at the north pole of the S 3 at I + , and the descendants fill out SO(3, 2) representations on both sides of the duality [11]. This paper is organized as follows. In section 2, we begin by reviewing the standard global S 3 modes and the construction of Euclidean modes and Green functions. In section 3, we show that the quasinormal modes comprise the highest-weight modes and their descendants, specializing for simplicity to the case of conformal mass m 2 ℓ 2 = 2. Then in section 4, the modified R-norm and its properties are presented. Next, in section 5 we prove that half the quasinormal modes are Euclidean modes and demonstrate their completeness by deriving the Euclidean Green function from a quasinormal mode sum. In section 6 we generalize these results to the case of light scalars with m 2 ℓ 2 ≤ 9/4. Finally, in section 7 we isolate quasinormal modes that vanish in the northern or southern diamonds -the analogues of Rindler modes in Minkowski space -and present a complete set of quasinormal modes in the southern diamond. These might eventually be useful for understanding the thermal nature of physics in a single dS causal diamond, but we do not pursue this direction further herein.
In addition, in Appendix A we provide the explicit forms of dS 4 Killing vectors as well as their commutation relations. This is followed by Appendix B, which computes the norm of spherically symmetric descendants using the SO(4, 1) algebra, and Appendix C, which provides details on the Euclidean two-point function evaluated on the south pole observer's worldline.
We expect our discussion to generalize to the case of heavy scalars with m 2 ℓ 2 > 9/4 as well as other dimensions and spin.

SO(4)-invariant global mode decomposition
In this section we describe the standard dS 4 mode decomposition in terms of the spherical harmonics of the S 3 spatial sections. These modes are regular everywhere on dS 4 and sometimes referred to as 'global modes'.
We will work in the dS 4 global coordinates x = (t, ψ, θ, φ) with line element where Ω = (ψ, θ, φ) are coordinates on the global S 3 slices. We denote the north and south pole by In this coordinate system, the dS-invariant distance function P (x; x ′ ) is given by where Θ 3 (Ω, Ω ′ ) denotes the geodesic distance function on S 3 and cos Θ 3 (Ω, Ω ′ ) = cos ψ cos ψ ′ + sin ψ sin ψ ′ [cos θ cos θ ′ + sin θ sin θ ′ cos(φ − φ ′ )] . (2.4) Following the notation of [11], solutions of the wave equation may be expanded in representations of the SO(4) rotations of the S 3 spatial slice at fixed t: These have total SO(4) angular momentum L and spin labeled by the multi-index j. The where Ω A denotes the antipodal point of Ω, while √ h = sin 2 ψ sin θ and D 2 are the measure and Laplacian on the unit S 3 , respectively. Here and hereafter, denotes summation over all allowed values of L and the multi-index j. The time dependence y L (t) is then governed by the differential equation The general solution has the I + falloff For the time being, we restrict our attention to the case m 2 ℓ 2 = 2, which corresponds to a conformally coupled scalar with The case of generic mass is qualitatively similar but with algebraic functions replaced by hypergeometric ones. We give the correspondingly more involved formulae in section 6.
The so-called Euclidean modes, which define the vacuum, are those which remain nonsingular on the southern hemisphere when dS 4 is analytically continued to S 4 . In other words, they are defined by the condition Explicitly, these modes are [11]: Note that they are singular on the northern hemisphere at t = iπ/2. In terms of the Klein-Gordon inner product on global S 3 slices, we have normalized the modes such that (2.14) Using these modes, one can define the Euclidean vacuum by the condition whereΦ is the quantum field operator. Since the modes Φ E Lj are not SO(4, 1)-invariant, it is not immediately obvious that the Euclidean vacuum is dS-invariant, but this can be checked explicitly. The Wightman function is Using the iǫ-prescription, this may be expressed in terms of the dS-invariant distance function If we rewrite P (x; x ′ ) in terms of the coordinates X on the embedding 5D manifold with Minkowski spacetime metric η (in which dS 4 is just the hyperboloid η µν X µ X ν = ℓ 2 ), then we can represent s(x; x ′ ) by [18] s Note that this is exactly the same as sending dS 4 has 10 real Killing vectors which, letting k ∈ {1, 2, 3}, we will refer to as the dilation

SO(4, 1)-invariant quasinormal modes
In this section we describe the SO(4, 1)-invariant mode decomposition in terms of (anti-) quasinormal modes. We begin by defining where x A denotes the antipodal point of x. These Green functions fall off like e −2h ± t as both arguments are taken to I + . Next we introduce 'Ω-modes' as follows: The normalization factor was chosen for future convenience.
In terms of the global coordinates, the Ω-modes take the explicit form The delta-functions above are normalized as one-dimensional delta-functions, that is, such The Ω-modes can be expanded in terms of the Euclidean global SO(4) modes as follows: The lowest-weight and highest-weight modes are respectively given by [11] Φ By construction, the modes Φ ± hw are eigenfunctions of L 0 with eigenvalues −h ± and are annihilated by M −k for each k ∈ {1, 2, 3}. The descendants of the highest-weight modes are obtained by acting with the M +k , for any k ∈ {1, 2, 3} (see Appendix A): where K is a multi-index denoting the set {k 1 , . . . , k n }. To emphasize this we adopt the notation At this point we have eight highest-weight representations of SO(4, 1), with elements We shall see below that this is an overcomplete set: only the first or second row of modes is needed to obtain a complete basis.

