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MASSLESS MINIMALLY COUPLED FIELDS IN DE SITTER SPACE:

arXiv:gr-qc/9305013v1 16 May 1993MASSLESS MINIMALLY COUPLED FIELDS IN DE SITTER SPACE:O(4)-SYMMETRIC STATES VERSUS DE SITTER INVARIANT VACUUMKlaus Kirsten∗, 1 and Jaume Garriga∗∗∗Department of Physics, Universit¨at Kaiserslautern,Erwin Schr¨odinger Straβe, Postfach 3049, D-6750 Kaiserslautern, Germany∗∗Tufts Institute of Cosmology,Department of Physics and Astronomy, Tufts University, Medford, MA 02155AbstractThe issue of de Sitter invariance for a massless minimally coupledscalar field is revisited. Formally, it is possible to construct a de Sitterinvariant state for this case provided that the zero mode of the field isquantized properly.

Here we take the point of view that this state is phys-ically acceptable, in the sense that physical observables can be computedand have a reasonable interpretation. In particular, we use this vacuum toderive a new result: that the squared difference between the field at twopoints along a geodesic observer’s space-time path grows linearly with theobserver’s proper time for a quantum state that does not break de Sitterinvariance.

Also, we use the Hadamard formalism to compute the renor-malized expectation value of the energy momentum tensor, both in theO(4) invariant states introduced by Allen and Follaci, and in the de Sitterinvariant vacuum. We find that the vacuum energy density in the O(4)invariant case is larger than in the de Sitter invariant case.1Present address: Universit`a degli Studi di Trento, Dipartimento di Fisica, 38050 Povo(Trento), Italy

1IntroductionQuantum field theory in curved spacetimes has been extensively studied duringthe past two decades or so (see e.g. ref.

[1] for a review) with the purpose ofunderstanding quantum effects in the presence of strong gravitational fields. Inparticular, a lot of attention has been devoted to de Sitter space, mainly becauseit has a high degree of symmetry and the wave equation can be exactly solved inthis background.

A 4-dimensional de Sitter space can be conveniently definedas a hyperboloid embedded in a 5-dimensional Minkowski space:ξµ(x)ξµ(x) = H−2,(1)where ξµ(x) denotes the position vector of the point x in the embedding space(µ = 0, ..., 4).The manifest invariance of equation (1) under 5-dimensionalLorentz transformations implies that de Sitter space has a 10 parameter groupof isometries, the de Sitter group O(4,1).Scalar fields of mass m with an arbitrary coupling ξ to the Ricci curvaturescalar [see eq. (4) below] can be easily quantized in de Sitter space, and quantumstates that respect the O(4,1) invariance of the background can be constructedfor these fields[2].

Physical quantities such as the two point function and therenormalized expectation value of the energy momentum tensor in the de Sitterinvariant states were computed exactly in early work [3].Later, interest in this subject was motivated by the inflationary cosmologyscenario [4] (since the geometry of spacetime during inflation is that of de Sitterspace). In this context, it was realized that the mean squared fluctuations ofa massless minimally coupled field (i.e.

m = ξ = 0) grow linearly with timeduring inflation [5],< φ2 >≈H34π2 t.(2)Note that this expression is not de Sitter invariant, essentially because the quan-tum state that was used in its derivation breaks the O(4,1) invariance explicitly.The massless minimally coupled case is peculiar in that the de Sitter invari-ant two point function becomes infrared divergent in the limit m →0, ξ →0.This led some authors [6, 7, 8] to the definition of various other ‘vacua’ with lesssymmetry than the full de Sitter group, but with a two point function whichis free from infrared divergences. In particular, here we will consider the twoparameter family of O(4)-invariant Fock ‘vacua’ introduced by Allen and Fo-lacci [7].

Note that the ξ0 = const. spatial sections of (1) are 3-spheres.

TheO(4) vacua are not invariant under all de Sitter transformations, but only underspatial rotations of these 3-spheres.In this paper we would like to reconsider the possibility of constructing a deSitter invariant state for the massless minimally coupled field. (This state is nota Fock vacuum state.) That such state can be formally constructed was alreadyimplicit in refs.

[9, 10], where the quantization was studied in the functional2

Schr¨odinger picture. Our emphasis here will be in the physical interpretation.For m = ξ = 0 the action [eq.

(4) below] has a zero mode: it is invariant underconstant shifts of the fieldφ →φ + const.The two point function is ill defined because all values of the spatially constantpart of the field are equally probable in the de Sitter invariant state (whichis analogous to an eigenstate of momentum in the quantum mechanics of afree particle). However, such ambiguity does not prevent us from computingthe expectation value of physical observables.

To illustrate this point here weshall use this vacuum to derive a more powerful result than the one given inequation (2). Namely that one may have a de Sitter invariant state |0 >, and inthis state, any freely-falling observer who picks a basepoint x in spacetime willsee < 0|(φ(x) −φ(y))2|0 > increasing with proper time along their path.

Weshall also compute the renormalized expectation value of the energy momentumtensor, both in the one-parameter family of O(4) invariant states and in the deSitter invariant vacuum. As we shall see, the vacuum energy density in the deSitter invariant case is lower than in the O(4)-symmetric case.The rest of the paper is organized as follows.

In Section 2 we briefly reviewthe quantization of a scalar field in de Sitter space, with the purpose of fixingthe notation. In Section 3 we compute the energy momentum tensor in thetwo parameter family of O(4) invariant vacua.

Section 4 discusses the de Sitterinvariant vacuum for the massless minimally coupled field. In Section 5 we usethis vacuum for the calculation of some observables.

