Towards a formal description of the collapse approach to the inflationary origin of the seeds of cosmic structure
TTowards a formal description of the collapse approach to theinflationary origin of the seeds of cosmic structure
Alberto Diez-Tejedor ∗ Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,Circuito Exterior C.U., A.P. 70-543, M´exico D.F. 04510, M´exico andDepartamento de F´ısica, Divisi´on de Ciencias e Ingenier´ıas,Campus Le´on, Universidad de Guanajuato, Le´on 37150, M´exico
Daniel Sudarsky † Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,Circuito Exterior C.U., A.P. 70-543, M´exico D.F. 04510, M´exico andInstituto de Astronom´ıa y F´ısica del Espacio,Casilla de Correos 67, Sucursal 28, Buenos Aires 1428, Argentina (Dated: V.1 November 5, 2018) a r X i v : . [ g r- q c ] A ug bstract Inflation plays a central role in our current understanding of the universe. According to thestandard viewpoint, the homogeneous and isotropic mode of the inflaton field drove an early phaseof nearly exponential expansion of the universe, while the quantum fluctuations ( uncertainties )of the other modes gave rise to the seeds of cosmic structure. However, if we accept that theaccelerated expansion led the universe into an essentially homogeneous and isotropic space-time,with the state of all the matter fields in their vacuum (except for the zero mode of the inflatonfield), we can not escape the conclusion that the state of the universe as a whole would remainalways homogeneous and isotropic. It was recently proposed in [A. Perez, H. Sahlmann and D.Sudarsky, “On the quantum origin of the seeds of cosmic structure,” Class. Quant. Grav. ,2317-2354 (2006)] that a collapse (representing physics beyond the established paradigm, andpresumably associated with a quantum-gravity effect `a la Penrose) of the state function of theinflaton field might be the missing element, and thus would be responsible for the emergence of theprimordial inhomogeneities. Here we will discuss a formalism that relies strongly on quantum fieldtheory on curved space-times, and within which we can implement a detailed description of such aprocess. The picture that emerges clarifies many aspects of the problem, and is conceptually quitetransparent. Nonetheless, we will find that the results lead us to argue that the resulting pictureis not fully compatible with a purely geometric description of space-time. ∗ Electronic address: alberto.diez@fisica.ugto.mx † Electronic address: [email protected] . INTRODUCTION Inflation represents an appealing framework to understand the “initial conditions” of theuniverse [1–7]. The early phase of accelerated expansion seems to resolve some of the clas-sical “naturalness problems” of the big bang model, such as the flatness, the horizon, theexotic relics or the entropy ones [8–11]. Moreover, it is generally accepted that inflation alsoaccounts for the small anisotropies observed in the cosmic microwave background (CMB),necessary for the subsequent emergence of cosmic structure [12–16]. The accelerated ex-pansion makes it possible to “push” the short wavelength, causally connected modes of theinflaton field to the largest scales of the observable universe. If, in addition, the acceler-ated expansion was (nearly) de-Sitter, and the state of the quantum fields was a featurelessBunch-Davies “vacuum” (except for the zero mode of the inflaton field), a single physicalscale would appear in the primordial spectrum of the quantum fluctuations (uncertainties):the Hubble parameter. Roughly speaking, this is what lies behind the appearance of ascale-free (Harrison-Zel’dovich) primordial power spectrum of density perturbations in thestandard model of inflation.However, upon a deeper examination, one finds that there is something rather strange inthat picture: if the observable universe starts from the Bunch-Davies vacuum for the quan-tum fields in a Robertson-Walker space-time (both of which are completely homogeneousand isotropic), and considering their evolution according to the rules of standard physics,the state of the quantum fields, and the universe as a whole, will remain both homogeneousand isotropic at all times. But, how can this be compatible with the highly complex struc-ture of the universe we are inhabiting? In fact, how can such proposal be said to explainthe emergence of the seeds of cosmic structure, which early traces we study in the CMB?(See the references [17, 18] for a full discussion of the problem and the various attemptsto deal with these issues within the standard physical paradigms.) The standard and phe-nomenologically successful accounts on this matter rely on the identification of the quantum fluctuations of certain observables associated to a homogeneous and isotropic universe, withthe averages over an ensemble of inhomogeneous universes of their analogue classical quan-tities: more specifically (or more crudely), on the identification of quantum uncertainties and classical perturbations . As discussed in [18], one can not avoid concluding that usingstandard physics there is no justification for such kind of identification. In fact, these short-3omings have started to be recognized by others in the literature: see for instance Section10.4 in [4], the end of Section 8.3.3 in [5], Section 10.1 in [6], Section 24.2 in [7], and Section30.14 in [19].On the other hand, it is undeniable that the standard approach has a great phenomeno-logical success: its “predictions” match exquisitely well the most accurate observations todate. Nonetheless, as it has been argued in [17, 18], some new element must be added tothe standard picture in order to have a physical description of the transition from the initialsymmetric stage of the universe to the late non-symmetric one, preserving at the same timethe phenomenological success of the usual approach.In this context, it was recently proposed that a new kind of effect, which at the phe-nomenological level would be analogous to a “self-induced” collapse in the state function ofthe inflaton field, could play the required role in the early universe [17] (see also [20–25]).The point, of course, is that such kind of collapses are a serious departure form the unitaryevolution of standard physics. It is worth mentioning that, in fact, conceptually similarideas have been considered by many physicists concerned with the measurement problemin quantum mechanics (see for instance [26] and references therein), and the proposal thatsome quantum aspects associated to the gravitational interaction might lie at the root ofthis effect only serves to make the idea even more attractive [27] (see also Chapter 30 inreference [19]). The suggestion that such a mechanism could play a determinant role in thebreakdown of the initial symmetry of the universe seems doubly appealing to us: On the onehand, as we will see, it turns out to have the features leading to the kind of effect requiredfor the emergence of the seeds of cosmic structure. On the other, when we take such collapseas a fundamental ingredient for the generation of primordial perturbations, these becomean observationally accessible stage where such a novel aspect of physics, presumably con-taining some hints about the nature of gravity at the quantum level, could be investigated.Interestingly enough, some of the resulting predictions seem to differ in various aspects fromthe standard accounts [17, 23]. The main objective of this work is to establish a formalismthat can serve as a basis to explore these ideas in a well defined and rigorous setting, whichwould, in the future, allow us to address questions such as the applicability of collapse theo-ries [26, 28–33] to the inflationary universe, and to unearth the problems that might arise ineach particular proposal. Our program is based on the joint use of quantum field theory incurved space-time and semiclassical gravity in self-consistent settings. However, as we will4ee, we will need to introduce a modification/addition to that general setting, in connectionwith the hypothetical collapse of the wave function. We will describe all that in detail inthe main parts of the manuscript.At this point we should note that inflation is expected to be associated with very highenergy scales, well beyond the regime of the physical processes we have otherwise explored.Nonetheless, it is generally expected that such a regime is not really close to those wherea quantum description of the gravitational interaction would be required (see for instanceChapter 14 in reference [34]). However, the fact that we are interested in a situation wherethe matter fields affecting the space-time geometry require a quantum description suggeststhat the appearance of some effects, ultimately traceable to a theory of quantum gravity,should not be too surprising. In fact, we should keep in mind that inflation, to the extentthat it is observationally accessible through its imprints in the seeds of cosmic structure,represents the unique situation available to us which requires a treatment which simultane-ously relies on both quantum theory and Einstein’s gravity. (There are several experimentswhich people often claim are related to the quantum/gravity interface, such as the famousneutron interference experiment [35] suitable for detecting the effects of the Earth’s Newto-nian potential. However, a detailed analysis [36] indicates that these experiments are notsensitive to curvature at all, and it is curvature where gravitation truly lies, according toour post-Einsteinian views.)The paper is organized as follows. In Section II we review the ideas regarding the collapseof the state function, attempting to frame the proposal, even if only schematically, withinour current understanding of physical theory. We also introduce some useful concepts to dealwith the physics of the collapse, describing the nature of the formalism to be employed andexplaining the extent to which it is taken to describe the relevant phenomena. In SectionIII we study a simplified version of a wave function collapse in the early universe, describingthe extent to which the transition from the symmetric to the inhomogeneous phase can betreated within the present formalism. Finally, we discuss our results in Section IV. We haveplaced in the appendices some details of the presentation which are not essential to followthe main text.The conventions we will be using include: ( − , + , + , +) signature for the space-time metric,Wald’s convention for the Riemann tensor, and natural units with c = 1. The Newton’sgravitational constant G and the reduced Planck’s constant (cid:126) are used to define the Planck5ass M p ≡ (cid:126) / πG and the Planck time t p = 8 πG (cid:126) . Space-time indexes are denoted byGreek letters ( µ = 0 , , , i = 1 , ,
3) associatedwith specific hypersurfaces.
II. THE NATURE OF OUR EFFECTIVE DESCRIPTION
Before we embark on the description and the general setting for our ideas, we would liketo remind the reader that, despite the claims to the contrary, the measurement problemis still an open question in quantum mechanics [26]. In fact, the problem becomes evenmore serious in the context of quantum field theory [37]. Even worse, the interpretationalframework of the quantum theory becomes intolerable when we want to apply it to theuniverse as a whole, because, in that case, we can not even rely on the practical usage ofidentifying an observer and/or a measuring device. Those problems have led researchers,such as J. Hartle, to argue for a modified or extended interpretational framework makingthe theory suitable for the cosmological applications [38, 39]. However, it seems that suchscheme does not go far enough, lacking the kind of feature we need to address the problemthat concern us here [18]. It is for that reason that we shall reconsider in some detailthe schematic view proposed in [17] for dealing with this conundrum. As we have justcommented in the Introduction, the proposal involves incorporating a “self induced collapseof the wave function of the matter fields”, and considering the resulting inhomogeneities andanisotropies in the energy momentum tensor generated during the collapses as the origin forthe inhomogeneities and anisotropies of the space-time metric of our universe.Now let us characterize the general nature of our approach. We should start by offeringa discussion of a plausible manner whereby the collapse hypothesis might be incorporatedwithin the general current understanding of a physical theory. However, for conciseness of thepresentation this is done in the Appendix A. Following the general arguments given there, wewill take the view that the fundamental degrees of freedom for the gravitational interactionare not the ones characterizing the standard metric variables, but some other variables thatare indirectly connected to those (consider, for example, the fluxes and holonomies in loopquantum gravity [40]). That is, we will consider that geometry is an emergent phenomena,and that general relativity should be regarded as some sort of hydrodynamical limit of a6ore fundamental theory. This point of view seems to be further supported by the analogybetween the behaviour of black holes and the laws of thermodynamics [41], and by the ideasdeveloped in [42] (see also [43–45]).The regimes where the metric description for the gravitational degrees of freedom holdswould correspond to the situations where the whole universe might be described by states | ξ (cid:105) = | ξ (cid:105) g ⊗ | ξ (cid:105) m (where the first factor corresponds to the gravitational degrees of freedom,and the second one to the matter fields) for which the gravitational sector | ξ (cid:105) g is such thatoperators corresponding to the metric variables have sharply peaked values (this would notbe a strict eigenstate of the metric, where the wave function would be a delta function,but only to states where the wave function uncertainties in the geometrical quantities arerather small at the level of precision one is working). Needless is to say that all our existingexperience with the gravitational interaction would be, according to this view, well describedby such hydrodynamical limit, and its non-geometrical behaviour would show up, generically,just in the extreme domains such as those normally associated with quantum gravity. Welloutside those elusive domains, the fundamental operators for the gravitational degrees offreedom would be, for the most part, well described in terms of the metric tensor g µν , and itsdynamics can be taken as controlled by the corresponding Einstein tensor g (cid:104) ξ | ˆ G | ξ (cid:105) g = G µν [ g ].At the same time, we will consider that the matter fields should be described in terms ofa quantum field theory in a curved space-time. Under these conditions one can expect torecover the semiclassical description of gravitation in interaction with quantum fields as Hydrodynamics is the effective theory describing the long wavelength, large frequency, low energy modesof highly coupled systems with a large number of “particles”. At those scales, only a small number ofdegrees of freedom are necessary to characterize the state of the system: the hydrodynamic modes . Inthe case of a perfect fluid, it can be described by the energy density, the entropy density and the field ofvelocities; all related by the Euler, the continuity and the thermodynamic equations. (Notice that it hasnot been necessary to introduce the concept of a fundamental constituent − the particles − for the fluid,of which, the 18 th century physicist who developed the theory was naturally unaware.) This descriptionwill be appropriate to describe the fluid at length scales larger than the mean free path, and for timescales larger than the mean free time. The view we will take here is that something similar occurs withthe gravitational interaction, where the characteristic physical scale should be probably determined nowby the Planck mass. That is, in fact, the situation in various approaches to quantum gravity, such as theloop quantum gravity program or the poset proposal. Continuing with our hydrodynamical analogy, we might consider the very complicated wave-functioncharacterizing the state of all the particles in a perfect fluid, but which for practical purposes can beeffectively described by some classical quantities such as the energy density, the entropy density, and thefield of velocities. The components of the space-time metric play now the role of the hydrodynamic modes. G µν [ g ] = 8 πG m (cid:104) ξ | ˆ T µν [ g ] | ξ (cid:105) m . (1)Here the right hand side stands for the expectation value of the energy-momentum ten-sor operator in the corresponding state of the quantum fields, constructed from the fieldoperators and the space-time metric g µν . The matter fields are described in terms of opera-tors acting on an appropriate Hilbert space H m , according to the standard construction of aquantum field theory on a background space-time, with the latter satisfying the semiclassicalequations given in (1).Given the fact that we do not have yet a fully workable quantum theory for the gravita-tional interaction, it is very difficult to analyse explicitly the regime of validity of this semi-classical description. However, it seems reasonable to assume that it would be appropriatefor the situations in which the matter fields | ξ (cid:105) m are sharply peeked around a classical fieldconfiguration ( ϕ i,ξ ( x ) , π i,ξ ( x )), with ϕ i,ξ ( x ) ≡ m (cid:104) ξ | ˆ ϕ i ( x ) | ξ (cid:105) m and π i,ξ ( x ) ≡ m (cid:104) ξ | ˆ π i ( x ) | ξ (cid:105) m ,and where there are no relevant Planck scale phenomena, i.e. all curvature scalars are wellbelow the Planck scale. Furthermore, we will be considering that the regime of validity ofsemiclassical gravity, with a relatively simple modification, can be extended to include theself-induced collapse (or dynamical reduction) of the wave function of matter fields. Ourgoal is to consider a precise formalism able to incorporate these ideas, that will allow us toexplore them in a detailed manner. It is worthwhile noting here that the general formalismto be presented in the first part of the present work should be useful for the detailed studiesof the dynamical wave function collapse theories (such as [26, 29–33]) in the situations wherethe gravitational back-reaction becomes important, as well as for certain aspects encoun-tered in related approaches. Each one of the discontinuous modifications of the evolutionequations, which are controlled by stochastic functions introduced in those schemes, wouldcorrespond to one single collapse as the one we will describe in this paper. The details ofsuch correspondence would clearly be different for each specific case. As an example, weconsider briefly the so called stochastic gravity proposal [46] in Appendix B.Next we turn to the development of the precise formalism we seek. The description ofthe system in the periods between collapses is considered in Section II A. In Section II B wedescribe the proposal for incorporating the collapses within the general formalism.8 . The Semiclassical Self-consistent Configurations (SSCs) It is clear that the semiclassical regime that we have described so far can not be but aneffective description with a limited range of applicability. However, we will assume that suchregime includes the cosmological setting at hand. Next, we seek to specify in some detailthe nature of the formal description we will be considering, even though we do not take itto be, in any sense, fundamental. Among the advantages of having such a precise structure,is that it allows us to uncover and, in principle, to start investigating, the places where adeparture from such scheme is in fact required. As we will see, this approach will allow usto discuss with precision some delicate questions, and to focus sharply on some of the lessunderstood aspects of the collapse ideas. (Consider for comparison the formal developmentsin classical general relativity, which include the famous singularity theorems, which as weknow indicate a limitation of the validity of that very same theory.)Under these assumptions (and at the above specified level) we will consider that theuniverse can be described by what we call a
Semiclassical Self-consistent Configuration (SSC). That is, a space-time geometry characterized by a classical space-time metric and astandard quantum field theory constructed on that fixed space-time background, togetherwith a particular state in that construction such that the semiclassical Einstein equationshold. In other words, we will say that the set { g µν ( x ) , ˆ ϕ ( x ) , ˆ π ( x ) , H , | ξ (cid:105) ∈ H } representsa SSC if and only if ˆ ϕ ( x ), ˆ π ( x ) and H correspond to a quantum field theory constructedover a space-time with metric g µν ( x ) (as described in, say [47]), and the state | ξ (cid:105) in H issuch that G µν [ g ( x )] = 8 πG (cid:104) ξ | ˆ T µν [ g ( x ) , ˆ ϕ ( x ) , ˆ π ( x )] | ξ (cid:105) (2)for all the points x in the space-time manifold. That is, we are basically relying on a strictinterpretation of semiclassical gravity, considered not as a fundamental theory, but as aneffective description (Note here that, as discussed in reference [48], it might even be possibleto take the alternative viewpoint.) It is worth noting that an analogous approach is some-times used in the field of quantum optics when one is not interested in some intrinsicallyquantum aspects of the electromagnetic field (see for instance the “self-consistent equations”in Section 1.5 of reference [49]). We should keep in mind the self-referential feature of thisapproach, and also the fact that it is very close, in spirit, to the Schr¨odinger-Newton descrip-tion [50–55], which seems to have a relationship with some kind of collapse-like behaviour929, 56]. From now on we will ignore the subindexes “ m ” and “ g ”, as it would be understoodthat the quantum description refers only to the matter fields. For simplicity, a single scalarfield ϕ has been assumed, where the evident generalization is understood.The actual construction of even one of such SSC’s is not trivial. One must somehow“guess” the appropriate space-time metric, construct the quantum theory for the matterfields “living” on that given background, and then, find an appropriate state | ξ (cid:105) in H (ifany) compatible with the selected space-time configuration. We should emphasize that, ingeneral, for a given SSC, most of the states in H together with the original g ( x ), ˆ ϕ ( x ), ˆ π ( x )and H do not represent a valid SSC. Only for a few states, if any (besides the original one),will the equation (2) hold. Explicit constructions of two SSCs will be given in Sections III Aand III B. The first one corresponds to a perfectly homogeneous and isotropic universe, andthe second to a slightly inhomogeneous space-time, where, for simplicity, just one Fouriercomponent is taken to be “excited”.In order to avoid possible future misunderstandings, we should note at this point variousimportant differences between this formalism and the one usually employed within the con-text of the inflationary universe. The standard approach is based on the consideration of aclassical description for both, the space-time metric (taken to admit a flat Robertson-Walkerdescription) and inflaton field (taken to be in a slow-rolling homogeneous and isotropic con-figuration) “backgrounds”, and the perturbations of both metric and inflaton field, whichare treated at the quantum mechanical level. In contrast, in the present formalism, thesplit between the quantum and classical descriptions is not tied to a particular perturbativeapproach, but to the gravity-matter distinction. This seems to be justified, as we have ar-gued above, not only by the lack of a workable theory of quantum gravity, but also by theconceptual problems that seem intrinsic to that program, and, in particular, to the so called“problem of time in quantum gravity”. One might be concerned with the fact that it seemsalways possible to move part of the degrees of freedom from the metric to the matter fields(and back) through a conformal transformation, and with the idea that changes of coordi-nates mix the gravity and matter field perturbations. These are common misunderstandings,and to their clarification we have devoted the Appendix C.10 . Beyond a single SSC: the collapse It should be clear that the Semiclassical Self-consistent Configurations introduced in theprevious subsection are not enough in order to deal with the problem at hand, i.e. thetransmutation of a symmetric universe into the actual inhomogeneous and anisotropic one.The extra element we must consider in order to represent our ideas is the quantum collapseof the wave function of the matter fields. That is, the proposal that the normal unitaryevolution characterizing the standard field theory (and then by definition the SSCs) shouldbe supplemented by instances of quantum collapse, thought to be triggered, somehow, bythe effects of the gravitational degrees of freedom which are not fully represented in themetric description (as it was indicated, the metric tensor should be regarded here as a mereeffective description − at the hydrodynamical level − of the average aggregate behaviour ofthe true gravitational degrees of freedom). One of the main purposes of this paper is topropose a formalism that would allow us to represent and explore such idea, and to uncoverits limitations. We will separate the evolution of the system into the standard part and the collapse,considering the first one as described in terms of the Heisenberg picture, while the collapsewill be treated in terms of the Schr¨odinger one. We can look at that distinction as an“interaction picture description”, where the role of the interaction is played by whateverphysics lies behind the collapse process, and the rest is absorbed into the evolution of theoperators, as it is done in the Heisenberg picture. Then, the space-time dependence encodedin the field operators reflects the standard unitary evolution, where the states will remainconstant except when a collapse occurs, which will be characterized by a random jump ofthe state | ξ (cid:105) to one among a set of suitable related states {| ζ (cid:105) , . . . , | ζ n (cid:105)} , in what we willcall a “self-induced” collapse of the wave function, | ξ (cid:105) → | ζ (cid:105) . (3) Note that, despite the previously mentioned indications of collapse-like behaviour in the relatedSchr¨odinger-Newton system, we are postulating the collapse as an additional feature, because it doesnot seem that those previously observed features are able to account for the breaking of the initial trans-lational and rotational symmetries of the configuration. That is, the Shr¨odinger-Newton system, onceprovided with an initial data possessing one such symmetry, would result in an equally symmetric solu-tion, simply due to the deterministic nature of the problem and the fact that the dynamics does not breakthose symmetries. → SSC-II . (4)In order to proceed, we need a rather precise prescription for the description of “thecollapse”. Consider first, within the Hilbert space associated to the given SSC-I, that atransition | ξ (I) (cid:105) → | ζ (I) (cid:105) target “is about to happen”, with both | ξ (I) (cid:105) and | ζ (I) (cid:105) target in H (I) .Generically, the set { g (I) , ˆ ϕ (I) , ˆ π (I) , H (I) , | ζ (I) (cid:105) target } will not represent a new SSC. We willthus say that the state | ζ (I) (cid:105) target is “not physical”. It represents a characterization (of sorts)of the state into which the collapse will take our matter fields, employing the mathematicallanguage of the H (I) , a language that would be inappropriate if the state of the matterfields were indeed | ζ (I) (cid:105) target , simply because in such case the space-time metric would haveto be different from the one used to make the construction of that Hilbert space. In orderto have a sensible picture, we need to connect this state | ζ (I) (cid:105) target with another one | ζ (II) (cid:105) “living” in a new Hilbert space H (II) for which { g (II) , ˆ ϕ (II) , ˆ π (II) , H (II) , | ζ (II) (cid:105)} is an actualSSC. We will denote the new SSC by SSC-II. Thus, first we need to determine the “target”(non-physical) state in H (I) to which the initial state is in a sense “tempted” to jump, andafter that, we need to relate such target state with a corresponding state in the Hilbert spaceof a new SSC, the SSC-II. We will define these notions more precisely below. Following ourprevious treatments on the subject (see for instance reference [17]), we will consider thatthe target state is chosen stochastically, guided by the quantum uncertainties of designatedfield operators, evaluated on the initial state | ξ (I) (cid:105) , at the collapsing time. We will maintainthe essence of such prescriptions here. On the other hand, regarding the identificationbetween the two different SSCs involved in the collapse, there seems to be in principle manynatural options. The usefulness of those depends, of course, on the degree that they actuallydetermine possible SSC’s. In this paper we will be focusing on the possibility offered by thefollowing prescription: Consider that the collapse takes place along a Cauchy hypersurfaceΣ c . A transition from the physical state | ξ (I) (cid:105) in H (I) to the physical state | ζ (II) (cid:105) in H (II) (associated to the target non-physical state | ζ (I) (cid:105) target in H (I) ) will occur in a way that target (cid:104) ζ (I) | ˆ T (I) µν [ g (I) , ˆ ϕ (I) , ˆ π (I) ] | ζ (I) (cid:105) target (cid:12)(cid:12) Σ c = (cid:104) ζ (II) | ˆ T (II) µν [ g (II) , ˆ ϕ (II) , ˆ π (II) ] | ζ (II) (cid:105) (cid:12)(cid:12) Σ c , (5)12.e. in such a way that the expectation value of the energy-momentum tensor associated tothe states | ζ (I) (cid:105) target and | ζ (II) (cid:105) evaluated on the Cauchy hypersurface Σ c coincide. Note thatthe left hand side in the expression above is meant to be constructed from the elements ofthe SSC-I (although | ζ (I) (cid:105) target is not really the state of the SSC-I), while the right hand sidecorrespond to quantities evaluated using the SSC-II. As we have relied for motivation ofour proposal involving the collapse of the wave function on some ideas related to quantumgravity, and given that, at the classical level, the energy-momentum tensor acts as the“source of the gravitational interaction”, we find it reasonable to assume that it is preciselythe expectation value of the energy-momentum tensor of the different states involved in thecollapse the determining aspect for such identification.That means that, in general, at the collapsing time we will have two different metrics g (I) and g (II) for a given Cauchy hypersurface. There could be some special situations for whichboth metrics g (I) and g (II) coincide over some neighbourhood of Σ c , but in general there isno reason to expect that there might be a suitable interpolating metric description for thespace-time during the process of collapse. At best we might hope to find a recipe where theinduced metric on the hypersurface would be continuous at Σ c , but its normal derivative(i.e. the extrinsic curvature), would not. Later we will find that, indeed, this would be whathappens in the case of interest when we use the matching prescription given in equation (5).By construction, the standard unitary evolution of quantum mechanics takes place withina given SSC, thus we need the collapses in order to jump between the different SSCs, andhave the possibility of describing the generation of the seeds of cosmic structure. After all,generation means that “something that did not exist at a certain time does exist at a laterone”. It is worth emphasizing that the dynamical collapse of the wave function does notbelong to what we could call “well established physics”, although there are several proposalsformulated within the community working on foundational aspects of quantum theory (seefor instance [26] and references therein) which could be connected with the general ideasoutlined in this section.We should note that the equation (1) would not, in general, hold through the collapses. At such times the excitation of the fundamental quantum gravitational degrees of freedom In this work we are considering the collapse as taking place instantaneously, i.e. on a space like hyper-surface Σ c . However, it is perhaps better to think of this as an approximated description of somethingtaking place very fast in comparison to the other time scales of the problem. Q µν in the semiclassical field equations, which is supposed to becomenonzero only during the collapse of the quantum mechanical wave function of the matterfields: G µν + Q µν = 8 πG (cid:104) ˆ T µν (cid:105) . (6)The setting is so far general and would allow in principle to consider various situations.In this work, we want to focus on the description of the emergence of the seeds of cosmicstructure in the context of a universe which was initially described by a homogeneous andisotropic state for the gravitational and matter degrees of freedom. The idea is that, at somepoint, the quantum state of the matter fields reaches a stage whereby the corresponding statefor the gravitational degrees of freedom leads to a quantum jump of the matter field wavefunction. The resulting state of the matter fields needs not share the symmetries of theinitial state, | symmetric (cid:105) → | non-symmetric (cid:105) , (7)and its connection to the gravitational degrees of freedom, which again is assumed to beaccurately described by the Einstein semiclassical equations, leads to a geometry that is nolonger homogeneous and isotropic. The matching at the collapsing time of the two SSCsconstructed in Sections III A and III B will be explicitly analysed in Section III C followingthe general ideas discussed here. III. FITTING INFLATION INTO OUR GENERAL FORMAL SCHEME
