Infrared problem in the Faddeev-Popov-ghost propagator in perturbative quantum gravity in de Sitter spacetime
aa r X i v : . [ g r- q c ] J a n Infrared problem in the Faddeev-Popov-ghost propagator in perturbative quantumgravity in de Sitter spacetime
Jos Gibbons, ∗ Atsushi Higuchi, † and William C. C. Lima ‡ Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom (Dated: January 20, 2021)The propagators for the Faddeev-Popov (FP) ghosts in Yang-Mills theory and perturbative grav-ity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. An IR cutoff in themomentum space to regularize these divergences breaks the de Sitter invariance. These IR diver-gences are due to the spatially constant modes in the Yang-Mills case and the modes proportionalto the Killing vectors in the case of perturbative gravity. It has been proposed that these IR diver-gences can be removed, with the de Sitter invariance preserved, by first regularizing them with anadditional mass term for the FP ghosts and then taking the massless limit. In the Yang-Mills case,this procedure has been shown to correspond to requiring that the physical states, and the vacuumstate in particular, be annihilated by some conserved charges in the Landau gauge. In this paper weshow that there are similar conserved charges in perturbative gravity in the covariant Landau gaugein de Sitter spacetime and that the IR-regularization procedure described above also correspond torequiring that the vacuum state be annihilated by these charges with a natural definition of theinteracting vacuum state.
PACS numbers: 04.62.+v
I. INTRODUCTION
Inflationary cosmological models [1–5] have been themain motivation for theoretical investigation into quan-tum field theory (QFT) in de Sitter spacetime. The ob-servation consistent with the assumption that the rate ofexpansion of our universe is accelerating [6, 7] providesanother motivation for this investigation. QFT in de Sit-ter spacetime has been investigated also in the contextof dS/CFT correspondence [8]. Perturbative quantumgravity is not renormalizable, but it is still a theory withpredictive power as an effective theory at each order ofperturbation theory [9].Perturbative quantum gravity in de Sitter spacetimehas many challenging features. Among them is the factthat the graviton propagator is infrared (IR) divergentin the physical gauge, with all gauge degrees of freedomfixed, natural to the spatially-flat (or Poincar´e) patch ofthis spacetime [10]. The source of the IR divergences isthe similarity of graviton modes in this coordinate patchto those of massless minimally-coupled scalar field [11–13]. However, it was found that these divergences do notmanifest themselves in the physical quantities studied bythe authors of Ref. [10]. This finding is consistent withthe fact that the IR-divergent part of the propagator canbe written in pure-gauge form [14–16], i.e. that the IR-divergent part of the gravitational perturbation can beexpressed as h µν = ∇ µ A ν + ∇ ν A µ . (See Refs. [17, 18]for an analogous result for single field inflation.) Someauthors have claimed to show that these IR divergenceswould lead to breakdown of de Sitter invariance (see, ∗ [email protected] † [email protected] ‡ [email protected] e.g. Refs. [19, 20]), but this has not been established ina gauge-invariant manner.The pure-gauge nature of the IR divergences in thesense explained above suggests that the graviton propa-gator may be IR finite in gauges natural to other coordi-nate patches. Indeed it is IR finite in the physical gaugenatural to global coordinates of de Sitter spacetime [21].Moreover, the covariant propagator in global coordinatesis also IR finite [22].Now, one also needs the Faddeev-Popov (FP)ghosts [23–25], which are fermionic vector fields, in thecovariant quantization of the gravitational field. Al-though the graviton propagator is IR finite in global coor-dinates, the FP-ghost propagator is IR divergent. TheseIR divergences for the FP-ghost propagator are due tothe modes proportional to the Killing vectors. However,the anti-ghost field, ¯ c µ ( x ), appear in the Lagrangian den-sity only in the form ∇ µ ¯ c ν + ∇ ν ¯ c µ , and for this reasonthe IR-divergent Killing-vector modes do not contributeto the interaction. It has been proposed that the IR-divergences for the FP ghosts should be first regularizedby the introduction of a small mass and that the mass-less limit should be taken at the end [26]. The resultingamplitude will be IR finite, i.e. it does not diverge in themassless limit, because the interaction terms are suchthat the IR divergences are eliminated because of theform of the interaction terms mentioned above. How-ever, this procedure would appear rather ad hoc and itneeds further justification, in particular, with regards toits compatibility with the BRST invariance [27, 28].The FP-ghost propagator is IR divergent also in Yang-Mills theory in de Sitter spacetime because the FP ghostsare massless minimally-coupled scalar field in this theory.(In fact its propagator is IR divergent in any spacetimewith compact Cauchy surfaces.) These IR divergencescan also be regularized with a small mass term and byaking the massless limit at the end. The IR divergencesin this case are due to the constant modes, and since onlythe derivatives of the anti-ghost field appear in the inter-action terms, the resulting amplitude is IR finite [26].For the Yang-Mills case, the procedure to eliminatethe IR divergences from the FP-ghost sector mentionedabove was shown to corresponds to requiring the vac-uum state to be annihilated by certain conserved chargesin the covariant Landau gauge [29]. It was also proposedthat all physical states be annihilated by these charges.(We note that a similar method has been used to elim-inate the IR divergences in massless minimally-coupledscalar field in de Sitter spacetime [30].) These chargestransform among themselves under BRST transforma-tion, and hence this requirement on the vacuum state iscompatible with, i.e. invariant under, the BRST trans-formation.In this paper, we show that this equivalence holdsalso for perturbative quantum gravity in global de Sitterspacetime in the covariant Landau gauge with a natu-ral definition of the interacting vacuum state. That is,there are similar conserved charges in perturbative quan-tum gravity in this spacetime and the regularization andelimination of the IR divergences through a small massterm corresponds to requiring that the vacuum state beannihilate by these charges in this gauge. These chargesagain transform among themselves under BRST trans-formation. Hence the requirement on the vacuum stateis compatible with BRST invariance of the theory. Someof the results we present in the next sections were antic-ipated in Ref. [31].The remainder of the paper is organized as follows. InSec. II we present a brief description of de Sitter space-time, with emphasis on its Killing vectors. In Sec. III wedescribe the IR divergences in the propagator of the FPghosts for perturbative gravity in the covariant Landaugauge using the Euclidean formulation. In Sec. IV wefind the conserved charges which play the central rˆole inthis paper. In Sec. V we identify the conserved chargesfound in Sec. IV essentially as the canonical momentaconjugate to cyclic variables. Then, in Sec. VI we showthat the regularization of the FP-ghost propagator witha mass term implies that the vacuum state is annihilatedby the conserved charges found in Sec. IV at tree level,i.e. in the free theory obtained by turning off the inter-action. In Sec. VII we discuss our definition of the inter-acting vacuum state in Hamiltonian perturbation theory.This definition is combined with the result in the previ-ous section to show that the interacting vacuum state isalso annihilated by these charges. Finally in Sec. VIII wesummarize and discuss our results. The Appendices con-tain some details omitted in the main text. Throughoutthis paper we employ units such that G = ~ = c = 1 andadopt the signature ( − + + · · · +) for the metric. II. KILLING VECTORS IN DE SITTERSPACETIME
In this section we discuss the Killing vectors in n -dimensional de Sitter spacetime, which cause the IR di-vergences in the FP-ghost propagator. Consider ( n + 1)-dimensional Minkowski spacetime with Cartesian coordi-nates X µ , µ = 0 , , . . . , n , and the metric ds = − ( dX ) + n X i =1 ( dX i ) . (2.1)Then, the hypersurface defined by − ( X ) + n X i =1 ( X i ) = 1 /H , (2.2)where H is the Hubble constant, is the n -dimensionalde Sitter spacetime. Let X = H − sinh Ht , (2.3a) X i = H − cosh Ht ˆ x j , ≤ j ≤ n , (2.3b)where t ∈ ( −∞ , ∞ ) and P ni =1 (ˆ x j ) = 1. Thus, the co-ordinates ˆ x j parametrize the unit ( n − S n − .By substituting these formulas into Eq. (2.1) we find themetric of de Sitter spacetime as ds = − dt + H − cosh Ht d Ω n − , (2.4)where d Ω n − is the metric on S n − . From now on we let H = 1 for simplicity.The n -dimensional de Sitter spacetime has the Killingsymmetries of ( n + 1)-dimensional Minkowski spacetimewith the origin fixed, i.e. so ( n, n ( n − / S n − . In addition there are n Killing vector fields gener-ating the boosts in n different directions. These Killingsymmetries are closely related to the IR divergences ofthe FP-ghost propagator as we find in the next section.It is useful to remind ourselves of the spherical harmon-ics on S n − . The scalar spherical harmonics Y ( ℓσ ) ( θ ), ℓ = 0 , , , . . . , on S n − , where θ denotes the angularcoordinates covering the sphere, satisfy [32] D i D i Y ( ℓσ ) ( θ ) = − ℓ ( ℓ + n − Y ( ℓσ ) ( θ ) , (2.5)where the label σ distinguishes between the scalar spheri-cal harmonics with the same angular momentum ℓ . Here,the covariant derivative D i is compatible with the metricon S n − and the indices are lowered and raised by themetric on S n − . We require Z S n − d Ω Y ∗ ( ℓσ ) ( θ ) Y ( ℓ ′ σ ′ ) ( θ ) = δ ℓℓ ′ δ σσ ′ , (2.6)where d Ω is the surface element of S n − . Thedivergence-free vector spherical harmonics Y i ( ℓσ ) ( θ ) sat-isfy D i Y i ( ℓσ ) ( θ ) = 0 and [32] D k D k Y i ( ℓσ ) ( θ ) = [ − ℓ ( ℓ + n −
