Effects of a scalar field on the thermodynamics of interuniversal entanglement
aa r X i v : . [ g r- q c ] N ov Effects of a scalar field on the thermodynamics of interuniversal entanglement
I˜naki Garay and Salvador Robles-P´erez Programa de P´os-Gradua¸c˜ao em F´ısica, Universidade Federal do Par´a, 66075-110, Bel´em, PA, Brazil. Departamento de F´ısica Te´orica, Universidad del Pa´ıs Vasco, Apartado 644, 48080 Bilbao, Spain. Estaci´on Ecol´ogica de Biocosmolog´ıa, Pedro de Alvarado, 14, 06411-Medell´ın, Spain. (Dated: April 16, 2018)We consider a multiverse scenario made up of classically disconnected regions of the space-timethat are, nevertheless, in a quantum entangled state. The addition of a scalar field enriches themodel and allows us to treat both the inflationary and the ‘oscillatory stage’ of the universe onthe same basis. Imposing suitable boundary conditions on the state of the multiverse, two differentrepresentations are constructed related by a Bogoliubov transformation. We compute the thermo-dynamic magnitudes of the entanglement, such as entropy and energy, explore the effects introducedby the presence of the scalar field and compare with previous results in the absence of scalar field.
PACS numbers: 98.80.Qc, 03.65.Yz
I. INTRODUCTION
The search of a satisfactory explanation of the current accelerating stage of the universe has entailed the studyof a wide variety of new cosmic scenarios. Among them, the multiverse stands out, probably, as one of the mostcontroversial since it appears to be an untestable proposal. However, this would not be the case if a particular theoryprovided us with observable and distinguishing predictions of the effects of other universes on the properties of oursingle universe. That would bring the multiverse into the physical scene of testable theories.Different multiverse scenarios can be found in the literature [1–6], so we need first specify the kind of multiverse weare dealing with in this paper. We shall consider a multiverse made up of causally disconnected regions of the space-time, each of which will be named throughout the paper with the word ‘universe’. The universes of the multiversecan be topologically disconnected, i.e., they can be simply-connected regions of a larger multiply-connected manifold,or they can be causally separated by the existence of event horizons that prevent them from any physical signaling.At first sight it would seem that we should just consider one of those regions and disregard the rest of them asphysically admissible. However, classical and quantum correlations may appear among different universes of thephysical multiverse [7] as well as residual interactions coming from the dimensional reduction of multi-dimensionaltheories [8]. In that case, other universes should be considered as well in order to describe physical reality [5].It is worth noticing that the multiverse opens the door to the possibility of having quantum effects with no classicalanalogue, like entanglement or squeezing [9, 10], in an otherwise large macroscopic universe. Thus, the quantumeffects of the space-time of a single universe may not be only restricted to the very early stage of the universe butthey could appear as well on macroscopic scales, becoming therefore testable.In the present work, the universes of the multiverse will quantum mechanically be described in the framework of theso-called third quantization formalism [11–13], which has recently received a renewed attention [14–17]. It basicallyconsists of considering the wave function of the universe as the field to be quantized (this field propagates along thevariables of the minisuperspace). Then, the general solution of the Wheeler-DeWitt equation can be given in termsof an orthonormal basis of number states that would give, in an appropriate representation, the number of universesof the multiverse.Such an appropriate representation of universes is, however, a difficult task to elucidate in the multiverse scenario,and it depends on the boundary condition that is imposed on the state of the whole multiverse. Furthermore, theexistence of quantum correlations in a composite state crucially depends on the representation chosen [18] and itmay indeed happen in the multiverse that universes which appear to be independent in a given representation mayinteract in another representation [8]. Therefore, the boundary conditions of the multiverse will