Lecture notes on interacting quantum fields in de Sitter space
aa r X i v : . [ h e p - t h ] J a n ITEP-TH-32/13
Lecture notes on interacting quantum fields in de Sitter space
E.T. Akhmedov Institute for Theoretical and Experimental Physics117218, ul. B.Cheremushkinskaya, 25, MoscowandMoscow Institute of Physics and Technology141700,Institutskii per. 9, Dolgoprudny, Moscow regionandMathematical Faculty of theNational Research University Higher School of Economics117312, Vavilova 7, Moscow
Abstract
We discuss peculiarities of quantum fields in de Sitter space on the example of the self-interactingmassive real scalar, minimally coupled to the gravity background. Non-conformal quantum fieldtheories in de Sitter space show very special infrared behavior, which is not shared by quantumfields neither in flat nor in anti-de-Sitter space: in de Sitter space loops are not suppressed incomparison with tree level contributions because there are strong infrared corrections. That istrue even for massive fields. Our main concern is the interrelation between these infrared effects,the invariance of the quantum field theory under the de Sitter isometry and the (in)stability of deSitter invariant states (and of dS space itself) under nonsymmetric perturbations.
Key words: de Sitter space, quantum fields in curved space-time, particle creation
PACS numbers: E–mail: [email protected]
Contents
I. Introduction
II. de Sitter geometry
III. Free scalar fields in de Sitter space
IV. Loops in de Sitter space QFT α -harmonics 38D. One-loop correction in the contracting Poincar´e patch 39E. One-loop correction in global de Sitter space 40 V. Summation of leading infrared contribution in all loops
VI. Sketch of the results for λφ theory (instead of conclusions) VII. Acknowledgements References I. INTRODUCTION D -dimensional de Sitter (dS) space solves the general relativity (GR) equations of motion withpositive vacuum energy [1]: G αβ = − ( D − D − H g αβ , α, β = 0 , . . . , D − . (1)Here G αβ is the Einstein tensor and H is the Hubble constant. The signature of the metric, g αβ ,is ( − , + , . . . , +). This space has a big isometry group, SO ( D, h T αβ i , to the right hand side of the GR equationsof motion. By h T αβ i we denote the quantum average of the energy-momentum tensor of whateverquantum fields are present on the dS background. What is the influence of this quantum averageon the background geometry? Our final goal is to answer this question. However, in these lectureswe concentrate on just a part of this program: We fix dS background and check whether theassumption of negligible or small effects of h T αβ i is selfconsistent.This quantum average, h T αβ i , contains the standard divergence due to zero-point fluctuations.It is proportional to g αβ and should be absorbed into the ultraviolet (UV) renormalization of thecosmological constant. Furthermore, quantum fields on the dS background are in a nonstationarysituation. Hence, nontrivial finite contributions to h T αβ i , such as, e.g., fluxes, can be present andare of interest for us. A. Motivation
It is natural to use the big isometry group of dS space in the formulation of quantum fieldtheory (QFT) on this background, i.e., it is appropriate to look for a dS-invariant state, if any,and quantize excitations of such a state. (This is what one does when one quantizes fields onMinkowski background: one explores the Poincar´e group.) But then, if the symmetry is respectedat all stages of quantization, correlation functions depend only on dS-invariant geodesic distancesrather than on each of their argument separately. Hence, in the free theory one can use the two-point correlation function to find that all contributions to h T αβ i are proportional to the metric, g αβ . That is just a consequence of the symmetry in question. Furthermore, with the use of thehigher-point correlation functions one can extend this calculation to the interacting fields with thesame conclusion, if the dS isometry is also respected on loop level.Thus, in the case of perfectly dS-symmetric situation, quantum effects just renormalize thecosmological constant and dS space remains intact. Perhaps that is true unless exact correlators,as functions of geodesic distances, show an explosive behavior. The main goal of these notes,however, is to question the stability of dS space and to investigate if the dS isometry is brokenor the dS-invariant state is unstable with respect to nonsymmetric perturbations. It is probablyworth stressing here that in these notes we study just the behavior of correlation functions in dSspace and avoid using the notion of particle, unless this notion is meaningful and useful for theinterpretation of the obtained equations.At tree level the situation is as follows: For massless scalar fields, which are minimally coupledto the dS background, there is no Fock dS-invariant state [2], [3]. Also the very possibility todefine a dS-invariant state for gravity is still under discussion (see [4], [5]–[11], [12], [13], [14] and[15]–[18]). All these interesting and important issues will not be touched in these lectures. We willconcentrate on the study of the real massive minimally coupled scalar field, φ . Then the situationis conceptually simpler because in this case there is a one-parameter family of so-called α -vacua,which respect the dS isometry at tree level [2], [19].First, we would like to understand whether or not the dS isometry is respected by loop contri-butions, if one quantizes over such a dS-invariant state: We will see that in some circumstancesthe dS isometry is indeed broken. Second, in those circumstances, in which the dS isometry isrespected, we would like to understand whether or not the corresponding state is stable undernonsymmetric perturbations. (Note that the vacuum is still dS-invariant. We just consider a nontrivial density matrix.) We find it physically inappropriate to consider the stability of any systemin a state in which all its symmetries are preserved. We will see that the exact dS-invariant statedoes exist, but it is unstable under sufficiently strong nonsymmetric perturbations.A few points are worth stressing. First, Minkowski space is stable under nonsymmetric particledensity perturbations over the Poincar´e invariant vacuum. That is just a consequence of theenergy conservation and of the H-theorem, neither of which is straightforwardly applicable in dSspace. Second, in the presence of nonsymmetric density perturbations, even tree level two-pointcorrelation functions depend on each of their arguments separately rather than on dS-invariantdistances between them. In view of what we have said here one can conclude that argumentation,which is based on analytical properties of the correlators as functions of geodesic distances [20]–[24],[25], [26], is not sufficient to support the stability of dS space.For the other IR issues in dS space please see, e.g., [27]–[51]. B. General physical explanation of our main statements
To explain our statements let us briefly describe the dS geometry. dS space can be totalycovered by the so-called global metric: ds = − dt + cosh ( Ht ) H d Ω D − , with d Ω D − being the line element on the unit ( D − ds = − dτ + e Hτ + d~x = 1( Hη + ) (cid:2) − dη + d~x (cid:3) , Hη + = e − Hτ + . It covers only geodesically incomplete half of the whole space. The latter is referred to as theexpanding Poincar´e patch (EPP). The other half is referred to as the contracting Poincar´e patch(CPP) and is covered by the metric ds − = − dτ − + e − Hτ − d~x − = 1( Hη − ) (cid:2) − dη − + d~x − (cid:3) , Hη − = e Hτ − . The boundary between these patches ( η ± = + ∞ ), which is simultaneously the initial Cauchysurface of the EPP and the final one of the CPP, is light-like. One can obtain the CPP metric fromthe EPP one by reflecting the direction of the conformal time, η + ∈ (+ ∞ , −→ η − ∈ (0 , + ∞ ).The EPP and CPP have a peculiarity in their geometry. The spatial part of their metric hasthe conformal factor 1 /η ± . Due to its presence every wave experiences strong blue shift towardsthe past (future) infinity of the EPP (CPP). I.e., these regions of the Poincar´e patches correspondto the UV limit.In loop integrals on the EPP background the vertex integration goes over the half of dS space.Hence, naively the dS isometry should be broken because there are generators of this symmetrywhich can move the EPP within the whole dS space. However, following the original work [52], inthese lectures we will show that the dS isometry can indeed be respected in the loops, but only if one starts exactly with the so-called Bunch–Davies (BD) state at the past infinity of the EPP.BD is such a state that there are no positive energy excitations at the past (future) infinity ofthe EPP (CPP) [53], [54]. In the region of dS space near the boundary between the EPP andCPP one can define what one means by particle and what one means by positive energy. Thisis possible because, as we have explained above, every momentum experiences infinite blue shifttowards the past (future) of the EPP (CPP). In fact, high energy harmonics are not sensitive tothe comparatively small curvature of the background space and behave as if they are in flat space.After a Bogolyubov rotation from the BD harmonics to other modes, corresponding to otherdS-invariant states from the aforementioned α -family, one mixes the positive and negative energystates. That spoils the UV behavior of the correlation functions. As a result, any vacuum differentfrom the BD one violates the dS isometry in the loops, even though it respects the symmetry attree level.Should one conclude then that the problem of the influence of massive scalar fields on the dSgeometry is solved? It seems that in the EPP one just has to quantize fields over the BD state. Butthe solution of this problem is not yet complete because of large infrared (IR) effects: IR correctionsmay become destructive in the presence of nonsymmetric density perturbations because they arenot suppressed in comparison with tree level contributions.As we will explain, all QFTs in dS space, which are not conformally invariant , share thesame characteristic property, which is not present in massive QFTs in flat or anti-dS spaces [55].Obviously the UV limit of any meaningful QFT over the dS or anti-dS backgrounds should be thesame as in flat space, hence, all the differences appear in the IR limit. Large IR effects, due to I.e., such QFTs, which can feel the difference between the flat space and conformally flat dS one. their large scale nature, are sensitive to the boundary and initial conditions in various patches ofthe entire space. Hence, they should be separately considered in the EPP, CPP and in global dSspace.Also, the character of these IR contributions in dS space crucially depends on the relationbetween the mass and the Hubble constant: If the mass of the scalar field is big, m/H > ( D − / oscillate and decay to zero, as η →
0. Scalar fields withsuch masses are composing the so-called principal series of theories. At the same time, harmonicfunctions of the scalars with small masses, 0 < m/H ≤ ( D − /
2, homogeneously decay to zero,as η →
0. The corresponding theories are composing the so-called complementary series.We first describe the situation in the EPP and then continue with the CPP and global dS space.Due to the spatial homogeneity of the EPP and also of the initial states that we consider, it isnatural to perform the partial Fourier transformation along ~x + directions. Then, the two-pointcorrelation function of interest for us acquires the form D ( p | η + , η ′ + ) = (cid:10)(cid:8) φ ( η + , ~p ) , φ ( η ′ + , − ~p ) (cid:9)(cid:11) ,where {· , ·} is the anticommutator. In the nonstationary situation it is the appropriate object tostudy and is referred to as the Keldysh propagator. As we review below, in the limit when thesystem approaches future infinity, p η = p p η + η ′ + →
0, the correlation function in question receiveslarge corrections for any mass of the field.E.g., for the λ φ scalar field theory from the principal series the first loop contains terms whichare proportional to λ η D − log( pη ) [56], [57]. The same linear logarithmic corrections are alsopresent in the second loop of λφ theory [58]. (The only difference between the φ and φ theoriesis in the mass-dependent coefficients of these IR contributions.) At the same time, the fields fromthe complementary series receive powerlike corrections, which are proportional to λ η D − ( pη ) − ν ,where parameter ν depends on the mass and 0 < ν < ( D − / η D − in front of every contribution, which is dueto the expansion of the spatial sections of the EPP, such loop corrections do not make the Keldyshpropagator singular in the limit η →
0. But, on the other hand, even if λ is very small thecorrections in question become comparable to tree level contributions, as pη →
0. Hence, one hasto sum the unsuppressed leading IR contributions from all loops.Of course one should not worry about the stability of the EPP, if the dS isometry is respected.But what if we consider some finite nonsymmetric density perturbation over the BD state at thepast infinity of the EPP? Due to the rapid expansion of the EPP it is usually believed that suchdensity perturbations would quickly fade away and, thus, one should not care about their negligibleinfluence on the background geometry. However, the situation is quite counterintuitive because theloop contributions go into the factor multiplying η D − and marginally depend on the expansionand/or contraction of spatial sections.Let us clarify this observation here. The common wisdom is that excitations in the EPP canreproduce themselves only very slowly. (It is believed that the reproduction should be at mostlinear in time, e.g., due to the constant particle creation caused by the background field.) At the Note that the only possible exception is given by the massless minimally coupled scalar field theory [33], [46], forwhich ν = ( D − / same time, the expansion of the spatial sections is exponential and, hence, will rapidly win oversuch a reproduction.However, let us look more carefully at what is actually happening. Suppose we are in a situationwhere one can give a meaning to the notion of particle in the future infinity of the EPP. (We willsee that this is possible under some conditions.) Then, consider, e.g., a particle dust in the EPP.Its density per physical volume is indeed rapidly decreasing. However, the density per comovingvolume remains constant. (The comoving density is actually one of the quantities contributing tothe factor multiplying η D − in the two-point correlation function under consideration.) Supposenow there is some constant particle production process, i.e., it is the density per comoving volumewhich is linearly growing. (That is what we will actually see.) Then, sooner or later it will becomevery large and one will not be able to neglect the nonlinear particle self-reproduction processes.We will see that the non-linearities are proportional to the density per comoving volume ratherthan to that per physical one. (It may sound as very weird, but note that we are talking aboutwaves whose size is growing with time and even redshifted outside the cosmological horizon in theprogress towards future infinity.) As we will see, the nonlinear self-reproduction can actually causethe destruction and win over the expansion.Although we just perform the calculation of loop contributions to correlation functions, thesummation of unsuppressed IR loop corrections allows the particle interpretation and is relatedto the above described particle kinetics. That happens to be true at least for the fields fromthe principal series. We are not yet able to perform the summation of the IR divergences for thecomplementary series because their physical meaning is not yet clear to us. But on general physicalgrounds we expect that the IR effects in this case will be even stronger.To sum unsuppressed IR loop corrections one has to find an IR solution of the system of Dyson–Schwinger (DS) equations for the vertices, propagators and self-energies. We show, however, thatin the limit under study only the equation for the Keldysh propagator is relevant. Furthermore,for the principal series it reduces to a kinetic equation of Boltzmann type, where plane waves aresubstituted by exact dS harmonics. (As is known in condensed matter theory, loop effects maybecome classical. That is related to the fact that loop corrections are not suppressed in comparisonwith tree level contributions (see, e.g., [59], [60]).)All elements of the kinetic equation that we obtain have a clear physical interpretation anddescribe various particle decay and creation processes in the future infinity of the EPP [57], [61],[62]. Moreover, we can find solutions of this kinetic equation for various initial conditions whichare set up by the initial (tree level) Keldysh propagator. One of the solutions shows an explosivebehavior of the two-point correlation function.What about IR effects in the CPP and global dS? If we consider an exactly spatially homoge-neous initial state, the loop calculation in the CPP follows straightforwardly from the one in theEPP. Note, however, that unlike the EPP, the spatially homogeneous state in the CPP is unstablewith respect to the inhomogeneous perturbations. But it is still instructive to study loop effects insuch an ideal situation.The metric in the CPP is identical to the one in the EPP if one reverses the direction of theconformal time. But then, for the same reason as we have observed large IR contributions in theEPP, there are IR divergences in the CPP: For the scalar fields from the principal series with the λφ self-interaction the corrections are proportional to λ η D − log( η/η ), if pη ≪
1. At the sametime, if pη ≫
1, the corrections are proportional to λ η D − log( pη ). Here η ≡ p η − η ′− , η is themoment of time at past infinity, pη →