Finally, a discussion ofthe results is given in Section 6. The quantization of the scalar field in thefunctional Schr¨odinger picture is summarized in the Appendix.2Scalar field in de Sitter spaceIn this section we summarize the quantum theory of a scalar field of mass mand arbitrary coupling to the scalar curvature in de Sitter space, which wasdeveloped in Refs.

[2, 11, 3, 6, 9].The line element in de Sitter space readsds2 = gabdxadxb = H−2 sin−2 η[−dη2 + dΩ2],(3)where we are using the closed coordinate system xa = (η, Ω), (a = 0, ..., 3) thatcovers the whole hyperboloid (1). Here η ∈(0, π) is the so-called conformaltime, Ωis a set of angles on the 3-sphere and dΩ2 denotes the line element onthe unit 3-sphere.The action for the scalar field is given byS = −12Z √−g[∂aφ∂aφ + (m2 + ξR)φ2]d4x,(4)3

where g is the determinant of the metric, R = 12H2 is the Ricci scalar and ξ isan arbitrary coupling. It is convenient to expand the field asφ =XLMχLM(η)YLM(Ω),(5)where YLM are the usual spherical harmonics on the 3-sphere, normalized asZYLM(Ω)Y ∗L′M′(Ω)dΩ= δLL′δMM′.

(6)They are eigenfunctions of the Laplacian on the 3-sphere∆(3)YLM = −JYLM,(7)with J = L(L + 2), L = 0, ..., ∞. The index M, M = 0, ..., (L + 1)2, labels thedegeneracy for given L.Introducing (5) in (4) one findsS = 12XLMZ(H sin η)−2[( ˙χLM)2 −ω2L(η)χ2LM]dη,(8)whereω2L(η) ≡J + m2 + ξR(H sin η)2 ,and the overdot indicates derivative with respect to η.

In going from (4) to (8)the term ∂iYLM∂jYLM has been integrated by parts and the relations (6) and(7) have been used. Equation (8) can be seen as the action for a collection ofharmonic oscillators with time dependent frequencies.

The classical equationsof motion for the modes χLM(η) read,¨χLM −2 cot η ˙χLM + ω2L(η)χLM = 0. (9)To quantize the theory, the field variables χLM and their canonically conju-gate momentaπLM ≡∂L∂˙χLM= (H sin η)−2 ˙χLM,(10)are promoted to operators satisfying the canonical commutation relations[ˆχLM, ˆπL′M′] = iδLL′δMM′.

(11)In the Heisenberg picture, these are time dependent operators, and it is cus-tomary to expand them in terms of (time independent) creation and anihilationoperators aLM and a†LM,ˆχLM = ULMaLM + U ∗LMa†LM(12)4

ˆπLM = (H sin η)−2[ ˙ULMaLM + ˙U ∗LMa†LM].Here ULM(η) are solutions of the field equation (9) (with χLM ↔ULM) nor-malized according to the Wronskian conditionULM ˙U ∗LM −U ∗LM ˙ULM = i(H sin η)2. (13)The commutation relations (11) follow from (13) and the usual commutationrelations for the creation and anihilation operators[aLM, a†L′M′] = δLL′δMM′,[aLM, aL′M′] = [a†LM, a†L′M′] = 0.A “vacuum” state |0 > can be defined byaLM|0 >= 0,∀L, M,(14)and the complete Hilbert space of states can be generated by repeated operationon |0 > of the creation operators a†LM.

As it is usual in curved space (see e.g. [1]), the definition of this vacuum is somewhat arbitrary, since it depends onwhat particular choice we make for the set of modes {ULM}.

However, de Sitterspace is a maximally symmetric space, invariant under a 10 parameter groupof isometries [the de Sitter group O(4,1)], and it is natural to choose a vacuumstate which also has the same symmetry. Actually, there exists a one-parameterfamily of de Sitter invariant quantum states.

Among them, we shall concentrateon the so-called Euclidean vacuum as the only one whose two point function hasHadamard form and so the ultraviolet behavior is the same as for field theoryin flat spacetime. The mode functions corresponding to the Euclidean vacuumare given by [2],ULM = AL(sin η)3/2[P λν (−cos η) −2iπ Qλν(−cos η)],(15)where P λν and Qλν are Legendre functions on the cut, andλ =94 −m2 + ξRH21/2,ν = L + 12.

(16)The normalization constants are given byAL =√π2 Heiλπ/2Γ(L −λ + 3/2)Γ(L + λ + 3/2)1/2. (17)The de Sitter invariance of this state is manifest in the symmetric two pointfunctionG(1)(x, x′) =< 0|φ(x)φ(x′) + φ(x′)φ(x)|0 >=XLM[ULM(η)U ∗LM(η′)YLM(Ω)Y ∗LM(Ω′) + ULM(η′)U ∗LM(η)YLM(Ω′)Y ∗LM(Ω)],(18)5

which can be evaluated to yield [2]G(1)(Z) = 2H2(4π)2 Γ32 −λΓ32 + λF32 −λ, 32 + λ, 2; 1 + Z2. (19)Here F is the hypergeometric function, and Z is given by [7]Z(x, x′) = H2ξµ(x)ξµ(x′) = cos γ −cosη cos η′sin η sin η′,(20)where γ is the angle between Ωand Ω′.

Note that the two point function dependsonly on Z, which is a Lorentz invariant quantity in the embedding space, andtherefore G(1) is de Sitter invariant.The quantity Z(x, x′) can also be expressed as [7]Z = cosrRσ6 ,(21)where σ(x, x′) is defined as one half of the square of the geodesic distance be-tween x and x′. If x and x′ are time-like separated, then σ < 0 and Z > 1.On the other hand, if they are space-like separated, then Z < 1.