R-norm
We wish to expand the scalar field operator in the (anti-)quasinormal modes. Towards this end it is useful to introduce an inner product. The Klein-Gordon norms of the Ω-modes are denote the two-point functions for a CFT 3 operators with dimensions h ± . These satisfy The norm of a highest-weight quasinormal mode is obtained by setting Ω 1 = Ω 2 = Ω N P , which is evidently divergent. Hence the Klein-Gordon norm is not suitable for quantization of the quasinormal modes.
Alternate norms have been employed in de Sitter spacetime for a variety of reasons [7,8,9,10,11]. Here, following [11], a useful 'R-norm' can be defined by inserting a reflection R on S 3 across the equator: R : (ψ, θ, φ) → (π − ψ, θ, φ), With respect to this R-norm, In particular, the norms of the highest-weight quasinormal modes are simply while the R-inner product between a quasinormal mode and the complex conjugate of any quasinormal mode vanishes.
Changing the norm affects the hermiticity properties of the 10 real Killing vector fields which generate the dS 4 isometries. Under the Klein-Gordon norm, their adjoints are so that the Killing generators are all antihermitian. However, under the modified R-norm, To recover antihermitian generators in the R-norm, one must send M k +M −k → i(M k +M −k ) and L 0 → iL 0 while keeping the rest of the generators the same. The Lie bracket algebra of the antihermitian vector fields is then SO(3, 2) rather than SO(4, 1). See Appendix A for more details.
Interestingly SO(3, 2) is the symmetry group of a CFT in 2 + 1 dimensions. This suggests that the quantum states on which these generators act could belong to a 2 + 1-dimensional CFT, which fits in nicely with the dS 4 /CFT 3 conjecture.
Using (4.8) we can compute the norm of the descendants. For example, the norm of the first descendant is (not summing over k) Observe that under this R-norm, the descendants of Φ + QN are orthogonal to those of Φ − QN . For the SO(3)-symmetric states, we provide the exact formula in Appendix B.

Completeness of quasinormal modes
In this section we show that the quasinormal modes form a complete set in the sense that the Euclidean Green function can be written as a simple sum over such modes. In particular, the antiquasinormal modes are not needed.
First we note from (3.5) that the quasinormal modes can be written as linear combinations of the global Euclidean modes, without using their complex conjugates. Therefore they are themselves Euclidean modes, and the Euclidean vacuum obeys Note that this relation, unlike the corresponding one for the global Euclidean modes, is manifestly dS-invariant because the quasinormal modes lie in representations of SO(4, 1).
Let us now assume that we can expand the field operator in the presumably complete basis (5.1): where the N ± KK ′ are defined through Then, using (5.2), the quasinormal mode Green function is given by where Φ ± K ≡ M +K Φ ± QN . A demonstration that the function G(x; x ′ ) so obtained is indeed the standard Euclidean Green function G E (x; x ′ ) implies that the quasinormal modes in (5.1) form a complete basis, in the sense that they satisfy The norm for such states is calculated in Appendix B and is given by while the modes at Ω = Ω SP are given by the full sum (5.5) is wheres Noting that for small ǫ,s(x; x ′ ) is equivalent to s(x; x ′ ) defined in (2.18), it follows that this Green function agrees with that in (2.17) on the south pole observer's worldline. Since the construction of our Green function is dS-invariant ‡ , agreement on this worldline implies that this Green function equals the Euclidean one on any two timelike separated points.
For spacelike separated points, we find from (5.5) that By dS-invariance, we can extend this to any two spacelike-separated points. This concludes the proof that the quasinormal Green function (5.5) is indeed the Euclidean Green function.
6 Results for general light scalars (m 2 ℓ 2 < 9/4) In the general case of a light scalar with m 2 ℓ 2 < 9/4, we can write out the explicit form of the Euclidean two-point function as (see for instance [9]) The asymptotic behaviors of the Euclidean Green function are: Note that in dealing with the branch-cut of G E (t, Ω; t ′ , Ω ′ ), we go under (above) it when t > t ′ (t < t ′ ) in accordance with the iǫ-prescription. ‡ This follows from the fact that the Green function is just a position-space representation of the projection operator onto the highest-weight representation of the three-dimensional conformal group characterized by the highest-weight −h, as can be seen by writing out this projection as a sum over complete states of the representation and using the definition of SO(4, 1) generators.
Let us define G ± as These satisfy the future boundary conditions in Ref. [19] in the region P < −1. Now, we define the highest-weight modes as The Φ ± QN are explicitly given by The asymptotic behavior of the modes as t → ∞ is We have defined § which satisfy The norm is easily evaluated at I + to be As such, we find that the R-norms of the quasinormal modes are The rest of the discussion on the induced norms of the descendants carries over from the m 2 ℓ 2 = 2 case.
Next, we follow our previous strategy of showing that the mode sum and the Euclidean Green function agree on the south pole observer's worldline. Again, we evaluate As before, the norm for such states is Note that The quasinormal mode Green function (5.5) is then . (6.17) In Appendix C, we show that on the south pole observer's worldline, this Green function is equal to the Euclidean Green function. Thus by dS-invariance of both Green functions, they agree for any two timelike separated points in dS 4 .
For spacelike separated points, we consider the Euclidean Green function with one point at the south pole and the other point at the north pole. One notices that Since these points are spacelike separated, we do not have to worry about the iǫ-prescription and the Green function is real. If we had evaluated the quasinormal mode Green function G(t, Ω SP ; t ′ , Ω N P ), then we would have obtained the same sum as in (6.16), provided that we sent t ′ → −t ′ and removed the phase e −iπh ± from the coefficients (6.17). This is equivalent to sending t ′ → −t ′ + iπ and hence by dS-invariance we have proved that for any two spacelike separated points, the quasinormal mode Green function is the Euclidean Green function.