In this section we will show a detailed realization of the ideas described above. Themotivation for this is twofold: On the one hand it will serve as a proof of concept, and onthe other it will help us clarify the description of the cosmological situation that gave riseto the general ideas explored in [17] and subsequent works.14s we have argued, in order to have a sensible picture for the problem at hand we shoulddetermine the SSC’s suitable for describing the cosmological evolution of the universe. Ourpoint of view is that, in principle, something like this could be done in most situations.However, it seems clear that, in practice, the complexity of the problems one is commonlyinterested in would make this task simply unmanageable. Nevertheless, as we are mainlyinterested in the study of the relatively simple case of the very early universe, we can restrictourselves to the subset of the nearly (as characterized by the parameter ε in expression (16))homogeneous and isotropic SSCs. That is, even though the problem is well defined in general,we will approach its solution through a practical perturbative approach. However, we willalways have under control the exact nature of the approximations we will be using, providingfor the first time a clear interpretative picture which is well defined from the beginning anddoes not change as the discussion progresses.We will start now to be rather specific. As it was indicated in the previous section thematter fields will be treated in the language of a quantum field theory on curved space-times [47, 58–60]. We will concentrate here on the case of a single scalar field, the inflaton.The other fields, and in particular all the fields of the standard model of particle physics,are assumed in their vacuum state and will be ignored in the present work. In order toclarify our ideas, let us remind the reader that inflation is supposed to occur in a patchwhich emerges from the Planck era (at time t IS , where inflation starts), corresponding toa situation where the scalar field is in a regime where the inflaton potential is sufficientlylarge, the kinetic term sufficiently small, and the space-time geometry sufficiently close to ahomogeneous and isotropic one for inflation dynamics to take place. After the onset of suchdynamics, that patch exponentially inflates, leading to a region of space-time which is veryclose to a homogeneous and isotropic flat Robertson-Walker universe, with the remnants ofthe inhomogeneities that were present at t IS reduced by an exponential factor in the numberof e-folds, N ≡ ln( a/a IS ), with a the value for the scale factor (see expression (15) below).Thus, there would be at any time during inflation remnants of inhomogeneity that are atmost of order e −N . These remanent inhomogeneities are not supposed to be relevant atall in any physical consideration during much of the inflationary regime itself, and are thussaid to be essentially erased by inflation. This is the situation where our analysis will befocussed on, and this is indeed necessary if we want to consider as justified any deductionthat relies on the use of the vacuum state of the inflaton field to characterize the seeds of the15osmological structures we observe today. Alternatively, if one wanted to claim that suchremnants from the pre-inflationary era are tied to the generation of the seeds of structure,we would have to accept that it is impossible to predict any features of such structure,because we do not know anything about the form of the spectrum that might characterizesuch remnants.The fact that we do not even have a truly non-perturbative method for dealing withinteracting quantum fields in curved spaces leads us to set our attention on the case of amassive, non-interactive scalar field. In fact, such a model applied to the early phase ofaccelerated expansion has been extensively analysed in the literature, see for instance [61].At the classical level the inflaton field satisfies the Klein-Gordon equation, g µν ∇ µ ∇ ν φ − m φ = 0 , (8)with g µν the inverse of the space-time metric, and ∇ µ the covariant derivative. Given φ ( x )and φ ( x ) two solutions to the classical equation of motion (8), the symplectic product isdefined by ( φ , φ ) Sympl ≡ − i (cid:90) Σ [ φ ( ∂ µ φ ∗ ) − ( ∂ µ φ ) φ ∗ ] d Σ µ . (9)Here d Σ µ ≡ n µ d Σ, with n µ a time-like, future-directed, normalized 4-vector orthogonal tothe 3-dimensional Cauchy hypersurface Σ, and d Σ = √ g Σ d x its volume element. As usualthe expression (9) does not depend on the selected Cauchy hypersurface. In terms of theconjugate momentum associated to the inflaton field, π ( x ) ≡ √ g Σ ( n µ ∂ µ φ ), the symplecticproduct can be re-written in the form(( φ , π ) , ( φ , π )) Sympl ≡ − i (cid:90) Σ [ φ π ∗ − π φ ∗ ] d x. (10)At the quantum level the inflaton field and its conjugate momentum are promoted to fieldoperators acting on a Hilbert space H . These operators must satisfy the standard equaltime commutation relations between them, which, upon an appropriate choice of space-timecoordinates (to be further specified shortly), take the form,[ ˆ φ ( η, (cid:126)x ) , ˆ π ( η, (cid:126)y )] = i (cid:126) δ ( (cid:126)x − (cid:126)y ) , [ ˆ φ ( η, (cid:126)x ) , ˆ φ ( η, (cid:126)y )] = [ˆ π ( η, (cid:126)x ) , ˆ π ( η, (cid:126)y )] = 0 . (11)The standard way to proceed now is to decompose ˆ φ ( x ) in terms of the time-independentcreation and annihilation operators,ˆ φ ( x ) = (cid:88) α (cid:0) ˆ a α u α ( x ) + ˆ a † α u ∗ α ( x ) (cid:1) , (12)16ith the functions u α ( x ) a complete set of normal modes orthonormal with respect to thesymplectic product, i.e.( g µν ∇ µ ∇ ν − m ) u α = 0 , ( u α , u α (cid:48) ) Sympl = (cid:126) δ αα (cid:48) . (13)A similar expression to that given in (12) can be obtained for the conjugate momentum ˆ π ( x ),replacing the functions u α by √ g Σ ( n µ ∂ µ u α ). As we will be only interested in configurationscorresponding to space-times very close to a flat Robertson-Walker one, we will be able tochoose the labels α specifying the different mode solutions to be the wave vectors (cid:126)k , despitethe fact that the spatial sections are not, in general, exactly flat. Clearly, one should alwayskeep in mind that, in general, the functions u (cid:126)k ( x ) would not correspond exactly to thestandard Fourier modes in flat space (the simple functional form of the modes u (cid:126)k ( x ) ∝ e i(cid:126)k · (cid:126)x will be appropriate only for the exactly homogeneous and isotropic case).With all these conventions (expressions (12) and (13) above) the commutators (11) trans-late into the standard [ˆ a (cid:126)k , ˆ a † (cid:126)k (cid:48) ] = δ (cid:126)k(cid:126)k (cid:48) , [ˆ a (cid:126)k , ˆ a (cid:126)k (cid:48) ] = [ˆ a † (cid:126)k , ˆ a † (cid:126)k (cid:48) ] = 0 (14)for the creation and annihilation operators. As usual, the vacuum is defined to be thestate that is annihilated by all the ˆ a (cid:126)k ’s, (i.e. ˆ a (cid:126)k | (cid:105) = 0 for all (cid:126)k ). The Hilbert space canbe constructed ( `a la Fock) by successive applications of creation operators on the vacuum.However, the relations (13) do not determine the set of mode solutions unequivocally, andthe particular choice corresponds to an election of the vacuum.This construction for the Hilbert space of the inflaton field is generic and applies for any(globally hyperbolic) space-time configuration. An exact solution will not be possible ingeneral, but an approximate one suitable for the study of the very early universe could beobtained perturbatively. The study of the seeds of cosmic structure depends essentially onthe scalar sector of the perturbations. Ignoring for simplicity the so called vector and tensormodes, we will choose a coordinate system (the so called conformal Newtonian gauge) inwhich the space-time metric takes the form ds = a ( η ) (cid:2) − (1 + 2 ψ ) dη + (1 − ψ ) δ ij dx i dx j (cid:3) , with ψ ( η, (cid:126)x ) (cid:28) . (15)Here a is the scale factor, ψ an analogue to the Newtonian potential, and η the cosmologicaltime in conformal coordinates. The coordinates x i label the observers co-moving with the17xpansion. Note that, in accordance with our approach, both a ( η ) and ψ ( η, (cid:126)x ) are (dimen-sionless) classical fields. Setting ψ = 0 we recover a spatially flat homogeneous and isotropicRobertson-Walker universe. In order to avoid problems in the infrared we will work withperiodic boundary conditions over a box of size L . We can take the limit L → ∞ at theend of the calculations. In the present context we can describe the Newtonian potential interms of a Fourier decomposition and write generically ψ ( η, (cid:126)x ) = ε (cid:88) (cid:126)k (cid:54) =0 ˜ ψ (cid:126)k ( η ) e i(cid:126)k · (cid:126)x , (16)where the sum is over all vectors (cid:126)k with k n = 2 πj n /L , j n = 0 , ± , ± , . . . and n = 1 , , ψ ( η, (cid:126)x ) we should demand ˜ ψ (cid:126)k ( η ) = ˜ ψ ∗− (cid:126)k ( η ), and we willbe assuming ˜ ψ (cid:126)k ( η ) (cid:46) O (1) and ε (cid:28)
1, so we can guarantee ψ ( η, (cid:126)x ) (cid:28)
1. Note that as thespace-independent part of the Newtonian potential can be reabsorbed in the scale factor, itdoes not appear in (16). Recall also that we were working in co-moving coordinates, wherethe (cid:126)k ’s are fixed in time and label each particular mode (related to their physical valuesthrough (cid:126)k/a ).Working up to the first order in ε , the equations (13) simplify to(1 − ψ )(¨ u (cid:126)k + 2 H ˙ u (cid:126)k ) − (1 + 2 ψ )∆ u (cid:126)k − ψ ˙ u (cid:126)k + a m u (cid:126)k = 0 , (17a) (cid:90) η =const. (cid:2) u (cid:126)k ( ∂ η u ∗ (cid:126)k (cid:48) ) − ( ∂ η u (cid:126)k ) u ∗ (cid:126)k (cid:48) (cid:3) (1 − ψ ) d x = i (cid:126) a − δ (cid:126)k(cid:126)k (cid:48) . (17b)For the general case even the construction of that SSC will not be trivial, although, inprinciple, it would be doable. However, in order to proceed, we will consider a simpleexample: the transition from an exactly homogeneous and isotropic universe, ψ ( η, x ) = 0,to a situation where, as a result of one of our collapse events, a single nontrivial plane-waveis excited. We characterize that by the wave vector (cid:126)k , and write ψ ( η, x ) = ε ˜ ψ (cid:126)k ( η ) e i(cid:126)k · (cid:126)x + c.c. , with ε (cid:28) c.c. denoting the complex conjugate. Next, we proceed to explicitlyconstruct the two SSC’s: the first corresponding to the homogeneous and isotropic pre-collapse situation, and the second to the post-collapse one, characterized by a SSC wherethere is an actual fluctuation with wave vector (cid:126)k . Then, we will consider in some detail thecharacter of the description of the transition between those two. Indeed, we recover that situation even if take ψ = ψ ( η ), as can be seen by a simple change of the space-timecoordinates. Here we will impose (cid:82) d (cid:126)x ψ ( η, (cid:126)x ) = 0 in order to remove those ambiguities.
18 small comment on notation is in order here. The objects that are specific to theconstructions we will be discussing in Sections III A and III B will bear an index (I or II)indicating the specific SSC construction to which they belong. In reading those two sectionsthe reader can simply ignore that index, as they will not change within each section. How-ever, we have left the index in place in order to avoid confusion when the two constructionsare brought together in the discussion of their matching in Section III C.
A. A homogeneous and isotropic SSC
At a time corresponding to few e-foldings after inflation started the relevant region of theuniverse is thought to be described by a homogeneous and isotropic SSC. The Newtonianpotential vanishes at that time, ψ ( η, (cid:126)x ) = 0. We will call it the first SSC, or simply the SSC-I. In order to carry out that construction we will take the space-time metric to be that of aRobertson-Walker universe, with a pre-established (nearly) de Sitter scale factor. The smalldeviation from the exact de Sitter expansion will be parametrized by (cid:15) (I) ≡ − ˙ H (I) / H ,where H (I) ≡ ˙ a (I) /a (I) is a measure of the expansion rate of the universe, related to thestandard Hubble parameter by H (I) = H (I) /a (I) . During slow-roll 0 < (cid:15) (I) (cid:28) ε (cid:28) a (I) ( η ) = ( − /H (I)0 η ) (cid:15) (I) and the Hubbleparameter in conformal coordinates H (I) = − (1 + (cid:15) (I) ) /η to the first order in the slow-rollparameter (remember that −∞ < η < u (I) (cid:126)k ( x ) for the previously givenhomogeneous and isotropic space-time configuration. Naturally, given the symmetries of thespatial background, we can look for solutions of the form u (I) (cid:126)k ( x ) = v (I) (cid:126)k ( η ) e i(cid:126)k · (cid:126)x /L / . (18)Introducing ψ = 0 and the ansatz (18) into the equations (17) we obtain¨ v (I) (cid:126)k + 2 H (I) ˙ v (I) (cid:126)k + (cid:0) k + a m (cid:1) v (I) (cid:126)k = 0 , (19a) v (I) (cid:126)k ˙ v (I) ∗ (cid:126)k − ˙ v (I) (cid:126)k v (I) ∗ (cid:126)k = i (cid:126) a − . (19b)We will find that a m is first order in the slow-roll parameter, a m = 3 (cid:15) (I) H tothe first nonvanishing order in (cid:15) (I) (see equation (27) bellow), so equation (19a) should be19onsidered to this same order, i.e. we should take H (I) = − (1 + (cid:15) (I) ) /η . For the modeswith k (cid:54) = 0 the most general solution to the equation (19a) is a linear combination ofthe functions η / (cid:15) (I) H (1) ν ( − kη ) and η / (cid:15) (I) H (2) ν ( − kη ), with H (1) ν ( − kη ) and H (2) ν ( − kη ) theHankel functions of first and second kind, and ν (I) = 3 / (cid:15) (I) − m / H to the first orderin the slow-roll parameter. A choice of modes corresponds to an election of the vacuum.For a space-time background without a time-like killing vector field there is no preferentialchoice, but following the standard literature on the subject we will take the Bunch-Davisconvention: i.e. use modes such that in the asymptotic past they behave as purely “positivefrequency solutions”, normalized according to (19b). Working to the lowest non-vanishingorder in the slow-roll parameters ( ν (I) = 3 / v (I) (cid:126)k ( η ) = (cid:114) (cid:126) k (cid:16) − H (I)0 η (cid:17) (cid:18) − ikη (cid:19) e − ikη . (20)These functions coincide with the standard mode solutions of a massless ( m = 0) scalarfield in a de Sitter ( (cid:15) (I) = 0) universe. We note, however, the Hankel functions are notwell behaved at the origin, and thus the zero mode is not included in (20). For k = 0 thegeneral solution to the equation (19a) is a linear combination of the functions η (3+2 (cid:15) (I) − ν ) / and η (3+2 (cid:15) (I) +2 ν ) / . The choice is arbitrary, provided it has positive symplectic norm. Herewe take v (I)0 ( η ) = (cid:115) (cid:126) H (I)0 (cid:20) − i (cid:16) − H (I)0 η (cid:17) (cid:21) (cid:16) − H (I)0 η (cid:17) m / H , (21)which has been normalized using the condition (19b). Note that, contrary to what we didfor the k (cid:54) = 0 modes, we have worked now to the first order in the slow-roll parameter. Wewill find that this is necessary in order to accommodate a slow-rolling expectation value forthe zero mode.So far we have given a prescription to construct the mode solutions u (I) (cid:126)k ( x ) for the SSC-I,and thus we have a construction of the Hilbert space and a representation of the fundamentalfields as operators acting on it. However, we still need to find a state | ξ (I) (cid:105) in H (I) suchthat its expectation value for the energy-momentum tensor leads to the desired nearly deSitter, homogeneous and isotropic cosmological expansion. As expected, the state in theSSC-I can just have the zero mode excited. The expectation value of the field operator isthen homogeneous and isotropic, φ (I) ξ, ( η ) ≡ (cid:104) ξ (I) | ˆ φ (I) ( x ) | ξ (I) (cid:105) = ξ (I)0 v (I)0 ( η ) /L / + c.c., (22)20ith ξ (I)0 ≡ (cid:104) ˆ a (I)0 (cid:105) . The subindex “0” in φ (I) ξ, ( η ) makes reference to this fact, whereas thesuper-index “(I)” indicates that the field operators involved are those of the SSC-I, and alsothat the expectation values are taken in the state | ξ (I) (cid:105) . We should note that as the potentialis quadratic in the field operator the Ehrenfest theorem guarantees the classical equation ofmotion for the expectation value of the inflaton field.As it was just argued, the symmetries of the space-time background lead us to considera state in which all the modes with k (cid:54) = 0 are in their vacuum state, while the zero mode isexcited. Thus, we consider a state of the form | ξ (I) (cid:105) = F (ˆ a (I) † ) | (I) (cid:105) , (23)where F ( ˆ X ) stands for a suitable generic function acting on the operators ˆ X (as we will seeit will be sufficient for our purposes in this paper to use the function that is associated withthe “coherent” states, namely F ( ˆ X ) ∝ exp( ˆ X )). For those particular states with only thezero mode excited, the components 00 and i = j of Einstein’s field equations simplify to H = 4 πG (cid:16) ( ˙ φ ) ξ, + a m ( φ ) ξ, (cid:17) , (24a) H + 2 ˙ H (I) = − πG (cid:16) ( ˙ φ ) ξ, − a m ( φ ) ξ, (cid:17) . (24b)The two sides of the 0 i and the i (cid:54) = j equations vanish identically for that particular metricand state. Here ( ˙ φ ) ξ, ≡ (cid:104) ξ (I) | : ( ∂ η ˆ φ (I) ) : | ξ (I) (cid:105) and ( ˆ φ ) ξ, ≡ (cid:104) ξ (I) | : ( φ (I) ) : | ξ (I) (cid:105) arefunctions of η . The equations (24) are analogous (but not exactly equal) to those obtainedin the context of a classical field theory, with the squares of the scalar field and its timederivative replaced now by the expectation value of their corresponding operators. Thisdifference could, in principle, introduce important departures from the classical behaviour,but fortunately this problem will not affect us in the simple case treated here. As we havejust mentioned, Ehrenfest’s theorem guarantees the classical equations of motion for theexpectation value of the inflaton field. In general, the “classical relations” ( ˙ φ ) ξ, = ( ˙ φ (I) ξ, ) and ( φ ) ξ, = ( φ (I) ξ, ) will not hold (that is, (cid:104) ˆ a (I)0 (cid:105) = ξ (I)0 does not necessarily imply (cid:104) ˆ a (cid:105) = There is also an infinite contribution to the expectation value of the energy-momentum tensor coming fromthe vacuum of the theory, but this is an issue related to the well known cosmological constant problemand it will not be considered any further here. We could assume that the energy-momentum tensor isrenormalized, say `a la