eventually determinethe correlations in the composite state of different universes.In this paper, we shall consider two main representations. First, we shall use an invariant representation whichis consistent with the general boundary condition that we impose: the global properties of the multiverse do notdepend on the value of the scale factor of a particular single universe. However, an observer inside a single universewould describe her universe in the asymptotic representation of a large parent universe like ours. In this paper, weshall show that these two representations are related by a Bogoliubov transformation and, thus, we can computethe thermodynamic properties of inter-universal entanglement much in a parallel way as they are computed in thecontext of a quantum field theory in a curved space-time [19–23]. There, the existence of an event horizon –a blackhole horizon in the case of the Schwarzschild metric or a cosmological horizon in the case of a de-Sitter space-time–makes inaccessible a part of the whole space-time. Then, the observable sector appears to be filled with thermalradiation. In the case of a pair of entangled universes, one of the universes becomes inaccessible to an observer insidethe partner universe. Then, it will be shown that, as a result of the inter-universal entanglement, such an observerwould perceive her universe as being in a thermal state which is indistinguishable from a classical mixture [24] butwhose properties depend on the rate of entanglement between the universes. In fact, it actually comes from a sharpquantum state.The aim of this paper is to study the effects that realistic matter fields may have on the quantum correlations of anentangled pair of universes. These universes can be seen as coming from a double instanton of the Euclidean regimethat gives rise to a pair of Lorentzian universes whose global properties are correlated [25].Furthermore, the thermodynamics of entanglement is expected to provide us with a quantum generalization of thecustomary formulation of thermodynamics [26, 27], and quantum entanglement may be seen as a novel source ofthermodynamic properties [28, 29]. The development of such an ambitious program would entail a major achievementfor the multiverse proposal we are dealing with, considering that it implies that the thermodynamic properties ofinter-universal entanglement could eventually be related to the customary thermodynamic properties of the universe,such as its energy or entropy. If that were the case, we should consider as well inter-universal entanglement in thegeneral thermodynamic picture of the universe. Particularly, it might have significant consequences in the vacuumenergy and in the arrow of time of our universe [25, 30].The outline of the paper is the following. First we describe in section II the details of the model we are consideringand we give the semiclassical solutions of the Wheeler-DeWitt equation for two different scenarios: the slow roll stageof the scalar field, where the potential is approximately constant and contributes to a large value of the cosmologicalconstant, that is, an inflationary stage of the universe; and, on the other hand, we consider the oscillatory regime ofthe scalar field. In section III we deal with the boundary conditions and with the construction of the two quantumrepresentations relevant for our results. Then, in section IV we compute some thermodynamic magnitudes of theentanglement between universes and we discuss the role played by the scalar field and compare with other scenarioswithout a scalar field. After the conclusions (section V) we added an appendix (A) where we show the standardprocedure of the quantization of a scalar field in a de-Sitter space-time and the thermal bath derived from it, thathas some analogies with the procedure followed in our case.
II. SEMICLASSICAL STATE OF THE UNIVERSE