Southern modes and T-norm
In this section we find quasinormal modes that vanish in the northern or southern diamonds -the analogues of Rindler modes in Minkowski space. We begin with the expression (6.6) for the lowest-weight mode For generic mass Φ ± QN (x) has a branch cut on the past horizon of the southern observer at tanh t = − cos ψ. We have chosen the phase convention so that the denominator is real above the past horizon. Crossing the past horizon gives an extra phase of e iπh ± . It follows that the southern mode vanishes below the past horizon. Similarly the northern mode vanishes above the past horizon. The R-norms between these modes are On the other hand, the R-norm for the global quasinormal modes is closely related to timereflection: While the R-norm has no analogue in the static patch, the T-norm is easily generalizable to the southern diamond as where B 3 S denotes the integral over a complete slice in the southern diamond. We have In the T-norm, Thus, the T-norm of the descendants is induced by the SO(4, 1) algebra.

Acknowledgements
It has been a great pleasure discussing this work with Tatsuo Azeyanagi and Dionysios Anninos. This work was supported in part by DOE grant DE-FG02-91ER40654 and the Fundamental Laws Initiative at Harvard.

A Appendix: dS 4 Killing vectors
In global coordinates, the 10 Killing vectors of dS 4 are given by: Their non-zero commutators are: As expected, these are the relations which define the SO ( The scalar Laplacian is a Casimir operator. It reads: The convention is that J 2 = −L(L + 1) on the spherical harmonics Y Lj . The conformal Killing vectors of the S 3 are given by the restriction of dS 4 Killing vectors on I + : M ∓1 = (1 ± cos ψ) sin θ sin φ ∂ ψ + (cot ψ ± csc ψ)(cos θ sin φ ∂ θ + csc θ cos φ ∂ φ ), M ∓2 = (1 ± cos ψ) sin θ cos φ ∂ ψ + (cot ψ ± csc ψ)(cos θ cos φ ∂ θ − csc θ sin φ ∂ φ ), To relate the above de Sitter generators to the embedding coordinates X defined by where η has signature (4, 1) and the usual Lorentz generators are given by with commutators we have for i, j, k ∈ {1, 2, 3} The standard Klein-Gordon adjoint acts as: where the adjoint is defined in the standard way as f, M † g KG ≡ Mf, g KG . The action of R on the Killing vectors is: or equivalently, for j, k ∈ {1, 2, 3}, In the R-norm, if we define M † R as f, M † R g R ≡ Mf, g R then
Also, observe that (C.9) implies that Away from the singularity at t = t ′ , we can set ǫ to zero so that G(t, Ω SP ; t ′ , Ω SP ) = e −iπh + H + (P ) − e −iπh − H − (P ) 2 sin(πµ) = G E (t, Ω SP ; t ′ , Ω SP ). (C.12) The singularity structure for G can be also analyzed from (C.6). The correction factor R is regular near the singularity, while the G ± (P ) have poles when P approaches 1. Expanding (C.11) to first order in ǫ yields . (C.14) The singularity structure therefore is the same as in definition (6.1) for the Euclidean Green function up to redefinition of ǫ by some positive function.