Haddamard, and that the cosmological constant is set to zero. In practice we willsimply impose normal ordering on the energy-momentum tensor operator, : ˆ a (cid:126)k ˆ a † (cid:126)k := ˆ a † (cid:126)k ˆ a (cid:126)k . ). However, as we want to consider a state | ξ (I) (cid:105) that is sharply peeked around a classicalfield configuration, φ (I) ξ, ( η ) and π (I) ξ, ( η ), we take it to correspond to a “highly excited coherent”state (i.e. ˆ a (I)0 | ξ (I) (cid:105) = ξ (I)0 | ξ (I) (cid:105) , with ξ (I)0 ∈ C ), for which those classical relations do hold. Inthat case we recover precisely the same Friedmann equations of the standard treatments.Requiring that the universe is characterized by a regime of slow-roll inflation withexpansion rate H (I)0 and slow-roll parameter (cid:15) (I) ≡ − ˙ H (I) / H implies 3( ˙ φ (I) ξ, ) = (cid:15) (I) a m ( φ (I) ξ, ) . This expression is obtained, working up to the first order in the slow-roll parameter (cid:15) (I) , by combining the equations (24a) and (24b) (once we have assumed acoherent state). Solving this equation for φ (I) ξ, ( η ) we obtain φ (I) ξ, ( η ) ∝ η (cid:113) (cid:15) (I) m / H . (25)Comparing with expressions (21) and (22), and taking the parameter ξ (I)0 as real, (cid:104) ξ (I) | ˆ φ (I) ( x ) | ξ (I) (cid:105) = 2 ξ (I)0 L / (cid:115) (cid:126) H (I)0 (cid:16) − H (I)0 η (cid:17) m / H . (26)Thus, we find compatibility of equations (25) and (26), corresponding to a period of slow-rollinflation with (cid:15) (I) = m H , H (I)0 = 2 (cid:15) (I) t p ( ξ (I)0 ) L , (27)where t p = 8 πG (cid:126) stands for the Planck time. As it is well known, in order to have (cid:15) (I) (cid:28) m (cid:28) H (I)0 . Inflation is expected to take place at very high energies, sothis requirement is not generally taken as problematic. On the other hand, given the valuesof the parameters H (I)0 and (cid:15) (I) characterizing the cosmological expansion during inflation,we can read from the second expression in (27) the state for the zero mode, ξ (I)0 (or moreprecisely, ξ (I)0 /L / ). This fixes the state for the SSC-I, as all other modes are taken to bein their vacuum state (with respect to the Bunch-Davies convention). We thus have themetric, the quantum field theory construction, and the specific state that is compatible withEinstein’s equation. This therefore completes the construction of the SSC-I. We now turn tothe more complex case corresponding to a slightly inhomogeneous and anisotropic situation. The presence in the last expression of the factor L − might seem strange at first sight, as the size ofthe artificial box should have no impact on the expansion rate of the universe. However, we must recallthat in our notation the expectation value of the scalar field is proportional to L − / , as can be seen inexpression (22). Thus, what we have here is simply the fact that H (I)0 is proportional to (cid:104) ˆ φ (cid:105) . . A SSC with the (cid:126)k -mode excited Here we want to carry out the construction of a new SSC corresponding to an excitation inthe Newtonian potential characterized by the wave-vector (cid:126)k . We will denote this new SSCby SSC-II. We must consider a slight deviation from the previous homogeneous and isotropiccosmological background, characterized now by the parameters H (II)0 and (cid:15) (II) (which might,in principle, differ slightly from those corresponding to the SSC-I discussed in the previoussection), and a Newtonian potential with one single term excited in the expression (16),described by an (in principle) arbitrary function ˜ ψ (cid:126)k ( η ) = P ( η ). We will latter see that thesemiclassical field equations will allow us to obtain this function and the quantum state ofthe SSC, leading to a complete determination of this construction.However, the first step in the building of the SSC-II will be the construction of thequantum theory for the inflaton field on the new type of space-time configuration we areconsidering now (with any given P ( η )). That is, we need a complete set of normal modes u (II) (cid:126)k ( x ) appropriate for the new construction. As we are working up to the first order in ε ,and since the Newtonian potential is given by ψ ( η, (cid:126)x ) = εP ( η ) e i(cid:126)k · (cid:126)x + c.c. , we can considerthe ansatz: u (II) (cid:126)k ( x ) = εδv (II) − (cid:126)k ( η ) e i ( (cid:126)k − (cid:126)k ) · (cid:126)x /L / + v (II)0 (cid:126)k ( η ) e i(cid:126)k · (cid:126)x /L / + εδv (II)+ (cid:126)k ( η ) e i ( (cid:126)k + (cid:126)k ) · (cid:126)x /L / . (28)Introducing (28) into (17) we find that, to the zeroth order in the ε , the evolution equationis given by ¨ v (II)0 (cid:126)k + 2 H (II) ˙ v (II)0 (cid:126)k + (cid:0) k + a m (cid:1) v (II)0 (cid:126)k = 0 , (29a)with normalization condition v (II)0 (cid:126)k ˙ v (II)0 ∗ (cid:126)k − ˙ v (II)0 (cid:126)k v (II)0 ∗ (cid:126)k = i (cid:126) a − , (29b)while at first order in ε the corresponding evolution equation takes the form δ ¨ v (II) ± (cid:126)k + 2 H (II) δ ˙ v (II) ± (cid:126)k + (cid:104) ( (cid:126)k ± (cid:126)k ) + a m (cid:105) δv (II) ± (cid:126)k = F ± (cid:126)k ( η ) , (30a)where, F + (cid:126)k ( η ) ≡ P ˙ v (II)0 (cid:126)k − (cid:0) k + a m (cid:1) P v (II)0 (cid:126)k , (30b) F − (cid:126)k ( η ) ≡ P ∗ ˙ v (II)0 (cid:126)k − (cid:0) k + a m (cid:1) P ∗ v (II)0 (cid:126)k . (30c)23he normalization condition is given by˙ v (II)0 ∗ (cid:126)k + (cid:126)k δv (II)+ (cid:126)k − v (II)0 ∗ (cid:126)k + (cid:126)k δ ˙ v (II)+ (cid:126)k − ˙ v (II)0 (cid:126)k δv (II) −∗ (cid:126)k + (cid:126)k + v (II)0 (cid:126)k δ ˙ v (II) −∗ (cid:126)k + (cid:126)k = 4 (cid:16) v (II)0 (cid:126)k ˙ v (II)0 ∗ (cid:126)k + (cid:126)k − ˙ v (II)0 (cid:126)k v (II)0 ∗ (cid:126)k + (cid:126)k (cid:17) P. (30d)The zeroth order problem coincides with the situation considered in the previous section,equations (19), and we can simply write: v (II)0 (cid:126)k ( η ) = (cid:114) (cid:126) k (cid:16) − H (II)0 η (cid:17) (cid:18) − ikη (cid:19) e − ikη , for k (cid:54) = 0 , (31a) v (II)00 ( η ) = (cid:115) (cid:126) H (II)0 (cid:20) − i (cid:16) − H (II)0 η (cid:17) (cid:21) (cid:16) − H (II)0 η (cid:17) m / H , for k = 0 . (31b)Next, we consider the first order problem. The functions δv (II) ± (cid:126)k ( η ) satisfy the dynamicalequation (30a), similar to that for the v (II)0 (cid:126)k ± (cid:126)k ( η ), but with a source term determined by theNewtonian potential (which would be specified once we provide the function P ( η ), somethingthat we will do shortly) and the zeroth order functions v (II)0 (cid:126)k ( η ) (given by the expressions (31)above). It is a second order linear equation, therefore, the solution is univocally determinedin terms of the corresponding initial data: δ ˙ v (II) ± (cid:126)k ( η c ) and δv (II) ± (cid:126)k ( η c ). (We take the initialtime for the SSC-II to be the collapsing time, η c .) The normalization relation (30d) is theonly constrain on the initial data. Two simple (but convenient) choices that satisfy thenormalization constraints are δ ˙ v (II) ± (cid:126)k ( η c ) = 0 , δv (II) ± (cid:126)k ( η c ) = 4 v (II)0 (cid:126)k ( η c ) P ( η c ) , (32a)and δ ˙ v (II) ± (cid:126)k ( η c ) = 4 ˙ v (II)0 (cid:126)k ( η c ) P ( η c ) , δv (II) ± (cid:126)k ( η c ) = 0 . (32b)We will pick one of them, for definiteness, the first one. As we have indicated this choicecompletely determines the solution (assuming the function P ( η ) is given). As we will seenext, we will not need to calculate the functions δv (II) ± (cid:126)k ( η ) explicitly in order to finish ouranalysis.So far, we have provided the basic ingredients defining the construction of the quantumfield theory for the SSC-II, but we still need to find a state | ζ (II) (cid:105) ∈ H (II) such that itsexpectation value for the energy-momentum tensor leads to the desired nearly de Sitter,slightly inhomogeneous cosmological expansion, characterized by the Newtonian potential ψ ( η, (cid:126)x ) in the space-time metric (15). The task of constructing that state is rather cumber-some, and the reader is encouraged to skip the details in the first reading. The important24esult is that this construction is carried out in general and, at the end, full compatibilityis ensured by a judicious choice of the function P ( η ) controlling the time dependence of theNewtonian potential.Let us first concentrate on the state. Looking at the symmetries of the space-time back-ground it seems natural to assume that such a state should be of the form: | ζ (II) (cid:105) = . . . | ζ (II) − (cid:126)k (cid:105) ⊗ | ζ (II) − (cid:126)k (cid:105) ⊗ | ζ (II)0 (cid:105) ⊗ | ζ (II) (cid:126)k (cid:105) ⊗ | ζ (II)2 (cid:126)k (cid:105) . . . . (33)Here we are making an evident abuse of notation: The vector in Fock space is characterizedby parameters indicating the specific modes that are excited (all other modes are assumedto be in the vacuum of the corresponding oscillator), and the parameters ζ (II) (cid:126)k are meantto indicate exactly how the mode (cid:126)k has been excited. We could take these parameters todescribe, for instance, a particular coherent state for each different mode, and then the statewe are considering in the expression above would be precisely | ζ (II) (cid:105) = . . . F ( ζ (II) − (cid:126)k ˆ a (II) †− (cid:126)k ) F ( ζ (II) − (cid:126)k ˆ a (II) †− (cid:126)k ) F ( ζ (II)0 ˆ a (II) † ) F ( ζ (II) (cid:126)k ˆ a (II) † (cid:126)k ) F ( ζ (II)2 (cid:126)k ˆ a (II) † (cid:126)k ) . . . | (II) (cid:105) , (34)where F ( ˆ X ) again stands for the object F ( ˆ X ) ∝ exp( ˆ X ). Needless is to say that we canconsider similar excitations of each mode which are not necessarily coherent, but the latterwill be sufficient for our purposes here.The expectation value of the field operator in such a state is given by φ (II) ζ ( x ) = φ (II) ζ, ( η ) + (cid:16) δφ (II) ζ,(cid:126)k ( η ) e i(cid:126)k · (cid:126)x + c.c. (cid:17) + (cid:16) δφ (II) ζ, (cid:126)k ( η ) e i (cid:126)k · (cid:126)x + c.c. (cid:17) + . . . . (35)Contrary to what happens for the SSC-I, expression (22), the excitation in each modeleads now to space-dependencies with three characteristic wavelengths. For instance, evenif just the k = 0 mode is excited, the field expectation value will include terms with thecharacteristic behaviour e ± i(cid:126)k · (cid:126)x . In fact, for a general state of the form (33) we have L / φ (II) ζ, ( η ) = ζ (II)0 v (II)00 ( η ) + ε [ ζ − (cid:126)k δv (II)+ − (cid:126)k ( η ) + ζ (cid:126)k δv (II) − (cid:126)k ( η )] + c.c., (36a) L / δφ (II) ζ,(cid:126)k ( η ) = ζ (II) (cid:126)k v (II)0 (cid:126)k ( η ) + ζ (II) ∗− (cid:126)k v (II)0 ∗− (cid:126)k ( η ) + ε [ ζ (II)0 δv (II)+0 ( η ) + ζ (II) ∗ δv (II) −∗ ( η )+ ζ (II) ∗− (cid:126)k δv (II)+ ∗− (cid:126)k ( η ) + ζ (II)2 (cid:126)k δv (II) − (cid:126)k ( η )] , (36b) L / δφ (II) ζ, (cid:126)k ( η ) = ζ (II)2 (cid:126)k v (II)02 (cid:126)k ( η ) + ζ (II) ∗− (cid:126)k v (II)0 ∗− (cid:126)k ( η ) + ε [ ζ (II) (cid:126)k δv (II)+ (cid:126)k ( η ) + ζ (II) ∗− (cid:126)k δv (II) −∗− (cid:126)k ( η )+ ζ (II) ∗− (cid:126)k δv (II)+ ∗− (cid:126)k ( η ) + ζ (II)3 (cid:126)k δv (II) − (cid:126)k ( η )] , (36c)25ith similar expressions for the other δφ (II) ζ,n(cid:126)k ( η ) (for positive integers n ). We are consideringa coherent state for all the different modes. However, all that is required for the validity ofthe expressions above is that ζ (II) ± n(cid:126)k ≡ (cid:104) ˆ a (II) ± n(cid:126)k (cid:105) . For convenience, we have considered here astate with a very simple form, and in particular we have assumed there is no entanglementbetween the different modes in (33). Note that we could set δφ (II) ζ,n(cid:126)k ( η ) = 0 for all n ≥ ζ (II) ± (cid:126)k , ζ (II) ± (cid:126)k , ζ (II) ± (cid:126)k , etc. Inprinciple, the details behind such relations could be used to determine the exact values ofthe undetermined parameters, and it is easy to see that | ζ (II) ± n(cid:126)k | ∼ ε n | ζ (II)0 | . Recalling thatwe are interested here just on a first order in ε calculation, it is clear that all the terms in εζ ± (cid:126)k and ζ ± n(cid:126)k (with n ≥
2) in the expressions (36) can be disregarded. Thus, in practiceand to the order we are working here, we can simply write | ζ (II) (cid:105) = | ζ (II) − (cid:126)k (cid:105) ⊗ | ζ (II)0 (cid:105) ⊗ | ζ (II) (cid:126)k (cid:105) .In particular that implies | δφ (II) ζ,(cid:126)k | ∼ εφ (II) ζ, , and, from now on, we will write φ (II) ζ ( x ) = φ (II) ζ, ( η ) + ε (cid:16) δ ˜ φ (II) ζ,(cid:126)k ( η ) e i(cid:126)k · (cid:126)x + c.c. (cid:17) , (37)where we have used the notation εδ ˜ φ (II) ζ,(cid:126)k ≡ δφ (II) ζ,(cid:126)k . Thus we have, up to order ε , L / φ (II) ζ, ( η ) = ζ (II)0 v (II)00 ( η ) + c.c., (38a) L / εδ ˜ φ (II) ζ,(cid:126)k ( η ) = ζ (II) (cid:126)k v (II)0 (cid:126)k ( η ) + ζ (II) ∗− (cid:126)k v (II)0 ∗− (cid:126)k ( η ) + ε [ ζ (II)0 δv (II)+0 ( η ) + ζ (II) ∗ δv (II) −∗ ( η )] . (38b)The conditions above are necessary in order to ensure that there are no terms in e ± in(cid:126)k · (cid:126)x (with n ≥
2) appearing in the expectation value of the energy-momentum tensor. That,in turn, is necessary to ensure compatibility of our state ansatz with Einstein’s equations,and the assumption that those terms do not appear to the order ε in the expression for theNewtonian potential of the SSC-II.To the first order in ε the 00, 0 i and i = j components of Einstein’s field equations take26he form:3 H − ε (cid:104)(cid:16) k P + 3 H (II) ˙ P (cid:17) e i(cid:126)k · (cid:126)x + c.c. (cid:105) = (39a)4 πG (cid:110) ( ˙ φ (II)0 ,ζ ) + a m ( φ (II)0 ,ζ ) +2 ε (cid:104)(cid:16) ˙ φ (II) ζ, δ ˙˜ φ (II) ζ,(cid:126)k + a m φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k + a m ( φ (II) ζ, ) P (cid:17) e i(cid:126)k · (cid:126)x + c.c. (cid:105)(cid:111) ,ε (cid:104)(cid:16) ˙ P + H (II) P (cid:17) e i(cid:126)k · (cid:126)x + c.c. (cid:105) = 4 πG (cid:110) ε (cid:104)(cid:16) ˙ φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k (cid:17) e i(cid:126)k · (cid:126)x + c.c. (cid:105)(cid:111) , (39b) −H − H (II) + 2 ε (cid:104)(cid:16) ¨ P + 3 H (II) ˙ P + 2( H + 2 ˙ H (II) ) P (cid:17) e i(cid:126)k · (cid:126)x + c.c. (cid:105) = (39c)4 πG (cid:110) ( ˙ φ (II)0 ,ζ ) − a m ( φ (II)0 ,ζ ) +2 ε (cid:104)(cid:16) ˙ φ (II) ζ, δ ˙˜ φ (II) ζ,(cid:126)k − a m φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k + a m ( φ (II) ζ, ) P −
2( ˙ φ (II) ζ, ) P (cid:17) e i(cid:126)k · (cid:126)x + c.c. (cid:105)(cid:111) . Here (39a) and (39b) are two constraints due to the diffeomorphism invariance of the theory,and (39c) a dynamical equation. The i (cid:54) = j equations vanish to this order.Let us concentrate first on the space-independent part of the SSC-II. Integrating the 00and the i = j components of Einstein equations over a spatial hypersurface of constant η we obtain 3 H = 4 πG (cid:16) ( ˙ φ (II)0 ,ζ ) + a m ( φ (II)0 ,ζ ) (cid:17) , (40a) H + 2 ˙ H (II) = − πG (cid:16) ( ˙ φ (II)0 ,ζ ) − a m ( φ (II)0 ,ζ ) (cid:17) . (40b)Again, these are the standard Friedmann equations. The spatial integral of the equations0 i and i (cid:54) = j vanishes (to the first order in ε ). The equations (40) are analogous to thoseobtained in Section III A for the SSC-I, and the state of the zero mode get fixed like in theprevious case (see expression (27) and recall that, to the order we are working here, theterms in εζ − (cid:126)k and εζ (cid:126)k must be neglected in (36a), i.e. see expression (38a) above). Notethat, in contrast with the homogeneous and isotropic case corresponding to the SSC-I, in thepresent case the system (40) represents just the first contribution in a series expansion, andalthough here we will limit ourselves to the first order, it is clear however that, in principle,we could extend the analysis to any desired order.Introducing the equations (40) into the space-dependent first order system (39) we obtain27 P + 3 H (II) ˙ P = − πG (cid:16) ˙ φ (II) ζ, δ ˙˜ φ (II) ζ,(cid:126)k + a m φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k + a m ( φ (II) ζ, ) P (cid:17) , (41a)˙ P + H (II) P = 4 πG (cid:16) ˙ φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k (cid:17) , (41b)¨ P + 3 H (II) ˙ P + 2 (cid:16) H + 2 ˙ H (II) (cid:17) P =4 πG (cid:16) ˙ φ (II) ζ, δ ˙˜ φ (II) ζ,(cid:126)k − a m φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k + a m ( φ (II) ζ, ) P −