0, after which self-interactions are adiabatically turnedon. The same linear logarithmic contributions, but with different mass-dependent coefficients,appear also in the second loop in the theory with λφ self-interaction [58]. For the scalars fromthe complementary series the divergences are powerlike.If it were not for the presence of the IR cutoff, η , the loop integrals in the CPP would beexplicitly divergent: η cannot be taken to past infinity. Thus, one has to have an initial Cauchysurface at some finite η . But such a surface can be moved within the dS space by an isometrytransformation. Then, holding η fixed breaks the dS isometry and correlation functions start todepend separately on each of their time arguments. Furthermore, the summation of the leadingIR contributions in the CPP is performed similarly to the EPP case. In fact, the kinetic equationin the CPP is obtained from the one in the EPP just by the time reversal.To study the situation in global dS space one has to keep in mind that it is the union of theCPP and EPP. Hence, loops in global dS also have explicit IR divergences that break the dSisometry [56], [62]. In this case it is also possible to derive the kinetic equation for the fields fromthe principal series. But unlike the EPP and CPP case, this kinetic equation does not possessan obvious quasi-particle description in the IR limit and we do not expect to have a stationarysolution in global dS, unless one considers a small enough part of it.To conclude, the statements that we are going to advocate in these lectures are as follows. First,we will observe that the only way to respect the dS isometry in loop contributions is to start atthe past infinity of the EPP exactly with the BD state. Any other dS-invariant vacuum, i.e., thatstate which respects the dS isometry at tree level, does break it at loop level. Second, we will seethat any invariant initial state, including the so-called Euclidian one, in global dS space violatesthe isometry in loops, due to IR divergences of loop integrals. Similarly, due to IR divergences,any invariant initial state in the CPP also violates the dS isometry in loop contributions. Andfinally, after the summation of unsuppressed IR contributions for the principal series in all loops,we will proceed to show that even the BD state in the EPP is unstable under sufficiently strongnonsymmetric perturbations. For the complementary series we expect even more destructive IReffects, but we are not yet able to perform the loop summation.The new content of these notes is mostly based on our previous work [57], [58], [61], [62]. II. DE SITTER GEOMETRY D -dimensional dS space can be realized as hyperboloid, − (cid:0) X (cid:1) + (cid:0) X (cid:1) + · · · + (cid:0) X D (cid:1) ≡ g AB X A X B = H − , A, B = 0 , . . . , D, (2)placed into the ambient ( D + 1)-dimensional Minkowski space, with metric ds = g AB dX A dX B .One way to see that a metric on such a hyperboloid solves equation (1) is to observe that it can0 Figure 1: Each constant X slice of this two-dimensional dS space is a circle of radius ( H − + X ). be obtained from the sphere via the analytical continuation X D +1 → i X . For illustrative reasonswe depict two-dimensional dS space on fig. 1.Another way to see that the hyperboloid in question has a constant curvature is to observe thateq. (2) is invariant under SO ( D,
1) Lorentz transformations of the ambient space. The stabilizer ofany point obeying (2) is SO ( D − ,
1) group. Hence, dS is homogeneous, SO ( D, /SO ( D − , SO ( D + 1) /SO ( D ).) Thus, SO ( D,
1) Lorentzgroup of the ambient Minkowski space is the dS isometry group.The geodesic distance, L , between two points, X A and X A , on the hyperboloid can be con-veniently expressed via the so-called hyperbolic distance, Z , as follows:cos ( H L ) H ≡ Z H ≡ g AB X A X B , where g AB X A , X B , = H − . (3)To better understand the meaning of this expression, it is instructive to compare it to the geodesicdistance, l , on a sphere of radius R : R cos (cid:18) l R (cid:19) ≡ R z ≡ (cid:16) ~X ~X (cid:17) , where ~X , = R . While the spherical distance, z , is always less than unity, the hyperbolic one, Z , can acquireany value because of the Minkowskian signature of the metric.1All geodesics on the hyperboloid of Fig. (1) are curves that are cut out on it by planes goingthrough the origin of the ambient Minkowski space-time. (Compare this with the case of thesphere.) Hence, space-like geodesics are ellipses, time-like ones are hyperbolas and light-like arestraight generatrix lines of the hyperboloid.For every point X A on dS space, g AB X A X B = H − , there is the antipodal one X A = − X A ,which is just its reflection with respect to the origin in the ambient Minkowski space. Note thatthen Z = − Z . A. Global de Sitter metric
To define a metric on dS space, which is induced from the ambient space, one has to find asolution of eq. (2). One possibility is as follows X = sinh( Ht ) H , X i = n i cosh( Ht ) H , i = 1 , . . . , D (4)where n i is a unit, n i = 1, D -dimensional vector. One can choose: n = cos θ , − π ≤ θ ≤ π n = sin θ cos θ , − π ≤ θ ≤ π . . . (5) n D − = sin θ sin θ . . . sin θ D − cos θ D − , − π ≤ θ D − ≤ π n D − = sin θ sin θ . . . sin θ D − cos θ D − , − π ≤ θ D − ≤ πn D = sin θ sin θ . . . sin θ D − sin θ D − . Then, the induced metric is: ds = − dt + cosh ( Ht ) H d Ω D − , (6)where d Ω D − = D − X j =1 j − Y i =1 sin θ i ! dθ j (7)is the line element on the unit ( D − t slices are compact ( D − D -dimensional sphere after the analytical continuation, H t → i (cid:0) θ D − π (cid:1) .The hyperbolic distance in these coordinates is given by:2 Figure 2: The standard quadratic Penrose diagram of D -dimensional dS space, when D >
2. The straightthin line is the constant t and/or θ slice. Z = − sinh( Ht ) sinh( Ht ) + cosh( Ht ) cosh( Ht ) cos( ω ) , (8)where cos( ω ) = ( ~n , ~n ). B. Penrose diagram
To understand the causal structure of dS space it is convenient to transform the global coordi-nates as follows: cosh ( Ht ) = 1cos θ , − π ≤ θ ≤ π ds = 1 H cos θ (cid:2) − dθ + d Ω D − (cid:3) , (9)which is conformal to that of the Einstein static universe, ds ESU = − dθ + d Ω D − , with compacttime θ . The causal structure of the latter universe coincides with that of dS space because signof ds coincides with that of ds ESU . Hence, one can drop the conformal factor and depict the3compact space. Such a procedure is just a variant of the stereographic projection. In the modernlanguage the result of the projection of a space-time is referred to as Penrose diagram.If
D >
2, then to draw the diagram on the two-dimensional sheet we should choose, in additionto θ , one of the angles θ j , j = 1 , . . . , D −
1. The usual choice is θ because the metric in questionhas the form dθ + dθ + sin ( θ ) d Ω D − , i.e., it is flat in the ( θ − θ )-plain. The Penrose diagramfor dS space, whose dimension is grater than 2, is depicted on fig. 2.Note that when D > θ is taking values in the range (cid:2) − π , π (cid:3) . At the same time, when D = 2 we have that θ ∈ [ − π, π ]. When D >
2, the problem with the choice of θ in the Penrosediagram is that then cylindrical topology, S D − × R , of dS space is not transparent. At the sametime, the complication with the choice of θ D − ∈ [ − π, π ], instead of θ , appears form the factthat the metric in the ( θ − θ D − )-plain is not flat. For this reason we prefer to consider just thestereographic projection in the two-dimensional case because it is sufficient to describe the causalstructure and also clearly shows the topology of dS space.The Penrose diagram of the two-dimensional dS space is shown on fig. 3. This is the stereo-graphic projection of the hyperboloid from fig. 1. What is depicted here is just a cylinder becausethe left and right sides of the rectangle are glued to each other. The fat solid curve is a world lineof a massive particle. Thin straight lines, which compose 45 o angle with both θ and θ axes, arelight rays. From this picture one can see that every observer has a causal diamond within whichhe can exchange signals. Due to the expansion of dS space there are parts of it that are causallydisconnected from the observer. C. Expanding and contracting Poincar´e patches
Another possible solution of (2) is based on the choice: − (cid:0) H X (cid:1) + (cid:0) H X D (cid:1) = 1 − (cid:0) H x i + (cid:1) e H τ + , (cid:0) H X (cid:1) + · · · + (cid:0) H X D − (cid:1) = (cid:0) H x i + (cid:1) e H τ + . (10)Then, one can define H X = sinh ( H τ + ) + (cid:0) H x i + (cid:1) e H τ + ,H X i = Hx i + e H τ + , i = 1 , . . . , D − ,H X D = − cosh ( H τ + ) + (cid:0) H x i + (cid:1) e H τ + . (11)With such coordinates the induced metric is ds = − dτ + e H τ + d~x . (12)Note, however, that in (11) we have the following restriction: − X + X D = − H e H τ + ≤
0, i.e.,metric (12) covers only half, X ≥ X D , of the entire dS space. It is referred to as the expanding4 Figure 3: The rectangular Penrose diagram of the two-dimensional dS space. Note that the left and rightsides of this rectangle are glued to each other. Thus, while on fig. 2 the positions θ = ± π sit at the oppositepoles of the spherical time slices, on the present figure the positions θ = ± π coincide. Poincar´e patch (EPP). Another half of dS space, X ≤ X D , is referred to as the contractingPoincar´e patch (CPP) and is covered by the metric ds − = − dτ − + e − H τ − d~x − . (13)In both patches it is convenient to change the proper time τ ± into the conformal one. Then, theEPP and CPP both possess the same metric: ds ± = 1( H η ± ) (cid:2) − dη ± + d~x ± (cid:3) , H η ± = e ∓ H τ ± . (14)However, while in the EPP the conformal time is changing form η + = + ∞ at past infinity ( τ + = −∞ ) to 0 at future infinity ( τ + = + ∞ ), in the CPP the conformal time is changing from η − = 0 atpast infinity ( τ − = −∞ ) to + ∞ at future infinity ( τ − = + ∞ ). Both the EPP and CPP are shownon fig. 4.The hyperbolic distance in the EPP and CPP has the form: Z = 1 + ( η − η ) − | ~x − ~x | η η (15)5 Figure 4: The boundary between the EPP and CPP is light-like and is situated at η ± = + ∞ . We also showhere the constant conformal time slices. It is worth mentioning here that it is possible to cover simultaneously the EPP and CPP with theuse of the metric (14), if one makes the changes ~x ± → ~x and η ± → η ∈ ( −∞ , + ∞ ). Then, while atthe negative values of the conformal time, η = − η + <
0, it covers the EPP, at its positive values, η = η − >
0, this metric covers the CPP. The inconvenience of such a choice of global metric is dueto that the boundary between the EPP and CPP simultaneously corresponds to η = ±∞ . D. Other de Sitter metrics and patches
There is another commonly used metric on dS space: ds = − h − ( H r ) i dT + dr h − ( H r ) i + r d Ω D − , ≤ r ≤ /H. (16)This metric covers only quoter of the entire space and is referred to as static. We are not goingto consider quantum fields on this background. The reason for that is as follows. The metricin question contains nontrivial time component, g ( r ) = 1, and, hence, is seen by noninertialobservers, unless the position of the observer is at r = 0. As we explain below, we would like tounderstand the physics in dS space as seen by such observers which are not affected by any otherforce except the gravitational one. Hence, we restrict our considerations to the global, EPP andCPP metrics which are seen by inertial observers.6Note here that the transformation from the proper time, τ ± , to the conformal one, η ± , (whichmakes g time-dependent) is nothing but the change of the clock rate rather than a transition tothe noninertial motion. E. Spatial volume in de Sitter space
For our future considerations it is important to define here the physical and comoving spatialvolumes in global dS space and in its Poincar´e patches. The spatial sections in all aforementionedmetrics contain conformal factors, cosh ( H t ) H or H η ± ) . Then, there is the volume form, d D − V , withrespect to the spatial metric, which is multiplying the corresponding conformal factor. This formremains constant during the time evolution of the spatial sections and is referred to as comovingvolume.It is important to observe that if one considers a dust in dS space, then its density per comovingvolume remains constant independently of whether spatial sections are expanding or contracting.At the same time if one takes into account the conformal factor, i.e., the expansion (contraction)of the EPP (CPP), then he has to deal with the physical volume, d D − V ± ( H η ± ) D − . In global dS space thephysical volume is cosh D − ( H t ) d D − V sphere H D − . Of course the density of the dust with respect to such avolume is changing in time. III. FREE SCALAR FIELDS IN DE SITTER SPACE
We start our discussion with free massive real scalar fields which are coupled to the dS back-ground in the minimal way. From now on we set the curvature of dS space to one, H = 1, andassume it to be fixed. In the following sections we will question this assumption in the presence ofquantum effects in interacting theories. A. Free waves in Poincar´e patches
The action of the free theory under consideration is as follows: S = Z d D x p | g | h g αβ ∂ α φ ∂ β φ + m φ i . (17)In the EPP or CPP the Klein–Gordon (KG) equation is: (cid:2) − η ∂ η + ( D − η ∂ η + η ∆ − m (cid:3) φ ( η, ~x ) = 0 , (18)where ∆ is the ( D − ± ” indexes of η and ~x . The harmonics, which solve this equation,can be represented as φ p ( η, ~x ) = g p ( η ) e ∓ i ~p ~x . If one assumes the ansatz g p ( η ) = η D − h ( pη ),7where p = | ~p | , then (18) reduces to the Bessel equation for h ( pη ). The index of this equation is iµ = i q m − (cid:0) D − (cid:1) .Generic solution of the Bessel equation with such an index behaves as follows: h ( x ) = ( A e i x √ x + B e − i x √ x , x → ∞ C x iµ + D x − iµ , x → A, B, C, D are some complex constants. Taking into account that ( pη ) ± iµ ∼ e ± i µ τ , one caninterpret x ± i µ , if µ is real, as a single wave in the future (past) of the EPP (CPP).If the field is heavy, m > ( D − /
2, then, it belongs to the so-called principal series. Thecorresponding g p harmonics oscillate and decay to zero as η ( D − / ± iµ , when η →
0. At the sametime, for the light field from the complementary series, m ≤ ( D − /
2, the harmonic functionshomogeneously decay to zero as η ( D − / ± √ ( D − / − m , when η →
0. The only exception is themassless field, m = 0, for which harmonics approach a non-zero constant in future infinity.Bessel functions of the first, h ( x ) = q π sinh( πµ ) J iµ ( x ), or second, h ( x ) ∝ Y iµ ( x ), kinds correspondto those modes for which either C or D in (19) is vanishing. But then both A and B are not zero.Thus, Bessel harmonics represent single free waves only in the future (past) infinity of the EPP(CPP). Correspondingly they are referred to as out- (in-)harmonics in the EPP (CPP).Performing a Bogolyubov transformation, one can consider also other possibilities for h ( x ). Forexample, Hankel functions of the first, h ( x ) = √ π e − π µ H (1) iµ ( x ), or second, h ( x ) = √ π e π µ H (2) iµ ( x ),kinds are such that either A or B in (19) is vanishing. But then both C and D are not zero. Thus,Hankels, which are referred to as Bunch–Davies (BD) modes [53], represent single free waves onlyin the past (future) infinity of the EPP (CPP).For the beginning we do not specify our choice for h ( pη ) because none of them behaves as singlewave simultaneously at past and future infinity. Then, quantum field can be mode expanded inthe usual way: φ ( η, ~x ) = Z d D − ~p h a ~p g p ( η ) e − i ~p ~x + h.c. i , g p ( η ) = η D − h ( pη ) . (20)Corresponding annihilation, a ~p , and creation, a + ~p , operators obey the proper Heisenberg commu-tation relations. They follow from the commutation relations of φ with its conjugate momentumand are the corollary of the time-independence of the Wronskian, η − D (cid:0) g p ˙ g ∗ p − ˙ g p g ∗ p (cid:1) = ± i , whichfollows from the equations of motion. This observation allows to fix the proper normalization ofthe harmonic functions.For the illustrative reasons let us consider the free Hamiltonian in D = 4. In an arbitrarydimension the formulas are similar. The Hamiltonian can be found from (17), using the machinerypresented, e.g., in Ref. [63]. The energy-momentum tensor is T αβ = ∂ α φ ∂ β φ − g αβ L , where L is the Lagrangian density. Then, the free Hamiltonian (before the normal ordering) is H ( η ) = η R d x T ( η, ~x ) and can be expressed via the creation and annihilation operators as:8 H ( η ) = Z d p h A p ( η ) a + ~p a ~p + B p ( η ) a ~p a − ~p + h.c. i ,A p ( η ) = 12 η (cid:26) | ˙ g p | + (cid:20) p + m η (cid:21) | g p | (cid:27) ,B p ( η ) = 12 η (cid:26) ˙ g p + (cid:20) p + m η (cid:21) g p (cid:27) , ˙ g p ≡ dg p dη . (21)The main characteristic feature of this Hamiltonian is that one cannot diagonalize it once andforever. That is because there is no solution of the KG equation which coincides with the functionthat solves equation B p ( η ) = 0. (In flat space the simultaneous solution of the corresponding KGequation and of B p = 0 is the plane wave.) Moreover, one cannot use such g p ( η ) which solve B p ( η ) = 0 equation in place of the mode functions in (20) because then the corresponding creationand annihilation operators will not obey the appropriate Heisenberg algebra.However, B p ( η ) can be set to zero as η → + ∞ . This can be done if one chooses the BD modesbecause they behave as single plane waves when η → + ∞ . Then, we have a clear meaning of thepositive energy and of the particle in this region of space-time because the Hamiltonian is diagonal.Recalling that the past (future), η → + ∞ , of the EPP (CPP) corresponds to the UV limit of thephysical momentum, pη , one can see that other harmonics have wrong UV behavior. In fact, aftera Bogolyubov rotation of the BD modes to other harmonics one mixes positive and negative energyexcitations.There are no modes that allow to set B p ( η ) to zero as η →
0. That is because the gravitationalfield is never switched off in this limit. In fact, B p ( η ) does not asymptotically approach a constantand one has to re-diagonalize the free Hamiltonian at each new value of η , as η →
0. (Note that inthe above UV limit the gravitational field is effectively switched off because high energy harmonicsare not sensitive to the comparatively small curvature of the background space.) Hence, naively even the Bessel functions, out- (in-)modes, do not provide a proper quasi-particle description inthe future (past) infinity of the EPP (CPP).