2) + 1] Y i ( ℓσ ) ( θ ) . (2.7)2e require Z S n − d Ω Y ∗ ( ℓσ ) i ( θ ) Y i ( ℓ ′ σ ′ ) ( θ ) = δ ℓℓ ′ δ σσ ′ , (2.8)where the spatial index is lowered with the S n − metric.The Killing vectors ξ µ ( σ, R) on de Sitter spacetime thatgenerate the rotations are given by ξ σ, R) = 0 , (2.9a) ξ i ( σ, R) = Y i (1 σ ) , (2.9b)i.e. Y i ( ℓσ ) with ℓ = 1. The Killing vectors ξ µ ( σ, B) on de Sit-ter spacetime that generate the boosts are given by ξ σ, B) = Y (1 σ ) , (2.10a) ξ i ( σ, B) = tanh t D i Y (1 σ ) , (2.10b)where the index i is raised by the metric on S n − .The metric (2.4) on de Sitter spacetime (with H = 1)becomes that of the unit n -sphere ( S n ), d Ω n = dτ + sin τ d Ω n − , (2.11)by the following complex coordinate transformation: τ = π it . (2.12)Upon this coordinate transformation, both types ofKilling vectors ξ µ ( σ, R) and ξ µ ( σ, B) become, up to constantnormalization factors, the rotation Killing vectors V µ (1 ρ ) ,where V µ ( Lρ ) , L = 1 , , . . . , are the divergence-free vec-tor spherical harmonics on S n , satisfying the eigenvalueequation, ∇ ν ∇ ν V µ ( Lρ ) ( τ, θ ) = [ − L ( L + n −
1) + 1] V µ ( Lρ ) ( τ, θ ) , (2.13)and the normalization condition Z S n d Ω V ∗ ( Lσ ) µ ( τ, θ ) V µ ( L ′ ρ ′ ) ( τ, θ ) = δ LL ′ δ ρρ ′ , (2.14)where d Ω is the volume element on S n . III. IR DIVERGENCES IN THE FP-GHOSTPROPAGATOR
In this section we discuss the structure of the IR di-vergences of the FP-ghost propagator. The Lagrangiandensity for perturbative gravity in the covariant Landaugauge reads L = L GR + √− g L FP + √− g L gf , (3.1)where L GR is the diffeomorphism invariant Lagrangiandensity describing the gravitational field and where g is the determinant of the background metric tensor g µν . The gauge-fixing and FP-ghost Lagrangian densities aregiven by L gf = −∇ µ B ν ( h µν − kg µν h αα ) , (3.2a) L FP = − i ∇ µ ¯ c ν ( ∇ µ c ν + ∇ ν c µ − kg µν ∇ α c α + £ c h µν − kg µν g αβ £ c h αβ ) , (3.2b)where £ X denotes the Lie derivative in the direction ofthe vector X µ . That is, £ c h µν = c α ∇ α h µν + ( ∇ µ c α ) h αν + ( ∇ ν c α ) h µα . (3.3)The field h µν is the gravitational perturbation: the fullmetric is given by ˜ g µν = g µν + h µν , where g µν is thebackground de Sitter metric. The indices in Eq. (3.2)are raised and lowered by g µν . The ghost and anti-ghostfields c µ and ¯ c µ , respectively, are anti-commuting Hermi-tian fields [33, 34].The gauge-fixing term would fail to provide a time-derivative of h if k = 1. This value is excluded for thisreason, and it is often convenient to write k = 1 + 1 β . (3.4)The parameter β will be taken to be positive in this pa-per unless otherwise stated. The gauge-fixing Lagrangiandensity L gf is the α → L ( α )gf = α B µ B µ − ∇ µ B ν H µν , (3.5)where we have defined H µν ≡ h µν − kg µν h , (3.6)with h ≡ h αα . By definingˆ B µ ≡ B µ + 1 α ∇ ν H µν (3.7)and neglecting total-divergence terms, we have L ( α )gf = α B µ ˆ B µ − α ∇ ν H µν ∇ λ H µλ . (3.8)The field ˆ B µ can be neglected because it is decoupledfrom other fields. The remaining term is the gauge-fixing term more commonly used. Note that the Euler-Lagrange equation from varying B µ in the Lagrangiandensity L gf reads ∇ ν H µν = 0 . (3.9)Thus, in this gauge the gauge condition is a result of afield equation. In this paper the quantity obtained by dividing a Lagrangiandensity by √− g is also called a Lagrangian density. L gf + L FP is BRST invariant, followingthe well-known general procedure (see, e.g., Ref. [35]).The BRST transformation is given as follows: δ B h µν = ∇ µ c ν + ∇ ν c µ + £ c h µν , (3.10a) δ B c µ = c α ∇ α c µ , (3.10b) δ B ¯ c µ = iB µ , (3.10c) δ B B µ = 0 , (3.10d)where £ c h µν is given by Eq. (3.3). The transform δ B h µν can be understood as the Lie derivative with respect to c µ of the full metric ˜ g µν = g µν + h µν . Thus, the actionfor the gravitational field obtained by integrating L GR over the spacetime is invariant under this transformation.It can readily be verified that the BRST transformationgiven by Eq. (3.10) is nilpotent, i.e. δ = 0 [34]. Indeed,we find that δ c µ = ( δ B c α ) ∇ α c µ − c α ∇ α δ B c µ = − R µαβγ c α c β c γ = 0 , (3.11)where we have defined δ B to act from the left so that δ B (Ω Ω ) = ( δ B Ω )Ω − Ω δ B Ω if Ω is fermionic. Theequality δ h µν = 0 follows from £ c £ c ˜ g µν = £ X ˜ g µν , (3.12)where X µ ≡ c α ∇ α c µ . It can readily be seen that L FP + L gf = iδ B [( ∇ µ ¯ c ν ) H µν ] , (3.13)The BRST invariance of L FP + L gf follows from the nilpo-tency of δ B .Now, let us discuss the IR divergences in the FP-ghostpropagator. The free-field equation, i.e. the equation ob-tained by dropping the interaction terms, for the ghostfield is ∇ ν ( ∇ µ c ν + ∇ ν c µ − kg µν ∇ α c α ) = 0 . (3.14)From here to the end of this section, the fields c µ and¯ c µ are assumed to satisfy the free field equation. Thefree anti-ghost field ¯ c µ satisfies the same equation. It isconvenient to rewrite Eq. (3.14) by interchanging somederivatives and using R µν = ( n − g µν as L µν c ν = 0 , (3.15)where the differential operator L µν is given by L µν = − δ νµ ∇ α ∇ α + ∇ ν ∇ µ + 2 β − ∇ µ ∇ ν − n − δ νµ + m δ νµ . (3.16)We have inserted a mass term m δ νµ as an IR regulator.By writing the tree-level Feynman propagator as G (FP) µµ ′ ( x, x ′ ) ≡ − iT h | c µ ( x )¯ c µ ′ ( x ′ ) | i , (3.17) one finds that the function G (FP) µµ ′ ( x, x ′ ) satisfies L µν G (FP) νν ′ ( x, x ′ ) = g µν ′ δ ( n ) ( x, x ′ ) , (3.18)where the delta function δ ( n ) is defined to have the prop-erty Z d n x p − g ( x ) f ( x ) δ ( n ) ( x, x ′ ) = f ( x ′ ) , (3.19)for any compactly-supported smooth function f ( x ). Thedifferential operator L µν acts on x in Eq. (3.18).The IR divergences of the FP-ghost propagator in theBunch-Davies (or Euclidean) vacuum state [36–39] in the n -dimensional de Sitter background is best understoodin the Euclidean approach. The Feynman propagator G (FP) µµ ′ ( x, x ′ ) on de Sitter spacetime in the Euclidean vac-uum state can be obtained by finding the (unique) so-lution to Eq. (3.18) on the n -dimensional sphere, S n ,and then analytically continuing it to de Sitter space-time by the relation (2.12). Since Eq. (3.18) shows that G (FP) µµ ′ ( x, x ′ ) is the inverse of the differential operator L µν , it can be expressed in terms of the eigenfunctionsof this differential operator on S n . Any smooth vectorfield on S n can be expressed as a linear combination ofthe divergence-free vector eigenfunctions V µ ( Lρ ) discussedin the previous section and the gradient eigenfunctions ∇ µ φ ( Lρ ) , L = 1 , , , . . . of the Laplace-Beltrami oper-ator ∇ ν ∇ ν . The vectors V µ ( Lρ ) satisfy Eqs. (2.13) and(2.14) whereas the functions φ ( Lρ ) satisfy ∇ ν ∇ ν φ ( Lρ ) = − L ( L + n − φ ( Lρ ) . (3.20)It is convenient to normalize these functions as follows: Z S n dS φ ∗ ( Lρ ) φ ( L ′ ρ ) = 1 L ( L + n − δ LL ′ δ ρρ ′ . (3.21)One readily finds L µν V ( Lρ ) ν = (cid:2) ( L − L + n ) + m (cid:3) V ( Lσ ) µ , (3.22a) L µν ∇ ν φ ( Lρ ) = (cid:2) − β − L ( L + n − − n −
1) + m (cid:3) ∇ µ φ ( Lρ ) . (3.22b)Hence, G (FP) µµ ′ ( x, x ′ )= ∞ X L =1 X σ V ( Lσ ) µ ( x ) V ∗ ( Lσ ) µ ′ ( x ′ )( L − L + n ) + m − β ∞ X L =1 X σ ∇ µ φ ( Lσ ) ( x ) ∇ µ ′ φ ( Lσ ) ∗ ( x ′ )2 { L ( L + n −
1) + ( n − β } − m = 1 m X σ V (1 σ ) µ ( x ) V ∗ (1 σ ) µ ′ ( x ′ ) + G (FP , reg) µµ ′ ( x, x ′ ) , (3.23)4here the function G (FP , reg) µµ ′ ( x, x ′ ) remains finite in thelimit m →
0. (Recall that β > V µ (1 σ ) are the Killing vectors on S n . Hence, upon analyticcontinuation, one finds that the Feynman propagator forthe FP ghosts takes the form G (FP) µµ ′ ( x, x ′ ) = 1 m X A c A ξ Aµ ( x ) ξ Aµ ′ ( x ′ )+ G (FP , reg) µµ ′ ( x, x ′ ) , (3.24)where ξ µA ( x ) with A = ( σ, R) or ( σ, B) are the Killingvectors in de Sitter spacetime and c A are constants. Al-though the Feynman propagator G (FP) µµ ′ ( x, x ′ ) is IR di-vergent, i.e. it diverges as m →
0, if one uses the reg-ularized propagator in perturbative calculations, the IRdivergences cancel out. This is because all interactionterms in L FP involve the factor ∇ µ ¯ c ν + ∇ ν ¯ c µ . Indeed,from the Killing equation ∇ µ ξ Aν + ∇ ν ξ Aµ = 0, one finds ∇ µ ′ G (FP) µν ′ ( x, x ′ ) + ∇ ν ′ G (FP) µµ ′ ( x, x ′ )= ∇ µ ′ G (FP , reg) µν ′ ( x, x ′ ) + ∇ ν ′ G (FP , reg) µµ ′ ( x, x ′ ) , (3.25)where the derivative ∇ µ ′ acts on x ′ . Thus, the use ofthe regularized FP-ghost propagator will lead to IR-finiteamplitudes.For this reason, it was proposed in Ref. [26] that oneshould use the regularized FP-ghost propagator and takethe massless limit after the calculation, thus preservingthe de Sitter invariance, rather than breaking it by intro-ducing a momentum cut-off. However, since a mass termbreaks the BRST invariance, it was not clear whethersuch a procedure leads to a consistent theory. The pur-pose of this paper is to establish that the use of the reg-ularized FP-ghost propagator corresponds to requiringthat the vacuum state be annihilated by certain con-served charges in a BRST-invariant manner and, hence,that such a procedure is consistent with the BRST in-variance of the theory. This equivalence is an analogue ofthat for Yang-Mills theory in de Sitter spacetime demon-strated in Ref. [29]. IV. CONSERVED CHARGES INPERTURBATIVE GRAVITY IN THE LANDAUGAUGE