2( ˙ φ (II) ζ, ) P (cid:17) . (41c)The key result, and the aspect that enables us to carry out the construction in a completemanner is the following fact: the equations (41) can be combined into a single dynamicalequation for the Newtonian potential, which is independent of the matter fields first orderquantities δ ˜ φ (II) ζ,(cid:126)k and δ ˙˜ φ (II) ζ,(cid:126)k ,¨ P +2 (cid:34) H (II) + a m φ (II) ζ, ˙ φ (II) ζ, (cid:35) ˙ P + (cid:34) k + 2 H + 4 ˙ H (II) + 8 πG ( ˙ φ (II) ζ, ) + 2 a m φ (II) ζ, ˙ φ (II) ζ, H (II) (cid:35) P = 0 . (42)In fact, we can now use Friedmann equations to write (42) with coefficients that de-pend on the the scale factor and its first and second time derivatives alone, by expressing a (II) mφ (II) ζ, / ˙ φ (II) ζ, as − [(2 H + ˙ H (II) ) / ( H − ˙ H (II) )] / (we have taken the negative signbecause during slow-roll ˙ φ (II) ζ, and φ (II) ζ, must have opposite signs). We can go further byusing the definition of the slow-roll parameter, ˙ H (II) / H = 1 − (cid:15) (II) , and express the aboveequation in the simpler looking way¨ P + 2 (cid:2) H (II) − A (II) (cid:3) ˙ P + (cid:2) k + 2(3 − (cid:15) (II) ) H − A (II) H (II) (cid:3) P = 0 . (43)Here we have used that [(2 H + ˙ H (II) ) / ( H − ˙ H (II) )] / = [(3 − (cid:15) (II) ) /(cid:15) (II) ] / , and defined A (II) ≡ (cid:112) /(cid:15) (II) ma (II) (1 − (cid:15) (II) / / √ (cid:15) (II) , which would be extremely largeduring a phase of slow-roll inflation. However, we should note that this factor appearsmultiplied by the scalar field mass m , and, just as in the case of the SSC-I (rememberexpression (27) in Section III A), we would have H (II)0 = m (1 + r(cid:15) (II) ) / √ (cid:15) (II) . Althoughlarge, this is simply the natural inflationary scale. (Note that we have kept the higher ordercorrection term r that was not explicit in equation (27) to be consistent with the expansionbeing first order in (cid:15) (II) .) Making use of the expression for the scale factor a (II) = H (II) /H (II) ,we find that the equation (43) becomes simply¨ P + (cid:15) (II) (1 + 6 r ) H (II) ˙ P + (cid:2) k − (cid:15) (II) (1 − r ) H (cid:3) P = 0 . (44)28he general solution to the equation (43) depends only on the zero mode (the space-independent part of the universe) and the initial conditions for the Newtonian potential, P c ≡ P ( η c ) and ˙ P c ≡ ˙ P ( η c ), P ( η ) = C η [1+(6 r +1) (cid:15) (II) ] J α ( − kη ) + C η [1+(6 r +1) (cid:15) (II) ] Y α ( − kη ) , (45)where J α ( − kη ) and Y α ( − kη ) are the Bessel functions of first and second kind, α = [1 + 3(1 − r ) (cid:15) (II) ] / (cid:15) (II) , and C and C two constants that will be determinedby the initial conditions. We will not be making use of this explicit solution in the rest ofthe analysis. However, it is worth noting that this represents a damped oscillation of theNewtonian potential. Once we have an expression for the function P ( η ), equation (45), themode solutions determining the quantum field theory construction become fully determinedas well, as we already noted bellow equation (32b).Regarding the initial values for the function P ( η ), we note that the problem has a fun-damental symmetry φ → − φ , and, for definiteness, we will be assuming from now on that φ (II) ζ, >
0. Making use of the two constraints (41a) and (41b), we can express the initialvalues that would determine the specific solution P ( η ) in the form, P ˙ P = √ πG(cid:15) (II) H (II) k − H (cid:15) (II) H (II) − A (II) − k − (3 − (cid:15) (II) ) H + A (II) H (II) −H (II) · δ ˜ φ (II) ζ,(cid:126)k δ ˙˜ φ (II) ζ,(cid:126)k . (46)The equations (46) apply for η ≥ η c , and in particular they are valid at the SSC-II sideof collapsing time, so the system (46) can be used in order to infer the initial conditionsfor the Newtonian potential in terms of the characteristics of the collapse, δ ˜ φ (II) ζ,(cid:126)k ( η c ) and δ ˙˜ φ (II) ζ,(cid:126)k ( η c ). Given the values for P c and ˙ P c at the collapsing time (or equivalently, δ ˜ φ (II) ζ,(cid:126)k ( η c )and δ ˙˜ φ (II) ζ,(cid:126)k ( η c )), we have thus a completely determined space-time metric, and, as discussedin connection with (32b), this then determines the set of normal modes for the SSC-II. Wecan also use the values for δ ˜ φ (II) ζ,(cid:126)k ( η c ) and δ ˙˜ φ (II) ζ,(cid:126)k ( η c ), together with the expression (36b) andthe identities (32a), to determine the value of the parameters ζ (II) (cid:126)k and ζ (II) − (cid:126)k (we have twoequations for two unknowns leading to definite values for the latter). A simple and natu-rally expected conclusion can be seen: the homogeneous and isotropic part of the universe In order to study what happens when k − (cid:15) (II) H = 0 (or close to that point) one would need toinclude the next order in the series expansion. However, we are not going to analyse this here. | ζ (II)0 (cid:105) (or vice versa), whereas the values for P c and˙ P c at the collapse time help determine the quantum state for the modes ± (cid:126)k (i.e. the “modestates” | ζ (II) − (cid:126)k (cid:105) and | ζ (II) (cid:126)k (cid:105) ).It is worth commenting at this point that the seemingly “strange” connection that wehave found between the excitation of the mode (cid:126)k and that of its higher harmonics ( (cid:126)k = n(cid:126)k ,with n natural), is intimately connected with the nonlinearity of the entire proposal (which,as we have explained, is the general relativistic version of what occurs in the Shr¨odinger-Newton system), and is very similar, at the mathematical level, to the effect known as“parametric resonance” that occurs in quantum optics with non-linear media [62, 63].This finalizes the construction of the SSC-II. We have seen that it is fully determinedonce given the values δ ˜ φ (II) ζ,(cid:126)k ( η c ) and δ ˙˜ φ (II) ζ,(cid:126)k ( η c ). Next we need to study the possibility ofmatching this SSC to the SSC-I on the hypersurface corresponding to the collapse time (seeAppendix C for a discussion of the aspect of this matching that is connected to the gaugeissues). C. The matching at the collapse time
As we have discussed before, when trying to describe the emergence of the seeds of struc-ture in our universe, we need to consider the transition from a homogeneous and isotropicSSC to another one lacking such symmetries. Here we will give an explicit constructionfor such a transition using the general ideas developed in Section II B. As we have alreadystressed, we are interested in discussing the formalism, rather than the completely realisticand evidently quite complicated case. We will only consider the transition from the SSC-I with ψ ( η, (cid:126)x ) = 0 we discussed in Section III A, to a situation where a single nontrivialmode (cid:126)k in the Newtonian potential is excited. That is, the SSC-II with ˜ ψ (cid:126)k ( η ) = P ( η ) δ (cid:126)k(cid:126)k described in Section III B. We note that there are two general issues to be treated in thiscontext: a) the actual matching conditions between the SSCI and the SSCII, and b) thecharacterization of the target state in the Hilbert space of the SSC-I , which in turn will beused to characterize the state of the SSC-II.As indicated before we will be considering that at time η c the mode (cid:126)k of the state | ξ (I) (cid:105) undergoes a collapse. (Remember that we are attempting to formalize the description ofthat novel − and evidently unknown − aspect of physics we have argued should be taking30lace in the early universe, described in this paper as the collapse of the wave function).Following the ideas developed in Section II B, we assume that, first, the characteristics of thecollapse are determined within the initial SSC. We will assume that such characterization isencoded in a state within the initial Hilbert space. We will often refer to such state (using avery loose, but heuristically helpful language) as the state the system “is tempted to jumpinto” or the “target state”. In our case that will be a state belonging to the Hilbert space H (I) corresponding to the homogeneous and isotropic SSC-I, but will not be the state thatmakes up the SSC-I. In accordance with the comments above, we will assume that such statecorresponds to a tendency of excitation in the (cid:126)k mode, related in some way to the collapseprocess that we are about to describe, | ξ (I) (cid:105) = | ξ (I)0 (cid:105) → | ζ (I) (cid:105) target = | ζ (I) − (cid:126)k (cid:105) ⊗ | ξ (I)0 (cid:105) ⊗ | ζ (I) (cid:126)k (cid:105) , (47)with all the other modes remaining in their previous state. We will consider how thisparticular target state | ζ (I) (cid:105) target in H (I) is chosen later on. However, as we have anticipated,what we need to face is the fact that the target state in H (I) , together with g (I) µν ( x ), ˆ φ (I) ( x )and ˆ π (I) ( x ), can not represent a new SSC. Thus, we need a new SSC corresponding to aspecific version of the SSC-II, { g (II) µν ( x ) , ˆ φ (II) ( x ) , ˆ π (II) ( x ) , H (II) , | ζ (II) (cid:105)} , such that the state | ζ (II) (cid:105) in H (II) is in some way related to the state | ζ (I) (cid:105) target in H (I) . In the simple caseconsidered here we can focus on the operator ˆ φ (I) (cid:126)k ( η ), with ˆ φ (I) ( x ) = (cid:80) (cid:126)k ˆ φ (I) (cid:126)k ( η ) e i(cid:126)k · (cid:126)x , andwrite φ (I) ζ t , ( η c ) ≡ target (cid:104) ζ (I) | ˆ φ (I)0 ( η c ) | ζ (I) (cid:105) target and εδ ˜ φ (I) ζ t ,(cid:126)k ( η c ) ≡ target (cid:104) ζ (I) | ˆ φ (I) (cid:126)k ( η c ) | ζ (I) (cid:105) target ,even though, as we have stressed, the target state is an element of H (I) but is not part ofany SSC. Here, the subindex “ t ” in ζ t refers to the fact that the quantity corresponds to theexpectation value in such a target state.As it was anticipated in Section II B, we will consider that the identification of the SSC-Iand the SSC-II is guided by the expectation value of the energy-momentum tensor, expres-sion (5). We will see this in detail in the following and, in particular, we will see how thathelps determining the state corresponding to the SSC-II in terms of the target state. To thezeroth order in ε the requirement (5) gives:( ˙ φ (I) ζ t , ) + a (I)2 m ( φ (I) ζ t , ) = ( ˙ φ (II) ζ, ) + a (II)2 m ( φ (II) ζ, ) , (48a)( ˙ φ (I) ζ t , ) − a (I)2 m ( φ (I) ζ t , ) = ( ˙ φ (II) ζ, ) − a (II)2 m ( φ (II) ζ, ) . (48b)From these equations it is easy to conclude that ( ˙ φ (I) ζ t , ) = ( ˙ φ (I) ζ, ) and a (I)2 ( φ (I) ζ t , ) = a (II)2 ( φ (II) ζ, ) . We will assume that there is no jump in the scale factor, and also that there is31o jump in sign in the respective field expectation values. Under these assumptions we areled to ˙ φ (I) ζ t , = ˙ φ (I) ζ, and φ (I) ζ t , = φ (II) ζ, .Now, let us proceed to discuss the matching to the first order in ε . At that order,expression (5) gives:˙ φ (I) ζ t , δ ˙˜ φ (I) ζ t ,(cid:126)k + a (I)2 m φ (I) ζ t , δ ˜ φ (I) ζ t ,(cid:126)k = ˙ φ (II) ζ, δ ˙˜ φ (II) ζ,(cid:126)k + a (II)2 m ( φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k + ( φ (II) ζ, ) P ) , (49a)˙ φ (I) ζ t , δ ˜ φ (I) ζ t ,(cid:126)k = ˙ φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k , (49b)˙ φ (I) ζ t , δ ˙˜ φ (I) ζ t ,(cid:126)k − a (I)2 m φ (I) ζ t , δ ˜ φ (I) ζ t ,(cid:126)k = ˙ φ (II) ζ, δ ˙˜ φ (II) ζ,(cid:126)k − a (II)2 m ( φ (II) ζ, δ ˜ φ (II) ζ,(cid:126)k + ( φ (II) ζ, ) P ) −
2( ˙ φ (II) ζ, ) P. (49c)First, we can use expression (49b) together with our previous results to conclude that δ ˜ φ (I) ζ t ,(cid:126)k = δ ˜ φ (II) ζ,(cid:126)k . Next, subtracting equation (49c) from (49a) and using again the previ-ous results we find that ( ˙ φ (II) ζ, ) P = 0. Thus, as we will assume that after the collapse theuniverse remains in a slow-roll expansion, ˙ φ (II) ζ, (cid:54) = 0, we can conclude that P = 0. This mightseem problematic, but let us recall that the matching conditions are supposed to hold onlyat the time of collapse, η = η c , thus P ( η c ) = 0, but P at later times need not vanish. Onthe other hand, as we will see, this will drastically simplify our analysis. Finally, using theseresults and adding equations (49c) and (49a) we find that δ ˙˜ φ (I) ζ t ,(cid:126)k = δ ˙˜ φ (II) ζ,(cid:126)k .It is worth mentioning that equations (48a) and (48b) have been obtained from( T (I)00 ) ζ t , = ( T (II)00 ) ζ, and ( T (I) ii ) ζ t , = ( T (II) ii ) ζ, , whereas (49a), (49b) and (49c) from( δT (I)00 ) ζ t , = ( δT (II)00 ) ζ, , ( δT (I)0 i ) ζ t , = ( δT (II)0 i ) ζ, and ( δT (I) ii ) ζ t , = ( δT (II) ii ) ζ, . For a scalarfield, and up to the first order in ε , the other components of the energy-momentum tensordo not contain additional information. Of course the identities (48) and (49) are only validat the collapsing time. What is more, the target state only plays a role at that particulartime.Recapitulating, we will ask that at the matching a (II) ( η c ) = a (I) ( η c ), and found φ (II) ζ, ( η c ) = φ (I) ζ t , ( η c ) , ˙ φ (II) ζ, ( η c ) = ˙ φ (I) ζ t , ( η c ) , (50a) δ ˜ φ (II) ζ,(cid:126)k ( η c ) = δ ˜ φ (I) ζ t ,(cid:126)k ( η c ) , δ ˙˜ φ (II) ζ,(cid:126)k ( η c ) = δ ˙˜ φ (I) ζ t ,(cid:126)k ( η c ) , (50b)and P ( η c ) = 0. We can use now expressions (40) to conclude that H (I) = H (II) and (cid:15) (I) = (cid:15) (II) ,i.e. we have continuity for the space-independent part of the space-time background (thespatial metric and extrinsic curvature). 32ithin each SSC, the expectation values of field and momentum operators satisfy theEhrenfest theorem (recall that within each SSC there is nothing exotic going on, and eachmode of the field is essentially a harmonic oscillator). This can be used to compute thequantities φ (II) ζ, ( η c ) and δ ˜ φ (II) ζ,(cid:126)k ( η c ) (see the expression (36) above and remember that, to theorder we are working here, the parameters εζ ± (cid:126)k and ζ ± n(cid:126)k (with n ≥
2) can be disregarded,as we did for instance in equation (38)). The values for ˙ φ (II) ζ, ( η c ) and δ ˙˜ φ (II) ζ,(cid:126)k ( η c ) can beobtained from the time derivatives of the expressions given in (36). However, note that φ (I) ζ t , ( η c ) and δ ˜ φ (I) ζ t ,(cid:126)k ( η c ) correspond to expectation values in a target state, which, as we havebeen emphasising, despite being an element of the Hilbert space of the SSC-I, is not the state characterizing the SSC-I. Thus, in general, such quantities can exhibit spatial dependences.In fact, these are given by the expressions L / φ (I) ζ t , ( η c ) = ξ (I)0 u (I)0 ( η c ) + c.c., (51a) L / εδ ˜ φ (I) ζ t ,(cid:126)k ( η c ) = ζ (I) (cid:126)k u (I) (cid:126)k ( η c ) + ζ (I) ∗− (cid:126)k u (I) ∗ (cid:126)k ( η c ) , (51b)and ˙ φ (I) ζ t , ( η c ) and δ ˙˜ φ (I) ζ t ,(cid:126)k ( η c ) by their time derivatives. Comparing the identities (50) withthe expressions (38) and (51), we arrive to ζ (II)0 = ξ (I)0 , ζ (II) − (cid:126)k = ζ (I) − (cid:126)k and ζ (II) (cid:126)k = ζ (I) (cid:126)k (recallthat, according to the choice given in (32a), for the case P ( η c ) = 0 we have δ ˙ v (II) ± (cid:126)k ( η c ) = δv (II) ± (cid:126)k ( η c ) = 0). That is, using the mode solutions chosen in Sections III A and III B, theparameters ζ (II) (cid:126)k characterizing the state of the inflaton field in the SSC-II can be directlyread from the parameters ζ (I) (cid:126)k characterizing the target state in H (I) . Thus, once the targetstate is determined, the complete SSC-II gets fixed, because as we showed in Section III Beverything is determined there once we specify the SSC state.From P ( η c ) = 0 and the equation (46) we find that, at the time of collapse, we shouldhave (3 H (II) − A (II) ) δ ˜ φ (II) ζ,(cid:126)k ( η c ) + δ ˙˜ φ (II) ζ,(cid:126)k ( η c ) = 0 . (52)Combining (46) and (52) we obtain the explicit expression for ˙ P ( η c ), which shows that, eventhough the Newtonian potential is continuous at η c , its time derivative is not,˙ P ( η c ) = √ πG(cid:15) (II) H (II) δ ˜ φ (II) ζ,(cid:126)k ( η c ) . (53)This jump in the time derivative of the Newtonian potential at the collapsing time givesrise to the primordial perturbation in the (cid:126)k mode. That is, the spatial metric is continuous33t the transition, but its time derivative is not, i.e. at the collapsing time we will have acontinuous but non-smooth description for the space-time manifold.Finally, we must give a prescription for the election of the target state | ζ (I) (cid:105) target involvedin the collapse, expression (47). As it was anticipated in Section II B, and as it has been usualin our previous treatments of the subject, we will consider that the target state is chosenstochastically, guided by the quantum uncertainties, at the time of collapse, of some fieldoperators evaluated in the pre-collapse state | ξ (I) (cid:105) . These operators have been usually takento be the corresponding modes of the inflaton field or their conjugate momenta, or somecombination thereof, and sometimes even both. We had rather large freedom in what wechose in that regard. However, in the present analysis, we find that the field and momentumcan not be assumed to change their expectation value during the collapse in an arbitraryway: The condition (5) imposes P c = 0, and then the relation (52) above. Here, we willconsider that the collapse is guided by the quantum uncertainties associated to the modesof the field operator, and that the immediate post-collapse expectation value of momentumoperator is such that the condition P c = 0 is satisfied.We need to clarify one more point before analysing the determination of the target state.As discussed in [17], in order for the collapse to resemble as much as possible the imple-mentation of the reduction postulate (which is connected with Hermitian observables), wedecompose the operators ˆ φ (I) (cid:126)k ( η ) in its real and imaginary parts, ˆ φ (I) (cid:126)k ( η ) = ˆ φ (I)R (cid:126)k ( η ) + i ˆ φ (I)I (cid:126)k ( η ),and focus the collapse on those. The operators we must consider are thenˆ φ (I)R,I (cid:126)k ( η ) = 1 √ (cid:16) u (I) (cid:126)k ( η )ˆ a (I)R,I (cid:126)k + u (I) ∗ (cid:126)k ( η )ˆ a (I)R,I † (cid:126)k (cid:17) (54)and ˆ a (I)R (cid:126)k = 1 √ (cid:16) ˆ a (I) (cid:126)k + ˆ a (I) − (cid:126)k (cid:17) , ˆ a (I)I (cid:126)k = − i √ (cid:16) ˆ a (I) (cid:126)k − ˆ a (I) − (cid:126)k (cid:17) . (55)With these definitions ˆ φ (I)R,I (cid:126)k ( η ) are Hermitian operators (i.e. ˆ φ (I)R,I (cid:126)k ( η ) = ˆ φ (I)R,I † (cid:126)k ( η )), butthe commutation relations between ˆ a (I)R (cid:126)k and ˆ a (I)I (cid:126)k are