(However ageodesic joining the two points exists only if −1 ≤Z, hence σ is undefined forZ < −1. )3The massless minimally coupled case:O(4)invariant vacuumIt should be noted that the two-point function (19) is ill defined in the masslessminimally coupled case (m = ξ = 0), since one of the gamma functions has apole at λ = 3/2.

This divergence has led some authors [6, 7, 8] to the definitionof other vacua with less symmetry than the full de Sitter group, but with a welldefined two point function.In the closed coordinate system that we are using, one such natural vacuumis the O(4) invariant vacuum [7], which is symmetric under rotations of theη = const. spatial sections (which are 3-spheres).

The set of modes that definesthe O(4) invariant quantum state is given by (15) for L > 0 but, in order toavoid the infrared divergence, the L = 0 mode solution is chosen as [7]U0 = H[A(η −12 sin 2η −π2 ) + B],(22)withA = −iα,α ∈(0, ∞),B = 1α(14 + iβ),β ∈(−∞, ∞).6

The two complex parameters A and B have been reduced to two real parametersα and β because an overall phase is irrelevant and because (13) must be satisfied.In addition, requiring time reversal invariance fixes β = 0 [7], which leaves uswith just one parameter, α. In what follows we take β = 0.The two point function in this state isG(1)α (x, x′) = ˆG(x, x′) +12π2 [U0(η)U ∗0 (η′) + U0(η′)U ∗0 (η)],(23)where ˆG is defined as a sum over modes similar to (18) but without the L = 0term.

This sum is given (up to some irrelevant constant) in closed form by [7]ˆG(x, x′) =R48π211 −Z −log(1 −Z) −log(4 sin η sin η′) −sin2 η −sin2 η′,(24)with Z defined in (20).We will be interested in constructing the energy momentum tensor using theHadamard formalism [13, 16]. For this we need to study the two point functionin the coincidence limit, that is, we have to bring G(1) into the Hadamard form[14, 23]G(1)(x, x′) =14π2∆1/2(x, x′)σ+ V (x, x′) log σ + W(x, x′),(25)where σ(x, x′) was defined in eq.

(21) and ∆(x, x′) is the Van Vleck-Morettedeterminant. In de Sitter space it is given by [14]∆(σ) =Rσ63/2 "sinrRσ6#−3.

(26)In eq. (25), V (x, x′) and W(x, x′) are symmetric functions of x and x′ which aresmooth in the coincidence limit.Using (21) and (26), one can compare expressions (25) and (23-24) to findV (x, x′) = −R12,(27)W(x, x′) = F(σ) −R12[log(4 sin η sin η′) + sin2 η + sin2 η′]+2[U0(η)U ∗0 (η′) + U ∗0 (η)U0(η′)].

(28)HereF(σ) ≡R1211 −cos X −21(X sin3 X)1/2 −log R6X2 [1 −cos X],(29)7

withX =rRσ6 .One can check that W is well behaved at σ = 0 (as expected from the generaltheory) by expanding each term in (29) in powers of σ. We find that the negativepowers of σ cancel out and we haveF(σ) = −R12log R12+ 13 −Rσ480 + O(σ2).

(30)As usual, the singular part in (25) is purely geometrical, and all the dependenceof G(1) on the quantum state is contained in the function W(x, x′).The two point function is now in a form ready for the computation of therenormalized expectation value of the energy momentum tensor.Using theHadamard formalism, this is given by [13, 16]8π2 < Tab >ren= τab[W] −τab[V ] log µ2 + 2v1gab −m416 gab,(31)whereτab[f] ≡limx→x′ Dab′(x, x′)[f(x, x′)].Here D is the differential operator associated with the point splitted expressionof the formal energy-momentum operator. In the massless minimally coupledcaseDab′ ≡∇a∇b′ −12gab′gdd′∇d∇d′,where gb′a is the bivector of parallel transport [15].

In eq. (31), µ2 is a renormal-ization scale (arbitrary, in principle), and v1 is the ‘trace anomaly’ scalar, whichin de Sitter space is equal to [16]v1 = 29R28640 .From (27) it is clear that in our caseτab[V ] = 0,and the dependence on the renormalization scale disappears.

This is fortunate,since in the massles case there is no natural mass parameter in the problem.Also, the last term in (31) vanishes for m = 0.All that we need to evaluate is τab[W], with W given by (28). The termτab[F(σ)] can be easily computed by noticing thatlimx→x′ ∇a∇b′F(σ) = limx→x′[F ′′(σ)σ,aσ,b′ + F ′(σ)σ;ab′] = −F ′(σ)|σ=0gab,(32)8

where a prime indicates derivative with respect to σ and we have used (seee.g. [13])limx→x′ σ,a = 0,limx→x′ σab′ = −gab.The value of F ′(σ) at σ = 0 can be read offfrom (30), and using (32) we haveτab[F] = R25760gab.Also, it is clear thatτab[log(2 sin η) + log(2 sin η′) + sin2 η + sin2 η′] = 0,and one can check thatτab[U0(η)U ∗0 (η′) + U ∗0 (η)U0(η′)] = 136R2α2(1 −2δaη) sin6 η gab.Substituting the previous expressions in (31), we have< α|Tab|α >ren=119R2138240π2gab +R2144π2 α2 sin6 η gab(1 −2δaη).

(33)Therefore, the energy momentum tensor is not de Sitter invariant, but onlyO(4) invariant, because of the explicit time dependence. Notice also that theterm which is not de Sitter invariant decays with the expansion of the Universeas a−6, where a is the scale factor (compare with radiation, which behaves asa−4 or with the vacuum energy itself which behaves as a0) and therefore it isunlikely to have any cosmological consequences.