B. Digression on particle interpretation in de Sitter space
It is probably worth stressing here that throughout these notes we will avoid using the notionof particle, unless this notion is meaningful. We will concentrate on the behavior of the correlationfunctions. However, let us make here some comments about the particle interpretation.In curved space-times one usually avoids the use of the notion of particle because it is believedto be an observer dependent phenomenon. In fact, different observers may detect different particlefluxes. However, one should separate the Unruh effect [64] from what we would like to call as thereal particle production. In Minkowski space, both inertial and noninertial observers see the samestate — Minkowski (Poincar´e invariant) vacuum. However, while an inertial observer sees it as theempty space, a noninertial one sees it as the thermal state. That is due to the specific correlation ofthe vacuum fluctuations along its worldline [65], [66]. Note that there is no nontrivial gravitational9field present in the circumstances under consideration because in flat space the Riemanian tensoris exactly zero.The real particle creation is due to a change of the ground state under the influence of quantumeffects in a non-trivial background field. Then a flux is seen by all sorts of observers. That isexactly what happens in the strong electric field, in dS space and in the collapsing background.Rephrasing that, we would like to say here that, while in Minkowski space there is one type ofobservers that does not see any particle flux, in dS space there is no such an observer that seesnothing. On general grounds, we expect that the least flux is seen by inertial observers — theydo not see the extra Unruh type of flux, so to say. (Note here that the least possible flux in agiven space-time is an observer independent/invariant notion, i.e. is a characteristic feature of thegiven space–time.) Apart from that, if we see strong backreaction in a noninertial frame it is notclear whether it should be attributed to the background gravitational field or to the extra non-gravitational force acting on the corresponding observer. While in an inertial frame the situationis unambiguous.In any case, independently of whatever name we use for different quantities that are calculatedbelow, all of them are just components of correlation functions. And at the end of the day theobjects that we calculate are just correlation functions. However, the notion of particle sometimesis convenient for the physical interpretation of various equations that will appear below.So, what do we mean by particle? In general if the free Hamiltonian of a theory is diagonalthen one indeed can have a particle interpretation. Furthermore, if for some reason the anomalousquantum average h a p a − p i is strongly suppressed in comparison with h a + p a p i then one also can givea meaning to particle like excitations, as we will see below.Then, in principle at every given moment of time, η , one can make an instantaneous Bogolyubovrotation as follows [68]: b ~p ( η ) = α p ( η ) a ~p + β p ( η ) a + − ~p , b + ~p ( η ) = α ∗ p ( η ) a + ~p + β ∗ p ( η ) a − ~p , (22)with α p ( η ) = s A p + Ω p p and β p ( η ) = B ∗ p A p + Ω p α p ( η ) , (23)where Ω p ( η ) = q A p − | B p | . The Hamiltonian (21) becomes diagonal H ( η ) = Z d ~p Ω p ( η ) h b + ~p ( η ) b ~p ( η ) + h.c. i . (24)The rotated harmonics are¯ g p ( η ) = α ∗ p g p − β ∗ p g ∗ p = i η h p + m η i ˙ g p − i q p + m η g p (cid:12)(cid:12)(cid:12) ˙ g p − i q p + m η g p (cid:12)(cid:12)(cid:12) . (25)0This allows to have a particle interpretation around any given moment of time η .Note that the new creation and annihilation operators, b ~p and b + ~p , depend on time η . But a ~p and a + ~p are time independent in the free theory — all their time dependence is, then, gone into theharmonics g p ( η ). They start to depend on time if one turns on interactions.One reason to avoid using b ’s is that if one knows all expressions in terms of a ’s then it is nothard to restor their form in terms of b ’s. Another reason is that as we will see, if m > ( D − / a ’sdo provide a proper quasi-particle description in the IR limit. In fact, it will happen that for somechoice of harmonics, h a p a − p i is suppressed in comparison with h a + p a p i , as the system approachesfuture infinity. Hence, one does not really need to make the rotation to b ’s. C. Free waves in global de Sitter space
In global dS space the KG equation is as follows: (cid:20) − ∂ t + ( D −
2) tanh( t ) ∂ t + ∆ D − (Ω)cosh ( t ) − m (cid:21) φ ( t, Ω) = 0 . (26)Here ∆ D − (Ω) is the ( D − φ j, ~m ( t, Ω) = g j ( t ) Y j, ~m (Ω), where ∆ D − (Ω) Y j, ~m (Ω) = − j ( j + D − Y j, ~m (Ω),and Y j, ~m (Ω) are ( D − ~m is the multi-index enumerating themin the dimension grater than two.The equation for g j ( t ), following from (26), can be reduced to the hypergeometric one. Onepossible its solution is [67]: g ( in ) j ( t ) = 2 j + D − √ µ cosh j ( t ) e ( j + D − ± i µ ) t F (cid:18) j + D − , j + D − ∓ i µ ; 1 ∓ i µ ; − e t (cid:19) . (27)Where, from now on, F ( a, b ; c ; x ) is the hypergeometric function of the (2 ,
1) type and µ was definedin the previous subsection. These harmonics behave as single waves at the past infinity, t → −∞ ,of global dS space: g ( in ) j ( t ) ∼ e D − t e ∓ i µ t . That is the reason why they are referred to as in-harmonics. At the same time, as t → + ∞ , thesemodes behave as g ( in ) j ( t ) ∼ e D − t (cid:0) C e ∓ i µ t + C e ± i µ t (cid:1) , where C , are both non-zero complex constants in even dimensional dS space. In odd dimensionaldS spaces C = 0 [67]. The out-harmonics in global dS are as follows g ( out ) j ( t ) = h g ( in ) j ( − t ) i ∗ . Theirname is justified by the observation that they behave as single waves at future infinity.1Another peculiar type of harmonics in global dS space is given by the so-called Euclidian ones[2], [19], [67]. They are defined as: g ( E ) j ( t ) = 2 j + D − i − j + D − √ µ cosh j ( t ) e ( j + D − ± i µ ) t ×× F (cid:18) j + D − , j + D − ± i µ ; 2 j + D −
1; 1 + e t (cid:19) (28)and are regular on the lower hemisphere after the analytical continuation, t → i ( θ − π/ φ ( E ) j, ~m (cid:0) X (cid:1) = h φ ( E ) j, ~m ( X ) i ∗ , where X is theantipodal point of X .All the aforementioned mode functions in global dS space belong to the one-parameter familyof the so-called α -harmonics [2], [67]: φ ( α ) j, ~m ( X ) = 1 √ − e α + α ∗ (cid:20) φ ( E ) j, ~m ( X ) + e α φ ( E ) j, ~m ( X ) (cid:21) , (29)where α is a complex number.Note the coincidence, after the identification η ∓ = e ± t , of the above behavior of the global dSharmonic functions with that of the modes at the past and future of the CPP and EPP. In fact,under such an identification the metric of global dS space at its past and future infinity can bewell approximated by those of the CPP and EPP, correspondingly. Hence, there is a one-to-onecorrespondence between harmonic functions in the EPP (CPP) and α -modes in global dS.The free Hamiltonian of the four-dimensional theory can be written as: H ( t ) = 12 X j, ~m h A j ( t ) a + j, ~m a j, ~m + B j ( t ) a j, ~m a j, − ~m + h.c. i ,A j ( t ) = cosh ( t )2 (cid:26) | ˙ g j | + (cid:20) L ( L + 2)cosh ( t ) + m (cid:21) | g j | (cid:27) ,B j ( t ) = cosh ( t )2 (cid:26) ˙ g j + (cid:20) L ( L + 2)cosh ( t ) + m (cid:21) g j (cid:27) × (a phase) . (30)There is no choice of harmonics which allows to diagonilize this Hamiltonian neither in the pastnor in the future infinity because in global dS space the background field is never switched off. Infact, B j ( t ) does not approach a constant neither in the past nor in the future infinity of global dSspace. D. Green functions in de Sitter space
We continue with the construction of two-point correlation functions. The Wightman function, h φ ( X ) φ ( X ) i , is a solution of the homogeneous KG equation: (cid:0) (cid:3) − m (cid:1) G ( X , X ) = 0. Becauseof the dS isometry invariance, it should be a function of the invariant distance between X and X .The KG operator, (cid:0) (cid:3) − m (cid:1) , when acting on a function of Z = Z can be reduced to [19], [2]:2 (cid:2)(cid:0) Z − (cid:1) ∂ Z + D Z ∂ Z + m (cid:3) G ( Z ) = 0 . (31)This equation coincides with that for the Wightman function on the D -dimensional sphere. We justhave to keep in mind that in the latter case Z is the spherical distance rather than the hyperbolicone. The same equation is also valid in anti-dS and Euclidian anti-dS (Lobachevsky) space. Onejust has to change the sign of m term, due to the change of the sign of the curvature H , andkeep in mind that Z is the hyperbolic distance in the corresponding space.Eq. (31) has three singular points Z = ± , ∞ in the complex Z -plain. Hence, it is not hardto recognize in it the hypergeometric equation. After the transformation to the new variable, z = (1 + Z ) /
2, one puts the singular points into their standard positions, z = 0 , , ∞ . Then, thegeneric solution of this equation is: G W ( Z ) = A F (cid:18) D −
12 + i µ, D − − i µ ; D Z (cid:19) ++ A F (cid:18) D −
12 + i µ, D − − i µ ; D − Z (cid:19) . (32)Here A , are some complex constants and µ was defined above. The two hypergeometric functionsin (32) behave, when Z → ±
1, as follows: F (cid:18) D −
12 + i µ, D − − i µ ; D ± Z (cid:19) ∼ ∓ Z ) D − . (33)Also they have the branching point at Z → ∞ : G W ( Z ) ∼ B + Z − D − + i µ + B − Z − D − − i µ . Here B ± are some complex constants which depend on A , . Thus, G W ( Z ) is an analytical functionon the complex Z -plain with two branch cuts going from Z = ± G W ( Z ) has? Is there any state for which G W ( Z ) looks as in (32)? We are going to address these questions now.
1. Two-point correlation functions in global de Sitter space
We first scetch the situation in global dS and then continue with a bit more extensive discussionof the EPP (CPP) case. We choose some solution of the corresponding KG equation and, thus,specify α -modes. Then, we define the α -vacuum as the state which is annihilated by the corre-sponding annihilation operators: a ( α ) j, ~m | α i = 0. For every choice of α there is the corresponding α -vacuum [2]. Then, one can construct the two point Wightman function as G α ( X , X ) ≡ h α | φ ( X ) φ ( X ) | α i , α -harmonic expansion of φ ( X ). It is straightforward to show (using,e.g., [70]) that away from its singularity points this function looks as (32) with such A , whichdepend on α . At the same time Z = Z is the hyperbolic distance, which is expressed via theglobal coordinates of X and X .For the Euclidian vacuum we have that A = 0 and the singularity point of the correspondingWightman function G E ( Z ) is at Z = 1. From (33) one can see that it is the standard UVbehavior, when X is sitting on the light-cone whose apex is at X . The demand that this UVsingularity should be the same as in flat space allows to fix simultaneously the A coefficient in(32) and the ǫ -prescription (the resolution of the singularity): G E ( Z ) = G ( Z − i ǫ sign ∆ t ) , where G ( Z ) ≡ Γ (cid:0) D − + i µ (cid:1) Γ (cid:0) D − − i µ (cid:1) (4 π ) D Γ (cid:0) D (cid:1) F (cid:18) D −
12 + i µ, D − − i µ ; D Z (cid:19) . (34)The same value of A also follows from the proper normalization of the Euclidian harmonics.The ǫ -prescription for the α -harmonics follows after the Bogolyubov rotation (29). The resultis as follows [67]: G α ( Z ) = 11 − e α + α ∗ h G ( Z − i ǫ sign ∆ t ) + e α + α ∗ G ( Z + i ǫ sign ∆ t ) ++ e α ∗ G ( − Z + i ǫ sign ∆ t ) + e α G ( − Z − i ǫ sign ∆ t ) i , (35)where G ( Z ) is defined in (34). If one puts ǫ = 0 in this expression, he can reproduce (32) withgeneric A , .At the same time ǫ -prescription for the T-ordered (Feynman) propagator in the Euclidian vac-uum is G T ( Z ) = G ( Z − i ǫ ). This correlation function is just the analytical continuation of thepropagator on the sphere in the complex Z -plain.
2. Two-point correlation functions in the Poincar´e patches
To define the Wightman function in the EPP or CPP, one also has to pick up a solution of theBessel equation and define the corresponding vacuum, a ~p | vac i = 0. Then, doing the same as itwas done above, one obtains the Fourier expansion of the correlation function: G W ( X , X ) = Z d D − ~p e i ~p ( ~x − ~x ) ( η η ) D − h ( pη ) h ∗ ( pη ) . (36)Using [70], one can calculate this integral for different choices of h ( pη ). 6.672.1-4 of [70] can be usedfor the calculation in the case of 2D, and generalizations to higher dimensions are straightforward.There are two conclusions that follow. First one is that G W depends on the invariant hyperbolicdistance Z expressed via η , and ~x , as (15). And the second one is that different solutions of4the Bessel equation, h ( pη ), are in one–to–one correspondence with the concrete values of A , in(32) [2].In particular, for the BD state we have that A = 0. Thus, this state in the EPP and CPPleads to the same propagator as the Euclidian state in global dS space. Then, for the BD statethe singularity of G W ( Z ) is at Z = 1 and corresponds to the UV limit of the physical momentum, pη → ∞ . Hence, similarly to the flat space case, for the integral in (36) to be properly defined atthe singularity, there should be an appropriate shift as follows: η − η → η − η ± i ǫ . The signof this shift depends on which one among η or η is grater than the other. Thus, the Wightmanfunction in the BD state is also defined by (34). The difference is that now Z is expressed viathe EPP (CPP) coordinates of X and X and ∆ t should be substituted by ∓ ∆ η ± . Here the “ − ”sign for the EPP is due to the reverse order of the time flow.After a Bogolyubov rotation to other modes in the EPP (CPP) A becomes nonzero and A is changed. Thus, the residue of the singularity at Z = 1 is changed and also appears anothersingularity at Z = −
1. Recalling that Z = − Z , one can conclude that another singularitycorresponds to the situation when X is sitting on the light-cone with the apex at X — antipodalpoint of X . This singularity is causally disconnected from the one at Z = 1. In fact, a light raypassing through any point X on the hyperboloid on fig. 1 is a generatrix of this hyperboloid. Twogeneratrix lines crossing at X never intersect those which are crossing at X . This can be seenfrom the Penrose diagram.Thus, for the other harmonics in the EPP and CPP the Wightman functions are also given by(35), where instead of the Euclidian harmonics in (29) one should use the BD ones. E. Digression on an alternative quantization
There is a different way of field quantization in dS space . While in flat space this procedureleads to the same result, in dS space it provides an alternative quantization.Consider for the illustrative reasons two-dimensional scalar field in global dS space: S = 12 Z dt Z π dθ cosh( t ) (cid:20) − ˙ φ + m φ + 1cosh ( t ) ( ∂ θ φ ) (cid:21) . (37)Let us Fourier expand the field in the spatial direction φ ( t, θ ) = P + ∞ p = −∞ g p ( t ) e i p θ , where g p obeysthe condition g p = g ∗− p because φ is real. In terms of g p the Lagrangian is: L = π + ∞ X p =0 cosh( t ) (cid:20) ˙ g p ˙ g ∗ p − (cid:18) m + p cosh ( t ) (cid:19) g p g ∗ p (cid:21) . (38)If one defines g p = √ π ( q p + iQ p ), M ( t ) = cosh( t ) and ω p ( t ) = m + p cosh ( t ) , then he can writethe corresponding Hamiltonian as follows: I would like to thank V.Losyakov and A.Morozov for the discussions on this issue. H = + ∞ X p =0 " M ( t ) (cid:0) p p + P p (cid:1) + M ( t ) ω p ( t )2 (cid:0) q p + Q p (cid:1) . (39)Here p p and P p are momenta conjugate to q p and Q p , correspondingly. Then, via the definition ofthe operators a p = 1 √ p p − iq p ) and A p = 1 √ P p − iQ p ) , one can rewrite the Hamiltonian as: H = 12 + ∞ X p =0 (cid:26)(cid:20) M ( t ) + M ( t ) ω p ( t ) (cid:21) (cid:2) a + p a p + A + p A p (cid:3) + (cid:20) M ( t ) − M ( t ) ω p ( t ) (cid:21) (cid:2) a p + A p (cid:3) + h.c. (cid:27) . (40)It is straightforward to show, however, that this way of quantization leads to propagators thatare not dS-invariant because a ’s and A ’s here depend on time. This makes such a procedureinappropriate for our considerations. In fact, our goal is to understand if the dS isometry can berespected at all stages of quantization. Hence, we would not like to break it by the choice of anon-invariant vacuum state. However, otherwise this way of quantization is perfectly sensible andis also worth studying. IV. LOOPS IN DE SITTER SPACE QFT
In this section we study one-loop contributions to propagators and vertices in the Poincar´epatches and in global dS space.