non-standard,[ˆ a (I)R (cid:126)k , ˆ a (I)R † (cid:126)k (cid:48) ] = ( δ (cid:126)k,(cid:126)k (cid:48) + δ (cid:126)k, − (cid:126)k (cid:48) ) , [ˆ a (I)I (cid:126)k , ˆ a (I)I † (cid:126)k (cid:48) ] = ( δ (cid:126)k,(cid:126)k (cid:48) − δ (cid:126)k, − (cid:126)k (cid:48) ) , (56)with all the other commutators vanishing.Now we turn to specifying the target state in SSC-I, and thus the relevant state ofSSC-II. As discussed in [17], we will be assuming, in a loose analogy with standard quantum34echanics, that the collapse is somehow similar to an imprecise measurement of the operatorsˆ φ (I)R,I (cid:126)k ( η ), and that the final results will be guided by εδ ˜ φ (II)R,I ζ t ,(cid:126)k ( η c ) = x R,I (cid:126)k (cid:114) (cid:104) (I) (cid:126)k | (cid:104) ∆ ˆ φ (I) (cid:126)k ( η c ) (cid:105) | (I) (cid:126)k (cid:105) = x R,I (cid:126)k (cid:114) (cid:12)(cid:12)(cid:12) v (I) (cid:126)k ( η c ) (cid:12)(cid:12)(cid:12) , (57)with x R,I (cid:126)k taken to be two independent random variables distributed according to a Gaussianfunction centred at zero with unit-spread. The expression (57), together with the relation(52), determines the values for εδ ˜ φ (I) ζ t ,(cid:126)k ( η c ) and εδ ˙˜ φ (I) ζ t ,(cid:126)k ( η c ) at the collapsing time (in termsof the random variables x R,I (cid:126)k ), and then the state | ζ (I) (cid:105) target , and thus | ζ (II) (cid:105) ζ (II) (cid:126)k = ζ (I) (cid:126)k = − i (cid:126) − a ( η c ) ε (cid:104) ˙ v (I) ∗ (cid:126)k ( η c ) δ ˜ φ (I) ζ t ,(cid:126)k ( η c ) − v (I) ∗ (cid:126)k ( η c ) δ ˙˜ φ (I) ζ t ,(cid:126)k ( η c ) (cid:105) , (58a) ζ (II) − (cid:126)k = ζ (I) − (cid:126)k = − i (cid:126) − a ( η c ) ε (cid:104) ˙ v (I) ∗ (cid:126)k ( η c ) δ ˜ φ (I) ∗ ζ t ,(cid:126)k ( η c ) − v (I) ∗ (cid:126)k ( η c ) δ ˙˜ φ (I) ∗ ζ t ,(cid:126)k ( η c ) (cid:105) . (58b)That is, the state of the inflaton field in the SSC-II will be given by | ζ (II) (cid:105) = | ζ (II) − (cid:126)k (cid:105) ⊗ | ζ (II)0 (cid:105) ⊗| ζ (II) (cid:126)k (cid:105) , with ζ (II)0 = ξ (I)0 , and ζ (II) (cid:126)k and ζ (II) − (cid:126)k determined in terms of the values for εδ ˜ φ (I) ζ t ,(cid:126)k ( η c )and εδ ˙˜ φ (I) ζ t ,(cid:126)k ( η c ) at the collapsing time, expressions (58). Regarding the metric tensor, itsspace-independent part will not be affected by the collapse, H (I) = H (II) and (cid:15) (I) = (cid:15) (II) .On the other hand, the Newtonian potential will be given by the solution of the differentialequation expression (42), with initial conditions P c = 0 and ˙ P c determined in terms of thevalues for εδ ˜ φ (I) ζ t ,(cid:126)k ( η c ) and εδ ˙˜ φ (I) ζ t ,(cid:126)k ( η c ) by (46) (see also the expression (53) above).One should avoid being deceived by the close relationship between the target state | ζ (I) (cid:105) target and the state | ζ (II) (cid:105) . While, as we have seen, their corresponding expectationvalues for the basic field operators at the collapse time are the same, their subsequent evo-lution will in general deviate from one another. In particular, the Newtonian potentialwill become non-vanishing, and thus the modes of the SSC-II construction will involve non-vanishing δv (cid:126)k ’s (see equation (28) above). These aspects might become relevant in a detailedanalysis of the resulting primordial spectrum, something that lies well beyond the scope ofthe present manuscript.We emphasize that we have found that, although the Newtonian potential is continuousat the collapse time, its time derivative is not. This means we do not really have a truespace-time description of the process. In the next section we will briefly discuss our viewson this problematic aspect of our results, and in future works we hope to investigate waysin which this aspect of our formalism might be improved.35 V. DISCUSSION
We have considered the generic joint description of gravitation in interaction with aquantum field, to the extent that this can be done without a fully workable theory ofquantum gravity. Our proposal is formalized in terms of what we call a Semiclassical Self-consistent Configuration (SSC), which is nothing but a combination of quantum field theoryon a background space-time, with the requirement that the state of the matter fields and thespace-time geometry be connected through the semiclassical Einstein equations. We haveapplied this approach to a simple inflationary cosmological model, describing both a perfectlyhomogeneous and isotropic configuration, and a situation that deviates from the former oneby a slight excitation of a particular inhomogeneous and anisotropic perturbation. We haveconsidered in detail a proposal for describing the “space-time” where such a perturbationactually emerges from the initial homogeneous and isotropic configuration as the result of thecollapse of the wave function of the inflaton field. To our knowledge, this represents the firsttime such a detailed description of the process of emergence of structure is ever presented.We believe that the treatment developed in this paper can be useful in uncovering how thecollapse of a wave function can be made compatible with a fundamental theory of gravity.If and when we would be in possession of a fully workable and satisfactory quantum theoryof gravitation, and are able to describe in detail its semiclassical regime, we should beable to explore the exact behaviour of the gravitational degrees of freedom on the collapsehypersurface. However, even in the absence of such a theory, a study of these issues couldproduce interesting insights into some of the features that such theory should contain. Wehave argued that the general formalism developed in this work should be useful in detailedstudies of the various theories involving wave function collapse (such as [26, 28–33]), and inparticular in their applications to situations where the gravitational back-reaction becomesimportant, as well as in proposals such as the stochastic gravity of Hu and Verdaguer [46](see the Appendix B for more details).Regarding the inflationary regime (which provided the motivation for the development ofthis formalism), it is clear that what we have done here is just a starting point, as we havelimited ourselves to the study of a single collapse. In order to analyse the problem of theemergence of the seeds of cosmic structure in a complete fashion we would need to considermultiplicity of collapses, occurring in multiple times and involving all the different modes,36s we have done, schematically, in previous works [17, 20–22]. However, in contrast withwhat was done there, the study can now, in principle, be carried out using a well definedand precise formalism as presented here. That formalism would allow us to consider issuessuch as the degree to which “energy conservation” is violated during a collapse, and possiblyto analyse its effects on the evolution of the universe during and immediately after inflation.Moreover, as discussed in section III B, the formalism suggests the existence of correlationsbetween the excitation level of the modes (cid:126)k and that of its higher harmonics, a feature thatis reminiscent of the so called “parametric resonances” occurring in quantum optics withnonlinear materials [62, 63].It is worthwhile to contrast the formalism we have developed in this work with thestandard treatment of inflation, which is, at the fundamental level, essentially perturbative.Such a treatment is based on the separation of a background (involving both the space-time metric and the inflaton field), which is described at the classical level, reserving thequantum treatment just for the linear perturbations. The formalism we have considered hereis, in principle, amenable to a non-perturbative treatment, even though in practice we areoften impeded from carrying that out simply because of the usual limitations that prevent ageneral non-perturbative treatment of a quantum field theory. Nonetheless, if one managesto overcome this limitation on the quantum field side, for instance through treatments basedon lattice approaches, or simply by considering some quantum field solvable model, thescheme could be appropriate to the consistent inclusion of gravity. One obvious advantageof such a setup is that the path to considering higher order perturbations (i.e. anythingbeyond first order perturbations) is clear and well defined from the beginning. On the otherhand, it is quite evident that this cannot be considered as a fully satisfactory description ofnature because of all the well known arguments indicating that we need a quantum theoryof gravitation, see for instance references [48, 64, 65]. Nevertheless, it seems reasonable toassume that a theory like this can be suitable for a description of a situation where themeasures and estimates (i.e. classical and quantum mechanical) of the space-time curvatureare well below the Planck regime, and where the matter fields energy-momentum tensor haveuncertainties that are “not too large” (i.e. see Penrose’s arguments). We will be workingunder the assumption that this would be the case for most of the inflationary regime we areinterested in, and for the post-inflationary cosmological regimes that follow it.However, as we have previously argued, this cannot be the full story if we want to be able37o account for the transition from the completely homogeneous and isotropic universe whichis usually associated with the early and mid stages of inflation (and that we have consideredin Section III A within a very simple model described by the SSC-I), to the late situationwhere inhomogeneities and anisotropies are present (and which was described in a simplifiedmanner involving just one excited mode by the SSC-II of Section III B). Accounting for thisphenomena requires a departure from the established unitary evolution in the form of somesort of “collapse of the wave function” (as considered by Diosi [29–31], Ghirardi, Riminiand Weber [32], or Pearle [33]; see also [26]), a feature that might presumably find its fulljustification in a deeper theory of quantum gravity, as it has been previously discussed byR. Penrose [19, 27]. In the Appendix A we have presented a speculation of how somethinglike that might be tied to the resolution of the problem of time in quantum gravity, basedon the findings of [86] that the usage of a physical clock can naturally introduce effectivedeviations from unitary evolution.Nonetheless, regardless of such speculations, the issue we need to face is how to modifythe SSC formalism to include such a transition. Here, we have considered an attempt todo so, which involves the selection of a state within the Hilbert space of the SSC-I (calledthe “target state”) to which the system “wants to jump” (to which it would jump if theresult was also a SSC), and then finding a new SSC for which the associated state had, onthe collapse hypersurface (a space-like hypersurface taken for simplicity to coincide with ahomogeneous and isotropic one of the cosmological model described by the SSC-I), the sameenergy-momentum tensor as the target state. We can argue that this “matching recipe” ismore or less natural, although clearly has several aspects that can be considered rather adhoc. One can certainly consider other possible recipes. However, as it turns out a posteriori ,this option has some nice features, in the sense of limiting the degree of arbitrariness onthe specification of both the target state (through the requirement that the condition (52)is satisfied) and the SSC-II (in the sense of fixing the expectation values of the basic fieldoperators, and the values for the Newtonian potential and its first time derivative at thecollapsing time, see equation (50)). The resulting “space-time” turns out to be described by acontinuous but not smooth metric (there is a jump in the extrinsic curvature on the matchinghypersurface), and, as such, the result is not truly a space-time. In fact, we already knewthat a jump associated with a quantum collapse would imply that since, at the correspondingspace-time points, Einstein’s semiclassical equations would not hold, simply because of the38act that for any smooth space-time the Bianchi identity implies the vanishing of ∇ µ G µν ,while the expectation value of the energy momentum tensor would quite generally not bedivergence-less during the jump of the quantum state [17]. This is certainly an unappealingaspect of the formalism, and this is one of the reasons for which we can not take this as acomplete and satisfactory description of the problem at hand.There are some other aspects of the proposal that seem unsatisfactory, with one of themost problematic being the issue of general covariance. It is evident that the association ofa collapse with a particular space-like hypersurface brings up the very issue that is usuallycause of grave concern regarding the compatibility with special relativity of any theoryinvolving an instantaneous reduction of the wave function. At this point, we should mentiona related (but different) issue of gauge dependence (or independence) of the proposal. Asthis issue has led to considerable confusion we have devoted the Appendix C and turnthe reader to it for a careful discussion. Turning back to the character of the collapsehypersurface, the explicit analysis we have presented here ascribes to such particular space-like hypersurface a very particular physical role : It separates the space-time region which isperfectly homogeneous and isotropic from the one where a particular kind of anisotropy hasset in. It is then clear that the selection of such hypersurface is not a simple gauge choice,but it corresponds to part of the characterization of the proposal regarding the collapseprocess itself.On the other hand, the evident tension between the existence of space-like hypersurfaceswith particular physical properties and relativistic ideas is clearly a very serious issue, as itseems to imply the physical breakdown of cherished and well established principles, and assuch, the problem can certainly be grave. Of course, this was not unexpected, as one of themost troublesome aspects of the notion “collapse of the wave function” is precisely that itseems intrinsically associated with a global and instantaneous process, and at this point wecan only hope that the idea might be eventually reconciled with the principles of relativity.This issue is in a sense close to the core of the famous EPR gedankenexperiment , and ofcourse the modern developments including its experimental realization. The problem isthus not hopeless by any means, but, of course, the discussion of alternatives is well beyondthe scope of the present paper. It is nevertheless worth mentioning that our view in this We should mention here the ideas of Rovelli regarding a relational wave function [66], the proposals but of course it wouldneed to have features that prevent it from being used (and here the important word is used , which implies the possibility of external manipulation by conscious beings) to sendinformation faster than light. It is worth noting that there are various works suggestingthat non-locality might play an even more fundamental role in a complete theory than thatit plays in standard quantum theory (see for instance [69–72]). We should, however, keepin mind that this is meant only as an effective description of limited validity, and that atruly developed theory of quantum gravity can naturally be expected to be needed in orderto obtain something completely satisfactory. It is perhaps also worth pointing out thatthe issue of compatibility with Lorentz invariance is a difficulty that seems to emerge oftenin connection with bringing together the quantum theory and gravitation, such as in loopquantum gravity [73–77] (similar problems appear also in the so called Liouville approachto string theory [78, 79]), and that in those cases, it often provides important constraintsthat have not been dealt with in full yet [80, 81].Perhaps, a simple analogy would be helpful in order to convey what is precisely what wehave in mind. Let us imagine for a moment that we do not have a mathematical descriptionof curved surfaces (say, for concreteness, 2-surfaces embedded in 3-dimensional Euclideanspace), and that we only know how to characterize planes. Let us assume that we want todescribe a certain smooth rock. Clearly we would not be able to do that unless the rockwas completely flat. However, we can obtain what would be for most purposes a reasonabledescription of the rock by wishing a large number of tangent planes and indicating thatthe rock is what lies within the volume that the planes define (adding all other relevantinformation about where certain planes would end, etc). It is clear that at the intersectionof the planes we would have certain singular behaviour (at those points we would not havea unique normal characterizing the rock), but we should not be surprised by that. We knowthat although the sharp vertexes associated with the intersections are not to be taken se- designed to make the collapse models applicable to field theories, which assume that the collapse istriggered at a single event and then propagates on the past light cone [67], or the analysis in [68]. Relativity requires the laws of physics to be the same in any reference frame, but that of course does notprevent the existence of special frames associated with a particular state of a physical system. Say a bodydetermines a spacial frame in which it is at rest.