We consider a multiverse made up of large homogeneous and isotropic regions of the space-time with closed spatialsections, which are endorsed with a cosmological constant Λ i and a set of n scalar fields, ~ϕ ( i ) ≡ ( ϕ ( i )1 , ϕ ( i )2 , . . . , ϕ ( i ) n ),that represent the matter content of the i -universe. The index i labels the different types of universes that can exist inthe multiverse, which in the present case are all homogeneous and isotropic universes. For instance, in the landscapeof the string theories Λ i would run over all the values of the vacua of the landscape. Also notice that homogeneity andisotropy are assumable conditions as far as we work with universes of a length scale well above from the Planck length,where the quantum fluctuations of the space-time can be disregarded. Thus, the homogeneity and isotropic conditionsare even valid for most of the Euclidean regime (classically forbidden region) provided that the energy scale for whichthe universe crosses to the Lorentzian region (classically allowed region) is far from the Planck mass, M P ∼ GeV.Furthermore, potential observers would presumably inhabit large homogeneous and isotropic regions of the space-timelike our single universe that have undergone an inflationary stage, so that anisotropies and inhomogeneities can bedisregarded in a first approach in the state of the single universes, although they may play an important role in theglobal picture of the multiverse [31].For a large homogeneous and isotropic region of the space-time general relativity is effectively valid and theFriedmann-Robertson-Walker (FRW) metric can locally describe the geometry of the space-time. Then, followingthe canonical quantization procedure, the single i -universe of the multiverse would quantum mechanically be de-scribed by a wave function φ i defined in the minisuperspace with the set of variables { q A } ≡ { a, ~ϕ ( i ) } , where a is thescale factor and ~ϕ ( i ) ≡ ( ϕ ( i )1 , ϕ ( i )2 , . . . , ϕ ( i ) n ) are the n -fields that represent the matter content of the i -universe. In thatcase, the wave function φ i ≡ φ i ( a, ~ϕ ) is the solution of the Wheeler-DeWitt equation n −∇ LB + V ( i ) ( a, ~ϕ ) o φ i ( a, ~ϕ ) = 0 , (1)where the ‘Laplace-Beltrami operator’ ∇ LB is the covariant generalization of the Laplace operator [32] given by ∇ LB ≡ √−G ∂∂q A (cid:18) √−G G AB ∂∂q B (cid:19) , (2)and V ( i ) ( a, ~ϕ ) is the potential of each of the fields. The minisupermetric G AB possesses a Lorentzian signature [32],namely G AB ≡ diag( − a, a , . . . , a ) in appropriate units. It allows us to set down a formal analogy between theWheeler-DeWitt equation (1) and the wave equation for a field that propagates in a curved space-time. The scalefactor, a , formally plays the role of an intrinsic time variable and the matter fields, ~ϕ ( i ) of the i -universe, the role ofthe spatial components. Then, we can study the quantum state of the multiverse in the framework of a quantum fieldtheory in the n + 1-dimensional minisuperspace determined by the minisupermetric G AB .The general quantum state of the multiverse is given by a wave function, Ψ ~N ( a, ~φ ), which is a linear combinationof product states like [25] Ψ ~α N ( a, φ )Ψ ~α N ( a, φ ) · · · Ψ ~α m N m ( a, φ m ) , (3)where ~φ ≡ ( φ , φ , . . . , φ m ), and ~N ≡ ( N , N , . . . , N m ), with N i being the number of universes of type i representedby the wave function φ i ≡ φ ( a, ~ϕ i ) that corresponds to a universe which is described in terms of ~ϕ i matter fields and ~α i ≡ ( α i, , . . . , α i,k ) parameters. In the context of the landscape, for instance, the functions Ψ Λ i N i ( a, φ i ) in Eq. (3)would be the solutions of the third quantized Schr¨odinger equation [25] i ∂∂a Ψ Λ i N i ( a, φ i ) = H ( i ) ( a, φ, p φ )Ψ Λ i N i ( a, φ i ) , (4)where H ( i ) ( a, φ, p φ ) is the third quantized Hamiltonian [13, 17] that corresponds to each kind of universe, with p φ ≡ √− GG B ∇ B φ and ∇ B being the third quantized momentum and the covariant derivative in the minisuperspacerespectively. Throughout this paper we shall work in units for which ~ = 1 = c . Let us notice that we could consideras well Hamiltonians of interaction between different species of universes adding a more exhaustive phenomenologyto the model of the multiverse [8].In the present model, considering for simplicity only one scalar field, the Wheeler-DeWitt equation (1) for the i -universe can be written as [31, 32] ¨ φ + ˙ M ( a ) M ( a ) ˙ φ − a φ ′′ + ω ( a, ϕ ) φ = 0 , (5)where the scalar field has been rescaled as ϕ → M P p π ϕ , with M P the Planck mass, ˙ φ ≡ ∂φ∂a and φ ′ ≡ ∂φ∂ϕ , with φ ≡ φ ( a, ϕ ) and M ( a ) ≡ a , and therefore ˙ M ( a ) = 1 (we choose this notation in order to ease the analogy, afterwards,with the harmonic oscillator). In the units we are working with, ϕ has units of mass and it thus turns out tobe dimensionless after rescaling. For clarity, the index i of the i -universe