In the limit η →0 or η →π,which corresponds to cosmological time going to +∞or −∞, eq. (33) reducesto the result (3.6) in Ref.

[7] as corrected in Ref. [24].4De Sitter invariant vacuum for the masslessminimally coupled caseSometimes it is said [6, 8] that the infrared divergence in G(1) indicates that deSitter invariance is broken in the massless minimally coupled case.

However, itis still possible to define a de Sitter invariant vacuum for this case, and here wewill take the point of view that this state is physically acceptable in the sensethat physical quantities can be computed and have a reasonable interpretation.However, as we shall see, the space of states can not be simply represented asa Fock space built by applying creation operators to this vacuum state. Thequantization of φ in the case m = ξ = 0 is peculiar because the field contains azero mode: the action is invariant under the transformationφ →φ + const.

(34)9

It is well known that an expansion in terms of creation and anihilation operators,such as (12), is not adequate for the variables associated to the zero modes[17, 18, 19, 20].The situation is analogous to that of a quantum mechanical harmonic os-cillator: the expansion of the position and momentum operators, x and p, interms of creation and anihilation operators breaks down in the limit when thefrequency of the oscillator, ω, goes to zero (the free particle case). In the Heisen-berg picture we havex(t) = (2Mω)−1/2(ae−iωt + a†e+iωt)p(t) = −i(Mω/2)1/2(ae−iωt −a†e+iωt),where M is the mass of the particle.

Of course, these expressions are not valid inthe limit ω →0. The physical reason is that for a free particle the spectrum ofthe Hamiltonian becomes continuous and the number operator loses its meaning.Instead, we can consider the expansionsx(t) = x0 + p0t(35)p(t) = p0,where the new operators satisfy the commutation relation [x0, p0] = i.

At theclassical level, x0 and p0 have the interpretation of the initial position andmomentum (and are therefore constants) so (35) can be seen as a Hamilton-Jacobi canonical transformation in which the new variables are constants ofmotion.The first equation in (35) is obviously the general solution of theequations of motion if we think of x0 and p0 as constants of integration. In thissense this equation is analogous to (22).The simplest example of a field theory with zero modes is the massless scalarfield in a flat compact space with finite volume V and topology of a torusS1 × S1 × S1, discussed in Ref.

[18]. In that case, a complete set of solutions ofthe wave equation is given byfk = (2V ω)−1/2 exp i(kx −ωt),(k ̸= 0)(36)f0 = At + B.Here ω = |k| and the momenta k have the usual discrete spectrum due to finitevolume.

The Klein-Gordon normalization requires A∗B−B∗A = i/V . While themodes fk (k ̸= 0) are the classical solutions for a harmonic oscillator of frequencyω, the mode f0 is the classical solution for a free particle.

Therefore, although itis formally possible to define creation and anihilation operators associated withf0, in a manner analogous to the construction of the O(4) invariant vacuumof the previous section, it is more natural to define position and momentum10

operators analogous to p0 and x0 above. With this the field expansion reads[18]φ = x0 + p0t√V+Xk̸=0(akfk + h.c.).It can be checked that the equal time commutation relation for φ and its conju-gate momentum are satisfied if [x0, p0] = i and the usual commutation relationsfor the creation and anihilation operators are satisfied.Note that in the limit of infinite volume the special treatment of the zeromode becomes irrelevant, as it makes a contribution of zero measure in theexpansion of the field.

An equivalent statement is that the set of modes withk ̸= 0 becomes complete in the limit of infinite volume. However, for finitevolume the zero mode is important and makes a finite contribution to the energy.Indeed, it is straightforward to see thatE = p202 +Xk|k|a†kak + 12.One can define the ground state for this system through the equations p0|0 >=0, ak|0 >= 0.

This ground state is not normalizable, in the same way thatthe ground state of a quantum mechanical free particle is not (for a detaileddiscussion on this issues, see Ref. [17], Section 9).

The field operator is seen tobe equivalent to a collection of harmonic oscillators plus a free particle [whoseposition, in the Heisenberg picture, would be given by the operator x(t) =x0 +p0t]. The space of states is equivalent to the direct product of a Fock spacecorresponding to the oscillators and an ordinary Hilbert space correspondingto the free particle.

Since the energy is an observable, in addition to the usualFock space operators, the momentum p0 is also an observable.The above construction can be generalized to arbitrary curved backgrounds[17]. Of course, in general, there is the usual caveat that for non-stationarybackgrounds the energy is not conserved and the definition of a ground state isambiguous.

This is nothing new, it is the same problem that we encounteredin Section 2 when discussing the massive field: the definition of a “vacuum” innon-stationary backgrounds is always a matter of choice. Here, as in Section 2,we will be guided by considerations of symmetry in making this choice.In the case of a massless minimally coupled field in De Sitter space, thezero mode associated to (34) is in the homogeneous sector (L = 0), and that isthe reason why the coefficient A0 [see eq.

(17)] becomes infinite for λ →3/2.Instead of defining creation and anihilation operators for L = 0 we replace theexpansions (12) by [7]χ0 = H√2[Q + (η −12 sin 2η −π2 )P](37)π0 =√2H P.11

The coefficients of Q and P in the expansion of χ0 are solutions of the fieldequation (9), and the expression for π0 follows from (10). Moreover, the com-mutation relation between χ0 and π0 implies[Q, P] = i,so, again, (37) can be seen as a Hamilton-Jacobi transformation in which thenew canonical variables are constants of motion.We define a vacuum state byP|0 >= 0,(38)aLM|0 >= 0,L > 0,where aLM were defined in Section 2.The ambiguity in the choice of a vacuum corresponds to the freedom in thechoice of the mode functions ULM for L ̸= 0 [which we take to be the same asfor the O(4) vacuum], plus the freedom in choosing the mode solutions whichappear as coefficients of Q an P in Eq.(37).