A. Brief introduction to the Schwinger–Keldysh diagrammatic technique
Because free Hamiltonians in global dS and in the EPP (CPP) depend on time, the systemunder consideration is in a nonstationary state and one has to apply Schwinger–Keldysh (SK) (akain-in, aka nonstationary) diagrammatic technique. The systematic introduction to it can be foundin [59] or [60]. To set the notations we will sketch here the general physical motivation for thistechnique.Suppose one would like to calculate the expectation value of an operator O at some moment oftime t : hOi ( t ) ≡ D Ψ (cid:12)(cid:12)(cid:12) T e i R tt dt ′ H ( t ′ ) O T e − i R tt dt ′ H ( t ′ ) (cid:12)(cid:12)(cid:12) Ψ E , (41)6where H ( t ) = H ( t )+ H int ( t ) is the full Hamiltonian of a theory; while T denotes the time-ordering, T is the reverse time-ordering; t is an initial moment of time; | Ψ i is an initial state. The initialvalue of hOi ( t ) is supposed to be given in the setup of the problem. The expression (41) isvalid both in the Heisenberg picture, when the evolution operators are attributed to O , and in theSchrodinger one, when they are attributed to the bra and ket states. The generalization of ourconsiderations to multiple operators under the average is straightforward.After the transformation to the interaction picture, we get [59]: hOi ( t ) = (cid:10) Ψ (cid:12)(cid:12) S + ( t, t ) O ( t ) S ( t, t ) (cid:12)(cid:12) Ψ (cid:11) = (cid:10) Ψ (cid:12)(cid:12) S + ( t, t ) T [ O ( t ) S ( t, t )] (cid:12)(cid:12) Ψ (cid:11) == (cid:10) Ψ (cid:12)(cid:12) S + ( t, t ) S + (+ ∞ , t ) S (+ ∞ , t ) T [ O ( t ) S ( t, t )] (cid:12)(cid:12) Ψ (cid:11) == (cid:10) Ψ (cid:12)(cid:12) S + (+ ∞ , t ) T [ O ( t ) S (+ ∞ , t )] (cid:12)(cid:12) Ψ (cid:11) , (42)where S ( t, t ) = T e − i R tt dt ′ H int ( t ′ ) ; O ( t ) and H int ( t ) are the same operators as were defined above,but written in the interaction picture.To perform the first step in (42) we have used the Baker-Hausdorff formula: e A + B = T exp (cid:26)Z dt e − t B Ae t B (cid:27) e B (43)which follows from the logarithmic t derivative of the operator G ( t ) = e t ( A + B ) e − t B : G ( t ) − d t G ( t ) = e − t B A e t B . To perform the step on the second line of (42) we had inserted the following resolution of the unitoperator: 1 = S + (+ ∞ , t ) S (+ ∞ , t ). That allows one to extend the original evolution (from t to t and back) to that which goes from t to future infinity and back. We put the operator O ( t ) onthe forward going part of the time contour.To understand the meaning of the technique in question, let us slightly change the problem.We adiabatically turn on the interaction term, H int , after t , i.e., | Ψ i does not evolve before t .Then, one can rewrite the expectation value (42) as follows: hOi t ( t ) = (cid:10) Ψ (cid:12)(cid:12) S + (+ ∞ , −∞ ) T [ O ( t ) S (+ ∞ , −∞ )] (cid:12)(cid:12) Ψ (cid:11) . (44)A good question is if one can take t to past infinity, t → −∞ , i.e. to get rid of the dependenceof hOi t ( t ) on t . The seminal example when one can do so is as follows: The free Hamiltonian, H , does not depend on time and | Ψ i coincides with its ground state | vac i , H | vac i = 0. One alsoassumes that the interaction term is adiabatically switched off at future infinity — after the time t . If | vac i is the true vacuum state of the free theory, then, by adiabatic turning on and then switch-ing off the interactions, one cannot disturb such a state, i.e., h vac | S + (+ ∞ , −∞ ) | excited state i = 0,while |h vac | S + (+ ∞ , −∞ ) | vac i| = 1. Hence,7 hOi ( t ) = X state (cid:10) vac (cid:12)(cid:12) S + (+ ∞ , −∞ ) (cid:12)(cid:12) state (cid:11) h state | T [ O ( t ) S (+ ∞ , −∞ )] | vac i == (cid:10) vac (cid:12)(cid:12) S + (+ ∞ , −∞ ) (cid:12)(cid:12) vac (cid:11) h vac | T [ O ( t ) S (+ ∞ , −∞ )] | vac i == h vac | T [ O ( t ) S (+ ∞ , −∞ )] | vac ih vac | S (+ ∞ , −∞ ) | vac i . (45)To perform the first step in (45), we have inserted the resolution of unity 1 = P state | state i h state | ,where the sum is going over the complete basis of eigen-states of H . To perform the second step,we have used that | vac i is the only state from the sum which gives a non-zero contribution.Thus, the dependence on t disappears and we arrive at the expressions which contain only T-ordering (and no any T-orderings), i.e., we obtain the standard Feynman diagrammatic technique.Note that one can also apply the SK technique in the stationary situation because then the T-ordered expressions just cancel out vacuum diagrams.However, if | Ψ i is not a ground state and/or H depends on time, one cannot use the abovemachinery and has to deal directly with (44) or (42). If one knows the matrix element A = (cid:10) Ψ (cid:12)(cid:12) S + ( t , t ) (cid:12)(cid:12) Ψ (cid:11) for arbitrary t , and generic states | Ψ , i (which do not have to belong to the same Fock space),then one can calculate (42) with the use of such a generalized Feynman technique. Unfortunately,usually there are no algorithmic tools to calculate such matrix elements as A or even to deal withtheir unusual divergences. In this case the efficient method is the so-called SK technique, whereone has to perturbatively expand both S and S + under the quantum average. Many comparativelysimple and interesting examples of the application of this technique are presented in [60].We continue with the concrete example of real massive scalars with λ φ self-interaction. Wehave chosen the theory with such an unstable potential just to simplify the equations becauseeffects, that we consider below, are not affected by such an instability. The situation in the stable λ φ theory is described in [58] and is similar to the case under consideration (see the last section).The functional integral for the theory in question can be derived in the standard manner. Thefunctional integral form of (42) is: hOi ( t ) = Z D ϕ + ( x ) D ϕ − ( x ) (cid:10) ϕ + | ρ ( t ) | ϕ − (cid:11) Z ϕ − ϕ + D φ + ( t, x ) D φ − ( t, x ) O ( t ) ×× exp (cid:26) i Z + ∞ t dt Z d D − x p | g | (cid:20) ( ∂ µ φ − ) + m φ − + λ φ − − ( ∂ µ φ + ) − m φ − λ φ (cid:21)(cid:27) , (46)where h ϕ + | ρ ( t ) | ϕ − i is the matrix element of the initial density matrix, which in our case is ρ ( t ) = | Ψ i h Ψ | . While φ + is defined on the direct side of the time contour, φ − belongs to itsreverse side; ϕ ± are initial/final values of φ ± at t , correspondingly. All these complications aredue to the simultaneous presence of S and S + in (42).8Also it is worth stressing here that in the nonstationary situation one usually cannot take t → −∞ , unless the system has a stationary state, towards which it has to evolve inevitably. Wewill encounter such a strongly nonstationary situation at one loop in the CPP and in global dS.At the same time, in the EPP we will encounter an unusual stationary situation — there t canbe taken to past infinity.From the functional integral (46) one can deduce that in the SK technique every vertex carriesthe “ ± ” index, depending on whether it belongs to the “+” or “ − ” sides of the time contour. Asa result, if every particle is described by the propagator matrix,ˆ G ( X , X ) = G ( X , X ) G − ( X , X ) G − + ( X , X ) G −− ( X , X ) ! , (47)all loop expressions can be written in a matrix form (see, e.g., [69] and the next subsection forthe details). The mixed “ ± ” propagators appear because of the presence of the non-trivial initialdensity matrix [60].With the use of the Wightman function all of the constituents of the propagator matrix can bewritten as follows G − + ( X , X ) = i h φ − ( X ) φ + ( X ) i = i h ˆ φ ( X ) ˆ φ ( X ) i ,G − ( X , X ) = i h φ + ( X ) φ − ( X ) i = i h ˆ φ ( X ) ˆ φ ( X ) i ,G ( X , X ) = h T ˆ φ ( X ) ˆ φ ( X ) i = θ ( t − t ) G − + ( X , X ) + θ ( t − t ) G − ( X , X ) ,G −− ( X , X ) = h T ˆ φ ( X ) ˆ φ ( X ) i = θ ( t − t ) G − ( X , X ) + θ ( t − t ) G − + ( X , X ) . (48)They obey one relation G − + G − + = G + G −− . To reduce the number of propagators it isconvenient to perform the Keldysh rotation [59]: φ cl φ q ! = [ φ + + φ − ] φ + − φ − ! = ˆ R φ + φ − ! , where ˆ R =
12 12 − ! . (49)Then, the action becomes: S = Z + ∞ t dt Z d D x p | g | (cid:20) ∂ µ φ cl ∂ µ φ q + m φ cl φ q + λ (cid:18) φ q φ cl + 14 φ q (cid:19)(cid:21) . (50)At the same time the propagator matrix gets converted into:ˆ D ( X , X ) ≡ ˆ R ˆ G ( X , X ) ˆ R T = i D K ( X , X ) D R ( X , X ) D A ( X , X ) 0 ! , (51)where9 Figure 5: While the solid line corresponds to φ cl , the dashed one — to φ q . D RA ( X , X ) = G ( X , X ) − G ±∓ ( X , X ) == θ ( ± ∆ t ) [ G ∓± ( X , X ) − G ±∓ ( X , X )] == ± θ ( ± ∆ t ) h [ φ ( X ) , φ ( X )] i , ∆ t = t − t (52)are the retarded and advanced propagators. They define the spectrum of excitations in the theoryunder consideration. At tree level they do not depend on the state with respect to which theaveraging is done because the commutator, [ · , · ], of φ ’s is just a c-number. The Keldysh propagatoris: D K ( X , X ) = − i (cid:2) G − + ( X , X ) + G − ( X , X ) (cid:3) = − i h{ φ ( X ) , φ ( X ) }i , (53)where {· , ·} is the anti-commutator. Note that while the retarded propagator is given by D R ∼h φ cl φ q i in the rotated variables, the Keldysh one is as follows D K ∼ h φ cl φ cl i . Although it is notobvious from the action (50), the Keldysh propagator is not trivial because of the initial densitymatrix in (46) (see, e.g., [60] for the explanations). The Feynman rules for (50) are depicted onfig. 5.As we will see below, the Keldysh propagator is sensitive to the time evolution of the backgroundstate and essentially is a classical quantity. That in particular, explains the adapted in condensedmatter theory notations that φ cl is the “classical” field, while φ q is the “quantum” one [60].One last point which should be stressed here is that the SK technique, unlike the Feynman one,is strictly causal. In terms of, e.g., Eq. (46) that means that all contribution to the expectationvalue in question comes from the causal past of time t . Rephrasing this, the result of the calculation0of a quantity with the use of the SK technique is a solution of a Cauchy problem whose initial dataare set up by the tree level value of the quantity under study. B. On de Sitter isometry invariance at loop level
To start with, we show that loop corrections to the BD state in the EPP are dS isometryinvariant . For this problem we prefer to use the SK technique before the Keldysh rotation (49).Then, e.g., the one-loop correction to the propagator matrix can be written as:ˆ G ( Z XY ) = λ Z [ dW ] Z [ dU ] ˆ G ( Z XW ) ˆΣ ( Z W U ) ˆ G ( Z UY ) (54)where ˆ G , ( Z ) = G , ( Z ) G , − ( Z ) G , − + ( Z ) G , −− ( Z ) ! and ˆΣ ( Z ) = (cid:2) G ( Z ) (cid:3) (cid:2) G − ( Z ) (cid:3) (cid:2) G − + ( Z ) (cid:3) (cid:2) G −− ( Z ) (cid:3) ! , (55)and the measure is written in terms of the embedding coordinates of the ambient Minkowski space[ dW ] = d ( D +1) W δ (cid:0) W A W A − (cid:1) θ (cid:0) W − W D (cid:1) . It is equivalent to dη + η D + d D − ~x + after the substitution of the EPP coordinates of W A . This formulafor ˆ G is valid for any dS-invariant α -vacuum. Note that in (54) the dS isometry is naively brokenby the presence of the Heavyside θ -function, which restricts to the EPP.As follows from the discussion in the previous section, for the BD state we have that [35]: G [ Z ] = G [ Z + i ǫ ] , G − [ Z ] = G [ Z − i ǫ sign ( η − η )] ,G −− [ Z ] = G [ Z − i ǫ ] , G − + [ Z ] = G [ Z + iǫ sign ( η − η )] . (56)Here G ( Z ) is defined in (34). (Note the reverse time flow in the EPP.)Several comments are in order at this point. First, it is not hard to check that for the BD statethe UV divergence in (54) is the same as in flat space. For the other vacua this is not the case.Second, all the arguments of the present subsection are valid only if there are no IR divergences inthe loop integrals. We will see in the next subsection that in the EPP there are no IR divergencesin the field theory under consideration. There are only large IR contributions.Let us examine now a variation of ˆ G under a transformation of SO ( D,
1) isometry group whichchanges arguments of θ -functions in [ dW ] and [ dU ]. (Here we reproduce the arguments of [52].)Let us perform an infinitesimal rotation around X towards, e.g., X : X D → X D − ψ X . Taylorexpanding the integration measure up to the first order in ψ , we get: I would like to thank A.Polyakov for explaining this point to me. δ ψ Z [ dW ] · · · = Z d ( D +1) W δ (cid:0) W A W A − (cid:1) δ (cid:0) W − W D (cid:1) ψ W · · · == Z d (cid:0) W + W D (cid:1) d ( D − W δ (cid:0) W A W A − (cid:1) ψ W . . . and similarly for [ dU ] integration.Consider, e.g., the situation with d (cid:0) W + W D (cid:1) integral. Its integrand is a function of Z XW = −
12 ( X − X D ) ( W + W D ) −
12 ( X + X D ) ( W − W D ) + X a W a and of Z UW . Here W − W D = 0 because of the presence of δ (cid:0) W − W D (cid:1) in the integrationmeasure for the variation δ ψ ˆ G . Also (cid:0) X − X D (cid:1) ≥ G (cid:2) W + W D (cid:3) ≡ G (cid:2) Z (cid:0) W + W D (cid:1)(cid:3) , when considered as the function of (cid:0) W + W D (cid:1) , has the sameanalytical properties as those in the complex Z XW -plain. Furthermore, because of δ (cid:0) W − W D (cid:1) , η w in sign ( η x − η w ) goes to past infinity. Hence, we have a definite sign of the ǫ -prescription insideall propagators. The same is true for the functions of Z UW .As a result, due to the ǫ -prescription for the BD state the integrand of d (cid:0) W + W D (cid:1) is ananalytical function on the complex (cid:0) W + W D (cid:1) -plain with the cut going from 1 to infinity andslightly shifted to either the upper or lower plane (depending on whether it is + or − vertexin the contribution under consideration). Then, because propagators have a powerlike decay, as (cid:0) W + W D (cid:1) → ∞ , one can close the integration contour by an infinite semicircle in either lower orupper half of the complex (cid:0) W + W D (cid:1) -plain, correspondingly. As we just explained, the integrandis analytical function inside the contour, hence, the integral is zero.The same arguments work for the d ( U + U D ) integral and also for the infinitesimal rotations inthe other directions around X . Furthermore, it is straightforward to extend these arguments tohigher loops and higher-point correlation functions. Hence, in the case of the BD state the exactmatrix propagator ˆ G ( X , X ) is a function of Z only.The arguments of this subsection do not work for the other α -vacua because then tree levelpropagators have another cut going from Z = − ǫ -prescription. Thatspoils the analytical properties of the corresponding Wightman function in the complex Z -plain.Furthermore, because of the IR divergences, which we will discuss below, these arguments also donot work in the CPP for any dS-invariant vacuum.It is tempting to propose, however, that the dS isometry will be also respected in loops, if onewould consider that half of the entire dS space, which corresponds to t ≥ dW ] = d D +1 W δ (cid:0) W A W A − (cid:1) θ (cid:0) W (cid:1) . However, one of the possible transformation of SO ( D,