Acknowledgments
The work of ADT is supported by a UNAM postdoctoral fellowship and the CONACYTgrant No 101712. The work of DS is supported in part by the CONACYT grant No 101712,by the PAPIIT-UNAM grant IN107412-3, and by sabbatical fellowships from CONACYTand DGAPA-UNAM. DS thanks the IAFE-UBA for the hospitality during the sabbaticalstay.
Appendix A: The collapse within our current understanding of physical theory
Let us try to frame our proposal, even if only schematically, within the general currentunderstanding of physical theory. The basic idea underlying our current considerations, andwhich was initially proposed in [18], is to connect the problem at hand to that encounteredwhen trying to write a theory of quantum gravity through the canonical quantization proce-dure. As it is well known, when following approaches of this kind, such as the old Wheeler deWitt proposal [64], or its more modern incarnation in the form of loop quantum gravity [40],one ends up with an atemporal theory. This is known as “the problem of time in quantumgravity” [84]. That is, in both schemes one starts with a formulation in which the basic41anonical variables describe the geometry of a 3-spatial hypersurface Σ, and characterizethe embedding of this 3-surface in a 4-dimensional space-time. From those quantities one isled to the identification of a set of canonical variables, which we will denote here genericallyby ( G , Π). (In the Wheeler de Witt case this stands for the spatial metric h µν and a certainfunction of the extrinsic curvature K µν , i.e. ( h µν , K µν ), while in loop quantum gravity thesewill be the densitized triad E µi and connection A iµ variables, i.e. ( E µi , A iµ )). The problemis that time, or its general relativistic counterpart (a time function usually specified by thelapse function and the shift vector) simply disappears from the theory, given that the Hamil-tonian vanishes when acting on the physical states (those satisfying the diffeomorphism andHamiltonian constraints).The problem is then how would one recover a space-time description of our world, clearlyan essential element one would need in order to be able to connect the theory with ob-servations. One of the most favoured approaches towards addressing this problem is toconsider, simultaneously with the geometry, some matter fields, which we will describe hereschematically by a collection of ordered pairs of canonically variables, { ( ϕ , π ) , . . . , ( ϕ n , π n ) } , (A1)and to identify an appropriate variable (or combination of variables) in the joint mattergravity theory that could act as a physical clock T ( ϕ i , π i , G , Π). The next step consistsof characterizing the state for the remaining variables in terms of the correlations of theirvalues with those of the physical clock. That is, one starts from the wave function forthe configuration variables of the theory Φ( ϕ , . . . , ϕ n , G ), which must satisfy the so calledHamiltonian and momentum constraints H µ Φ( ϕ , . . . ϕ n , G ) = 0. Next, one needs to obtainan effective wave function Ψ for the remaining variables by projecting Φ into the subspacewhere the operator T ( ϕ i , π i , G , Π) takes a certain range of values. That is, let us denote by P T, [ t,t + δt ] the projector operator onto the subspace corresponding to the region between t and t + δt of the spectrum of the operator T . One then attempts to recover a Schr¨odinger-like evolution equation by studying the dependence of Ψ( t ) ≡ P T, [ t,t + δt ] Φ on the parameter t . (In the general relativistic setting this would be a global time function T defined onthe reconstructed approximate space-time). After obtaining, by the above procedure, awave function associated with the spectrum of the operator T we might use it to computethe expectation values of the 3-dimensional geometrical operators (say the triad E µi ( x ) and42onnection A iµ ( x ) variables of Ashtekar, or some appropriate smoothing thereof) for thewave function Ψ( t ). Such collection of quantities could be seen as providing the geometricaldescriptions of the “average” space-time in terms of the 3 + 1 decomposition. In otherwords, one would have constructed a space-time where the slicing would correspond tothe hypersurfaces on which the geometrical quantities are given by the expectations of theprojected wave functions Ψ( t ), and thus one would be able to characterize the space-timeand its slicing in terms of the lapse and shift functions.The precise realization of this procedure depends strongly on the situation and specifictheory for the matter fields one is considering, and such study is quite beyond the scope ofthe present paper, among other reasons because we do not have at this point a satisfactoryand workable theory of quantum gravity. On the other hand, several works along these linesexist in the literature [85]. The point we want to make here, however, is that in such asetting, the standard Schr¨odinger equation emerges only as an effective description, and it isonly approximately valid. Under these circumstances, small modifications to that equationwould not be unexpected. In fact, in a recent analysis [86] of a quantum mechanical system,it was found that describing it in terms of the time measured by a physical clock, ratherthan an idealized one, implied modifications representing departures from the quantum me-chanical unitary evolution. We consider that this could be the grounds where a modificationof the Schr¨odinger evolution, involving something akin to a collapse of the wave function,might find its ultimate explanation. There are, indeed, several proposals for such a modi-fication centering on the analysis of standard laboratory situations [26, 28–33]. Moreover,it should be noted that with a paradigm where the quantum jumps occur generically andspontaneously, rather than being thought as triggered by the decisions of observers to mea-sure particular quantities, one might avoid the kind of problem discussed in [37] (see alsothe idea of the objective wave function reduction developed by R. Penrose in Chapter 29 ofreference [19]). Appendix B: Conection with other approaches
Let us compare the general formalism introduced in Section II with the stochastic gravityproposal by Hu and Verdaguer [46]. The idea behind that approach to semiclassical gravityis to attempt to take into account the “fluctuating part of the energy momentum tensor”43f the (quantum) mater fields through the introduction of a stochastic field χ µν ( x ). Theproposal involves a modified version of Einstein equations written as: G µν ( x ) = 8 πG ( (cid:104) ˆ T µν ( x ) (cid:105) + χ µν ( x )) . (B1)The proposal then assumes that the ensemble statistics of the stochastic term are charac-terized by a certain measure of the uncertainties of the energy momentum tensor.Let us show that one of the instantaneous collapses, occurring say at t = t c , could beseen as corresponding in the above scheme to a particular contribution to the stochasticfield at that particular time, the time of collapse. As usual in quantum field theory we usethe Heisemberg picture. However we assume that the state of the field is not constant intime, but that as a result of the collapse process (which we will be considering to occurinstantaneously), it “jumps”. Thus, the state of the field is described by | ψ ( t ) (cid:105) = θ ( t c − t ) | ξ (cid:105) + θ ( t − t c ) | ζ (cid:105) , where θ ( . ) is the step function (it is 0 when the argument is negative and1 when it is positive). Einstein equations would then be given by G µν ( x ) = 8 πG (cid:104) ψ ( t ) | ˆ T µν ( x ) | ψ ( t ) (cid:105) = 8 πG ( (cid:104) ξ | ˆ T µν | ξ (cid:105) + χ µν ) , (B2)where the stochastic term reflecting the “jump” takes the form χ µν ≡ θ ( t − t c )( (cid:104) ζ | ˆ T µν | ζ (cid:105) −(cid:104) ξ | ˆ T µν | ξ (cid:105) ). As we have indicated the exact relationship between the formalism developed inthis work and the different approaches is outside of the scope of the present paper, but theabove considerations indicate the existence of a relatively close connection. In future workswe will explore such connections more closely in order to, on the one hand, use the formalismto better understand those proposals, and on the other hand to consider the possibility ofincorporating the inflationary issue that has motivated this and previous works within thecontext of such theories and proposals. Appendix C: Gauge conditions, change of variables and all that
In mathematical physics space-time is characterized by a differential manifold M with atype (0 ,
2) tensor field g defined on it. This characterization is independent of the coordi-nates. On the same manifold we can have other fields (of scalar, vector or tensor nature),denoted generically by ψ and representing matter. Again such characterization is inde-pendent of the choice of coordinates. Modification of coordinate choices can never mix up44ifferent fields. When we choose some specific coordinates, ( x µ ), the metric can be writtenin their components, using for instance the basis of one forms naturally associated with thatcoordinate chart, g = g µν dx µ dx ν . This is a tensor of type (0 ,
2) built with tensor productsof two one-forms. A change of coordinates will change the components g µν . However, themetric as a mathematical object will not change.Now let us see when and how the issue of “gauge” appears, and how it often leads toconfusion. Consider a situation where we have two space-times with metric and matterfields defined on them, ( M, g, ψ ) and ( ˜
M , ˜ g, ˜ ψ ), and assume we want to compare one withthe other. For that we need to use some diffeomorphism (which will only exist when thetwo differential manifolds are diffeomorphic), F : M → ˜ M , mapping one manifold to theother. Then one might want to consider the differences in metric and fields by looking at,say, δg ≡ ˜ g − F ∗ ( g ) and δψ ≡ ˜ ψ − F ∗ ( ψ ). When doing this, the result will evidently dependon the choice of F . This is what is often done when considering perturbations from, say,a homogeneous and isotropic space-time to one that deviates from the former by a smallamount. This is the setting often used to considerations involving inflationary cosmology.There ( M, g, ψ ) is taken as the “homogeneous and isotropic background”, and ( ˜
M , ˜ g, ˜ ψ ) thesituation representing somehow our universe. This is not an issue of coordinates, but it canbe confused with one. The fact is, however, that the issue is customarily solved by choosinga “gauge” that effectively uses the symmetries of the first space-time to determine (to thedesired perturbative order) the diffeomorphism F . In an alternative approach, which is oftenemployed in cosmology, one considers suitable combinations of certain components of δg and δψ (associated with suitable coordinate choices), looking for combinations that are invariantunder “small changes of F ”: these are the so called “gauge invariant perturbations”.The approach based on the use of gauge invariant quantities can be quite useful in somecalculations, but often can make things a bit more difficult when discussing interpretationalaspects. We can see this by noting that what is observed using our satellites is often describedusing coordinates, such as the angular coordinates on the celestial sphere to characterizethe CMB, just to give one example. Moreover, the fact that in our treatment the matterfields and the space-time metric are so clearly distinct forces us to avoid a gauge invarianttreatment and work with the approach based on fixing the gauge. In this paper we havechosen to work in the so called “Newtonian” (sometimes also known as “longitudinal” or“conformal-Newtonian”) gauge, introduced for instance in Chapter 9.2 of reference [5].45n the particular situation we want to consider in this paper we have a scalar field livingon a “space-time” which is the result of gluing together two pieces. The first piece is theregion characterized by η < η c of the SSC-I, i.e a perfectly homogeneous and isotropic space-time with the scalar field in a state where only the zero mode (which is also homogeneousand isotropic) is excited. This is described in detail in Section III A. The second piece is theregion η > η c of the construction corresponding to a slightly inhomogeneous and anisotropicspace-time, with a characteristic wave vector (cid:126)k and a scalar field in a state where there isa nontrivial excitation not only of the zero mode, but also of several other modes (cid:126)k , 2 (cid:126)k ,3 (cid:126)k , . . . . This is described in detail in Section III B. The two pieces are glued together atthe hypersurface Σ c , corresponding to the regions with coordinates η = η c in each of thetwo pieces. The matching of the two pieces makes up a space-time describing the emergence(in the traditional sense of the word: i.e. something that was not there “at a given time”is there “at a latter time”) of perturbations. This is described in Section III C. Regardingthe coordinates, we consider the whole manifold to be covered by a single coordinate chart,( η, x , x , x ). The construction we have given describes something that is almost a space-time (i.e is a space-time except for the fact that the extrinsic curvature is not continuousin the hypersurface Σ c ), and is, in the mathematical sense, analogous to the formalismemployed in considering infinitely thin matter shells (see for instance [82]), and are thus notrealistic in the very same sense. We expect that in a more realistic description the shellswould have a finite thickness, and that the collapse hypersurface would perhaps have somesmall (but finite) temporal extent. The similarity breaks down in the fact that, in the case ofthe thin shells, we have a workable and well defined theory capable of treating the problemto any desired degree of accuracy. In the situation at hand, the collapse is expected to bedescribed by some theory which does not exit yet, and as we have argued, such theory wouldprobably trace its origin to the quantum gravity regime. Moreover, as we have discussedin Section II, we would expect that the characterization of any collapse theory in termsof space-time language would only be achievable once the fundamental degrees of freedomfor the gravitational interaction have been given an approximate classical description. Wenote here that if we wanted to change coordinates, the hypersurface Σ c would in generalbe described in the new choice by some complicated function characterizing for instance its“time coordinate” in terms of the other three coordinates. Thus our “almost” space-timewould be the same but described in a more complicated form. The