In principle we could have chosenany two homogeneous solutions of the wave equation, say f1(η) and f2(η),χ0 = f1 ˜Q + f2 ˜Pπ0 = (H sin η)−2( ˙f1 ˜Q + ˙f2 ˜P),subject to the Wronskian condition ˙f2f1 −˙f1f2 = H2 sin2 η. With the choice(37) the equation P|0 >= 0 implies that the vacuum wave functional Ψ doesnot depend on χ0PΨ = H√2−i ∂∂χ0Ψ = 0.

(39)If we are interested in a de Sitter invariant vacuum, this turns out to be theright choice.In appendix A we review the quantization of the scalar field in the Schr¨odingerpicture. We show that in the limit m →0 and ξ →0, the de Sitter invariantwave functional becomes independent of χ0, and therefore it satisfies P|0 >= 0(the other equations in (38) are also satisfied by construction).

Note that thesolution of (39) is not normalizable, and that is the reason why G(1) is ill definedin the de Sitter invariant state. This should not be taken as an indication thatthe state is pathological: it simply means that all values of χ0 are equaly proba-ble.

The same problem would arise in the quantum mechanics of a free particleif we tried to compute < p|x2|p >, where |p > is an eigenstate of momentum.Apart from considerations about De Sitter invariance (the group of isome-tries of the background spacetime), there is another (aesthetic) reason for choos-ing a state with P|0 >= 0, based on the symmetry of the Lagrangian under12

φ →φ + const. The corresponding Noether current is jµ = ∂µφ.

The generatorof the symmetry is the “charge”ˆQ =Zη=constdΣµjµ,so the vacuum will be invariant under this symmetry if it is anihilated by thecharge, ˆQ|0 >= 0. Introducing jµ = ∂µφ in (4) we find ˆQ = 2πH−1P, so thecondition becomes P|0 >= 0.

Note that even though the current is linear inφ, the charge operator is non-vanishing and well defined precisely because thespace has compact spatial sections.As mentioned before, the vacuum state defined in this way is not simply aFock-space vacuum (in fact, this would be in contradiction to the work of Allen[6]) but the direct product of a Fock space and an ordinary Hilbert space cor-responding to the χ0 variable. As we shall see, in order that the energy densityT00 is a physical observable, in addition to the usual Fock space observables theoperator P is also a physical observable.A basis for the space of states is the direct product of the basis for the Fockspace times the basis for the Hilbert space of a particle in one dimension.

Thestructure of the Fock space corresponding to the modes L > 0 is identical tothe one corresponding to the 0(4) invariant vacuum, and we shall not discussit further. The Hilbert space for one particle in one dimension is isomorphic tothe usual space of square integrable complex functions of a real variable, anda convenient basis is formed by the eigenstates of the momentum operator Pwith eigenvalue p|p >≡eipQ|0 > .Since they form a continuous basis, these states are not normalized in the dis-crete sense, but they have the continuous normalization < p|p′ >= δ(p −p′).

Inthe ‘q’ representation they are the ordinary plane waves< q|p >= (2π)−1/2eipq(40)where |q > are eigenstates of Q with eigenvalue q, normalized as < q|q′ >=δ(q −q′).In this representation Q acts as a multiplicative operator and P as a deriva-tive operator< q|Q|ψ >= q < q|ψ >,< q|P|ψ >= −i ∂∂q < q|ψ > .To make a connection with the previous section, one can see that the L = 0sector of the O(4) invariant vacuum |α > discussed previously corresponds tothe normalized gaussian wave packet [7]< q|α >≡ψα(q) =√2απ1/4 e−2α2q2. (41)13

Indeed, the operator a0 of the O(4) vacuum can be expressed [using (37)] asa0 = i√2[B∗P −A∗Q], which clearly anihilates (41). Also, the “multi-particle”homogeneous (L = 0) excitations above |α > are obtained by repeated operationof a†0 on (41), which gives,ψnα ≡< q|(a†0)n√n!|α >=2απ1/22nn!1/2Hn(2αq)e−2α2q2,(42)where Hn are the Hermite polynomials.Throughout this section we have worked in the Heisenberg picture, andtherefore the states (42) are time independent.

To obtain the correspondingwave functions in the Schr¨odinger picture one can solve the Schr¨odinger equa-tion with initial conditions (42). As we show in Appendix A, this Schr¨odingerequation is just the one for a free particle, so the evolution of (41) is just thatof a minimal wave packet which spreads in time.5Dispersion of the field and energy momentumtensorIn order to gain intuition on the structure of the de Sitter invariant vacuumdefined in (38), let us consider the ‘dispersion’ of the field, defined byD2(x, y) ≡< 0|(φ(x) −φ(y))2|0 >,(43)which will give us an idea on how the value of the field fluctuates over space andtime.

Since D2 contains terms of the form < 0|φ(x)φ(x)|0 >, we will encounterthe usual ultraviolet divergences associated with the product of operators in thecoincidence limit. A convenient way of getting around such divergences is tosmear the field operator over a region of size s (see e.g.

[21, 22])φs(x) ≡1V ol(s)Zd(x,x′)

The ‘diameter’ s of the smearingregion should be less than 2πH−1, since there are no space-like geodesics longerthan that [see comments after eq. (21)], and we shall take s ∼H−1.