has been removed assuming that all theexpressions throughout this section will be given for a single universe unless otherwise indicated. The potential termin Eq. (5) can generally be written as, ω ( a, ϕ ) ≡ σ ( H a − a ) , (6)where σ ≡ πM P and H ≡ H ( ϕ ) is the Hubble function. The frequency ω has units of mass or, equivalently, unitsof the inverse of time or length, as it was expected. We shall consider two contributions to the Hubble function, i.e., H = H + H . The first one is caused by the existence of a cosmological constant, Λ = 3 H , which is assumed tobe very small. The second contribution is due to the potential of the scalar field, H = π M P V ( ϕ ) = m ϕ , where inthe last equality it has been assumed a quadratic potential.As it is well-known [31, 33], there are four scales of interest in the creation and subsequent inflationary pictureof a single universe (in the following we shall assume a quadratic potential although the picture is rather general,see Refs. [31, 33]). First, for values ϕ & λ − (i.e., V ( ϕ ) & M P ), with λ ≡ mM P ≪
1, there is no consistentdescription of the space-time because the quantum fluctuations of the metric tensor become of the same order than thecomponents of the metric. It corresponds to the realm of the space-time foam [34–36]. For the value λ − & ϕ & λ − ( M P & V ( ϕ ) & λM P ), the fluctuations of the space-time are weakened and the large value of the scalar field makes itto slowly roll down the potential during the time scale H − & M − P . The energy of the field is stored in the potentialterm, which is approximately a constant, and inflation then starts in the Lorentzian regions of the space-time. Besides,the fluctuations of the scalar field are large and give rise to new inflationary regions, i.e., new universes are nucleatingin an eternal ‘self-inflationary’ process [37, 38].For a value λ − & ϕ & λM P & V ( ϕ ) & λ M P ), inflation goes on but the fluctuations of the scalar field areweakened and the creation of new universes stops. Finally, for values 1 & ϕ & λ M P & V ( ϕ ) & FIG. 1: The creation of a large parent universe from a baby universe. actual structure of the universe. Most of the energy stored in the scalar field during the inflationary stage is nowtransmitted to the particles and the universe enters thus in the hot regime [31, 33, 39].Our model will quantum mechanically picture the following scenario. First, we shall consider the quantum creationof universes in entangled pairs. After being created, these universes undergo an inflationary stage in which theirquantum states are still correlated. The exponential expansion of the universe is then led by an effective value H ofthe cosmological constant, because Λ ≪
1. After the exit of inflation, the universe enters in the oscillatory regime ofa scalar field which is quantized in the curved background of a de-Sitter space-time with a value Λ of the cosmologicalconstant. The wave function of the universes may still retain some residual correlations that might be tested in thecurrent stage of the universe. A. Creation of universes in entangled pairs
Within a region slightly larger than the Planck scale the space-time can approximately be considered homogeneousand isotropic provided that the energy associated to the scalar field is smaller than the Planck energy density, i.e., V ( ϕ ) . M P . Then, for a sufficiently uniform and slowly varying field with a large initial value ( ϕ ≫ dadt = ωσa ≈ q a H − , dϕdt ≈ − H dVdϕ . (7)For a value a < H − there is no Lorentzian solution to the first of these equations. However, by performing a Wickrotation to Euclidean time τ , the Euclidean solution a E ( τ ) = 1 H cos( H τ ) , (8)with τ ∈ ( − π H , τ → − π H . Before reaching the singular value a E = 0 the Euclidean instanton would delveinto the space-time foam where the approximations of homogeneity and isotropy are, actually, no longer valid. Atthe Euclidean time τ = 0 the instanton finds the Lorentzian regime and the universe locally emerges as a Friedmann-Robertson-Walker space-time with a scale factor given by a ( t ) = 1 H cosh( H t ) , (9)with t ≥ φ ( a, ϕ ) = Z dk e ikϕ φ k ( a ) ˆ c ,k + e − ikϕ φ ∗ k ( a ) ˆ c † ,k , (10)where ˆ c † ,k and ˆ c ,k are constant operators that represent, respectively, the creation and annihilation of universes witha particular value k of the mode. Inserting the wave function (10) into the Wheeler-deWitt equation (5), it followsthat the probability amplitudes φ k ( a ) satisfy the equation of the damped harmonic oscillator¨ φ k + ˙ MM ˙ φ k + ω k φ k = 0 , (11)with a scale factor dependent frequency given by ω k ≡ ω k ( a ) = σ r a H − a + k σ a = σH a q ( a − a )( a − a − )( a + a ) , (12)where a + ≡ a + ( k ) = 1 √ H r α k , (13) a − ≡ a − ( k ) = 1 √ H r − α k + π , (14) a ≡ a ( k ) = 1 √ H r − α k − π , (15)with a + ≥ a − ≥ a and α k ≡ arccos(1 − k k m ) ∈ [0 , π ] , (16)considering k m ≡ σ H = π M P H and k m ≥ k ≥