1) group, which moves the boundary of thissubspace, is the infinitesimal Lorentz boost in the W D direction: W → W + ψ W D . Under sucha boost the measure is changed by2 δ ψ Z [ dW ] · · · = Z d ( D +1) W δ (cid:0) W A W A − (cid:1) δ (cid:0) W (cid:1) ψ W D · · · == Z d D W δ D X i =1 W i W i − ! ψ W D . . . But the integrand here, considered as the function of any one among W i , i = 1 , . . . , D , ( G [ W i ] = G [ Z ( W i )]) does not have the same analytical properties as those in the complex Z W X -plain: Thecut in the complex W i -plain ( i is fixed) does not coincide with the one in the Z W X -plain because,unlike ( X − X D ) in the EPP, W i does not have a definite sign. Hence, by cutting global dS spaceat its neck, one breaks the dS isometry with any initial state. C. One–loop correction in the expanding Poincar´e patch
In this subsection we calculate leading IR one-loop contributions to propagators D R,A,K ( X , X )and vertices. Due to spatial homogeneity of the EPP itself and due to spatial homogene-ity of background states that we are going to consider, we find it convenient to performthe Fourier transformation of all quantities along the spatial directions: D K,R,A ( p | η , η ) ≡ R d D − x e i ~p ~x D K,R,A ( η , ~x ; η , ± ” indexes thatdistinguish coordinates of the EPP from the CPP.Below we do not care about UV divergences, i.e., we assume some kind of UV renormalizationand also assume that masses of the fields and coupling constants possess their physical renormalizedvalues. But it is probably worth stressing here that mixed expressions, with the partial Fouriertransformation along only the spatial sections, are not sensitive to the UV divergences. In fact,to reveal the latter even in the flat space Feynman diagrammatic technique one has to eithertransform back to the configuration space or to make the remaining Fourier transformation overthe time direction. Also to observe the UV divergences in the Keldysh technique in dS space onehas to go back to the configuration, ( η, ~x ), space.We consider such an IR limit in which pη → , where η = √ η η = e − ( τ + τ ) / , and η /η = e − ( τ − τ ) = const. This is the limit when both arguments of the propagators are taken to the future infinity, whilethe time distance between them is kept fixed. The presence of large IR corrections in such a limitmeans that there are growing with time contributions, as the system progresses towards futureinfinity.Let us start with the Keldysh propagator. The retarded (advanced) propagator and the verticeswill be described below. One-loop diagrams contributing to the Keldysh propagator are depictedon fig. 6. (It is straightforward to see that tadpole diagrams do not contain large IR corrections.)The corresponding expression is:3
Figure 6: One loop correction to the Keldysh propagator D K . D K ( p | η , η ) = λ Z d D − ~q (2 π ) D − Z Z η dη dη ( η η ) D ×× " D R ( p | η , η ) D K ( q | η , η ) D K ( | ~p − ~q | | η , η ) D A ( p | η , η ) ++2 D R ( p | η , η ) D R ( q | η , η ) D K ( | ~p − ~q | | η , η ) D K ( p | η , η ) ++2 D K ( p | η , η ) D K ( q | η , η ) D A ( | ~p − ~q | | η , η ) D A ( p | η , η ) −− D R ( p | η , η ) D R ( q | η , η ) D R ( | ~p − ~q | | η , η ) D A ( p | η , η ) −− D R ( p | η , η ) D A ( q | η , η ) D A ( | ~p − ~q | | η , η ) D A ( p | η , η ) . (57)Here η > η , is an early, η → + ∞ , moment after which the self-interactions are adiabaticallyturned on; the Keldysh propagator is D K ( p | η , η ) = ( η η ) D − Re [ h ( pη ) h ∗ ( pη )] , the retarded and advanced propagators are D R ( p | η , η ) = θ ( η − η ) 2 ( η η ) D − Im [ h ( pη ) h ∗ ( pη )] , and D A ( p | η , η ) = − θ ( η − η ) 2 ( η η ) D − Im [ h ( pη ) h ∗ ( pη )] , h h ∗ ] = h h ∗ + h ∗ h and Im [ h h ∗ ] = i [ h h ∗ − h ∗ h ].We will show below, that the leading IR contribution to D K ( p | η , η ) is hidden within thefollowing expression D K ( p | η , η ) ≈ ( η η ) D − h h ( pη ) h ∗ ( pη ) n p ( η ) + h ( pη ) h ( pη ) κ p ( η ) + c.c. i , where n p ( η ) ≈ − λ (2 π ) D − Z d D − q Z d D − q Z Z ηη dη dη ( η η ) D − δ ( ~p + ~q + ~q ) ×× h ∗ ( pη ) h ( pη ) h ∗ ( q η ) h ( q η ) h ∗ ( q η ) h ( q η ) , and κ p ( η ) ≈ λ (2 π ) D − Z d D − q Z d D − q Z ηη dη Z η η dη ( η η ) D − δ ( ~p + ~q + ~q ) ×× h ∗ ( pη ) h ∗ ( pη ) h ∗ ( q η ) h ( q η ) h ∗ ( q η ) h ( q η ) . (58)In deriving this expression from (57) we have used that in the IR limit in question one can neglectthe difference between η and η and substitute the average conformal time, η = √ η η , instead ofboth η and η , in the upper limits of the integrations over η and η . I.e., in (58) we dropped offsubleading terms of the order of λ log ( η /η ) or smaller, which may arise from integrals between η , and √ η η .Before performing the explicit calculation of (58) for the concrete choice of h ( x ), it is worthchecking what is happening in the flat space limit. To take this limit in (58) we just change η → t ,substitute η ( D − / h ( pη ) by the plane wave, e − i ǫ ( p ) t / p ǫ ( p ), and exchange √ g = 1 /η D for one.Then, as the upper limits of the time integrals in (58) go to future infinity, while the lower onesare taken to past infinity, the time integrals are converted into the δ -functions establishing theenergy conservation: The integrands of RR dq dq in the expressions for n p and κ p will contain thetime-line integrals of δ [ ǫ ( p ) + ǫ ( q ) + ǫ ( q )]. The argument of this δ -function is never zero. Hence,the whole expressions for n p and κ p are vanishing.Thus, we see that in the flat space limit the expressions for n p and κ p do vanish, which is theconsequence of the energy conservation. This gives us a hint for the physical interpretation of (58).In fact, these expressions can arise in the presence of the particle creation process, when thereis no energy conservation due to the background gravitational field. Also this observation showsthat for the case of the EPP the contributions to n p and κ p from the past infinity, i.e., from theregion where pη ≫ µ and q , η ≫ µ , are negligible: When qη ≫ µ the function g p ( η ) can be wellapproximated by the plane wave with the minor pre-exponential factor. Furthermore, this allowsone to safely take η to past infinity, η → + ∞ . In fact, dη , integrals in (58) are converging inthe generalized sense on their lower limits of integration. The origin of these observations can betraced back to the presence of the infinite blue shift towards the boundary between the EPP andCPP.5
1. Bunch–Davies vacuum
From now on we talk about the scalar fields from the principal series, m > ( D − /
2, unlessotherwise stated. We comment on the complementary series in the last section.So, we would like to estimate leading contributions to n p and κ p in (58) for the BD modes.First, we take the integrals over ~q with the use of the δ -functions and rename ~q → ~q . Second, itis straightforward to check that the largest IR contribution to (58) comes from the region where q ≫ p inside the integral over dq . Hence, we neglect p in comparison with q under the integralsin (58). That changes, however, the lower limits of integration over η , from η → + ∞ to µ/p .Note that by doing this we just neglect the contributions to n p and κ p from the high energy region pη , ≫ µ .Third, we make the following change of the integration variables in (58): q to x = qη , η to x = qη and leave η unchanged. As a result, the integration measure is converted into Z Z Z dq q D − dη dη ( η η ) D − · · · = Z Z Z dη η dx dx ( x x ) D − . . . . Note that once the integral over q was taken and ~p was neglected in comparison with ~q ≡ ~q , theintegrand of d D − q depends only on | ~q | .Now we can harmlessly extend the lower limits of integration over x , to infinity both in n p and κ p . Furthermore, in the expression for n p we can extend the upper limits of integration over x , to zero. In the expression for κ p the latter change is done only for x . These changes are harmlessbecause after them the integrals remain finite and pre-factors of the expressions, that we will findbelow, are just slightly changed by the contributions from the high energy region.Fourth, the integrals for n p and κ p are saturated around qη , ∼ µ , while pη , ≪ qη , . Hence,we can expand h ( pη , ) ≈ A + ( pη , ) iµ + A − ( pη , ) − iµ , where A ± = √ π e ± πµ ± iµ + Γ (1 ± iµ ) sinh ( ± πµ ) . And finally, we perform the integration over η from µ/p to η and keep in the expressions for n p and κ p only divergent, as pη →
0, terms. The result is n p ( η ) ≈ λ S D − (2 π ) D − log (cid:18) µp η (cid:19) Z Z ∞ dx dx ( x x ) D − ×× " | A + | (cid:18) x x (cid:19) iµ + | A − | (cid:18) x x (cid:19) iµ h ( x ) [ h ∗ ( x )] ,κ p ( η ) ≈ − λ S D − (2 π ) D − log (cid:18) µp η (cid:19) A + A − Z ∞ dx Z x ∞ dx ( x x ) D − ×× "(cid:18) x x (cid:19) iµ + (cid:18) x x (cid:19) iµ h ( x ) [ h ∗ ( x )] , (59)6where S D − is the volume of the ( D − h T αβ i does not contain any fluxes . In fact, the inverseFourier transformation of (58), with η -dependent n p and κ p from (59), will be a function of theinvariant geodesic distance between ( η , ~x ) and ( η , ~x ). One just has to bear in mind that in(59) we have kept inside n p and κ p the leading contributions in the limit under consideration. I.e.,we have forgot that in principle n p and κ p depend on η and η separately rather than on theircombination η = √ η η .However, to understand the physical meaning of the obtained expressions, let us for a momentforget about the dS-invariance of (58), (59). That makes sense for the following reasons: At thepast infinity of the EPP n p and κ p have the clear interpretation as the particle density per comovingvolume, h a + p a p i , and as the anomalous quantum average, h a p a − p i , correspondingly. Then, if wetake some non-zero initial value of n p , the dS isometry is broken even at tree level (by the initialstate). In fact, for non-zero, η -independent value of n p (even when κ p = 0) the inverse Fouriertransform of D K ( p | η , η ) depends on each its time argument separately rather than on the invariantgeodesic distance. Moreover, below we will repeat the calculation in the situation when the dSisometry is broken by an initial perturbation. The result will be the same as (58), but with a bitdifferent expressions for n p and κ p .Thus, having in mind what we have just said, the physical interpretation of (58), (59) is asfollows. If we start from the vacuum, then both n p and κ p are zero at past infinity. As we haveseen above, in the absence of the gravitational field they also remain zero in future infinity. However,once the gravitational field is turned on, they are generated in loops and are comparable to one,even though they have been zero initially and λ is much less than one. So the presence of thesequantities signals the particle creation in the EPP. Moreover, below we give arguments favoringthat κ p is the measure of the strength of the backreaction of quantum effects on the backgroundstate. Hence, its significant presence signals that the backreaction on the initial BD state is strong,i.e., in future infinity the system should relax to a different state from the BD one.
2. Contributions to the retarded and advanced propagators and to the vertices
Once we have understood the origin of the large IR corrections to the Keldysh propagator, letus consider the situation in the case of the retarded (advanced) propagator and vertices. This willbe important for the summation of the leading IR corrections from all loops.The one-loop diagram, that contributes to D R , is depicted on fig. 7. The correspondingexpression is: It is interesting to note, however, that a free floating detector will click in such a stationary state. This is justa consequence of the fact that in the stationary state the Wightman function depends only on the hyperbolicdistance Z , which is equal to Z = cosh( τ − τ ) if the spatial position of the detector is not changing. The detectorwill click because according to (33) the Wightman function has poles on the imaginary axis in the complex propertime-plain [65], [66]. Figure 7: One-loop correction to D R . For D A the diagram is mirror symmetric to this one.Figure 8: Here we show some of the diagrams. The other diagrams are just complex conjugate of those thatare presented here. D R ( p | η , η ) = λ Z d D − ~q (2 π ) D − Z Z ∞ dη dη ( η η ) D ×× D R ( p | η , η ) D R ( q | η , η ) D K ( | ~q − ~p | | η , η ) D R ( p | η , η ) . (60)Here one can immediately see that due to the presence of D R inside the loop and in the externallegs the limits of integration over η , are such that η > η > η > η . As a result, the loop integral(60) does not receive large corrections in such a limit when η /η is held fixed. In the differentlimit, when say η → + ∞ and η →
0, this integral can receive at most a correction proportional to λ log ( η /η ) [71], but it is not of interest for us here. The situation with the advanced propagatoris the same.8The diagrams that contribute to the vertex renormalization are depicted on fig. 8. We find itmore convenient to consider loop contributions to vertices in the SK technique before the Keldyshrotation (49). Then the expressions for these diagrams are as follows : λ −−− ( η , , ; p , , ) = ( − i λ ) ( η η η ) D − Z d D − p (2 π ) D − ×× h θ ( η − η ) h ∗ ( pη ) h ( pη ) + θ ( η − η ) h ∗ ( pη ) h ( pη ) ih θ ( η − η ) h ∗ ( | ~p + ~p | η ) h ( | ~p + ~p | η ) + θ ( η − η ) h ∗ ( | ~p + ~p | η ) h ( | ~p + ~p | η ) ih θ ( η − η ) h ∗ ( | ~p − ~p | η ) h ( | ~p − ~p | η ) + θ ( η − η ) h ∗ ( | ~p − ~p | η ) h ( | ~p − ~p | η ) i , and λ + −− ( η , , ; p , , ) = − ( − i λ ) ( η η η ) D − Z d D − p (2 π ) D − ×× h θ ( η − η ) h ∗ ( pη ) h ( pη ) + θ ( η − η ) h ∗ ( pη ) h ( pη ) i h ∗ ( | ~p + ~p | η ) h ( | ~p + ~p | η ) h ( | ~p − ~p | η ) h ∗ ( | ~p − ~p | η ) , (61)up to the factor of δ ( ~p + ~p + ~p ). Now it is straightforward to check that in the limit p i η i → i = 1 , ,
3, there are no large IR contributions to these expressions. The fastest way to make thisobservation is to put all the external momenta p i to zero and to see that, then, λ ±±± are all finite.At the same time, the one-loop correction to D K is divergent in this limit.