Also, inorder to smear the field operator it is necessary to make a particular choice ofspace-like hypersurface at the point x. In what follows we shall always considersituations in which geodesic observers are involved, so the smearing regionscan be defined on the space-like sections orthogonal to these geodesics.

Forinstance, if x and y are time-like separated, we can consider the geodesic curvethat links x with y, and take space-like surfaces at x and y generated by the14

space-like geodesics orthogonal to this curve. Later, we shall also consider thefield measured by two observers moving along two different geodesics.

Eachobserver can smear the field on the space-like surface orthogonal to his or hergeodesic. In any case, in the limit of large separation between x and y, theleading term in the dispersion will not depend on the details of how we smearthe field.Now we can consider the dispersion of the smeared field,D2s(x, y) =< 0|(φs(x) −φs(y))2|0 > .

(44)Notice that this expression has no infrared divergences either, since the operatorQ (which causes trouble in the two point function because its expectation valueis ill defined in the de Sitter invariant vacuum) cancels out when we considerthe difference φ(x) −φ(y).As an intermediate step to compute (44) we ‘point-split’ and symmetrize theexpression (43)D2ǫ(x′, x′′; y′, y′′) ≡12 < 0|{[φ(x′) −φ(y′)], [φ(x′′) −φ(y′′)]}|0 > .Here x′ and x′′ are points within the smearing region surrounding x, separatedby a geodesic distance ǫ (ǫ < s) (similarly for y′ and y′′), and the brackets {, }denote the anticommutator. Since P|0 >= 0, this expression reduces toD2ǫ = 12[ ˆG(x′, x′′) + ˆG(y′, y′′) −ˆG(x′, y′′) −ˆG(y′, x′′)],with ˆG defined in (23-24).

It is convenient to rewrite it asD2ǫ = H28π2 [g(x′, x′′) + g(y′, y′′) −g(x′, y′′) −g(y′, x′′)],(45)whereg(u, v) ≡11 −Z(u, v) −log |1 −Z(u, v)|.Notice that (45) is a fully de Sitter invariant expression (as it should, since weare dealing with a de Sitter invariant state). All the terms in (24) that dependexplicitly on the conformal time η cancel out in the expression for Dǫ.Now we can easily estimate the smeared dispersion D2s(x, y) for the case whenthe separation between x and y is much larger than H−1 (that is |Z| >> 1).This can be done by smearing the expression (45) term by term.

We note thatwhen the two points x′ and x′′ lie within the same smearing region, the integrals1[V ol(s)]2Zd(x,x′)

give contributions of order 1, while if one of the points lies in the neighborhoodof x and the other lies in the neighborhood of y, we have1[V ol(s)]2Zd3x′d3y′′g(x′, y′′) ≈g(x, y) ≈−log |Z(x, y)|,where we have used |Z| >> 1. As a result,D2s(x, y) ≈H24π2 log |Z(x, y)|,(|Z| >> 1).

(46)If x and y are timelike separated, we can use (21) to write< 0|(φs(x) −φs(y))2|0 >≈H34π2 τ,(τ >> H−1)(47)where τ is the proper time measured by a geodesic observer travelling from xto y.Equation (47) embodies a familiar property of massless minimally coupledfields in de Sitter space, namely, that the mean squared fluctuations in the fieldgrow linearly with time [5, 22] [see eq.(2)]. Here we have been able to derive thisresult in an invariant way, without the need of using a quantum state that breaksde Sitter invariance and without the need of introducing a cosmological timecoordinate [τ in eq.

(47) is just the geodesic distance]. As noted by Vilenkin [22],the linear growth in time of the mean squared fluctuation can be interpreted interms of a random walk of the field φ.

The magnitude of φ smeared over theinterior of a Hubble-radius (H−1) two-sphere changes by ±(H/2π) per expansiontime H−1. Then, the average displacement squared is D2s ∼(H/2π)2N, whereN ∼Hτ is the number of steps.

To support this interpretation, Vilenkin studiedfield correlations between points which were at large space-like separations. Wecan repeat his arguments using the de Sitter invariant formalism.Since points separated by space-like distances greater than πH−1 cannotbe connected by geodesics (and we are interested in much larger separations),the discussion will require more work than in the case of time-like separations.Consider, to begin with, an arbitrary point x in de Sitter space and a time-like geodesic Cx passing through it.

We can think of Cx as the trajectory of aninertial observer. Without loss of generality (by using de Sitter transformations)we can take x to have coordinates (η = π/2, Ω), and Cx to be the curve Ω=const., while the metric still takes the form (3).

Let x′ be a second point on thespacelike hypersurface orthogonal to Cx at x, such that the geodesic distancebetween x and x′ is much smaller than H−1; and let Cx′ be a geodesic throughthe point x′ which is initially parallel to Cx.In our coordinate system, x′has coordinates (η = π/2, Ω′), Cx′ is the curve Ω′ = const., and the distancebetween x and x′ is γH−1, (γ << 1), where γ is the angle between Ωand Ω′.Parametrizing both geodesics by the proper time τ and taking τ = 0 at η = π/2,we can find what is the separation between points in Cx and C′x at any given τ.16

From (20) we have,Z(τ, γ) = 1 + [cos γ −1] cosh2 Hτ,where Z(τ, γ) means the invariant function Z between the two points on thegeodesics Cx and C′x at proper time τ and we have used (sinη)−1 = cosh Hτ.Two observers at x and x′ which were initially close and at rest relative to eachother (Z ≈1), are pulled apart by the expansion, so that eventually they reachlarge space-like separation Z << −1. The distance between both observers willbe equal to (π/2)H−1 at the time τ∗when Z(τ∗, γ) = 0 [see eq.