In this section we find some conserved charges in per-turbative gravity in the Landau gauge about a back-ground spacetime satisfying Einstein’s equations withcompact Cauchy surfaces and with Killing symmetries.Such spacetimes include global de Sitter spacetime. Wealso show how these charges are related to one anotherby BRST transformation.First, the field equation ∇ ν H µν = 0, where H µν isdefined by Eq. (3.6), and the Killing equation ∇ µ ξ Aν + ∇ ν ξ Aµ = 0 imply that ∇ µ ( ξ Aν H µν ) = 0. Hence thefollowing charges are conserved: Q ( H ) A ≡ Z Σ d Σ n µ ξ Aν H µν . (4.1)where n µ is the future-pointing unit normal to theCauchy surface Σ, i.e. n > n µ n µ = −
1, and where d Σ is the hypersurface element on Σ.Next, we note that the field equation coming fromvarying ¯ c µ is also the divergence of a symmetric tensor,i.e. ∇ µ S µν = 0 , (4.2)where S µν = ∇ µ c ν + ∇ ν c µ − kg µν ∇ α c α + £ c h µν − kg µν g αβ £ c h αβ . (4.3)Then, ∇ µ ( ξ Aν S µν ) = 0. Hence, the charges given by Q ( c ) A ≡ Z Σ d Σ n µ ξ Aν S µν (4.4)are conserved. It is clear that Q ( c ) A = δ B Q ( H ) A since S µν = δ B H µν .It is convenient to remove the derivatives on h µν in theexpression for Q ( c ) A for later purposes. We observe ξ Aν S µν = ξ Aν e S µν − ( ∇ α ξ Aν ) c α H µν + 2 ∇ α ( ξ Aν c [ α H µ ] ν ) , (4.5)where e S µν = ( ∇ µ c ν + ∇ ν c µ − kg µν ∇ α c α )(1 + kh )+ ( ∇ α c µ ) H αν − ( ∇ α c α ) H µν + ( ∇ µ c α ) H αν + ( ∇ ν c α ) H αµ − kg µν ( ∇ α c β ) H αβ . (4.6)Then these charges can be expressed as follows: Q ( c ) A = Z Σ d Σ n µ h ξ Aν e S µν − ( ∇ α ξ Aν ) c α H µν i , (4.7)because the integral of a vector of the form ∇ µ F µν ,where F µν is an antisymmetric tensor, over any (com-pact) Cauchy surface vanishes by the generalized Stokestheorem.There are also conserved charges arising from the fieldequation coming from varying c µ . To show this, it isconvenient to write the sum of the Faddeev-Popov andgauge-fixing Lagrangian densities in the following form: L FP + L gf = − i ( ∇ µ ¯ c ν + ∇ ν ¯ c µ − kg µν ∇ α ¯ c α + £ ¯ c h µν − kg µν g βγ £ ¯ c h βγ ) ∇ µ c ν − ∇ µ ¯ B ν H µν + i (1 − k )( ∇ µ ¯ c ν ∇ α c α − ∇ α ¯ c α ∇ µ c ν ) H µν , (4.8)5here ¯ B ν ≡ B ν + i [¯ c α ∇ α c ν − ( ∇ α ¯ c ν ) c α ] . (4.9)Equation (4.8) is derived in Appendix A. Notice that, if k = 1 /
2, then this equation will be equal to the negativeof the original one with B µ replaced by − ¯ B µ and with c µ and ¯ c µ interchanged. The field equation arising fromvarying the Lagrangian density (4.8) with respect to c µ is then ∇ ν T µν = 0 , (4.10)where T µν = ∇ µ ¯ c ν + ∇ ν ¯ c µ − kg µν ∇ α ¯ c α + £ ¯ c h µν − kg µν g βγ £ ¯ c h βγ +(2 k − g µν ∇ β ¯ c γ h βγ − h µν ∇ α ¯ c α ) . (4.11)The contribution from varying c µ through ¯ B µ in Eq. (4.9)has not been included here because it is proportional to ∇ ν H µν , which vanishes by a field equation. Then, since ∇ µ ( ξ Aν T µν ) = 0 for any Killing vector ξ µA , we have thefollowing conserved charges: Q (¯ c ) A ≡ Z Σ d Σ n µ ξ Aν T µν . (4.12)The similarity of S µν and T µν allows us to use thesame method to remove the derivative of h µν from thesecharges. Thus, by defining e T µν = ( ∇ µ ¯ c ν + ∇ ν ¯ c µ − kg µν ∇ α ¯ c α )(1 + kh )+( ∇ α ¯ c µ ) H αν − k ( ∇ α ¯ c α ) H µν +( ∇ µ ¯ c α ) H αν + ( ∇ ν ¯ c α ) H αµ − g µν ( ∇ α ¯ c β ) H αβ , (4.13)we find Q (¯ c ) A = Z Σ d Σ n µ h ξ Aν e T µν − ( ∇ α ξ Aν )¯ c α H µν i . (4.14)The BRST transforms of the charges Q ( c ) A = δ B Q ( H ) A vanish because δ = 0. However, the BRST transformsof the charges Q (¯ c ) A are nonzero. The conservation of Q (¯ c ) A and the BRST invariance of the theory imply thatthe charges δ B Q (¯ c ) A are also conserved. We show in Ap-pendix B that these charges are precisely the Noethercharges Q (st) A associated with the spacetime symmetriesgenerated by the Killing vectors ξ µA .We note in passing that the action is invariant underthe “anti-BRST” transformation if k = 1 / k = 1 /
2, the lastterm in Eq. (4.8) vanishes, and the Lagrangian density L FP + L gf is left unchanged if we replace ( c µ , ¯ c µ , ¯ B µ ) by (¯ c µ , − c µ , B µ ). Hence, for this value of the gauge param-eter the action is also invariant under the following anti-BRST transformation: δ ¯B h µν = ∇ µ ¯ c ν + ∇ ν ¯ c µ + £ ¯ c h µν , (4.15a) δ ¯B ¯ c µ = ¯ c α ∇ α ¯ c µ , (4.15b) δ ¯B c µ = − i ¯ B µ , (4.15c) δ ¯B ¯ B µ = 0 . (4.15d)Furthermore, Eq. (4.11) shows that for k = 1 / T µν corresponds to the tensor S µν given in Eq. (4.3)with c µ replaced by ¯ c µ . This implies that Q (¯ c ) A = δ ¯B Q ( H ) A .Hence, we conclude that the Noether charges associatedwith the background spacetime symmetries are the anti-BRST transforms of the charges Q ( c ) A . Thus, the de Don-der gauge would appear to be a natural one because allconserved charges found in this section can be derivedfrom Q ( H ) A by the BRST and anti-BRST transformationin this gauge. However, unfortunately, this value of k cor-responds to β = −
2, and the negative values of β turnsout to be problematic in perturbative gravity in de Sit-ter spacetime. For this reason we exclude this gauge,together with all other negative values of β . V. IDENTIFICATION OF THE KILLINGVECTOR MODES
Since the charges Q ( H ) A , Q ( c ) A and Q (¯ c ) A found in theprevious section are conserved, it is consistent to requirethat all physical state, particularly the vacuum state | Ω i ,be annihilated by these charges: Q ( H ) A | Ω i = Q ( c ) A | Ω i = Q (¯ c ) A | Ω i = 0 . (5.1)The main aim of this paper is to show that imposingthese conditions on | Ω i corresponds to using the FP-ghost propagator regularized by a finite mass term andthen taking the massless limit at the end as described inSec. III. We shall discuss this equivalence in the Hamilto-nian formulation in Sec. VII. For this purpose we need toidentify the components of the fields B µ , c µ and ¯ c µ thatare proportional to the Killing vectors at each time. Theconserved charges Q ( H ) A , Q (¯ c ) A and Q ( c ) A will be shown tobe (essentially) the canonical conjugate momenta of thesecomponents. This procedure may look rather artificial,but it is necessary for using the Hamiltonian formalismto discuss the conditions (5.1).We first extract the modes proportional to the Killingvectors ξ µA at each time t for V µ = c µ , ¯ c µ and B µ as V A (0) ( t ) = Z Σ d Σ V µ η Aµ , (5.2)where Σ is the hypersurface of constant t , which is an( n − t . The covectors η Aµ are chosen to satisfy Z Σ d Σ ξ µA η Bµ = δ BA . (5.3)6hese conditions do not determine η Aµ uniquely and thereis some freedom in choosing them. It is natural to choosethem for the rotation Killing vectors ascosh n − t η ( σ, R)0 = 0 , (5.4a)cosh n − t η ( σ, R) i = Y (1 σ ) i . (5.4b)We can choose them for the boost Killing vectors ascosh n − t η ( σ, B)0 = 22 + ( n − β sinh t Y (1 σ ) , (5.5a)cosh n − t η ( σ, B) i = β sinh t cosh t n − β sinh t D i Y (1 σ ) . (5.5b)With the help of these definitions, we can then expandthe field V µ as V µ ( x ) = X A V A (0) ( t ) ξ µA ( x ) + V µ (+) ( x ) . (5.6)In order to simplify the notation, we also define θ A ( t ) ≡ c A (0) ( t ) , (5.7a)¯ θ A ( t ) ≡ ¯ c A (0) ( t ) . (5.7b)In the ADM Hamiltonian formalism [47] the metric isgiven as follows: ds = − N dt + g ij ( dx i + N i dt )( dx j + N j dt ) , (5.8)where N and N i are called the lapse function and shiftvector, respectively. They are given in terms of the fullmetric components ˜ g µν as N = q − ˜ g + ˜ g ij ˜ g i ˜ g j , (5.9a) N i = ˜ g ij ˜ g j , (5.9b)where ˜ g ij is the inverse of the matrix ˜ g ij . As is wellknown, the Lagrangian density for the Einstein-Hilbertaction can be given in terms of ˜ g ij , N and N i up to atotal divergence, and this Lagrangian density containsno time derivatives of N or N i . (See, e.g. Appendix E ofRef. [48].)Since this Lagrangian density depends on h = ˜ g − g and h i = ˜ g i − g i only through N and N i , it does notcontain any time derivatives of h or h i . This allows usto identify H ν as the momentum variables conjugate to B ν as √− gH ν = ∂ L gf ∂ ˙ B ν , (5.10)with the notation ˙ f ≡ ∂ t f , if there were no terms con-taining ˙ h ν in the FP-ghost Lagrangian density L FP . Infact, the Lagrangian density L FP does contain terms in-volving ˙ h ν , but they can be removed by redefining theauxiliary field B µ . Thus, by definingˇ B µ ≡ B µ − i ( ∇ α ¯ c µ ) c α , (5.11) we find, up to a total divergence, L FP + L gf = −∇ µ ˇ B ν H µν + iR νβαµ ¯ c β c α H µν − i ∇ µ ¯ c ν ( ∇ µ c ν + ∇ ν c µ − kg µν ∇ α c α )(1 + kh ) − i [ − ( ∇ µ ¯ c ν )( ∇ α c α ) + ( ∇ α ¯ c µ )( ∇ ν c α )+( ∇ α ¯ c µ )( ∇ α c ν ) + ( ∇ µ ¯ c α )( ∇ α c ν )] H µν . (5.12)We present a derivation of this result in Appendix C.To identify the conserved charges found in the previoussection essentially as the canonical momenta conjugateto cyclic variables we need to redefine the auxiliary fieldfurther. Hence, let us define˜ B µ ≡ ˇ B µ + i ¯ θ A ( ∇ α ξ νA ) c α + i ¯ c α (+) ( ∇ α ξ νA ) θ A , (5.13)where θ A and ¯ θ A are the canonical variables multiplyingthe Killing vectors in the expansion of c µ and ¯ c µ , respec-tively, as defined in Eq. (5.7). Then, after a tedious butstraightforward calculation we find L FP + L gf = L (+)FP+gf + i ˙¯ θ A ˙ θ B (cid:26)(cid:20) g ij ξ iA ξ jB + 2 β ξ A ξ B (cid:21) (1 + kh )+ (1 − k ) ξ A ξ νB H ν + ξ µA ξ νB H µν (cid:27) + i ˙¯ θ A θ B ξ ν [ A,B ] H ν + i ˙¯ θ A h ξ νA e S (+)0 ν − ( ∇ α ξ νA ) c α (+) H ν i + i h ξ νA e T (+)0 ν − ( ∇ α ξ νA )¯ c α (+) H ν i ˙ θ A + ˙ B A (0) ξ νA H ν . (5.14)In Eq. (5.14), we have defined the Killing vector ξ µ [ A,B ] ≡ [ ξ A , ξ B ] µ , the variable B A (0) is the coefficient of the Killingvector mode of the field B µ as defined by Eq. (5.6), and L (+)FP+gf does not contain variables B A (0) , θ A or ¯ θ A . Thetensors e S (+) µν and e T (+) µν are obtained by replacing c µ and¯ c µ by c µ (+) and ¯ c µ (+) , which are defined by Eq. (5.6), in