0. It is worth noticing that the value k of the mode is related to themomentum p ϕ that is classically proportional to ∂ t ϕ . In the slow-roll approximation we should classically considerthe value k = 0 that corresponds to the ground state. However, as we have already pointed out, the fluctuations ofthe scalar field can be very large during the first stage of the inflationary period, when universes are created, so wehave to quantum mechanically consider as well other modes for the wave functions of the universes being created.The consideration of other modes different from zero entails a significant departure from the customary pictureof the creation of universes because of the quantum correction that appears in the frequency (12). Unlike in Eq.(7), where there is one Euclidean region and one Lorentzian region, in the equation of motion corresponding to thefrequency given by Eq. (12) there is one Euclidean region, for the value a + > a > a − , between two Lorentzian regions,for values a > a + and a < a − . Therefore, for k m > k > | k | so, therefore, the modes of the wave function of the corresponding universes are correlated.Let us also notice that, like in the customary case of a universe being created from nothing [41, 42], the ap-proximations of homogeneity and isotropy are still valid in the case of the creation of universes in entangled pairsprovided that a − is large enough with respect to the Planck length. Considering a plausible value of the mass ofthe scalar field of m ∼ GeV (i.e., the GUT scale), then λ ∼ − and the values of H for which the universesare created are M P > H & λM P . It implies that the size of the newborn universes is of order ℓ H ∼ H − , with ℓ H ∼ λ − ℓ P ∼ ℓ P . Thus, there is still room for the creation of a pair of homogeneous and isotropic universeswith, 10 ℓ P ≈ a + > a − ≫ ℓ P ; the range of possible values could be even larger for smaller values of λ . B. Inflationary stage of the universes
Once the universes have entered into the inflationary regime their scale factors exponentially grow by an overallfactor [31] P ∼ e λ − until the slow-roll approximation eventually fails at ϕ ∼
1. After the initial stages of inflationthe quantum correction term in Eq. (12) is quantitatively negligible. The solutions of the Wheeler-deWitt equation(11) can then be written in the WKB approximation as φ k ( a ) = 1 p M ( a ) ω k ( a ) e ± iσ S k ( a ) ≈ p M ( a ) ω ( a ) e ± iσ S ( a ) , (17) FIG. 2: The creation of a pair of entangled universes from a double instanton. where S k ( a ) ≡ σ Z a da ′ ω k ( a ′ ) ≈ σ Z a da ′ ω ( a ′ ) ≡ S ( a ) , (18)with ω ( a ) given by Eq. (6).Although quantitatively negligible, it is worth pointing out that the quantum correction term of Eq. (12) forthe k mode is the ultimate reason for the plausible creation of universes in entangled pairs and therefore for thecorrelations between universes. The classical evolution of each single universe is given however by a Friedman equation, ∂ t a = ω ( a ) σa ≈ H a , which is effectively independent of the value of the mode k , i.e., all the universes corresponding todifferent modes classically evolve in a similar way irrespective of the quantum mode from which they were created.They undergo an exponential expansion given by a ( t ) ≈ a e H t .While the universes are expanding, the field is slow-rolling down to the minimum of the potential until the scalarfield approaches the value ϕ ∼
1. Then, the slow-roll approximation fails, the Hubble parameter H is no longer largeenough to prevent the field from rapidly rolling down to the minimum of the effective potential where it starts tooscillate. The energy of the field, so far stored in the potential term V ( ϕ ), is now transferred to the particles that arecreated as a result of the oscillations of the field. These particles eventually collide between them and the universeheats up [31]. The hot universe cools down and the initial particles decay into new particles that will conform thestructure of the current universe. C. Oscillatory regime of the scalar field