(21)], so we canwriteZ(τ, γ) = 1 −cosh2 Hτcosh2 Hτ∗.For τ >> τ∗we have [from(46)]D2s ≈2 · H34π2 (τ −τ∗). (48)In the language of Ref.

[22], this result can be phrased as follows. The fieldmeasured by each one of the two observers undergoes a random walk of step∆φs = ±(H/2π).

As long as both observers lie within the same Hubble vol-ume their steps are correlated and the dispersion of the field does not grow.Aproximately after time τ∗, the Hubble volumes around the two observers stopoverlapping, this means the future light cones of the two observers fail to overlapand so the respective random walks of the field become uncorrelated. Therefore,the dispersion is proportional to (τ −τ∗).

The factor of 2 in Eq. (48) arisesbecause we have two independent random walks.Finally we should say that although (47) and (48) have been derived usingthe de Sitter invariant state, they would hold for any O(4) invariant state (inthe limit of large τ).

This is because the contribution of L = 0 to D2s isH22π (η −12 sin 2η + (η ↔η′)2) < P 2 > .This term remains bounded in time and eventually becomes subdominant withrespect to the vaccuum terms (47) and (48). Similarly, because the modes ULMare bounded in time, any finite number of particles in the modes L > 0 willmake a bounded contribution which will be irrelevant at late times.Another physical quantity that we can compute using |0 > is the expectationvalue of the energy-momentum tensor.

Since the differential operator Dab′ actingon a constant is zero, the operator Q will not be present in the formal expressionof Tab′. Also, since P|0 >= 0, it is clear that< 0|Dab′{φ(x), φ(x′)}|0 >= Dab′ ˆG(x, x′).17

The computation of < 0|Tab|0 > now reduces to the one presented in Section 4,replacing G(1)A,B by ˆG. Obviously, the result is given by eq.

(33) with A = 0< 0|Tab|0 >ren=119138240π2R2gab,(49)which is de Sitter invariant as expected.Since we chose a state with P|0 >= 0 there is no contribution from theL = 0 sector to < Tab >. The L = 0 contribution to the energy momentumtensor operator isˆT (L=0)ab=R2144π2 (1 −2δa0) sin6 η gabˆP 22 .In a state with non-vanishing momentum, the expectation value of this operatorhas to be added to the r.h.s.

of (49). In particular for the O(4) invariant states< P 2 >α= 2α2 and we recover (33).

Clearly, for the energy < T00 > to be anobservable, P has to be observable.It is interesting to compare eq. (49) with the general result for a massiveand non-minimally coupled field [3],< Tab >ren= −gab64π2 {m2[m2+(ξ−16)R]ψ32 −λ+ ψ32 + λ+ logR12m2−m2ξ −16R −118m2R −12ξ −162R2 + R22160 }.Notice that the limit of this expression as m →0 and ξ →0 is ambiguous,because the term−gab64π2 m2m2 +ξ −16Rψ32 −λ−→−gab1536π2R21 + ξRm2(50)gives different answers by approaching the origin of the (ξ, m2) plane in differentways.

It is intriguing that in order to recover the result (49), the limit m2, ξ →0has to be taken along a path such thatξRm2 →−2. (51)The origin of the ambiguity can be traced back to the contribution of themode L = 0 to < Tµν > in the Euclidean vacuum.

It is easy to see that thiscontribution is given by−R21536π2(ξR + 2m2)(m2 + ξR) + O(m2, ξ),and therefore it will vanish only if the limit is taken according to the path (51).This is equivalent to taking the limit m2, ξ →0 in the formal expression of theenergy momentum tensor operator before taking the vacuum expectation value.18

6Conclusions and discussionWe have used the Hadamard formalism to compute the renormalized expectationvalue of the energy momentum tensor for a massless minimally coupled field in deSitter space in the two parameter family of O(4) Hadamard vacua. We find thatthis tensor is not de Sitter invariant but only O(4) invariant (in disagreementwith the result of ref.

[7], which was subsequently corrected in ref. [24]).We have also studied the de Sitter invariant state for the massless minimallycoupled field.

It is worth noting that such state is not a Fock vacuum (indeed,Allen [6] has shown that for m = ξ = 0 there is no de Sitter invariant Fockvacuum): the discrete zero mode is not quantized in terms of creation andanihilation operators, but rather using the canonical position and momentumoperators. In particular, we have used it to derive a covariant version of eq.(2).

We find that the expectation value of the square of the difference φ(x) −φ(y) grows linearly with the geodesic distance between x and y, for time-likeseparations which are large compared with H−1 [see eq.(47)]. The linear growthD(x, y) ∝Hτ has the same physical origin as the linear growth in time of eq.

(2) and it can be interpreted, along the same lines, as a “Brownian motion” ofthe field due to quantum fluctuations (see e.g. ref.

[22]).We have computed the renormalized expectation value of the energy momen-tum tensor in the de Sitter invariant vacuum. We find that the renormalizedvacuum energy density ,< T00 >ren, is lower in this state than in any of theO(4) invariant states.

In this sense, only the de Sitter invariant state deservesto be called vacuum.The O(4) invariant < Tµν >ren, eq. (33), approaches the de Sitter invariantvalue (49) at time-like infinity.

Also, the dispersion D(x, y) computed in Section4 using the de Sitter invariant state coincides with the limit η →π of thedispersion computed in a O(4) invariant state. Therefore, the de Sitter invariantstate can be seen as the limit into which the O(4) invariant states evolve atsufficiently late times.

This behaviour is familiar from the massive case, andit corresponds to the fact that any excitations above the de Sitter invariantvacuum are redshifted away by the exponential expansion.After this paper was submitted, it was pointed out to us by the referee thatthe result of Ref. [7] had already been corrected in Ref.