e S µν in Eq. (4.6) and e T µν in Eq. (4.13), respectively. Thenwe find ∂∂ ˙ B A (0) ( L FP + L gf ) = ξ νA H ν , (5.15a) ∂∂ ˙¯ θ A ( L FP + L gf ) = i h ξ νA e S ν − ( ∇ α ξ νA ) c α H ν i , (5.15b) ∂∂ ˙ θ B ( L FP + L gf ) = − i h ξ νB e T ν − ( ∇ α ξ νB )¯ c α H ν i − i ¯ θ A ξ ν [ A,B ] H ν , (5.15c)where the derivatives with respect to ˙¯ θ A and ˙ θ B are left-derivatives. The integral of these over a Cauchy surfaceof constant time gives the canonical momenta conjugate7o B A (0) , ¯ θ A and θ B , which will be denoted by p A , ϕ A and¯ ϕ B , respectively. They satisfy (cid:2) p A , B B (0) (cid:3) = (cid:8) ϕ A , ¯ θ B (cid:9) = (cid:8) ¯ ϕ A , θ B (cid:9) = − iδ BA , (5.16)where { ω , ω } ≡ ω ω + ω ω . Equations (4.1), (4.7)and (4.14) then yield p A = Q ( H ) A , (5.17a) ϕ A = iQ ( c ) A , (5.17b)¯ ϕ A = − i (cid:16) Q (¯ c ) A + ¯ θ B Q ( H )[ B,A ] (cid:17) , (5.17c)where the charge Q ( H )[ B,A ] is the bosonic charge of Eq. (4.1)corresponding to the Killing vector ξ µ [ B,A ] . It is interest-ing to note that n Q (¯ c ) A , Q ( c ) B o = − iQ ( H )[ A,B ] . (5.18)By applying the BRST transformation and using δ B Q (¯ c ) A = 0, Q (¯ c ) A = δ ¯B Q ( H ) A and Q (st) A = iδ ¯B Q ( c ) A , wefind h Q (st) A , Q ( c ) B i = Q ( c )[ A,B ] , (5.19)which is the expected action of the spacetime-symmetrycharges Q (st) A on Q ( c ) B .The canonical conjugate momenta p A and ϕ A are thoseof cyclic variables B A (0) and ¯ θ A as can be seen fromEq. (5.14), and they are indeed time independent, beingproportional to conserved charges. The time derivativeof ¯ ϕ A can be found from the Lagrangian density (5.14)as ˙¯ ϕ A = ∂∂θ A Z Σ d Σ ( L FP + L gf )= − i ˙¯ θ B Q ( H )[ B,A ] , (5.20)which agrees with the result obtained by differentiatingEq. (5.17c) directly and using the conservation of thecharges Q (¯ c ) A and Q ( H )[ B,A ] . VI. THE CONDITIONS ON THE VACUUMSTATE AT TREE LEVEL
As we stated before, the main purpose of this paper isto show that the use of the regularized FP-ghost propaga-tor for perturbative gravity in de Sitter spacetime corre-sponds to the conditions Q ( H ) A | Ω i = Q ( c ) A | Ω i = Q (¯ c ) A | Ω i =0 on the vacuum state | Ω i . In this section, we show thatthe use of the regularized FP-ghost propagator impliesthat the non-interactive vacuum state | i is annihilatedby the tree-level charges. From here to the end of thissection, the charges Q ( H ) A , Q ( c ) A and Q (¯ c ) A are the con-served charges in the non-interacting theory with the in-teractions turned off, which are linear in h µν , c µ and ¯ c µ ,respectively. For the bosonic charges Q ( H ) A we show in Ap-pendix D that the result Q ( H ) A | i = 0 or, more precisely, h | ωQ ( H ) A | i = 0 for any canonical variable ω except B A (0) ,which are canonically conjugate to Q ( H ) A , follows auto-matically in the standard de Sitter-invariant quantizationof linearized gravity in the Landau gauge.To illustrate in what way the regularized propagatorcorresponds to the charges Q ( c ) A and Q (¯ c ) A annihilating thevacuum state at linear level in the massless limit, let usconsider a massive Hermitian scalar field φ on de Sitterspacetime. Expanding this field operator in terms of thescalar spherical harmonics, Eq. (2.5), we obtain φ ( t, θ ) = ∞ X ℓ =0 X σ h a ℓσ f ℓ ( t ) Y ( ℓσ ) ( θ ) + a † ℓσ f ∗ ℓ ( t ) Y ∗ ( ℓσ ) ( θ ) i , (6.1)where [ a ℓσ , a † ℓ ′ σ ′ ] = δ ℓℓ ′ δ σσ ′ , with other commutatorsnull, and f ℓ are normalized according to the Klein-Gordon inner product and chosen such that we have theBunch-Davies vacuum. The field time evolution is dic-tated by the Hamiltonian operator H = 12 (cid:20) π ( t ) π ( t )cosh n − t + m cosh n − tφ ( t ) φ ( t ) (cid:21) + H (+) , (6.2)where φ ≡ a f + a † f ∗ , π ≡ cosh n − t dφ /dt , thus[ φ , π ] = i , and H (+) is the Hamiltonian operator ofthe modes with ℓ >
0. We focus on the ℓ = 0 mode,as it is the one responsible for the IR divergence of thepropagator in this example. The form of f in the small- m limit can be found in Ref. [29] and reads f ( t ) = r V S n − c (cid:26) m − m [ g ( t ) + c + ic f ( t )] (cid:27) + O ( m ) , (6.3)where c and c are constants, V S n − ≡ π n/ / Γ (cid:0) n (cid:1) isthe volume of the unit S n − , and f ( t ) ≡ Z t dt ′ V ( t ′ ) , (6.4a) g ( t ) ≡ Z t dt ′ V ( t ′ ) Z t ′ dt ′′ V ( t ′′ ) , (6.4b)where we have defined V ( t ) ≡ V S n − cosh n − t . The con-tribution coming from the zero mode to the propagatorand its time derivatives in the de Sitter invariant vacuum | i has the form h | φ ( t ) φ ( t ′ ) | i = f ( t ) f ∗ ( t ′ )= V S n − c m − r V S n − c (cid:8) g ( t ) + g ( t ′ )+ 2 c + ic [ f ( t ) − f ( t ′ )] (cid:9) + O ( m ) , (6.5)8 | φ ( t ) π ( t ′ ) | i = f ( t ) cosh n − t ′ ˙ f ∗ ( t ′ )= − V ( t ′ )[ ˙ g ( t ′ ) − ic ˙ f ( t ′ )] √ V S n − c + O ( m ) , (6.6)and h | π ( t ) π ( t ′ ) | i = O ( m ) . (6.7)Hence, in the massless limit one finds h | π ( t ) π ( t ′ ) | i =0. This can be stated as h | ω ( t ) π ( t ′ ) | i = 0, where ω isany canonical variable except φ .If one expects the field φ to represent an observableor to couple with another field through its amplitude,then Eqs. (6.5) - (6.7) are just a manifestation of thewell-known fact that there is no de Sitter-invariant statefor a massless scalar field [12, 13]. On the other hand,if the field φ ( t ) itself is unobservable and only interactsvia its derivatives, which is precisely the case of the FPghosts, then the correlators h | φ ( t ) φ ( t ′ ) | i → ∞ and h | φ ( t ) π ( t ′ ) | i 6 = 0 in the limit m → φ is a cyclic variable in themassless limit. Its canonical conjugate momentum π isthe conserved charge associated with the conserved cur-rent J µ ≡ ∇ µ φ . This allows us to interpret the statement h | ω ( t ) π ( t ′ ) | i = 0 simply as π | i = 0 . (6.8)The condition (6.8) is the requirement that the state | i is invariant under φ → φ + constant, which is a gaugetransformation for the massless scalar field.We can turn the argument above around and show thatthe condition (6.8) in the massless theory corresponds todiscarding the contribution of the zero mode to the prop-agator. From the Hamiltonian (6.2) and the Heisenbergequation, we have that ddt (cid:20) V ( t ) dφ dt (cid:21) = 0 . (6.9)Therefore, the zero mode φ is analogous to a free quan-tum particle and we can expand the field operator φ as φ ( t, θ ) = ˆ q + ˆ pf ( t )+ ∞ X ℓ =1 X σ h a ℓσ f ℓ ( t ) Y ( ℓσ ) ( θ ) + a † ℓσ f ∗ ℓ ( t ) Y ∗ ( ℓσ ) ( θ ) i , (6.10)where f ( t ) was defined in Eq. (6.4a) and [ˆ q, ˆ p ] = i . Wenote that φ (0) = √ V S n − ˆ q and π = √ V S n − ˆ p . As in Interestingly enough, in the Euclidean theory it is possible toshow that a massless free scalar field on S n admits fully sym-metric states [49]. The idea is to treat the massless field as agauge theory symmetric under φ → φ + constant and add agauge-fixing term to the Lagrangian that effectively removes thezero mode. quantum mechanics, we can represent the operators ˆ q andˆ p on L ( R ) as the multiplication by q and the derivative − id/dq , respectively. We then consider the field state | Ψ i = ψ ( q ) ⊗ | (+) i , where ψ ( q ) is a normalized wave-function and | (+) i is the vacuum state for the modeswith ℓ >
0, i.e. a ℓσ | (+) i = 0 for all ℓ ≥
1. It is possibleto show (see, e.g. Ref. [29]) that the state | Ψ i is de Sit-ter invariant if, and only if, condition (6.8) is satisfied,i.e. ˆ p | Ψ i = 0. Hence, for | Ψ i to be de Sitter invariantwe must have the wavefunction ψ ( q ) constant. Again,if φ is observable or couples through its amplitude, thisis a manifestation of the fact that no such | Ψ i exists,since R + ∞−∞ dq | ψ ( q ) | = ∞ . However, if our field is un-observable and interacts via its derivatives, then, sincethe zero mode is spatially constant and its time deriva-tive annihilates the state, we can simply ignore it by re-defining the inner product of the field space of states.Thus, for two field states | Ψ i = ψ ( q ) ⊗ | α (+)1 i and | Ψ i = ψ ( q ) ⊗ | α (+)2 i , where | α (+)1 i and | α (+)2 i arestates in the Fock space built by applying a † ℓσ on | (+) i ,we define h Ψ | Ψ i ≡ h α (+)1 | α (+)2 i . The result of com-puting the propagator in the vacuum state annihilated byˆ p , with the redefined inner product, is the scalar coun-terpart of the use of the regularized propagator discussedin Sec. III.Let us now return to the analysis of the FP-ghostpropagator in perturbative quantum gravity. Whatwe shall demonstrate at tree level is that, if we reg-ularize the propagator with a small mass and thentake the massless limit, then h | Q ( c ) A ( t )Λ | i = 0 and h | ωQ (¯ c ) A ( t ) | i = 0 for any canonical variables ω unless ω = ¯ θ A for the former and unless ω = θ A for the lat-ter. (Note that ¯ θ A and θ A are canonically conjugate to Q ( c ) A and Q (¯ c ) A , respectively, at tree level.) It is sufficientto show that h | Q ( c ) A ( t ) Q (¯ c ) B ( t ′ ) | i , h | Q ( c ) A ( t )¯ c µ (+) ( x ′ ) | i and h | c µ (+) ( x ) Q (¯ c ) B ( t ′ ) | i all vanish in the massless limit.Some details for the following discussion will be delegatedto Appendix E.The FP-ghost field equation with mass m at tree levelreads ∇ ν ( ∇ ν c µ + ∇ µ c ν − kg µν ∇ α c α ) − m c µ = 0 . (6.11)There are two types of solutions to this equation. Bywriting c µ = V µ + ∇ µ Φ , (6.12)where ∇ µ V µ = 0, we find (cid:3) Φ − β (cid:2) n − − m (cid:3) Φ = 0 , (6.13a) (cid:3) V µ − (cid:2) m − ( n − (cid:3) V µ = 0 , (6.13b)where we have defined (cid:3) ≡ ∇ µ ∇ µ . The scalar sector ∇ µ Φ contributes to the charge Q ( c ) A at tree level onlyat order m , as shown in Appendix E. For positive β ,the field Φ is a scalar field of positive mass, and there9s no divergence from this sector in the limit m → h | ωQ (¯ c ) A | i and h | Q ( c ) A ¯ ω | i at tree level in the m → V µ ,are given by V (1; ℓ,σ )0 = 0 , (6.14a) V (1; ℓ,σ ) i = C ℓm cosh n − t P − µ ℓ − + λ ( i sinh t ) Y ( ℓσ ) i , (6.14b)and V (2; ℓ,σ )0 = − s ℓ ( ℓ + n − n − − m C ℓm × n t P − µ ℓ − + λ ( i sinh t ) Y ( ℓσ ) , (6.15a) V (2; ℓ,σ ) i = C ℓm p ℓ ( ℓ + n − n − − m ] × n t (cid:2) cosh t ∂ t + ( n −