3. Other α -harmonics Other α -modes can be relevant in the IR limit, although they show wrong behavior in the UVlimit. That is a quite frequent situation in condensed matter physics: One notable instance is theBCS theory for superconductivity, where, to observe the Cooper pairing, one has to perform theBogolyubov rotation into harmonics, which, however, are inappropriate in the UV limit. Becauseof that it is instructive to perform the one-loop calculation for them.The leading IR contribution in this case also can be expressed in such a form as (58). For themost values of α , both n p and κ p receive large contributions which are similar to (59), but withdifferent prefactors of log( pη ). The only exception is given by the out-modes. In the latter case h ( x ) ≈ A x iµ , as x →
0, where A is some complex constant following from the proper normalizationof the harmonic functions and the expansion of the Bessel functions around zero.To see the peculiarity of the out-modes and to calculate n p and κ p in this case, one has toperform the same manipulations as have been done during the one-loop calculation for the BDharmonics. However, if one substitutes the corresponding h ( pη ) into (58), he should take intoaccount that they behave at past infinity, x → ∞ , as follows h ( x ) = 1 √ x (cid:2) A e i x + A e − i x (cid:3) , (62) I would like to thank F.Popov and V.Slepukhin for the discussions on this point. A , are some complex constants. Due to the interference terms between e ix and e − ix , theintegrals over x , do not converge fast enough: they are saturated in the vicinity of px , /q ∼ µ rather than at x , ∼ µ . (Note that according to our approximations p/q ≪ h ( px , /q ) = h ( pη , ) around zero. However, if we are in D = 4, wecan subtract from and then add to h ( x ), under these integrals, the value of the interference term, A A | x | : h ( x ) = h ( x ) − A A | x | + A A | x | . Then, the x , integrals of h ( x ) − A A | x | are saturated around x , ∼ µ and, hence, one canTaylor expand h ( px , /q ) around zero inside the corresponding expressions. At the same time,the contributions from the additional integrals of A A | x | are suppressed in the IR limit. In fact,due to extra powers of η , the integrals over dη are not divergent in the limit as pη →
0. Thus,performing these manipulations we fetch out the leading IR correction. In higher dimensions theprocedure is the same, but may demand a subtraction of higher powers of 1 / | x | . It is interestingto note that this kind of a problem does not appear in φ theory [58].Thus, the contribution to the two-point function is as follows: n p ( η ) ≈ λ S D − (2 π ) D − | A | log (cid:18) µp η (cid:19) Z Z ∞ dx dx ( x x ) D − (cid:18) x x (cid:19) iµ V ( x ) V ∗ ( x ) , where V ( x ) ≡ h ( x ) − A A | x | − . . . , (63)and κ p does not receive any large IR correction! As we will explain below, κ p is the measure of thebackreaction strength. Hence, this observation is crucial for the proper quasi-particle interpretationin future infinity. D. One-loop correction in the contracting Poincar´e patch
Let us consider now one-loop corrections in the CPP. Similarly to the case of the EPP, it is nothard to show that D R,A and vertices do not receive large IR corrections. At the same time, in thelimit pη → D K ( p | η , η ) can be written as (58). The crucialdifference, however, is that now 0 < η < η , < + ∞ . Also one has to exchange the positive andnegative energy harmonics because of the time reversal.To estimate the largest contribution to the integrals in n p and κ p , one has to perform the samemanipulations as we have been doing in the calculation on the EPP background. The results areas follows. First, unlike the case of the EPP, now η cannot be taken to past infinity, η → n p and κ p will be explicitly divergent. Second, the answer for n p and κ p depends on the value of pη = p √ η η :For the BD modes, which are out-harmonics in the CPP, the IR divergence is present both in n p and κ p . If pη ≪ µ , it is proportional to log ( η/η ) = τ − τ . That is just the proper time elapsed0from η = e τ to η = e τ . If, however, pη ≫ µ , then the divergence is proportional to log ( µ/pη ).The coefficients of these divergences in both situations are the same as in (59).For the Bessel or in-harmonics in the CPP, n p ( η ) diverges as log ( η/η ), if pη ≪ µ . At the sametime, when pη ≫ µ , the divergence in n p is as log ( µ/pη ). (The coefficients in both cases are thesame as in (63).) At the same time, in this case κ p ( η ) does not have any divergence or large IRcontribution.For any other type of α -harmonics the IR behavior of n p and κ p is similar to that of the BDmodes. E. One-loop correction in global de Sitter space
We continue with the one-loop calculation in global dS. In this case the situation with theretarded (advanced) propagator and with vertices is the same as in the EPP and CPP. Thus, weconsider here the Keldysh propagator. Following [56], we would like to show that in global dS spacethe moment of turning on self-interactions cannot be taken to past infinity. Below we are going towork with even dimensional dS spaces. In odd-dimensional global dS spaces the situation is a bitdifferent due to some cancelations [52], but in essential sense it is similar to the even-dimensionalcase. The existence of the particle creation in the odd-dimensional case can be traced back to thefact that also in odd dimensions it is not possible to diagonalize the corresponding free Hamiltonianonce and forever.Unlike higher dimensional dS space, in the 2D case the one-loop calculation in global coordinatescan be made quite explicit. So let us present it before continuing with the IR divergences in generaldimensions. Again, due to spatial homogeneity of 2D global dS space together with backgroundstates under consideration, we find it convenient to perform the Fourier transformation of allquantities along the spatial direction ϕ ∈ [0 , π ]: D K,R,A ( p | t , t ) ≡ R π dϕ e i p ϕ D K,R,A ( t , ϕ ; t , D K is as follows: D K ( p | t , t ) = λ ∞ X q = −∞ Z Z + ∞ t dt dt cosh( t ) cosh( t ) ×× " D R ( p | t , t ) D K ( q | t , t ) D K ( | ~p − ~q | | t , t ) D A ( p | t , t ) ++2 D R ( p | t , t ) D R ( q | t , t ) D K ( | ~p − ~q | | t , t ) D K ( p | t , t ) ++2 D K ( p | t , t ) D K ( q | t , t ) D A ( | ~p − ~q | | t , t ) D A ( p | t , t ) −− D R ( p | t , t ) D R ( q | t , t ) D R ( | ~p − ~q | | t , t ) D A ( p | t , t ) −− D R ( p | t , t ) D A ( q | t , t ) D A ( | ~p − ~q | | t , t ) D A ( p | t , t ) . (64)Here the Keldysh propagator is D K ( p | t , t ) = Re (cid:2) g p ( t ) g ∗ p ( t ) (cid:3) , D R ( p | t , t ) = θ ( t − t ) 2 Im (cid:2) g p ( t ) g ∗ p ( t ) (cid:3) , and D A ( p | t , t ) = − θ ( t − t ) 2 Im (cid:2) g p ( t ) g ∗ p ( t ) (cid:3) , correspondingly; g p ( t ) is the temporal part of a solution of the KG equation, which we do notspecify for the beginning; t is the moment when self-interactions are turned on.The above one-loop expression can be rewritten as: D K ( p | t , t ) = − λ g p ( t ) g ∗ p ( t ) X q Z Z + ∞ t dt dt cosh( t ) cosh( t ) g ∗ p ( t ) g p ( t ) ×× ( − θ ( t − t ) θ ( t − t ) " g q ( t ) g ∗ q ( t ) g | p − q | ( t ) g ∗| p − q | ( t ) + c.c. + " θ ( t − t ) θ ( t − t ) + θ ( t − t ) θ ( t − t ) g q ( t ) g ∗ q ( t ) g | p − q | ( t ) g ∗| p − q | ( t ) − c.c. −− λ g p ( t ) g p ( t ) X q Z Z + ∞ t dt dt cosh( t ) cosh( t ) g ∗ p ( t ) g ∗ p ( t ) ×× ( θ ( t − t ) θ ( t − t ) " g q ( t ) g ∗ q ( t ) g | p − q | ( t ) g ∗| p − q | ( t ) + c.c. + " θ ( t − t ) θ ( t − t ) − θ ( t − t ) θ ( t − t ) g q ( t ) g ∗ q ( t ) g | p − q | ( t ) g ∗| p − q | ( t ) − c.c. + c.c. (65)We need to find the leading contribution from this expression in the limit t ≡ t + t → + ∞ , t − t = const and t → −∞ . (We take t → −∞ just to check if this limit is smooth.) As in thecase of the EPP and CPP, we can substitute t , by t in the arguments of the θ -functions.One can estimate directly the expression in Eq. (65). But instead of doing this we apply adifferent procedure which allows us to find the leading corrections in any dimension. If t → + ∞ and t → −∞ , then the leading IR contributions to D K appear from the regions of integration over t , in the vicinity of t and t . As we have explained above, in these regions the global dS metriccan be well approximated by those of the EPP and CPP. It is this observation which allows one toestimate the one-loop correction in an arbitrary dimension. In fact, then g p ( t ) can be approximatedby g p ( t ) ≈ η D − + h + ( pη + ) , η + = e − t , t → + ∞ η D − − h − ( pη − ) , η − = e t , t → −∞ (66)for non-zero p . (In an arbitrary dimension p coincides with the index of the temporal part of theharmonics.) Then, as pη → pη →
0, where η = e − t and η = e t , the leading IR contributionto D K is hidden within [56], [62]:2 D K ( p | t , t ) = g p ( t ) g ∗ p ( t ) n p ( t ) + g p ( t ) g p ( t ) κ p ( t ) + c.c., where n p ( t ) ≈ − λ S D − (2 π ) D − Z d D − q Z Z η ∞ dη dη ( η η ) D − ×× h ∗ + ( pη ) h + ( pη ) h ∗ + ( q η ) h + ( qη ) h ∗ + ( | ~q − ~p | η ) h + ( | ~p − ~q | η ) −− λ S D − (2 π ) D − Z d D − q Z Z ∞ η dη η ( η η ) D − ×× h ∗− ( pη ) h − ( pη ) h ∗− ( qη ) h − ( qη ) h ∗− ( | ~q − ~p | η ) h − ( | ~q − ~p | η ) , and κ p ( t ) ≈ λ S D − (2 π ) D − Z d D − q Z η ∞ dη Z η ∞ dη ( η η ) D − ×× h ∗ + ( pη ) h ∗ + ( pη ) h ∗ + ( qη ) h + ( qη ) h ∗ + ( | ~q − ~p | η ) h + ( | ~q − ~p | η ) ++ 2 λ S D − (2 π ) D − Z d D − q Z ∞ η dη Z η η dη ( η η ) D − ×× h ∗− ( pη ) h ∗− ( pη ) h ∗− ( qη ) h − ( qη ) h ∗− ( | ~q − ~p | η ) h − ( | ~q − ~p | η ) . (67)Thus, IR corrections in global dS space are just sums of the contributions from the contracting(past infinity, i.e. around t ) and expanding (future infinity, i.e. around t ) Poincar´e patches of thewhole space.We continue with the description of the character of the IR divergences for different choicesof the α -modes. That may help to specify the convenient choice of the harmonics for the properdefinition of quasi–particles in the IR limit.We start with the Euclidian harmonics. In this case, h ∗− ( x ) = h + ( x ) ∝ H (1) iµ ( x ). Using that h + ( x ) ≈ A + x iµ + A − x − iµ , as x →
0, and performing the same manipulations as in the EPP andCPP, we obtain: n p ( t ) ≈ λ S D − (2 π ) D − log (cid:18) µ p η η (cid:19) Z Z ∞ dx dx ( x x ) D − ×× " | A + | (cid:18) x x (cid:19) iµ + | A − | (cid:18) x x (cid:19) iµ h ( x ) [ h ∗ ( x )] ,κ p ( t ) ≈ − λ S D − (2 π ) D − log (cid:18) µ p η η (cid:19) A + A − Z ∞ dx Z x ∞ dx ( x x ) D − ×× "(cid:18) x x (cid:19) iµ + (cid:18) x x (cid:19) iµ h ( x ) [ h ∗ ( x )] . (68)Note that to calculate the expression for κ p we have used the identity A ∗ + A ∗− = A + A − . Thus,for the Euclidian vacuum the leading IR divergence is proportional to log (1 /ηη ) = ( t − t ) and ispresent both in n p and κ p .At the same time, for the in-harmonics h − ( x ) is proportional to the Bessel function Y iµ ( x ).Hence, h − ( x ) ≈ A ∗ x − iµ , as x → h + ( x ) ≈ C + x iµ + C − x − iµ ,as x → C ± are some complex constants related to A ± viathe corresponding Bogolyubov α -rotation.3Then, the leading IR contribution in this case looks as: n p ( t ) ≈ λ S D − (2 π ) D − log (cid:18) µp η (cid:19) Z Z ∞ dx dx ( x x ) D − ×× " | C + | (cid:18) x x (cid:19) iµ + | C − | (cid:18) x x (cid:19) iµ V + ( x ) V ∗ + ( x ) ++ λ S D − (2 π ) D − log (cid:18) µp η (cid:19) Z Z ∞ dx dx ( x x ) D − | A | (cid:18) x x (cid:19) iµ V ∗− ( x ) V − ( x ) ,κ p ( t ) ≈ − λ S D − (2 π ) D − log (cid:18) µp η (cid:19) C ∗ + C ∗− Z ∞ dx Z x ∞ dx ×× ( x x ) D − "(cid:18) x x (cid:19) iµ + (cid:18) x x (cid:19) iµ V + ( x ) V ∗ + ( x ) . (69)where V ± ( x ) = h ± ( x ) − A A | x | − . . . and was defined above. (Dots here stand for the higher powersof 1 / | x | .) Note that for the in-harmonics n p ( t ) depends separately on t and t rather than on theirdifference ( t − t ) and κ p ( t ) does not diverge as t → −∞ .Similar conclusions are true for the out-harmonics. The crucial difference, however, is that inthe latter case κ p does diverge as t → −∞ , but is finite as t → + ∞ . For the other α –harmonics theleading IR contributions to n p and κ p also depend on t and t separately and diverge simultaneouslywhen t → + ∞ and t → −∞ .The presence of the IR divergence means that one cannot take the regulator, t , to minusinfinity. Thus, one has to fix an initial Cauchy surface at some finite t . (Such a surface does notnecessarily have to be taken somewhere around past infinity. One can put it, e.g., around the neckof global dS space.) But holding t fixed, violates the dS-isometry. V. SUMMATION OF LEADING INFRARED CONTRIBUTION IN ALL LOOPS
Thus, we see that λ log( η ) and λ log( η ) contributions can become large, as η → η → λ is very small. I.e., loop corrections are not suppressed in comparison with classical treelevel contributions to the propagators. That is true even if the dS isometry is respected in loops.Hence, to understand the physics in dS space, one has to sum unsuppressed IR corrections fromall loops.We start with the discussion of the situation in the EPP and then continue with the CPPand global dS space. In the EPP there is a separate interesting problem to sum dS-invariant IRcontributions to the correlation functions of the exact BD state (see, e.g., [87], [88]). As we pointedout in the Introduction, solution of this problem is not sufficient to make a definite conclusion aboutthe stability of dS space: We find it as more physically sensible to question the stability of dS spaceunder nonsymmetric perturbations.Hence, we propose to consider an initial nonsymmetric perturbation on top of the BD state.But we cannot just put an initial comoving density at past infinity of the EPP, because thenthe physical density will be infinite. One has to put the density perturbation at an initial Cauchy4surface which is different from the past infinity + ∞ > η >
0. Moreover, due to the UV divergencesthe comoving momentum also should be cutoff at the UV scale. But, as was explained above n ( pη )and κ ( pη ) are attributed to the comoving volume and, hence, do not change before pη ∼ µ .Their behaviour for pη > µ is not much different from that in flat space–time. Thus, cuttingsimultaneously comoving momentum and conformal time integrals effectively amounts to cut thephysical momentum integrals at µ . On the other hand, due to the symmetries of the EPP we canput an initial comoving density at an initial value of the physical momentum ( pη ) ∼ ν and cutoffall the integrals over the physical momentum at this value. Then, the dS isometry is broken bythe presence of a density on top of the dS-invariant BD state. (In the case of the CPP and globaldS space we do not even have to do this because the dS isometry is broken in loops for any initialstate.) We would like to trace the destiny of such a density perturbation and to understand theeffect of the large IR contributions on the initial state, as the system progresses towards futureinfinity. A. From the Dyson–Schwinger to the kinetic equation in the Poincar´e patches