[24].AcknowledgementsWe would like to thank Larry Ford, Alex Vilenkin and Prof. V.F. M¨uller foruseful conversations.

The work of J.G. is supported by a Fulbright grant.19

Appendix AFor completeness, in this appendix we summarize the field quantization in theSchr¨odinger picture (see e.g. ref.

[9]).In the Schr¨odinger picture, ˆχLM and ˆπLM are time independent operatorssatisfying the commutation relations[ˆχLM, ˆπL′M′] = iδLL′δMM′,(A1)and acting on a Hilbert space of time dependent physical states Ψ. In the ‘q’representation, such states are described by wave functionals Ψ({χLM}, η) andthe action of the operators is given byˆχLMΨ = χLMΨ,ˆπLMΨ = −i∂∂χLMΨ.The time evolution is governed by the Schr¨odinger equationˆHΨ = −i ∂∂ηΨ,(A2)where ˆH is the Hamiltonian derived from the action (8), with χLM and πLMreplaced by its operator counterparts:ˆH =XLM12ˆπ2LM(H sin η)−2 + (H sin η)−2ω2Lχ2LM.

(A3)Note that throughout this appendix, χLM are not functions of η (as theywere in Section 2) but they are the time independent position operators ofthe Schr¨odinger picture (see e.g. [12]).Factorizing the wave functional asΨ =YLMΨLM(χLM, η),eq.

(A2) separates into a set of Schr¨odinger equations, one for each individualmode12−1(H sin η)−2∂2∂χ2LM+ (H sin η)−2ω2Lχ2LMΨLM = −i ∂∂ηΨLM. (A4)These can be solved by using the ansatzΨLM = gLM exp"i2(H sin η)−2 ˙VLMVLMχ2LM#,(A5)20

where gLM(η) and VLM(η) are unspecified functions. Substituting (A5) intoeq.

(A4) and collecting the terms proportional to χ2LM one finds¨VLM −2 cot η ˙VLM + ω2L(η)VLM = 0,(A6)so VLM must be a solution of the field equation (9). Collecting the terms whichare independent of χLM, one finds a differential equation for gLM which can besolved immediately to yieldgLM = CLMV −1/2LM ,where CLM is just an overall normalization constant.

Choosing one solution of(A6) for each L and M specifies a particular quantum state. In order to knowwhat set of solutions {VLM} corresponds to the de Sitter invariant quantum statedefined in Section 2, one has to impose that the wave functional be anihilated bythe operators aLM associated with the set of modes that defines such vacuum,eq.

(15):aLMΨ ="U ∗LM∂∂χLM−i˙U ∗LM(H sin η)2 χLM#Ψ = 0,where we have inverted (12) to express aLM in terms of ˆχLM and ˆπLM. Clearly,this conditions are satisfied if and only ifVLM = U ∗LM.In summary, the de Sitter invariant wave functional is given byΨ =YLM(2π)−1/4U −1/2LMexp"i2(H sin η)−2 ˙U ∗LMU ∗LMχ2LM#,(A7)with ULM given by (15).

It can be checked that this wave-functional is anihilatedby the operator generators of the de Sitter group [12], and is thus de Sitterinvariant. Note also that this wave functional is properly normalized, in thesense thatZ ∞∞YLMdχLM|Ψ({χLM}, η)|2 = 1.Note that the case m2 = ξ = 0 is special.

From (15) we find that U0 becomesconstant in the massless minimally coupled limit,U0 = A0 −r2π!,so the Wronskian condition can not be satisfied and the normalization constantA0 becomes infinite [see eq.(17)]. Such infinity can be understood by noticingthat, sincelimm2,ξ→0˙U0U0= 0,21

the wave functional becomes independent of χ0π0Ψ = −i ∂∂χ0Ψ = 0,(m2 = ξ = 0),and therefore Ψ is not normalizable in the discrete sense (which is natural foran eigenstate of momentum).To conclude, let us study in more detail the L = 0 term of the Schr¨odingerequation (A2). Using the notation ψ ≡ΨL=0 we have−12∂∂χ20ψ = −iH2∂∂˜tψ,where we have introduced the new time variable˜t ≡12(η −12 sin 2η −π2 ).In this notation the basic solutions are the eigenstates of momentum that wediscussed in Section 4,ψp(χ0) ∝ei(pq−p2˜t)with q ≡√2H−1χ0 [see Eq.

(37)]. For ˜t = 0 these are the Heisenberg wavefunctions (40).The wave packet (41) is just a superposition of these modes, and its timeevolution can be found in any elementary textbook.It represents a gaus-sian wave-packet that spreads in time.Noting that < α|P 2|α >= 2α2 and< α|Q2|α >= (2α2)−1 we have, from (37),< χ20 >α= H2 14α2 + 4α2˜t2.Since the range of ˜t is finite, ˜t ∈[−π/4, π/4], the expectation value of χ20 doesnot grow unbounded, but reaches a constant in the asymptotic past and future.Therefore the asymptotic growth in time of < φ2 > in de Sitter space is due tothe L > 0 modes.This behaviour is somewhat different from that of the theory of a masslessfield on a compact toroidal flat spacetime which we briefly discussed in Section4.There, the contribution of the L = 0 mode to < φ2 > also has a termproportional to < p20 > t2.

However, in that case, t is the Minkowski time.If we choose a state with < p20 ≯= 0 then < φ2 >∝t2 grows unbounded astime increases due to the L = 0 contribution alone. On the other hand, for theground state < p20 >= 0 but < x20 >= ∞and therefore < φ2 > is infinite, justlike in the de Sitter invariant state studied in this paper.22

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