1) sinh t cosh t (cid:3) × P − µ ℓ − + λ ( i sinh t ) D i Y ( ℓσ ) , (6.15b)where C ℓm ≡ s Γ (cid:0) ℓ + n − − λ (cid:1) Γ (cid:0) ℓ + n − + λ (cid:1) , (6.16a) λ ≡ s(cid:18) n + 12 (cid:19) − m , (6.16b) µ ℓ ≡ ℓ + n − . (6.16c)The function P − µν ( x ) is the associated Legendre functionof the first kind [50].The vector sector of the FP-ghost field can be ex-panded as V µ = X I,ℓ,σ h α ( I ) ℓσ V ( I ; ℓ,σ ) µ + α ( I ) † ℓσ V ( I ; ℓσ ) ∗ µ i . (6.17)The vector sector of the anti-ghost field is expanded inthe same way with the annihilation and creation opera-tors, α ( I ) ℓσ and α ( I ) † ℓσ , replaced by ¯ α ( I ) ℓσ and ¯ α ( I ) † ℓσ , respec-tively. The mode functions V ( I ; ℓ,σ ) µ are normalized sothat these annihilation and creation operators satisfy n α ( I ) ℓσ , ¯ α ( J ) † ℓ ′ σ ′ o = ( − I +1 iδ ℓ ℓ ′ δ σ σ ′ δ IJ , (6.18)with all other anti-commutators among α ( I ) ℓσ and ¯ α ( I ) ℓσ andtheir Hermitian conjugates vanishing.The de Sitter invariant tree-level vacuum state | i isannihilated by the annihilation operators α ( I ) ℓσ and ¯ α ( I ) ℓσ , I = 1 ,
2. It is useful to note that [50]P − µ ℓ − + λ ( i sinh t ) = (cosh t ) µ ℓ µ ℓ Γ (cid:0) ℓ + n (cid:1) × F (cid:18) b + ℓ , b − ℓ ; ℓ + n − i sinh t (cid:19) , (6.19)where we have defined b ± ℓ ≡ ℓ + n − ± λ . (6.20)The function F ( a, b ; c ; z ) denotes Gauss’s hypergeometricfunction.The contribution to the conserved charges Q ( c ) A and Q (¯ c ) A comes from the modes with ℓ = 1. For ℓ = 1, themassless limit yieldslim m → P − µ − + λ ( i sinh t ) = (cosh t ) n n Γ( n +22 ) , (6.21)since b − → m → F ( α, γ ; z ) = 1. Moreover,in this limit we also obtain C m ≈ r Γ( n + 2)2 m . (6.22)By substituting Eqs. (6.21) and (6.22) into Eqs. (6.14)and (6.15) we find for ℓ = 1 V (1;1 ,σ ) µ ≈ √ c m ξ ( σ, R) µ , (6.23a) V (2;1 ,σ ) µ ≈ √ c m ξ ( σ, B) µ , (6.23b)where the Killing vectors ξ µ ( σ, R) and ξ µ ( σ, B) are given byEqs. (2.9) and (2.10), respectively. We have used the dou-bling formula for the Γ-function to arrive at Eq. (6.23).The constant c , whose exact value is not important, isgiven by Eq. (D17).As we have stated before, only the ℓ = 1 modes con-tribute to the conserved FP-ghost charges at tree level.By substituting the mode functions V (1;1 ,σ ) µ and V (2;1 ,σ ) µ given by Eq. (6.23) in Eq. (4.7) at tree level, we find Q ( c )( σ, R) ( t ) ≈ √ c m cosh n +1 t × ddt F (cid:18) b +1 , b − ; n + 22 ; 1 − i sinh t (cid:19) α (1)1 σ + H.c. , (6.24a) Q ( c )( σ, B) ( t ) ≈ − √ c m cosh n +1 t × ddt F (cid:18) b +1 , b − ; n + 22 ; 1 − i sinh t (cid:19) α (2)1 σ + H.c. , (6.24b)where H.c. stands for the Hermitian conjugate of the pre-ceding terms.10he derivative of the hypergeometric function ap-pearing above is evaluated in the small- m limit in Ap-pendix E, and it yields ddt F (cid:18) b +1 , b − ; n + 22 ; 1 − i sinh t (cid:19) ≈ − m cosh n +1 t (cid:18) ic + Z t cosh n +1 τ dτ (cid:19) . (6.25)Hence, substituting this result in Eq. (6.24) yields Q ( c )( R,σ ) ( t ) ≈ − m √ c (cid:18) ic + Z t cosh n +1 τ dτ (cid:19) α (1)1 σ + H.c. , (6.26a) Q ( c )( B,σ ) ( t ) ≈ m √ c (cid:18) ic + Z t cosh n +1 τ dτ (cid:19) α (2)1 σ + H.c. . (6.26b)By combining these equations with Eq. (6.23) and theanti-commutators (6.18), we obtain for both the rotationand boost Killing vectors h | Q ( c ) A ( t )¯ c µ ( x ′ ) | i = 12 (cid:18) − ic − Z t cosh n +1 τ dτ (cid:19) ξ µA ( x ′ ) . (6.27)We similarly have h | c µ ( x ) Q (¯ c ) A ( t ′ ) | i = − ic − Z t ′ cosh n +1 τ dτ ! ξ µA ( x ) . (6.28)These equations imply that h | Q ( c ) A Q (¯ c ) B | i = 0 , (6.29a) h | Q ( c ) A ¯ c µ (+) ( x ) | i = 0 , (6.29b) h | c µ (+) ( x ) Q (¯ c ) A | i = 0 . (6.29c)These relations can be summarized as h | Q ( c ) A ¯ ω | i = h | ωQ (¯ c ) A | i = 0 , (6.30)for any canonical variables except for ¯ ω = ¯ θ A or ω = θ A . VII. HAMILTONIAN PERTURBATIONTHEORY
In the previous section we showed that the small-massregularization of the FP-ghost propagator correspondsto the vacuum state | i being annihilated by the con-served charges Q ( c ) A and Q (¯ c ) A at tree level in de Sitterspacetime. The analogous result for the bosonic charge Q ( H ) A , i.e. that it annihilates the vacuum state at treelevel in the standard de Sitter-invariant quantization of linearized gravity in the Landau gauge, can be found inAppendix D. In this section we show that these chargesannihilate the interacting vacuum state | Ω i to all ordersin Hamiltonian perturbation theory with | Ω i defined inthis framework. That is, we show that the conditions Q ( X ) A | i = 0 for X = H, c, ¯ c at tree level are inherited inthe interacting theory as Q ( X ) A | Ω i = 0.Let us first elaborate on the meaning of the conditions Q ( X ) A | Ω i = 0 for X = H, c, ¯ c . Since Q ( H ) A = p A are thecanonical momenta conjugate to B A (0) , if Ψ Ω ( B A (0) , · · · ) isthe Schr¨odinger representation of the state | Ω i , the op-erator Q ( H ) A is represented by − i∂/∂B A (0) . Hence, thecondition Q ( H ) A | Ω i = 0 means that the correspondingSchr¨odinger wave function Ψ Ω does not depend on thevariables B A (0) . The charge Q ( H ) A are the generators ofthe translation in the variables B A (0) . Hence, we may in-terpret the conditions Q ( H ) A | Ω i = 0 as the requirementthat the vacuum state | Ω i be invariant under the gaugetransformation B A (0) + constant. These are the naturalconditions because the Hamiltonian is invariant underthese gauge transformation, being independent of B A (0) .Once the conditions Q ( H ) A | Ω i = 0 are imposed, we mayset p A = Q ( H ) A = 0 in the Hamiltonian for the purposeof evaluating the expectation values of operators not in-cluding B A (0) in the vacuum state | Ω i . (We may excludethe variables B A (0) since these are “gauge-dependent vari-ables” breaking the gauge invariance generated by Q ( H ) A .)Then, Eq. (5.14), which shows that the only undiffer-entiated variables θ A is multiplied by Q ( H ) A in the La-grangian, and Eqs. (5.17b) and (5.17c) imply that, aftersetting Q ( H ) A = 0, the charges Q ( c ) A and Q (¯ c ) A are alsoeffectively the canonical momenta conjugate to the vari-ables ¯ θ A and θ A , respectively. Hence, these fermionicconserved charges can also be regarded as generatingthe gauge transformation of adding constant Grassmannnumbers to ¯ θ A and θ A .Since the conditions Q ( X ) A | Ω i = 0 with X = H, c, ¯ c enforces the gauge invariance of the Hamiltonian on thevacuum state, it is natural to expect that these conditionat tree level, Q ( X ) A | i = 0, will lead to the same conditionsafter including the interaction. We propose a definitionof the vacuum state in Hamiltonian perturbation theoryfor which this is indeed the case.The interaction Hamiltonian density in theories withderivative interactions, such as perturbative gravity, isnon-covariant. For this reason Hamiltonian perturbationtheory is not widely used, unlike Lagrangian perturbationtheory in the path-integral framework. The two pertur-bation schemes are equivalent in quantum electrodynam-ics (QED) with charged scalar field [51]. This equivalenceis explained in Appendix F. One can demonstrate theequivalence of the two schemes in a wide class of the-ories with derivative interactions including perturbative11ravity .In Hamiltonian perturbation theory in Minkowskispacetime the expectation value of the time-orderedproduct T ω ( t ) ω ( t ) · · · ω N ( t N ), where ω ( t ), ω ( t ),..., ω N ( t N ) are canonical variables, in the vacuum state | Ω i can be found in the interaction picture as T h Ω | ω ( t ) ω ( t ) · · · ω N ( t N ) | Ω i = 1 Z T h | ω ( I )1 ( t ) ω ( I )2 ( t ) · · · ω ( I ) ( t N ) × exp (cid:18) − i Z ∞−∞ H I ( t ) dt (cid:19) | i , (7.1)where H I ( t ) is the interaction Hamiltonian and ω ( I ) i ( t i ), i = 1 , , . . . , N , are the canonical variables ω i ( t i ) in theinteraction picture. Thus, the operators ω ( I ) i ( t i ) satisfythe free equations. Here, the state | i is the tree-levelvacuum state and Z = T h | exp (cid:18) − i Z ∞−∞ H I ( t ) dt (cid:19) | i . (7.2)The interacting vacuum state | Ω i in de Sitter space-time cannot be defined in the same way as in Minkowskispacetime since the integral in Eq. (7.1) would be di-vergent due to the exponential growth of the space tothe future and past. Instead, we propose to define it sothat the time-ordered N -point functions are the analyticcontinuation of those in the Euclidean theory obtainedby the coordinate transformation (2.12). Thus, for theEuclidean time the path-ordered product in the order ofdecreasing imaginary part of t to the left is defined by Ph Ω | ω ( t ) ω ( t ) · · · ω N ( t N ) | Ω i = 1 Z PE Ph | ω ( I )1 ( t ) ω ( I )2 ( t ) · · · ω ( I ) ( t N ) × exp (cid:18) − Z π H I ( t ) dτ (cid:19) | i , (7.3)where Z PE = Ph | exp (cid:18) − Z π H I ( t ) dτ (cid:19) | i . (7.4)The path-ordering of operators such that the imaginarypart of t decreases to the left corresponds to the orderingsuch that the variable τ increases to the left. The ana-lytic continuation of the N -point functions in Eq. (7.3)to the real-time variables is performed by deforming thetime-path as in the usual Schwinger-Keldysh perturba-tion theory [52, 53] (see, e.g. [54]). This analytic contin-uation appears to be a concrete realization of the vacuumstate proposed by Jacobson [55] in the context of generalspacetimes with bifurcate Killing horizons, which includede Sitter spacetime. A. Higuchi and W. C. C. Lima, in preparation.