To sum loop contributions one has to solve the system of Dyson-Schwinger (DS) equationsfor the propagators, self-energies and vertices. We would like to sum only powers of the leadingcontribution, λ log( pη ), and neglect suppressed terms, such as, e.g., λ log( pη ) or λ log( η /η )and etc.. As we have seen above, the retarded and advanced propagators do not receive large IRcontributions from the first loop. The same is true for the vertices. However, these quantities mayreceive large contributions from the second or higher loops. In fact, the latter may come from thosecorrections to the Keldysh propagator which appear at lower loops. But this just means that suchcontributions to D R,A and vertices will be suppressed by higher powers of λ . Thus, if we need toperform only the summation of the leading corrections, then we have to care only about the DSequation for the Keldysh propagator.In the following, we assume that D R,A and vertices take their tree level values (perhaps renor-malized by the UV contributions) and assume that all IR corrections go into IR exact D K . Then,the relevant DS equation acquires the form: D K ( p | η , η ) = D K ( p | η , η ) ++ λ Z d D − ~q (2 π ) D − Z Z ∞ dη dη ( η η ) D " D R ( p | η , η ) D K ( q | η , η ) D K ( | ~p − ~q | | η , η ) D A ( p | η , η ) ++2 D R ( p | η , η ) D R ( q | η , η ) D K ( | ~p − ~q | | η , η ) D K ( p | η , η ) ++2 D K ( p | η , η ) D K ( qη , η ) D A ( | ~p − ~q | | η , η ) D A ( p | η , η ) −− D R ( p | η , η ) D R ( q | η , η ) D R ( | ~p − ~q | | η , η ) D A ( p | η , η ) −− D R ( p | η , η ) D A ( q | η , η ) D A ( | ~p − ~q | | η , η ) D A ( p | η , η ) , (70)where D K ( p | η , η ) is the initial (tree level) value of the propagator. We propose the following5ansatz to solve this equation: D K ( p | η , η ) = ( η η ) D − d K ( pη , pη ) , where d K (cid:0) pη , pη (cid:1) = 12 h (cid:0) pη (cid:1) h ∗ (cid:0) pη (cid:1)(cid:20) n p (cid:0) η (cid:1)(cid:21) + h (cid:0) pη (cid:1) h (cid:0) pη (cid:1) κ p (cid:0) η (cid:1) + c.c.. (71)Here η = √ η η , n p ( η ) and κ p ( η ) are unknown functions to be defined by the equations underderivation. Below we also use the notation d − ( pη , pη ) = 2 Im [ h ( pη ) h ∗ ( pη )] to simplify theequations. The ansatz for the IR exact Keldysh propagator is inspired by the one-loop calculationand its physical interpretation.For general values of η and η the ansatz (71) does not solve (70). However, in the limit pη , → η /η = const one can neglect the difference between η and η in the expressionsthat follow and substitute the average conformal time η = √ η η instead of both η and η for thelimits of integrations over η and η . Then, the ansatz in question solves the DS equation if n p and κ p obey: n p ( η ) ≈ n (0) p − λ Z d D − q (2 π ) D − Z Z η ∞ dη dη ( η η ) D − ×× (" d K (cid:18) qη , qη (cid:19) d K (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) + 14 d − (cid:18) qη , qη (cid:19) d − (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) −− d − (cid:18) qη , qη (cid:19) d K (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) (cid:20) n (cid:0) pη (cid:1)(cid:21) h ∗ (cid:0) pη (cid:1) h (cid:0) pη (cid:1) ++4 θ ( η − η ) d K (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) Re (cid:20) d − (cid:18) qη , qη (cid:19) h (cid:0) pη (cid:1) h (cid:0) pη (cid:1) κ (cid:0) pη (cid:1)(cid:21)) (72)and κ p ( η ) ≈ κ (0) p − λ Z d D − q (2 π ) D − Z Z η ∞ dη dη ( η η ) D − ×× (" d K (cid:18) qη , qη (cid:19) d K (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) + 14 d − (cid:18) qη , qη (cid:19) d − (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) ++ d − (cid:18) qη , qη (cid:19) d K (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) (cid:20) n (cid:0) pη (cid:1)(cid:21) h ∗ (cid:0) pη (cid:1) h ∗ (cid:0) pη (cid:1) ++4 θ ( η − η ) d K (cid:18) | ~p − ~q | η , | ~p − ~q | η (cid:19) d − (cid:18) qη , qη (cid:19) h ∗ (cid:0) pη (cid:1) h (cid:0) pη (cid:1) κ (cid:0) pη (cid:1)) (73)where n (0) p and κ (0) p define the initial propagator D K ( p | η , η ) via Eq. (71).In the derivation of (72) and (73) we have used the following relations d − ( pη , pη ) = − d − ( pη , pη ) = − (cid:20) d − ( pη , pη ) (cid:21) ∗ and R d D − ~qf ( q, | ~p − ~q | ) = R d D − ~qf ( | ~p − ~q | , q ). Below weassume that n p ( η ) and κ p ( η ) are slow functions in comparison with h ( pη ). Then, one can safelychange arguments of all n ’s and κ ’s to the external time η . This is possible because of the usual6separation of scales, which lays in the basis of the kinetic theory [59]. In our case this approx-imation is correct at least for the fields from the principal series, m > ( D − /
2, because theirharmonics h ( pη ) oscillate at future infinity.Note that for the scalars from the complementary series, m ≤ ( D − / not oscillate at future infinity and, hence, can be asslow as n p and κ p . The main problem with the situation when h ( pη ) is slow function is that thenone cannot derive the kinetic equation of the usual form. More complicated integrodifferentialequations are available whose solution and physical interpretation is not yet known to us.To avoid such a situation when the equations themselves depend on their initial data, we wouldlike to convert the integral equations (72), (73) into their integrodifferential form, i.e., into theform of the kinetic equation [59]. This is done via a kind of the renormalization group procedureas follows [61], [62], [58]. In the given settings, n (0) p and κ (0) p are the particle density and anomalousaverage at some moment after η ∗ ∼ µ/p . In fact, as we will explain below, before this moment,all the kinetic processes in the theory under consideration are suppressed because of the strongblue shift. Hence, before η ∗ ∼ µ/p , n p ( η ) and κ p ( η ) remain practically unchanged, i.e. canbe set equal to their initial values, n (0) p and κ (0) p , correspondingly. Then, from (72), (73) it isstraightforward to find that due to the large IR corrections the difference between n p ( η ) , κ p ( η ) and n (0) p = n p ( η ∗ ) , κ (0) p = κ p ( η ∗ ) is proportional to the proper time elapsed from η ∗ to η . Also one canapproximate: n p ( η ) − n p ( η ∗ )log( η ) − log( η ∗ ) → dn p ( η ) d log( pη ) , and κ p ( η ) − κ p ( η ∗ )log( η ) − log( η ∗ ) → dκ p ( η ) d log( pη ) . The coefficients of the proportionality between the elapsed time and the change of n p and κ p arethe so-called collision integrals. With the use of the following matrixes: N p ( η , η ) = η D − n p ( η ) h ∗ ( pη ) κ p ( η ) h ( pη ) κ p ( η ) h ( pη ) n p ( η ) h ∗ ( pη ) ! , P = ! (74)the collision integrals for n p and κ p can be written compactly. The real one for n p has the form: dn p ( η ) d log( pη ) = 2 λ Z d D − q (2 π ) ( D − Z ∞ η dη ′ ( η η ′ ) D Re T r (cid:26) C p,q,p − q ( η ) (cid:20)(cid:0) N ∗ p (cid:1) N q N p − q − N ∗ p (1 + N q ) (1 + N p − q ) (cid:21) ( η, η ′ )++2 C q,q − p,p ( η ) h N ∗ q (1 + N q − p ) (1 + N p ) − (cid:0) N ∗ q (cid:1) N q − p N p i ( η, η ′ ) ++ D p,q,p + q ( η ) h (1 + N p ) (1 + N q ) (1 + N p + q ) − N p N q N p + q i ( η, η ′ ) o + [ N → P N ] . (75)At the same time the complex collision integral for κ p is as follows:7 dκ p ( η ) d log( pη ) = − λ Z d D − q (2 π ) ( D − Z η η dη ′ ( η η ′ ) D ( ~p → − ~p ) T r n C p,q,p − q ( η ) h (1 + N p ) (1 + N q ) (1 + N p − q ) − N p N q N p − q i ( η, η ′ )++2 C ∗ q,q − p,p ( η ) h N ∗ q (1 + N q − p ) (1 + N p ) − (cid:0) N ∗ q (cid:1) N q − p N p i ( η, η ′ ) ++ D ∗ p,q,p + q ( η ) (cid:20) (1 + N p ) N ∗ q N ∗ p + q − N p (cid:0) N ∗ q (cid:1) (cid:0) N ∗ p + q (cid:1)(cid:21) ( η, η ′ ) (cid:27) − [ N → P N ] . (76)The term [ N → P N ] means that we have to add to the explicitly written expressions in (75) and(76) the same quantities with every N substituted by the product P N ; Re
T r means that onehas to take the real part and the trace of the expression that stands after this sign; ( ~p → − ~p ) T r means that one has to take the trace and add to the expression standing after this sign thesame term with the exchange ~p → − ~p . Finally, to simplify the equations we use the notations: C k k k ( η ) = η D − h ∗ ( k η ) h ( k η ) h ( k η ) and D k k k ( η ) = η D − h ( k η ) h ( k η ) h ( k η ).Similarly to Eq. (58), the system of equations (75), (76) represents just a preliminary versionof the kinetic equations. It is not yet suitable to sum only the leading IR contributions in all loopsbecause on top of the latter it also accounts for some of the subleading corrections. In particular,one should restrict the dq integrals in (75) and (76) to the region q ≫ p and Taylor expand h ( pη )and h ( pη ′ ) around zero. As we discussed above, it is this region where the leading IR contributionscome from.
1. Physical meaning of infrared effects
Before going further let us explain the meaning of the system of equations (75) and (76). To dothat let us forget for the moment about the presence of κ p and consider the situation in flat space.If the time of the turning on self-interactions is t and the time of the observation is t , then thekinetic equation is: dn p dt ∝ − λ Z d D − qǫ ( p ) ǫ ( q ) ǫ ( p − q ) × (cid:26)Z tt dt ′ cos (cid:2) ( − ǫ ( p ) + ǫ ( q ) + ǫ ( p − q )) ( t − t ′ ) (cid:3) (cid:20) (1 + n p ) n q n | p − q | − n p (1 + n q ) (1 + n | p − q | ) (cid:21) ( t )+2 Z tt dt ′ cos (cid:2) ( − ǫ ( q ) + ǫ ( q − p ) + ǫ ( p )) ( t − t ′ ) (cid:3) h n q (1 + n | q − p | ) (1 + n p ) − (1 + n q ) n | q − p | n p i ( t )+ Z tt dt ′ cos (cid:2) ( ǫ ( q ) + ǫ ( q + p ) + ǫ ( p )) ( t − t ′ ) (cid:3) h (1 + n q ) (1 + n q + p ) (1 + n p ) − n q n q + p n p i ( t ) (cid:27) , (77)where ǫ ( p ) is the energy of the particle with the momentum ~p . There are many ways to derive thiskinetic equation (see, e.g., appendix of [57]). One of them is similar to that method which waspresented above for dS space.In the limit ( t − t ) → ∞ the dt ′ integrals in (77) are converted into (minus) δ -functions ensuringenergy conservation in some processes, which we will define now. The expression in the second line8of Eq. (77), which is multiplying dt ′ integral, describes the competition between the following twoprocesses. One of them corresponds to the term n p (1 + n q ) (1 + n | p − q | ) and is such a process inwhich a particle with momentum ~p decays into two excitations — ~q and ~p − ~q . This term appearswith the minus sign in the collision integral because it describes the loss of an excitation withthe momentum ~p . The inverse gain process, corresponding to the term (1 + n p ) n q n | p − q | with theplus sign, is such that two particles, ~q and ~p − ~q , are merged together to create an excitation withmomentum ~p .The third line in (77) also describes two competing processes. The first of them is such thata particle with momentum ~p joins together with another one, ~q − ~p , to create an excitation withmomentum ~q . This is the loss process. The inverse gain process is the one in which a particle withmomentum ~q decays into two, one of which is with momentum ~p . The coefficient 2 in front of thisterm is just the combinatoric factor.Similarly the fourth line describes two processes. The gain process is when three particles, oneof which is with momentum ~p , are created by an external field, if any. The loss process is whenthree such excitations are annihilated into vacuum.All these six types of processes are not allowed by the energy–momentum conservation for massive fields with φ self-interaction in flat space-time . Hence, the collision integral is vanishingas ( t − t ) → ∞ . However, in dS space these processes are allowed [72]–[86] because there is noenergy conservation.One can obtain from (77) the collision integral of Eq. (75) with κ p set to zero. To do thatit is necessary to exchanges dt ′ for dη ′ , multiply it by the proper weight, √ g , following fromthe nontrivial metric, and to use the dS harmonics, g p ( η ) = η D − h ( pη ), instead of the planewaves, e − i ǫ ( p ) t / p ǫ ( p ). Then, instead of the above mentioned δ -functions, which ensure energyconservation, there are some expressions, which define the differential rates (per given range of ~q )of the processes described in the above three paragraphs.Note that in the free theory ( λ = 0) n p remains constant. This is an indirect argument favoringthat n p is the particle density per comoving volume. (The direct argument follows from thecalculation of the density from the Wightman function.) Furthermore, due to the presence of κ p ,in (75) we have extra terms on top of those which are shown in (77). All of them can be obtainedfrom (77) via the simultaneous substitutions of some of (1 + n k )’s and n k ’s by κ k ’s or κ ∗ k ’s. E.g. in(75) we encounter terms of the type: (cid:2) (1 + n p ) κ p n | p − q | − n p κ q (1 + n | p − q | ) (cid:3) . (78)The meaning of (78) is that it describes the following two competing processes — a particle withmomentum ~p is lost (gained) in such a situation, in which instead of the creation (annihilation) oftwo particles, with momenta ~q and ~p − ~q , we obtain a single excitation ~p − ~q and missing momentum ~q is gone into (taken from) the background quantum state of the theory. For the massless flat space φ theory the collision integral in (77) is not zero. The second and the third lines in(77) describe nontrivial decay processes for the collinear momenta of the products of the reactions. In this case(77) will describe, e.g., phonons in solid bodies if instead of the speed of light one would use that of sound. (cid:2) (1 + n p ) κ q κ | p − q | − n p κ q κ | p − q | (cid:3) (79)which describe the processes in which both momenta ~q and ~p − ~q are coming from (going to) thebackground state. These observations, in particular, justify the interpretation of κ p as the measureof the strength of the backreaction on the background state of the theory.Now put both n p and κ p to zero inside collision integral (75). The only term which survivesthen is present in the last line of (75) or (77). It describes the particle creation process by the dSgravitational field. If one takes the integral of this term over the conformal time from η to η heobtains exactly the expression for n p ( η ) from (58). Using the collision integral for κ p , (76), onecan make similar observation about the origin of the corresponding expression in (58).Thus, in the nonstationary situations, i.e., when the collision integral is not zero, it is natural toexpect a linear time-divergence, ∝ ( t − t ), in n p and κ p . That indeed is true for the homogeneousin time background fields (such as, e.g., constant electric fields in QED), when particle productionrates are constant. But, in the time dependent metric of dS space one encounters a bit differentsituation. In particular, in the EPP the largest IR contributions have the form of λ log ( µ/pη ) andthere is no IR divergence as one takes the moment of turning on self-interactions to past infinity, η → + ∞ . That happens because the creation of particles with comoving momentum p effectivelystarts at η ∗ ∼ µ/p rather than at the moment when we turn on self-interactions, η → + ∞ . Forthe same reason in the CPP the particle creation process goes on until η ∗ ∼ µ/p and then stops,as we have seen from the one-loop calculation.We encounter such a situation because, first of all, the collision integral is not just a constantand depends on time. Second, the past (future) infinity of the EPP (CPP) corresponds to theUV limit of the physical momentum. In this limit g p ( η ) harmonics behave as plane waves. As aresult, all the rates inside the collision integral can be well approximated by δ -functions ensuringenergy conservation [57], [61], [62]. Hence, the collision integral becomes negligible as pη → ∞ both in the EPP and CPP. That is the reason why in the EPP, independently of the type of theharmonics, one can put the moment of turning on self-interactions, η , to past infinity. Similarlythese observations explain the time-independence of the leading IR contributions in the CPP, when η > µ/p .