construction has been46arried out in a specific gauge, but the result is gauge independent. This is essentially thesame when making some analysis in general relativity using some quite specific coordinates,say studying the perihelion of Mercury using Schwarzschild coordinates, one needs not worrythat the result is coordinate dependent.There is, on the other hand, a very different issue that might be confused with thatof gauge freedom. It is connected to the question: What is the physics that triggers thecollapse, and how does that mechanism determine that the surface where it should occuris the hypersurface η = η c ? We of course do not know the answer, simply because wedo not have a well developed and workable theory of quantum gravity, and much less atheory of collapse. The present work is only the first step in the development of a welldefined formalism that we hope could be useful in obtaining well defied but parametrizedpredictions that might be compared with observations (as done for intense in [20]), and as aresult would help us learn something about the physics of collapse. The only thing we cansay at this point is that, if something like the value of the uncertainties in the quantum stateof the matter fields is connected with triggering of collapse (as it would be for instance in atheory of collapse based on Penrose’s ideas), then the fact that the region described by theSSC-I is completely homogeneous leads us to conclude that these local uncertainties wouldreach the same level exactly at the same value of η , and thus it would be natural to expectthat the collapse would be associated with the corresponding hypersurface. But this is ofcourse just “educated” speculation in the absence of a detailed theory of collapse.Another source of confusion comes form the use of conformal transformations and thesubsequent mixing of variables. One might be concerned with the fact that it seems alwayspossible to move part of the degrees of freedom from the metric to the matter fields (andback) through a conformal transformation, and therefore the split between what ‘should’and ‘should not’ be treated at the quantum level would be completely artificial, and thusintrinsically arbitrary [83]. (Given a space-time metric g , one can introduce a new metric¯ g and a new scalar field χ related to the former metric by g = χ ¯ g , and then regard χ asa matter field. That is, one might be in doubt as to which of the two metrics one shoulddescribe at classical level.) We do not share such point of view, simply because it is basedon a classical treatment, where indeed such conformal transformations are well defined andmeaningful. Our view is that at a truly fundamental − and thus quantum − level, the space-time degrees of freedom are of a different nature than those of the matter degrees of freedom,47nd that at such level there would be no ambiguity whatsoever. Of course, at the practicallevel we must work without a satisfactory theory of quantum gravity, and the ambiguitywould have to be resolved in some other way. We will take the view that the resolutioncomes simply from considering the physical space-time metric to be the one for which thecorresponding geodesics are associated with the paths of the free particles, i.e. the metricto which the other fields are coupled in the minimal way. [1] J. A. Peacock, “Cosmological Physics,” Cambridge, UK: Cambridge Univ. Pr. (1998) 702 p. [2] A. R. Liddle and D. H. Lyth, “Cosmological inflation and large-scale structure,”
Cambridge,UK: Cambridge Univ. Pr. (2000) 400 p. [3] S. Dodelson, “Modern cosmology,”
Academic Press (2003) 440 p. [4] T. Padmanabhan, “Structure formation in the universe,”
Cambridge University Press, UK(1993) 500 p. [5] V. F. Mukhanov, “Physical foundations of cosmology ,”
Cambridge, UK: Cambridge Univ.Pr. (2005) pg. 421.[6] S. Weinberg, “Cosmology,” Section 10,
Oxford University Press, USA (2008) 544 p. [7] D. H. Lyth and A. R. Liddle, “The primordial density perturbation: Cosmology, inflation andthe origin of structure,”
Cambridge, UK: Cambridge Univ. Pr. (2009) 497 p. [8] A. H. Guth, “The inflationary universe: a possible solution to the horizon and flatness prob-lems,” Phys. Rev. D , 347-356 (1981)[9] A. D. Linde, “A new inflationary universe scenario: a possible solution of the horizon, flatness,homogeneity, isotropy and primordial monopole problems,” Phys. Lett. B , 389-393 (1982)[10] A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with radiativelyinduced symmetry breaking,” Phys. Rev. Lett. , 1220-1223 (1982)[11] A. A. Starobinsky, “A new type of isotropic cosmological models without singularity,” Phys.Lett. B , 99-102 (1980)[12] S. W. Hawking, “The development of irregularities in a single bubble inflationary universe,”Phys. Lett. B , 295 (1982)[13] A. A. Starobinsky, “Dynamics of phase transition in the new inflationary universe scenarioand generation of perturbations,” Phys. Lett. B , 175-178 (1982)
14] A. H. Guth and S. Y. Pi, “Fluctuations in the new inflationary universe,” Phys. Rev. Lett. , 1110-1113 (1982)[15] J. M. Bardeen, P. J. Steinhardt and M. S. Turner “Spontaneous creation of almost scale-freedensity perturbations in an inflationary universe,” Phys. Rev. D , 679 (1983)[16] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, “Theory of cosmological pertur-bations,” Phys. Rep. , 203-333 (1992)[17] A. Perez, H. Sahlmann and D. Sudarsky, “On the quantum origin of the seeds of cosmicstructure,” Class. Quant. Grav. , 2317-2354 (2006) [arXiv:gr-qc/0508100][18] D. Sudarsky, “Shortcomings in the understanding of why cosmological perturbations lookclassical,” Int. J. Mod. Phys. D , 509-552 (2011) [arXiv:0906.0315[gr-qc]][19] R. Penrose, “The road to reality : a complete guide to the laws of the universe,” Section 30.14, Vintage books, US (2004) 1136 p. [20] A. De Unanue and D. Sudarsky, “Phenomenological analysis of quantum collapse as source ofthe seeds of cosmic structure,” Phys. Rev. D , 043510 (2008) [arXiv:0801.4702[gr-qc]][21] G. Le´on and D. Sudarsky, “The slow roll condition and the amplitude of the primordial spec-trum of cosmic fluctuations: Contrasts and similarities of standard account and the ‘collapsescheme,” Class. Quant. Grav. , 225017 (2010) [arXiv:1003.5950[gr-qc]][22] G. Le´on, A. De Un´anue and D. Sudarsky, “Multiple quantum collapse of the inflaton field andits implications on the birth of cosmic structure,” Class. Quantum Grav. , 155010 (2011)[arXiv:1012.2419[gr-qc]][23] A. Diez-Tejedor, G. Le´on and D. Sudarsky, “The collapse of the wave function in the jointmetric-matter quantization for inflation,” (2011) [arXiv:1106.1176[gr-qc]][24] D. Sudarsky, “A signature of quantum gravity at the source of the seeds of cosmic structure?,”J. Phys. Conf. Ser. , 012054 (2007) [arXiv:gr-qc/0701071][25] D. Sudarsky, “The seeds of cosmic structure as a door to new physics,” J. Phys. Conf. Ser. , 012029 (2007) [arXiv:gr-qc/0612005][26] A. Bassi and G. C. Ghirardi, “Dynamical reduction models,” Phys. Rept. , 257 (2003)[arXiv:quant-ph/0302164][27] R. Penrose, “On gravity’s role in quantum state reduction,” Gen. Rel. Grav. , 581-600(1996)[28] F. Karolyhazy, “Gravitation and quantum mechanics of macroscopic objects,” Il Nuovo Ci- ento A , 390-402 (1966)[29] L. Di´osi, “Gravitation and quantum mechanical localization of macro-objects,” Phys. Lett. A , 199-202 (1984)[30] L. Di´osi, “A universal master equation for the gravitational violation of quantum mechanics,”Phys. Lett. A , 377-381 (1987)[31] L. Di´osi, “Models for universal reduction of macroscopic quantum fluctuations,” Phys. Rev.A , 1165-1174 (1989)[32] G. C. Ghirardi, A. Rimini and T. Weber, “A Unified dynamics for micro and macro systems,”Phys. Rev. D , 470 (1986)[33] P. Pearle, “Combining stochastic dynamical state-vector reduction with spontaneous localiza-tion,” Phys. Rev. A , 2277 (1989)[34] R. M. Wald, “General Relativity,” Chapter 14, University Of Chicago Press (1984) 506 p. [35] R. Colella, A. W. Overhauser and S. Werner, “Observation of gravitationally induced quantuminterference,” Phys. Rev. Lett. , 1472 (1975)[36] C. Chryssomalakos and D. Sudarsky, “On the geometrical character of gravitation,” Gen. Rel.Grav. , 605-617 (2003) [arXiv:gr-qc/0206030][37] R. D. Sorkin, “Impossible measurements on quantum fields,” [arXiv:gr-qc/9302018][38] J. B. Hartle, “Quantum cosmology: Problems for the 21st century,” [arXiv:gr-qc/9701022][39] J. B. Hartle, “Generalizing quantum mechanics for quantum gravity,” Int. J. Theor. Phys. ,1390-1396 (2006) [arXiv:gr-qc/0510126][40] C. Rovelli, “Quantum Gravity,” Cambridge, UK: Cambridge Univ. Pr. (2007) 484 p. [41] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D , 2333-2346 (1973)[42] T. Jacobson, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev.Lett. , 1260-1263 (1995) [arXiv:gr-qc/9504004][43] G. E. Volovik, “The Universe in a Helium Droplet,” Oxford University Press, USA (2003)536 p. [44] T. Padmanabhan, “Gravity and the thermodynamics of horizons,” Phys. Rept. , 49-125(2005) [arXiv:gr-qc/0311036][45] B. L. Hu, “Can spacetime be a condensate?,” Int. J. Theor. Phys. , 1785-1806 (2005)[arXiv:gr-qc/0503067][46] B. L. Hu and E. Verdaguer, “‘Stochastic Gravity: Theory and Applications,” Living Rev. Rel. , 3 (2008) [arXiv:0802.0658[gr-qc]][47] R. M. Wald, “Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics,” University Of Chicago Press (1994) 220 p. [48] S. Carlip, “Is Quantum Gravity Necessary?,” Class. Quant. Grav. , 154010 (2008)[arXiv:0803.3456 [gr-qc]][49] M. O. Scully and M. S. Zubairy, “Quantum Optics”, Cambridge University Press (1997) .[50] R. Harrison, I. M. Moroz, and K. P. Tod, “A numerical study of the Schr¨odinger-Newtonequation 1: Perturbing the spherically-symmetric stationary states,” [math-ph/0208045][51] R. Harrison, “A numerical study of the Schr¨odinger-Newton equations,” PhD/DPhil thesis,University of Oxford, (2001)[52] P. J. Salzman and S. Carlip, “A possible experimental test of quantized gravity,” (2006)[arXiv:gr-qc/0606120][53] P. J. Salzman, “Investigation of the Time Dependent Schr¨odinger-NewtonEquation,” Ph.D. dissertation, the University of California at Davis, (2005) [54] F . Guzm´an and L. Ure˜na-L´opez, “Newtonian Collapse of Scalar Field Dark Matter,” Phys.Rev. D , 024023 (2003) [arXiv:astro-ph/0303440][55] F . Guzm´an and L. Ure˜na-L´opez, “Evolution of the Schr¨odinger-Newton system for a self-gravitating scalar field,” Phys. Rev. D , 124033 (2004) [arXiv:gr-qc/0404014][56] R. Penrose, “Quantum computation, entanglement and state reduction,” Phil. Trans. R. Soc.Lond. A , 1927 (1998)[57] A. Ashtekar and M. Bojowald, “Quantum geometry and the Schwarzschild singularity,” Class.Quant. Grav. , 391 (2006) [arXiv:gr-qc/0509075][58] N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,” Cambridge, UK:Cambridge Univ. Pr. (1984) 352 p. [59] V. F. Mukhanov and S. Winitzki, “Introduction to Quantum Effects in Gravity ,”
Cambridge,UK: Cambridge Univ. Pr. (2007) 284 p. [60] L. Parker and D. Toms, “Quantum Field Theory in Curved Spacetime: Quantized Fields andGravity,”
Cambridge, UK: Cambridge Univ. Pr. (2009) 472 p. [61] V. A. Belinsky, I. M. Khalatnikov, L. P. Grishchuk and Y. B. Zeldovich, “Inflationary stagesin cosmological models with a scalar field,” Phys. Lett. B , 232-236 (1985)
62] D. C. Burnham and D. L. Weinberg “ Observation of Simultaneity in Parametric Productionof Optical Photon Pairs,” Phys. Rev. Lett. , 84 (1970)[63] C. K. Hong and L. Mandel “Theory of parametric frequency down conversion of light,” Phys.Rev. A , 2409 (1985)[64] B. S. de Witt, “Quantum theory of gravity. 1. The canonical theory,” Phys. Rev. D , 1113(1967)[65] J. A. Wheeler, “Superspace and the nature of quantum geometrodynamics,” in C. M. de Wittand J.A. Wheeler (eds.), Battelle Rencontres: 1967 Lectures in Mathematics and Physics , W.A. Benjamin, New York (1968), esp. pp. 284-289[66] C. Rovelli, “Relational quantum mechanics,” Int. J. Theor. Phys. , 1637 (1996)[arXiv:quant-ph/9609002][67] K. E. Hellwing and K. Kraus, “Formal description of measurements in local quantum fieldtheory,” Phys. Rev. D , 566-571 (1970)[68] W. C. Myrvold, “On peaceful coexistence: is the collapse postulate incompatible with rela-tivity?,” Studies in History and Philosophy of Modern Physics , 435-466 (2002)[69] S. Popescu and D. Rohrlich, “Quantum nonlocality as an axiom,” Foundations of Physics, ,379 (1994)[70] H. W. Hamber and R. M. Williams, “Nonlocal effective gravitational field equations and therunning of Newton’s G,” Phys. Rev. D , 106005(2006)[72] S. Deser and R. P. Woodard, “Nonlocal cosmology,” Phys. Rev. Lett. , 111301 (2007)[73] R. Gambini and J. Pullin, “Nonstandard optics from quantum spacetime,” Phys. Rev. D ,124021 (1999) [arXiv:gr-qc/9809038][74] J. Alfaro, H. A. Morales-T´ecotl and L. F. Urrutia, “Quantum gravity corrections to neutrinopropagation,” Phys. Rev. Lett. , 2318 (2000) [arXiv:gr-qc/9909079][75] H. A. Morales-T´ecotl and L. F. Urrutia, Phys. Rev. D , 103509 (2002)[76] J. Alfaro, “Quantum gravity and Lorentz invariance violation in the standard model,” Phys.Rev. Lett. , 221302 (2005) [arXiv:hep-th/0412295][77] J. Alfaro, “Quantum gravity induced Lorentz invariance violation in the standard model:hadrons,” Phys. Rev. D , 024027 (2005), [arXiv:hep-th/0505228]
78] J. R. Ellis, N. E. Mavromatos and D. V. Nanopoulos, “Quantum-gravitational diffusion andstochastic fluctuations in the velocity of light,” Gen. Rel. Grav. , 127 (2000) [arXiv:gr-qc/9904068][79] J. R. Ellis, N. E. Mavromatos and D. V. Nanopoulos, “A microscopic recoil model for light-cone fluctuations in quantum gravity,” Phys. Rev. D , 027503 (2000) [arXiv:gr-qc/9906029][80] J. Collins, A. Perez, D. Sudarsky, L. Urrutia and H. Vucetich, “Lorentz Invariance in QuantumGravity: A New fine tuning problem?,” Phys. Rev. Lett. , 191301 (2004)[81] J. Polchinski, “Comment on small Lorentz violations in quantum gravity: Do they lead to tounacceptably large effects?,” [arXiv:1106.6346][82] W. Israel, “Singular Hypersurfaces and Thin Shells in General Relativity,” Nuovo Cim.
B44 ,1 (1966); Erratum-ibid.
B48 , 463 (1967).[83] R. M. Wald, private communication.[84] C. J. Isham, “Canonical quantum gravity and the problem of time,” (1992) [arXiv:gr-qc/9210011][85] R. Gambini, R. Porto and J. Pullin, “A relational solution to the problem of time in quantummechanics and quantum gravity induces a fundamental mechanism for quantum decoherence,”New J. Phys. , 45 (2004) [arXiv:gr-qc/0402118][86] R. Gambini and J. Pullin, “Relational physics with real rods and clocks and the measure-ment problem of quantum mechanics,” Found. Phys. , 1074-1092 (2007) [arXiv:quant-ph/0608243], 1074-1092 (2007) [arXiv:quant-ph/0608243]