The expectation value of the path-ordered product inEq. (7.3) can be expressed as an integral of a productof two-point functions in the interaction picture by usingWick’s theorem as in Lagrangian perturbation theory.Thus, since h | ωQ ( X ) A | i = 0, X = H, c, ¯ c , where ω isany canonical variable, which is not any of the “gauge-dependent variables” B A (0) , θ A or ¯ θ A , at tree level, we have h Ω | Λ Q ( X ) A | Ω i = 0 for any string Λ of canonical variablesnot including B A (0) θ A or ¯ θ A . That is, Q ( X ) A | Ω i = 0. VIII. SUMMARY AND DISCUSSION
In this paper we found that there are conserved chargesassociated with the Killing vectors in perturbative grav-ity in the Landau gauge in spacetimes with compactCauchy surfaces. Our particular interest was perturba-tive quantum gravity in global de Sitter spacetime, wherethe de-Sitter-invariant FP-ghost propagator is IR diver-gent. We propose that the physical states, in particularthe vacuum state, should be annihilated by these charges.Then we showed, assuming a certain definition of thevacuum state, that the use of the regularized de-Sitter-invariant FP-ghost propagator corresponds to requiringthat the vacuum state be annihilated by the conservedcharges mentioned above. (We note that this correspon-dence follows as long as the vacuum state is defined insuch a way that the N -point function in Hamiltonian per-turbation theory is obtained as an integral of a sum ofproducts of the free-field two-point functions.) Since thegraviton propagator in global de Sitter spacetime is IR fi-nite [22] and that our FP-ghost propagator is effectivelyIR finite, we have a perturbation theory for quantumgravity in global de Sitter spacetime which is not plaguedby IR divergences coming from those in the propagators.We also found that the BRST transforms of the charges Q (¯ c ) A are the conserved charges associated with the back-ground spacetime symmetries. (Although the gauge-fixing and FP-ghost terms break the general covariance,the gauge-fixed perturbative gravity action is still invari-ant under the background symmetries.) Hence, a stateannihilated by Q (¯ c ) A must be de Sitter invariant in thecase of de Sitter spacetime. The vacuum state is natu-rally de Sitter invariant, but, since we propose that allphysical states be annihilated by Q (¯ c ) A (and the other con-served charges Q ( c ) A and Q ( H ) A ), we must require also thatthey be de Sitter invariant. This condition is reminiscentof the quantum linearization stability conditions arisingin linearized gravity quantized with the Dirac quantiza-tion method [56–58]. Non-vacuum de Sitter invariantstates have been constructed using “group-averaging” toimplement these conditions in Ref. [59]. We expect thatthe same method can be applied for the physical-stateconditions in this paper.Our definition of the vacuum state involves imaginarytime but it uses the Hamiltonian rather than the La-12rangian. The Euclidean action obtained as the inte-gral of the Einstein-Hilbert Lagrangian over an Euclideansection is not bounded from below (the conformal-modeproblem [60, 61]). This problem is usually dealt with viaa “conformal rotation”, which consists of changing thesign of the kinetic term of the conformal mode. It wouldbe interesting to investigate how this problem manifestsitself in our definition of the vacuum state in Hamiltonianperturbation theory. Finally, it would be interesting toinvestigate whether our proposal for the physical states ingauge theory and perturbative gravity gives any insightinto the discrepancy between the unitary and covariantformulations of these theories in de Sitter spacetime, orin curved spacetime in general [62–66]. ACKNOWLEDGMENTS
This work was supported by the grant no. RPG-2018-400, “Euclidean and in-in formalisms in static space-times with Killing horizons”, from the Leverhulme Trust.The work of JG was supported by a Studentship fromthe Engineering and Physical Sciences Research Council(EPSRC).
Appendix A: Derivation of Eq. (4.8)
In this Appendix we derive the form of the Lagrangiandensity (4.8) convenient for finding the conserved chargesinvolving the anti-ghost field. It is convenient to considerthe Lagrangian density obtained from L FP by interchang-ing the rˆoles of the ghost and anti-ghost fields: L FP ≡ − i ( ∇ µ ¯ c ν + ∇ ν ¯ c µ − kg µν ∇ λ ¯ c λ ) ∇ µ c ν − i £ ¯ c h µν ∇ µ c ν + ikg βγ £ ¯ c h βγ ∇ α c α . (A1)We find L FP − L FP = − i [( ∇ µ ¯ c ν ) c α − ¯ c α ( ∇ µ c ν )] ∇ α H µν − i [ ∇ µ ¯ c ν ∇ ν c α − ∇ ν ¯ c α ∇ µ c ν ] H αµ +2 ik [ ∇ α ¯ c α ∇ µ c ν − ∇ µ ¯ c ν ∇ α c α ] H µν , (A2)where we recall that H µν ≡ h µν − kg µν h . Next, we “in-tegrate by parts” the first term to remove the derivative ∇ α on H µν , and then commute the derivatives ∇ µ and ∇ α . The terms containing the Riemann tensors arisingfrom this procedure cancel out. Thus we obtain L FP = L FP + K µν H µν , (A3)up to a total divergence, where K µν ≡ i ∇ µ [( ∇ α ¯ c ν ) c α − ¯ c α ∇ α c ν ]+ i (1 − k )[ ∇ µ ¯ c ν ∇ α c α − ∇ α ¯ c α ∇ µ c ν ] . (A4)Then, we find Eq. (4.8) by adding L gf = −∇ µ B ν H µν tothe right-hand side of Eq. (A3). Appendix B: BRST Transform of the Charge Q (¯ c ) A In this Appendix we show that the charge δ B Q (¯ c ) A foreach A is proportional to the Noether charge for thespacetime symmetry generated by the Killing vector ξ µA .We write ξ µA simply as ξ µ , dropping the subscript A , inthe rest of this Appendix.First, we find the BRST transform of the conservedcurrent ξ ν T µν which corresponds to the conserved chargedefined by Eq. (4.12). Let us write δ B ( ξ ν T µν ) = J (B ,B ) µ + J (B , ¯ cc ) µ , (B1)where J (B ,B ) µ comes from the BRST transformation of¯ c α whereas J (B , ¯ cc ) µ comes from that of h µν . (Recall that δ B B µ = 0.) The current J (B ,B ) µ is obtained by replacing¯ c α by iB α in ξ ν T µν . It is convenient to write T µν = T µν + £ ¯ c H µν + kT µν h − g µν ∇ α ¯ c β H αβ + (1 − k ) ∇ α ¯ c α H µν , (B2)where we have defined T µν ≡ ∇ µ ¯ c ν + ∇ ν ¯ c µ − kg µν ∇ α ¯ c α . (B3)Thus, J (B ,B ) µ = iξ ν [( ∇ µ B ν + ∇ ν B µ − kg µν ∇ α B α )(1 + kh )+ £ B H µν − g µν ∇ α B β H αβ +(1 − k ) H µν ∇ α B α ] . (B4)To find the part of the current coming from the transfor-mation of h µν we use δ B H µν = S µν , (B5a) δ B h = 2 ∇ α c α + g αβ £ c h αβ . (B5b)We note that there will be an extra minus sign in thetransformation of T µν in Eq. (B2) because H µν and h are to the right of a fermionic variable ¯ c α . Thus, we findthis part of the current as J (B , ¯ cc ) µ = − ξ α (cid:2) ¯ c β ∇ β S µα + ∇ µ ¯ c β S βα + ∇ α ¯ c β S βµ − g µα ∇ β ¯ c γ S βγ + (1 − k ) ∇ β ¯ c β S µα + kT µα (cid:0) ∇ λ c λ + g βγ £ c h βγ (cid:1)(cid:3) . (B6)Next, we construct the conserved current for the space-time symmetry generated by ξ µ . Consider the diffeomor-phism transformation given by δ st h µν = ∇ µ ( αξ ν ) + ∇ ν ( αξ µ ) + £ αξ h µν = ∇ µ ( αξ ν ) + ∇ ν ( αξ µ ) + αξ λ ∇ λ h µν + ∇ µ ( αξ λ ) h λν + ∇ ν ( αξ λ ) h µλ , (B7a) δ st ψ µ = £ αξ ψ µ = α (cid:2) ξ λ ∇ λ ψ µ − ( ∇ λ ξ µ ) ψ λ (cid:3) − ( ∇ λ α ) ξ µ ψ λ , (B7b)13here ψ µ = B µ , c µ or ¯ c µ and where α is a compactly-supported function on the background spacetime. Let usrepresent the transformation (B7) as δ st Φ I = αX I + Y λI ∇ λ α , (B8)where Φ I represents h µν , B µ , c µ or ¯ c µ depending on theindex I . Let L ≡ L FP + L gf . Then δ st L = (cid:20) ∂ L ∂ Φ I − ∇ µ (cid:18) ∂ L ∂ ( ∇ µ Φ I ) (cid:19)(cid:21) ( αX I + Y λI ∇ λ α ) , (B9)where we have dropped a total divergence. In this equa-tion the index I is summed over. Since the transfor-mation (B8) with constant α would give the spacetimesymmetry transformation generated by the Killing vector ξ µ , we have δ st L| α =const. = ∇ µ ( ξ µ L )= ∂ L ∂ Φ I X I + (cid:18) ∂ L ∂ ( ∇ µ Φ I ) (cid:19) ∇ µ X I . (B10)By using this equation in Eq. (B9) we obtain δ st L = (cid:26)(cid:18) ∂ L ∂ ( ∇ µ Φ I ) (cid:19) X I − ξ µ L + (cid:20) ∂ L ∂ Φ I − ∇ λ (cid:18) ∂ L ∂ ( ∇ λ Φ I ) (cid:19)(cid:21) Y µI (cid:27) ∇ µ α , (B11)with a total divergence dropped.Now, let L GR = √− g L HE be the standard Hilbert-Einstein Lagrangian density for gravity. Then, since L EH + L is the total Lagrangian density, δ st L