2. The kinetic equation
The DS equation (70) is covariant under the simultaneous Bogolyubov rotation of the harmonicsand of n p and κ p . Hence, in principle a solution of (70) in terms of one type of harmonics can bemapped to another type.However, we would like to choose such harmonics, h ( pη ), for which there is a solution of (75),(76) with κ p being zero. From the one-loop calculations it is not hard to find that most of α -modes,including the BD ones, do not provide such a solution. That means that the backreaction on thecorresponding ground states is strong. Only for the out-harmonics, h ( x ) ∝ J iµ ( x ), the situation is0different. In particular, if one puts κ p ( η ) to zero it is not generated back in (72), (73). Or moreprecisely, contribution to it behaves as λ , i.e., is negligible in comparison with λ log( pη ).All in all, out-harmonics represent the proper quasi-particle states in future infinity. Whichmeans that for out-modes the ansatz (71) solves the DS equation with κ p ( η ) ≡
0. This is theargument which favors the interpretation that independently of the initial state at the past infinityof the EPP the field theory state flows in future infinity to the out-vacuum with some density ofparticles on top of it [57]. To support such a conclusion, we will show in a moment that for theout-harmonics κ p ( η ) indeed flows to zero in future infinity, even if originally it had some smallnon-zero value.The kinetic equation for out-modes is obtained from (72), (73), with κ p ( η ) set to zero, via thesame “renormalization group” procedure as was used above. The result is: dn ( x ) d log( x ) = − λ S D − | A | (2 π ) D − Z ∞ dx x D − Z ∞ dx x D − ×× n Re h x − iµ V ( x ) x iµ V ∗ ( x ) i n [1 + n ( x )] n ( x ) − n ( x ) [1 + n ( x )] o ++2 Re h x iµ W ( x ) x − iµ W ( x ) i n n ( x ) [1 + n ( x )] [1 + n ( x )] − [1 + n ( x )] n ( x ) n ( x ) o ++ Re h x iµ V ( x ) x − iµ V ∗ ( x ) i n [1 + n ( x )] [1 + n ( x )] − n ( x ) n ( x ) o (cid:27) , (80)where x = pη , V ( x ) = h h ( x ) − A A | x | − . . . i , W ( x ) = h | h ( x ) | − A A | x | − . . . i and h ( x ) = q π sinh( πµ ) J iµ ( x ). We have done here the same approximations as in the one-loop calculation.Now we have to check the stability of (80) under the linearized perturbations of κ p ( η ). If onekeeps only the linear in κ p terms, the integrodifferential form of the equation, which follows from(73), degenerates to: dκ p ( η ) d log( pη ) = Γ κ p ( η ) , Γ = 4 i λ S D − | A | (2 π ) D − Z Z ∞ dx dx x D − − iµ x D − + iµ θ ( x − x ) Im [ V ( x ) V ∗ ( x )] . (81)Below we will show that ReΓ > κ p ( η ) ∝ ( pη ) Γ ,reduces back to zero, as η → B. Solution of the kinetic equation in the Poincar´e patches
To find a solution of the kinetic equation (80) it is instructive to consider simultaneously thesituation in the EPP and CPP. Spatially homogeneous kinetic equations in EPP and CPP canbe trivially related to each other, because to perform the map from EPP to CPP one just hasto flip the limits of dη ′ integration inside the collision integrals in (75) and (76). On top of thatit is necessary to make the change of h ( x ) to h ∗ ( x ), because the positive energy harmonics are1exchanged with the negative ones under the flip of the time direction. As a result, solutions inboth patches also can be mapped to each other. That is physically meaningful because the systemof kinetic equations (75) and (76) is valid for the spatially homogeneous states and is imposed onthe quantities which are attributed to the comoving volume. It is worth mentioning that spatiallyhomogenous situation in the CPP is unstable under small inhomogeneous density perturbations.However, it is still instructive to consider such an ideal situation in the CPP .Thus, all the observations made for the EPP can be extended to the CPP. For the Bessel(in-)harmonics in the CPP the corresponding system (75) and (76) can be reduced to the singleequation: dn p ( η ) d log ( η/η ) = λ S D − | A | (2 π ) D − Z ∞ dq η ( qη ) D − Z ∞ dη ′ q (cid:0) qη ′ (cid:1) D − ×× ( Re h ( qη ) − iµ V ( qη ) (cid:0) qη ′ (cid:1) iµ V ∗ ( qη ′ ) i (cid:26) [1 + n p ] n q − n p [1 + n q ] (cid:27) ( η ) ++2 Re h ( qη ) iµ W ( qη ) (cid:0) qη ′ (cid:1) − iµ W ( qη ′ ) i n n q [1 + n q ] [1 + n p ] − [1 + n q ] n q n p o ( η ) ++Re h ( qη ) iµ V ( qη ) (cid:0) qη ′ (cid:1) − iµ V ∗ ( qη ′ ) i n [1 + n q ] [1 + n p ] − n q n p o ( η ) ) . (82)Neither (80) nor (82) possess the Plankian distribution as a stationary solution, because there is noenergy conservation. However, one can find a distribution which annihilates the collision integralof (80) or (82) in a different manner: while Plankian distribution in ordinary kinetic equationannihilates each term, describing aforementioned couples of competing processes, separately, inthe present case we will see the cancelation between different terms. The situation is somewhatsimilar to the one in which appears the turbulent stationary Kolmogorov like scaling. That isnatural to expect in a system with energy pumping in at one scale and its loss at another one.Thus, suppose we have started at the past infinity of the EPP with some very mild densityperturbation over the BD state. After the Bogolyubov rotation to the out-harmonics, one hassome initial values of n p and κ p . As can be understood from the discussion above, for the given p , the density n p and the anomalous average κ p practically do not change before η ∗ ∼ µ/p . Afterthis moment they begin to evolve according to (75), (76). If κ p is sufficiently small, it relaxes tozero, as we have shown above, and the problem is reduced to the solution of (80).Now, if the initial value of n p after the rotation to the out-harmonics is much smaller than one,we can use the following approximations:[1 + n ( x )] n ( x ) − n ( x ) [1 + n ( x )] ≈ − n ( x ) ,n ( x ) [1 + n ( x )] [1 + n ( x )] − [1 + n ( x )] n ( x ) n ( x ) ≈ n ( x ) , Actually it is not very hard to find the inhomogeneous extension of the kinetic equation (80). In the case in whichthe particle density starts to depend also on the spatial position n p = n p ( η, ~x ), one has to substitute η d/dη by η ∂ η + η ~p ~∂ x on the left hand side of (80). The extra η ~p~∂ x term is the simplest expression that is invariant underthe symmetries of the EPP and CPP metrics: such as, e.g., simultaneous scaling of all coordinates. n ( x )] [1 + n ( x )] − n ( x ) n ( x ) ≈ . (83)Because of the rapid oscillations of h ( x ) as x → ∞ , the integrals on the right hand side of (80) aresaturated at x , ∼ µ . At the same time pη ≪ µ and it is natural to assume that n ( x , ∼ ≪ n ( pη ) in the situation of the very small initial density perturbation and the vanishing productionof the high momentum modes. Hence, we can neglect the second contribution ( ∼ n ( x )) on theright hand side of (80), (83) in comparison with the first ( ∼ n ( x )) and the third ( ∼
1) ones.As a result, Eq. (80) is reduced to dn ( x ) d log( x ) ≈ Γ n ( x ) − Γ , Γ = λ S D − | A | (2 π ) D − (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ dy y D − − i µ (cid:20) h ( y ) − A A | y | (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) , Γ = λ S D − | A | (2 π ) D − (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ dy y D − + i µ (cid:20) h ( y ) − A A | y | (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) (84)Here, Γ and Γ are the particle decay and production rates, correspondingly. Note that x = pη isreducing to zero during the evolution towards the future infinity.The obtained equation (84) has the solution with the flat stationary point distribution n ( pη ) =Γ / Γ , which corresponds to the situation in which the production (gain) of particles on the level pη is equilibrated by the particle decay (loss) from the same level. The obtained solution is self-consistent for the large enough µ because then Γ / Γ ≈ e − πµ ≪
1, which is not hard to showusing the following property of the Hankel functions: Z ∞ dx x D − + iµ (cid:16) H (1) iµ ( x ) (cid:17) = e − πµ Z ∞ dx x D − − iµ (cid:16) H (2) iµ ( x ) (cid:17) . (85)(Also it is not hard to show that the introduced in the previous subsection Γ obeys ReΓ =Γ − Γ ≈ (cid:0) − e − πµ (cid:1) Γ > pη ≪ µ became big in comparison with one. Taking into account the flatness of thespectrum in dS space, it is natural to expect that, for the harmonics with the low physical momenta,the density very slowly depends on its argument. Hence, we can assume that n ( pη ) ≈ n ( qη ) whenboth pη ≪ µ and qη ≪ µ .Then, we can make the following approximations:3[1 + n ( pη )] n ( x ) − n ( pη ) [1 + n ( x )] ≈ − n ( pη ) ,n ( x ) [1 + n ( x )] [1 + n ( pη )] − [1 + n ( x )] n ( x ) n ( pη ) ≈ n ( pη ) , [1 + n ( x )] [1 + n ( pη )] − n ( x ) n ( pη ) ≈ n ( pη ) (86)and accept that, on the right hand side of (80), the main contribution to the x , integrals comesform the region in which x , ≪ µ because n ( x ) ≫ n ( y ) if x ≪ µ and y ≫ µ . As a result, thekinetic equation reduces to dn p ( η ) d log( η ) = ¯Γ n p ( η ) , where¯Γ ≈ λ (2 π ) D − µ Z µ dy ( y ) D − (cid:26) − Re (cid:20) y − i µ Z µ dy ′ ( y ′ ) D − + i µ (cid:21) ++2 Re (cid:20) y − i µ Z µ dy ′ ( y ′ ) D − + i µ (cid:21) ++ 3 Re (cid:20) y − i µ Z µ dy ′ ( y ′ ) D − +3 i µ (cid:21)(cid:27) > . (87)Note that ¯Γ is independent of p . This equation has the following solution: n p ( η ) ≈ η/η ⋆ ) , (88)where η ⋆ = µp e − C ¯Γ and C is an integration constant, which depends on initial conditions. Theobtained solution is valid if µ/p = η ∗ > η > η ⋆ .Thus, we see that there is a singular solution of the kinetic equation under consideration,which corresponds to the explosion of the particle number density per comoving volume within afinite proper time. Of course, such an explosion wins against the expansion of the EPP because D K ∝ η D − log η/η ⋆ . Hence, there are contributions to the energy-momentum tensor of the createdparticles which become huge, and the backreaction has to be taken into account. As a result, thedS space gets modified. But that is the problem for a separate study. At this point, we just wouldlike to stress that we see catastrophic IR effects even for the massive fields. It is natural to expectthat, for the light fields, the situation will be even more dramatic.In the CPP, we can find similar solutions of the kinetic equation (82). For example, (84) ismapped to dn p ( η ) d log( η/η ) ≈ − Γ n p ( η ) + Γ , (89)which shows a very peculiar phenomenon that, although in the first loop η cannot be taken tothe past infinity, after the summation of all loops, we can find the theory at the stationary pointstate, n p = Γ / Γ , which allows one to remove the IR cutoff, η , and restore the dS isometry.4At the same time, the solution (88) is mapped to n p ( η ) ≈ η ⋆ /η ) , (90)where η ⋆ = η e C/ ¯Γ < µ/p . Of course, whether the field theory state goes into (90) or to (89)depends on initial conditions. C. Kinetic equation in global de Sitter space
Having in mind the discussion of the EPP and CPP above, it is not hard to derive the systemof kinetic equations in global dS: dn j dt = [ N → P N ] + 2 λ X j , ~m ; j , ~m Re T r ( J j, ~m ; j , ~m ; j ~m Z + ∞−∞ dt ′ C ∗ j,j ,j ( t ) ×× (cid:20)(cid:0) N ∗ j (cid:1) N j N j − N ∗ j (1 + N j ) (1 + N j ) (cid:21) ( t, t ′ ) ++2 J j , ~m ; j , ~m ; j ~m Z + ∞−∞ dt ′ C ∗ j ,j ,j ( t ) h N ∗ j (1 + N j ) (1 + N j ) − (cid:0) N ∗ j (cid:1) N j N j i ( t, t ′ ) + I j, ~m ; j , ~m ; j ~m Z + ∞−∞ dt ′ D ∗ j,j ,j ( t ) h (1 + N j ) (1 + N j ) (1 + N j ) − N j N j N j i ( t, t ′ ) ) (91)and dκ j dt = − [ N → P N ] − λ X j , ~m ; j , ~m ( ~m → − ~m ) T r ( J j, ~m ; j , ~m ; j ~m Z t −∞ dt ′ C ∗ j,j ,j ( t ) ×× h (1 + N j ) (1 + N j ) (1 + N j ) − N j N j N j i ( t, t ′ ) ++2 J j , ~m ; j , ~m ; j ~m Z t −∞ dt ′ C ∗ j ,j ,j ( t ) h N ∗ j (1 + N j ) (1 + N j ) − (cid:0) N ∗ j (cid:1) N j N j i ( t, t ′ ) ++ I j, ~m ; j , ~m ; j ~m Z t −∞ dt ′ D ∗ j,j ,j ( t ) h (1 + N j ) N ∗ j N ∗ j − N j (cid:0) N ∗ j (cid:1) (cid:0) N ∗ j (cid:1)i ( t, t ′ ) ) (92)The notations [ N → P N ] and ( ~m → − ~m ) have been introduced above. In (91) and (92) insteadof the usual flat space δ -functions, which ensure energy-momentum conservation, we have J j ~m ; j ~m ; j ~m = Z d Ω Y ∗ j ~m (Ω) Y j ~m (Ω) Y j ~m (Ω) , C j ,j ,j = cosh D − ( t ) g ∗ j g j g j ,I j ~m ; j ~m ; j ~m = Z d Ω Y j ~m (Ω) Y j ~m (Ω) Y j ~m (Ω) , D j ,j ,j = cosh D − ( t ) g j g j g j . (93)5Unlike the case of EPP and CPP, we do not yet understand what is the appropriate choice of theharmonics for this system of kinetic equations. In fact, if one considers the entire dS space, then κ p is divergent either when t → −∞ or when t → ∞ , or in both cases simultaneously. In this casenone of the harmonics provides a proper quasi-particle description. However, if one will considerthat half of global dS space, which is above its neck, then the situation would be similar to that inthe EPP. I.e., in this case one can find the same solutions as we have found above. VI. SKETCH OF THE RESULTS FOR λφ THEORY (INSTEAD OF CONCLUSIONS)
Instead of conclusions let us present here the results for λφ theory both for the principal andcomplementary series (see [58] for grater details). It is straightforward to show that φ theory,unlike φ one, does not possess any large IR contributions to the Keldysh propagator at the first-loop level ( ∼ λ ). However, in the second-loop ( ∼ λ ), there are large IR contributions to D K ,which are of interest for us. They come from the so-called sunset diagrams. There are no IRcorrections to D R,A propagators and to the vertices.The kinetic equation for the principal series in the EPP can be derived in the same manner asin φ theory. For the out-modes it is as follows: dn pη d log( pη ) = − λ | A | Z d D − l (2 π ) D − d D − l (2 π ) D − Z ∞ dv v D − n h v iµ h ∗ ( l ) h ∗ ( l ) h (cid:16)(cid:12)(cid:12)(cid:12) ~l + ~l (cid:12)(cid:12)(cid:12)(cid:17) h ( l v ) h ( l v ) h ∗ (cid:16)(cid:12)(cid:12)(cid:12) ~l + ~l (cid:12)(cid:12)(cid:12) v (cid:17)i ×× h (1 + n pη ) n l n l (1 + n | ~l + ~l | ) − n pη (1 + n l )(1 + n l ) n | ~l + ~l | i +3Re h v iµ h ∗ ( l ) h ( l ) h (cid:16)(cid:12)(cid:12)(cid:12) ~l − ~l (cid:12)(cid:12)(cid:12)(cid:17) h ( l v ) h ∗ ( l v ) h ∗ (cid:16)(cid:12)(cid:12)(cid:12) ~l − ~l (cid:12)(cid:12)(cid:12) v (cid:17)i ×× h (1 + n pη ) n l (1 + n l )(1 + n | ~l − ~l | ) − n pη (1 + n l ) n l n | ~l − ~l | i +Re h v iµ h ∗ ( l ) h ∗ ( l ) h ∗ (cid:16)(cid:12)(cid:12)(cid:12) ~l + ~l (cid:12)(cid:12)(cid:12)(cid:17) h ( l v ) h ( l v ) h (cid:16)(cid:12)(cid:12)(cid:12) ~l + ~l (cid:12)(cid:12)(cid:12) v (cid:17)i ×× h (1 + n pη ) n l n l n | ~l + ~l | − n pη (1 + n l )(1 + n l )(1 + n | ~l + ~l | ) i +Re h v iµ h ( l ) h ( l ) h (cid:16)(cid:12)(cid:12)(cid:12) ~l + ~l (cid:12)(cid:12)(cid:12)(cid:17) h ∗ ( l v ) h ∗ ( l v ) h ∗ (cid:16)(cid:12)(cid:12)(cid:12) ~l + ~l (cid:12)(cid:12)(cid:12) v (cid:17)i ×× h (1 + n pη )(1 + n l )(1 + n l )(1 + n | ~l + ~l | ) − n pη n l n l n | ~l + ~l | io . (94)One can find solutions to this kinetic equation which are similar to those which we have foundabove.At the same time the two-loop corrections for the complementary series are as follows. Thisseries corresponds to the imaginary µ , i.e. to the real index of the solution of the Bessel equation h ( pη ). Below we use the notation ν = − iµ . Then, for the in harmonics we have that h = J ν + iY ν and both Bessel functions J ν and Y ν are real in the case of the real index ν . Expanding them nearzero, we get Y ν ( x ) ≈ D − x − ν + Bx − ν +2 and J ν ( x ) ≈ D + x ν . Because of the possible differencesbetween the behavior of h and h ∗ near zero, we have to pay attention separately to κ p and κ ∗ p .The contributions to n p and κ p , κ ∗ p can be expressed as6 n p ( η ) ≈ − λ π ) D − Z ∞ duu Z ∞ dvv D − F [ v ] h ( uv ) h ∗ (cid:16) uv (cid:17) θ [ uv − pη ] θ h uv − pη i ,κ p ( η ) ≈ λ π ) D − Z ∞ duu Z ∞ dvv D − F [ v ] h ∗ ( uv ) h ∗ (cid:16) uv (cid:17) θ [ uv − pη ] θ h uv − uv i ,κ ∗ p ( η ) ≈ λ π ) D − Z ∞ duu Z ∞ dvv D − F [ v ] h ( uv ) h (cid:16) uv (cid:17) θ [ uv − pη ] θ h uv − uv i , (95)with the use of the following notations F ( η , η ) = Z " Y i =1 d D − q i h ( q i η ) h ∗ ( q i η ) ( η η ) D − δ ( D − ( ~p − ~q − ~q − ~q ) , (96)which, after the change of integration variables u = p √ η η , v = q η η , can be expressed as F ( η , η ) = p ( uv ) D − u D − F [ v ], where F [ v ] = F ∗ [1 /v ] is some function of one variable v .The leading corrections to n p and κ p , κ ∗ p are given by Eq. (95), where from the Hankel functions, h ( uv ) and h ( u/v ), we single out only Y ’s. Such contributions give for n p and κ p , κ ∗ p the inversepowerlike behavior in pη , which, however, cancels out after the substitution into D K because Y isreal. The next order is obtained as follows. One also has to express h ( pη , ) through J ν and Y ν inthe full propagator D K . Then, from one of the four h ’s [ h ( pη , ) and h ( uv ), h ( u/v )], we have tosingle out J ν , while from the other three — Y ν ’s. This expression does not cancel out and providesthe leading IR contribution to D K : D K ( η , η , p ) = 8 λ D − D + η D − π ) D −
1) 0 Z ∞ du u − − ν Z ∞ dv v − D F [ v ] (" − (cid:18) upη v (cid:19) ν θ h uv − uv i θ [ − pη + uv ] + " − (cid:18) uvpη (cid:19) ν θ h − uv + uv i θ h − pη + uv i) . (97)After the straightforward manipulations, the obtained expression can be reduced to D K ( η , η , p ) = − λ D − D + η D − log( pη )3 (2 π ) D − ( pη ) ν ∞ Z dv v − D F [ v ] " − ν v ν + (cid:18) v (cid:19) ν − Z dv v − D F [ v ] (cid:20) ν ( v ) − ν + v ν (cid:21) . (98)The integral over v is convergent in the IR limit (as v → ∞ ) if D > ν . In the UV limit( v → h = J ν , h ∗ = Y ν . Thestraightforward calculation shows that7 n p ( η ) ∝ λ D − D + log( pη ) Z dv F [ v ] v ν +1 − D ,κ p ( η ) ∝ λ D − ( pη ) − ν Z dv F [ v ] v ν +1 − D ,κ ∗ p ( η ) ∝ λ D ( pη ) ν Z dv F [ v ] v − ν +1 − D . (99)After the substitution into the Keldysh propagator, the leading contribution comes from n p andis logarithmic. However, because of the character of these IR contributions for the light fields, wedo not yet understand their physical meaning and think that the kinetic equations obtained aboveare not applicable for the fields from the complementary series. In the case of the complementaryseries, we do not yet know how to perform the summation of the leading IR contributions from allloops. But on general physical grounds and from the two-loop result, we expect that light fieldsfrom the complementary series will show stronger IR effects than the heavy ones from the principalseries. VII. ACKNOWLEDGEMENTS
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