EH + δ st L must be a total divergence if the field equations are sat-isfied. But δ st L EH under the transformation (B7a) is atotal divergence (even if the field equations are not sat-isfied) since the corresponding Einstein-Hilbert action isdiffeomorphism invariant. Hence, δ st L must be a total di-vergence if the field equations are satisfied. This impliesthat the expression inside the curly brackets in Eq. (B11)must be divergence-free if the field equations are satisfied.Hence we identify the spacetime-symmetry current as J µ st = (cid:18) ∂ L ∂ ( ∇ µ Φ I ) (cid:19) X I − ξ µ L + (cid:20) ∂ L ∂ Φ I − ∇ λ (cid:18) ∂ L ∂ ( ∇ λ Φ I ) (cid:19)(cid:21) Y µI . (B12)Note that the term proportional to Y µI is absent forΦ I = B µ , c µ or ¯ c µ because it is proportional to the fieldequation for Φ I in these cases. This is not the case forΦ I = h µν , however, because its field equation comes from L EH + L , not just from L .Let us find the part of the current J µ st given byEq. (B12) coming from L gf . Since L gf depends on ∇ µ B ν and h µν but not on B µ or ∇ α h µν , this part of the current is J ( B ) µ st = ∂ L gf ∂h αβ h δ µα ξ β + δ µβ ξ α + δ µα ξ λ h λβ + δ µβ ξ λ h λα i + ∂ L gf ∂ ( ∇ µ B ν ) (cid:2) ξ λ ∇ λ B ν − ( ∇ λ ξ ν ) B λ (cid:3) + ξ µ ∇ α B β H αβ = − ( ξ α + H αλ ξ λ + khξ α )( ∇ µ B α + ∇ α B µ − kg µα ∇ β B β ) − H µν (cid:2) ξ λ ∇ λ B ν − ( ∇ λ ξ ν ) B λ (cid:3) + ξ µ ∇ α B β H αβ . (B13)Then we find J ( B ) µ st = iJ (B ,B ) µ + ∇ ν F (1) µν , (B14)where J (B ,B ) µ is given by Eq. (B4) and F (1) αµ ≡ B µ ξ β H αβ − B α ξ β H µβ , (B15)which is an anti-symmetric tensor.Next, the part of the current J µ st coming from the vari-ation of L FP with respect to ¯ c ν plus the term − ξ µ L FP reads J (¯ cc, ¯ c ) µ st = − i ( ξ α ∇ α ¯ c β S µβ − ∇ α ξ β ¯ c α S µβ − ξ µ ∇ β ¯ c γ S βγ ) , (B16)where S µν is defined by Eq. (4.3). Define F (2) µν ≡ iξ α (¯ c µ S ν α − ¯ c ν S µα ) , (B17)which is an anti-symmetric tensor. Then, we find, usingthe field equation ∇ α S αβ = 0 and the anti-symmetry ofthe tensor ∇ α ξ β , J (¯ cc, ¯ c ) µ st = − i (cid:2) ξ α ∇ α ¯ c β S µβ − ξ µ ∇ β ¯ c γ S βγ + ξ β ∇ α (¯ c α S µβ ) − ξ β ∇ α ¯ c µ S αβ (cid:3) , + ∇ ν F (2) µν . (B18)The part of J µ st coming from the variation of L FP withrespect to c α reads˜ J (¯ cc,c ) st µ = − iT µν ( ξ α ∇ α c ν − ∇ α ξ ν c α ) − iT µβ h βν ( ξ α ∇ α c ν − ∇ α ξ ν c α ) , (B19)where T µν is given by Eq. (B3). The contribution fromvarying h µν in T µν c α ∇ α h µν is˜ J (¯ cc,h µ st = i ∇ α ( T µν c α )( ξ ν + ξ β h βν ) − i T βγ c µ ξ α ∇ α h βγ − iT αβ c µ ( ∇ α ξ λ ) h λβ . (B20)Then we find J (¯ cc,c ) µ st + J (¯ cc,h µ st = − iT µν ξ α ∇ α c ν − iT µβ h βν ξ α ∇ α c ν − i T βγ c µ ξ α ∇ α h βγ − iξ ν T µβ c α ∇ α h βν + iξ ν ∇ α (cid:2) ( T αν + T αβ h βν ) c µ (cid:3) + ∇ ν F (3) µν , (B21)14here the anti-symmetric tensor F (3) µν is given by F (3) µν ≡ iξ λ h ( T µλ + T µβ h βλ ) c ν − ( T νλ + T νβ h βλ ) c µ i . (B22)Next, we note that the field equation obtained by varyingthe action with the Lagrangian density L = L FP + L gf with respect to c α can be written as ∇ β (cid:2) T βα + T βν h αν (cid:3) = 12 T νβ ∇ α h βν . (B23)(This equation can also be derived from ∇ ν T µν = 0 and ∇ ν H µν = 0.) By using this equation in Eq. (B21) wefind J (¯ cc,c ) µ st + J (¯ cc,h µ st = − iT µν ξ α ∇ α c ν − iT µβ h βν ξ α ∇ α c ν − iξ ν T µβ c α ∇ α h βν + iξ ν ( T αν + T αβ h βν ) ∇ α c µ + ∇ ν F (3) µν . (B24)Finally, the part coming from the variation of h µν in theterm − iT µν ∇ ν c α h αν in L FP is J (¯ cc,h µ = − iξ α ( T βα ∇ β c µ + T βµ ∇ β c α + T βγ ∇ β c µ h αγ + T γµ ∇ γ c β h αβ ) . (B25)Then J (¯ cc,ch ) µ st ≡ J (¯ cc,c ) µ st + J (¯ cc,h µ st + J (¯ cc,h µ st = − iξ α T µβ ( ∇ α c β + ∇ β c α + £ c h αβ )+ ∇ ν ( F (2) µν + F (3) µν ) . (B26)The Noether current for the spacetime symmetries gen-erated by the Killing vector ξ µ is J µ st = J ( B ) µ st + J (¯ cc, ¯ c ) µ st + J (¯ cc,ch ) µ st , (B27)where the currents on the right-hand side are given byEqs. (B13), (B18) and (B26). By a straightforward cal-culation we can show that J (¯ cc, ¯ c ) µ st + J (¯ cc,ch ) µ st = iJ (B , ¯ cc ) µ + ∇ ν ( F (2) µν + F (3) µν ) . (B28)This equation and Eq. (B14), together with Eq. (B1) and(B27), imply that J µ st = iδ B ( ξ ν T µν )+ ∇ ν ( F (1) µν + F (2) µν + F (3) µν ) . (B29)Thus, by defining the Noether charges for the spacetimesymmetries generated by the Killing vector ξ µA by Q (st) A ≡ Z Σ d Σ n µ J µ st , (B30)with ξ µ = ξ µA , we indeed have Q (st) A = iδ B Q (¯ c ) A , by thegeneralized Stokes theorem. Appendix C: Derivation of Eq. (5.12)
What we need to show is that the non-linear terms in L FP given by Eq. (3.2b) equals the non-linear terms inEq. (5.12) with ˇ B µ = ˇ B µ | B α =0 = − i ( ∇ α ¯ c µ ) c α , whereˇ B µ is defined by Eq. (5.11). That is, we need to showthat the part involving h µν in L FP plus ∇ µ ˇ B ν | B α =0 H µν equals the terms involving h µν in Eq. (5.12). The formerreads L ( h )FP = − i ∇ µ ¯ c ν [ c α ∇ α H µν + ∇ µ c α H αν + ∇ ν c α H αµ + k ( ∇ µ c ν + ∇ ν c µ − kg µν ∇ α c α ) h ] − i ∇ µ [( ∇ α ¯ c ν ) c α ] H µν . (C1)The first term contains the time derivative of H µν . Itcan be combined with the last term as − i ( ∇ µ ¯ c ν ) c α ∇ α H µν − i ∇ µ [( ∇ α ¯ c ν ) c α ] H µν = − i ∇ α [( ∇ µ ¯ c ν ) c α H µν ] + i ∇ α [( ∇ µ ¯ c ν ) c α ] H µν − i (cid:8)(cid:2) ( ∇ α ¯ c ν ) ∇ µ c α + ( ∇ α ∇ µ ¯ c ν ) c α − R νβαµ ¯ c β c α (cid:3)(cid:9) H µν . (C2)By substituting this formula into Eq. (C1) we find af-ter some simplification that L ( h )FP is equal to the termsinvolving h µν in Eq. (5.12) up to a total divergence. Appendix D: The Bosonic Condition on the VacuumState at Tree Level
The solutions to the field equations for the gravitons atlinearized level have been studied in de Sitter spacetimein global coordinates in Ref. [22]. The graviton field witha small mass term is expressed as h µν = h ( T ) µν + h ( V ) µν + h ( S ) µν .The tensor sector h ( T ) µν contains the mode functions com-posed of the tensor, vector or scalar spherical harmonicswith angular momentum ℓ ≥
2. This implies, by orthog-onality of spherical harmonics with respect to the spaceintegral, that h ( T ) µν does not contribute to the conservedcharge Q ( H ) A . The reason for this is that Q ( H ) A is de-fined by a space integral of the product of h µν and theKilling vector ξ µA composed of the scalar or vector spher-ical harmonic with ℓ = 1. (Although linearized gravitywas studied only in 4 dimensions in Ref. [22] these factshold in n dimensions as well.)The conserved bosonic charge Q ( H ) A has no contribu-tion from the scalar sector, either. To show this, we firstexpress the scalar sector h ( S ) µν as [67] h ( S ) µν = ∇ µ ∇ ν Φ + g µν Ψ , (D1)where (cid:3) Φ = ( n − β Φ − (cid:20) n + ( n − β − n − αβ (cid:21) Ψ , (D2a) (cid:3) Ψ = ( n − β Ψ , (D2b)15n the covariant gauge given by the gauge-fixingterm (3.8). We will show that the scalar contributionto the charge Q ( H ) A , Q ( H,S ) A ≡ Z Σ d Σ n µ ξ Aν (cid:16) h ( S ) µν − kg µν h ( S ) (cid:17) , (D3)vanishes in the Landau-gauge limit α → h ( S ) µν − kg µν h ( S ) = ∇ µ ∇ ν Φ − g µν ( (cid:3) + n − g µν n − α Ψ . (D4)Then, by using (cid:3) ξ µA = − ( n − ξ µA , which readily followsfrom the identities ∇ ρ ∇ µ ξ νA = R νµρσ ξ σA and R νµρσ = g νρ g µσ − g νσ g µρ , we find ξ Aν ( h ( S ) µν − kg µν h ( S ) )= ξ νA ∇ ν ∇ µ Φ − ξ µA ∇ ν ∇ ν Φ + ( ∇ ν ∇ ν ξ µA )Φ + n − αξ µA Ψ= ∇ ν [ ξ νA ∇ µ Φ − ξ µA ∇ ν Φ + ( ∇ ν ξ µA )Φ] + n − αξ µA Ψ . . (D5)Substituting this formula into Eq. (D3) yields Q ( H,S ) A = n − α Z Σ d Σ n µ ξ µA Ψ (D6)after using the generalized Stokes theorem. Thus, Q ( H,S ) A → α →
0. [Thescalar field Ψ, which satisfies the massive Klein-Gordonequation (D2b), has a finite limit as α → Q ( H ) A at tree level.The vector sector of the graviton field can be expressedas h ( V ) µν = ∇ µ V ν + ∇ ν V µ , (D7)where ∇ ρ V ρ = 0. Since the linearized Einstein-Hilbertaction is invariant under linearized gauge transforma-tions, h µν → h µν + ∇ µ Λ ν + ∇ ν Λ µ , the field equationfor the vector sector of the linearized gravity comes onlyfrom the gauge-fixing term (3.8). This equation reads ∇ µ ∇ ρ h ( V ) νρ + ∇ ν ∇ ρ h ( V ) µρ = 0 , (D8)because h ( V ) αα = 2 ∇ α V α = 0. Notice that this equa-tion is independent of the gauge parameter α . It can bewritten as ∇ ρ h ( V ) νρ ∝ ξ ν , (D9)where ξ ν is a Killing vector.The part of the field h ( V ) µν relevant to the charge Q ( H )( σ, R) corresponding to the rotation Killing vector ξ µ ( σ, R) , whichwill be given as h ( σ, R) µν = ∇ µ W ( σ, R) ν + ∇ ν W ( σ, R) µ , (D10) can be obtained by postulating W ( σ, R) µ = F ( σ, R) ( t ) ξ µ ( σ, R) , (D11)and solving Eq. (D9). Here, F ( σ, R) ( t ) is a time-dependentoperator. We find F ( σ, R) ( t ) = αa ( σ, R) f ( t ) + b ( σ, R) f ( t ) , (D12)where a ( σ, R) and b ( σ, R) are constant Hermitian operators,˙ f ( t ) = 1cosh n +1 t Z t cosh n +1 τ dτ , (D13a)˙ f ( t ) = 1cosh n +1 t , (D13b)with ˙ f i ( t ) = df i ( t ) /dt , i = 1 , Recalling that B µ = − α − ∇ ν H µν , the α → B ( σ, R) µ = − lim α → α ∇ ν h ( σ, R) µν = a ( σ, R) ξ µ ( σ, R) , (D14a) Q ( H )( σ, R) = lim α → Z Σ d Σ n µ ξ ν h ( σ, R) µν = b ( σ, R) . (D14b)Thus, from Eq. (5.6) we find a ( σ, R) = B ( σ, R)(0) , whileEq. (5.17a) leads to b ( σ, R) = p ( σ, R) , the canonical mo-mentum conjugate to B ( σ, R)(0) . That is,[ a ( σ, R) , b ( σ ′ , R) ] = iδ σσ ′ . (D15)Now, the de Sitter-invariant Bunch-Davies vacuumstate | i is annihilated by the operator A ( σ, R) , i.e. wehave A ( σ, R) | i = 0, where A ( σ, R) is a linear combinationof a ( σ, R) and b ( σ, R) . The operator F ( σ, R) ( t ) in Eq. (D12)is then expressed as F ( σ, R) ( t ) = [ f ( t ) + ic f ( t )] A ( σ, R) +[ f ( t ) − ic f ( t )] A † ( σ, R) . (D16)That is, the function f ( t ) + ic f ( t ) corresponds to thepositive-frequency mode for the Bunch-Davies vacuumstate | i . The constant c can be found as in Ref. [22],and it reads c = √ π Γ( n +22 )2Γ( n +32 ) . (D17)The exact value of this constant is not important; whatmatters is that Eq. (D16) is independent of α . The α -dependence of Eq. (D12) was chosen so that the com-mutator (D15) found from the symplectic product between themodes f ( t ) ξ µ ( σ, R) and f ( t ) ξ µ ( σ, R) is α -independent. The opera-tor F ( σ, R) ( t ) is defined only up to addition of a constant opera-tor. Note that addition of a constant to F ( σ, R) ( t ) does not alterthe field h ( σ, R) µν . B ( σ, R)(0) = 1 α (cid:16) A ( σ, R) + A † ( σ, R) (cid:17) , (D18a) Q ( H )( σ, R) = ic (cid:16) A ( σ, R) − A † ( σ, R) (cid:17) , (D18b)and h A ( σ, R) , A † ( σ, R) i = − α c . (D19)Thus, we find h | Q ( H )( σ, R) Q ( H )( σ ′ , R) | i = − αc → α → . (D20)That is, h | ωQ ( H )( σ, R) | i = 0 for all canonical variables ω except for ω = B ( σ, R)(0) .We now turn to the conserved charge Q ( H )( σ, B) associatedwith the boost Killing vector ξ µ ( σ, B) , with ξ σ, B) = Y (1 σ ) and ξ i ( σ, B) = tanh t D i Y (1 σ ) . The relevant part of h ( V ) µν tothis charge is denoted by h ( σ, B) µν and reads h ( σ, B) µν = ∇ µ W ( σ, B) ν + ∇ ν W ( σ, B) µ , (D21)where W ( σ, B)0 = F ( σ, B) ( t ) Y (1 σ ) , (D22a) W ( σ, B) i = − cosh tn − h ˙ F ( σ, B) ( t )+( n −
1) tanh tF ( σ, B) ( t ) i D i Y (1 σ ) . (D22b)The operator F ( σ, B) ( t ) is again given as F ( σ, B) ( t ) = αa ( σ, B) f ( t ) + b ( σ, B) f ( t ) , (D23)where f ( t ) and f ( t ) are given by Eqs. (D13a) and(D13b), respectively, and a ( σ, B) and b ( σ, B) are constantoperators.Following the same procedure as in the case for therotation Killing vector, we find a ( σ, B) = B ( σ, B)(0) and b ( σ, B) = − Q ( H )( σ, B) / F ( σ, B) ( t ) is expressed in terms of the annihi-lation and creation operators, A ( σ, B) and A † ( σ, B) , with A ( σ, B) | i = 0, in exactly the same way as in Eq. (D16).Hence, we find B ( σ, B)(0) = 1 α (cid:16) A ( σ, B) + A † ( σ, B) (cid:17) , (D24a) Q ( H )( σ, B) = − ic (cid:16) A ( σ, B) − A † ( σ, B) (cid:17) , (D24b)(D24c)and h A ( σ, B) , A † ( σ, B) i = αc . (D25)Then, we again have h | Q ( H )( σ, B) Q ( H )( σ ′ , B) | i = 0 in theLandau-gauge limit. Thus, h | ωQ ( H )( σ, B) | i = 0 for allcanonical variable ω except for ω = B ( σ, B)(0) . Appendix E: Some Details of the Calculations forthe Ghost Fields in Sec. VI
In this section we provide some details of the calcu-lations in Sec. VI, where it is shown that the linearizedghost charges Q ( c ) A and Q (¯ c ) A annihilate the tree-level vac-uum state. We show first that the scalar sectors of theFP ghosts do not contribute to the tree-level charges.The scalar sector of the field c µ is given, at tree level,by ∇ µ Φ, where the field Φ satisfies Eq. (6.13a). Thenthe contribution of this sector to the charge is Q ( c,S ) A = 2 Z Σ d Σ n µ ξ νA (cid:20) ∇ µ ∇ ν Φ − (cid:18) β (cid:19) g µν (cid:3) Φ (cid:21) = 2 Z Σ d Σ n µ ∇ ν [ ξ νA ∇ µ Φ − ξ µA ∇ ν Φ + ( ∇ ν ξ µA )Φ]+ m Z Σ d Σ n µ ξ µA Φ= m Z Σ d Σ n µ ξ µA Φ , (E1)where we have used Eq. (6.13a), the equation (cid:3) ξ µA = − ( n − ξ µA and the generalized Stokes theorem. Hencethe contribution of the scalar sector to the conserved FP-ghost charge at tree level vanishes in the limit m → d/dz ) F ( a, b ; c ; z ) = ( ab/c ) F ( a + 1 , b + 1; c + 1; z ), that ddt F (cid:18) b +1 , b − ; n + 22 ; 1 − i sinh t (cid:19) = − i b +1 b − n + 2 cosh tF (cid:18) b +1 + 1 , b − + 1; n + 42 ; 1 − i sinh t (cid:19) ≈ − m i cosh tn + 2 F (cid:18) n + 2 , n + 42 ; 1 − i sinh t (cid:19) , (E2)for small m . Then the formula [50] F (cid:18) a, b ; a + b + 12 ; 1 − √ z (cid:19) = AF (cid:18) a, b ; 12 ; z (cid:19) + B √ z F (cid:18) a + 12 , b + 12 ; 32 ; z (cid:19) , (E3)with the constants A = √ π Γ (cid:0) a + b + (cid:1) Γ (cid:0) a + (cid:1) Γ (cid:0) b + (cid:1) , (E4a) B = − √ π Γ (cid:0) a + b + (cid:1) Γ( a )Γ( b ) , (E4b)17llows us to write ddt F (cid:18) b +1 , b − ; n + 22 ; 1 − i sinh t (cid:19) ≈ − ic m cosh tF (cid:18) n + 22 ,
12 ; 12 ; − sinh t (cid:19) − m cosh t sinh tF (cid:18) n + 32 ,
1; 32 ; − sinh t (cid:19) = − m cosh n +1 t (cid:18) ic + Z t cosh n +1 τ dτ (cid:19) , (E5)where the constant c is given by Eq. (D17). We haveused F ( a, b ; c ; z ) = (1 − z ) c − a − b F ( c − a, c − b ; c ; z ) , (E6)to find F (cid:18) n + 32 ,
1; 32 ; − sinh t (cid:19) = 1sinh t (cosh t ) n +2 Z t (cosh τ ) n +1 dτ . (E7) Appendix F: Equivalence of Hamiltonian andLagrangian Perturbation Theories for Scalar QED
The Lagrangian density for QED with charged scalarfield φ in Minkowski spacetime is L = − ( ∂ µ φ † + ieA µ φ † )( ∂ µ φ − ieA µ φ ) − m φ † φ − F µν F µν − α ( ∂ µ A µ ) , (F1)where A µ is the gauge potential and F µν = ∂ µ A ν − ∂ ν A µ .The interaction Lagrangian density consists of the non-quadratic terms in the Lagrangian density: L I = ieA µ ( φ∂ µ φ † − φ † ∂ µ φ ) − e A µ A µ φ † φ . (F2)The canonical momentum density conjugate to φ † is π = ˙ φ − ieA φ , (F3)and the canonical momentum density conjugate to φ is π † . The interaction Hamiltonian density, i.e. the non-quadratic part of the Hamiltonian density is, H I = − ieA ( φπ † − φ † π ) − ieA i ( φ∂ i φ † − φ † ∂ i φ )+ e A i A i φ † φ . (F4) In both Lagrangian and Hamiltonian perturbation theo-ries in the interaction picture, the field operators satisfythe free-field equations. Thus, we have π = ˙ φ . Thisallows a direct comparison between the interaction La-grangian and Hamiltonian densities as L I = −H I − e A A φ † φ . (F5)Thus, L I = −H I .The difference between L I and −H I is accounted forby the fact that in Lagrangian perturbation theory thetime derivatives are applied to the propagator as follows[with x = ( t, x ) and x ′ = ( t ′ , x ′ )]: ∂ t ∂ t ′ T h | φ ( x ) φ † ( x ′ ) | i = T h | ˙ φ ( x ) ˙ φ † ( x ′ ) | i + δ ( t − t ′ ) h | (cid:2) φ ( x ) , φ † ( x ′ ) (cid:3) | i = h | π ( x ) π † ( x ′ ) | i + iδ (4) ( x − x ′ ) . (F6)Thus, in Lagrangian perturbation theory there is an ex-tra interaction term − e A A φ † φ and the field π = ˙ φ isreplaced by π + γ , where γ has the propagator T h | γ ( x ) γ † ( x ′ ) | i = iδ (4) ( x − x ′ ) . (F7)Integrating out the fictitious field γ ( x ) generates the fol-lowing effective interaction term:∆ L I = − i Z d x ′ eA ( x ) φ † ( x ) T h | γ ( x ) γ † ( x ′ ) | i× (cid:2) − eφ ( x ′ ) A ( x ′ ) (cid:3) = e A ( x ) A ( x ) φ † ( x ) φ ( x ) . 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