Each single universe of the entangled pair (in the scenario represented in Fig. 2) leaves the inflationary stage whenthe slow-roll approximation fails. The scalar field enters then in the oscillatory regime and it becomes meaningful todescribe it in terms of particles. In the oscillatory regime, the semiclassical solutions of the Wheeler-DeWitt equationfor each single universe, Eq. (5) with the frequency of the Eq. (6), can be written as [43, 44] φ sc ( a, ϕ ) = 1 p M ( a ) ω ( a ) e ± iσ S ( a ) ∆( a, ϕ ) , (19)with a prefactor that depends only on the gravitational degrees of freedom (only on the scale factor), where ω isgiven by Eq. (6) with H = H , S ( a ) ≡ σ Z a da ′ ω ( a ′ ) = ( a H − H , (20)and the function ∆( a, ϕ ) contains all the information about the scalar field. In the semiclassical regime, this functionsatisfies the Schr¨odinger equation [43] ∓ i ω ( a ) σa ˙∆( a, ϕ ) = (cid:18) − σa ∂ ∂ϕ + 2 π a V ( ϕ ) (cid:19) ∆( a, ϕ ) , (21)where ˙∆ ≡ ∂ ∆ ∂a . The expansion (or contraction) of the universe, given by the Friedmann equation ∂a∂t = ∓ ω ( a ) σa , where t is the Friedmann time, provides us with a time variable, t , that is well-defined in each branch of the universe [45].In terms of the Friedmann time, t , the Schr¨odinger equation (21) takes the customary form i ∂ ∆ ∂t ( ϕ, t ) = h ( ϕ, t )∆( ϕ, t ) , (22)where h ( ϕ, t ) is the corresponding Hamiltonian for the matter field ϕ . We can observe that this equation turns out tobe the Schr¨odinger equation for a scalar field in the semiclassical background of a de-Sitter space-time with a valueΛ ≪ H + H ≫ H ≫ i -type of universes in the multiverse (the generalization of Eq. (19) to spinorial fields isstraightforward). Therefore, the effects of other types of fields can be studied following the same general procedureused in this paper. The most general quantum state of the multiverse would be given by a linear combination ofproduct states (3) relating all types of universes of the multiverse with different kind of fields that represent thematter-energy content of the i -universe. For the sake of concreteness, let us particularize the general semiclassicalsolution (19) to a scalar field with mass m and V ( ϕ ) = m ϕ (notice that V ( ϕ ) = σ π m ϕ after rescaling thefield), with H ≫ m ≫ H . Then, h in Eq. (22) reads h ≡ M ( t ) p ϕ + M ( t ) m ϕ , (23)where M ( t ) ≡ σa ( t ) and p ϕ ≡ − i ∂∂ϕ . Notice that the Hamiltonian (23) leads to the customary quantization of ascalar field with mass in the curved background of the de-Sitter space-time (see Appendix A).The function ∆( t, ϕ ) in Eq. (22) can be expressed in terms of the eigenfunctions ∆ n ( t, ϕ ) of the harmonic oscillatorwith time dependent mass [46–51], i.e., ∆( t, ϕ ) = P n B n ∆ n ( t, ϕ ), with B n constant coefficients and ∆ n ( t, ϕ ) thenormalized eigenfunctions that can be expressed as (see, for instance, Sec. 4.2 of Ref. [47])∆ n ( t, ϕ ) = n n ! r M ( t ) e mπ ! e − i ( n + ) e mt e iM ( t )2 ( − H + i e m ) ϕ H n ( p M ( t ) e m ϕ ) , (24)with M ( t ) ≈ σ H e H t , H n ( x ) the Hermite polynomial of degree n , and e m = q m − H , ( e m ≈ m for the subdampedregime of the harmonic oscillator, i.e., for values m ≫ H ). In the previous equation, we have made use of theapproximation a ≫ H − , that is valid in the semiclassical approximation that we are considering. With such anapproximation, the value of the scale factor coming from the Friedmann equation ∂a∂t = ∓ ω ( a ) σa takes the form: a ( t ) = H − cosh H t ≈ H e H t . Then, by applying the inverse relation t ≈ H − ln(2 H a ), we can finally write thesemiclassical solutions of the wave function of the universe in terms of the scale factor and the scalar field as φ sc ( a, ϕ ) = X n B n φ scn ( a, ϕ ) ≡ X n B n | φ scn ( a, ϕ ) | e ± iσS n ( a,ϕ ) , (25)where S n ( a, ϕ ) ≈ H a (1 + ϕ ) + e mσH ( n + ) ln( H a ), and | φ scn ( a, ϕ ) | ≈ √ n n ! (cid:18) e mπσH a (cid:19) e − σ f m a ϕ H n ( √ σ e m a ϕ ) . (26)Furthermore, with the help of Eq. (19), the Wheeler-DeWitt equation (5) can be rewritten for each mode, in thesemiclassical regime as ¨ φ n + ˙ M ( a ) M ( a ) ˙ φ n + Ω n ( a, ϕ ) φ n = 0 , (27)with Ω n ( a, ϕ ) = ω + i σa ∆ n ∂ ∆ n ( t, ϕ ) ∂t (cid:12)(cid:12)(cid:12) t = t ( a ) = ω ∓ iω ˙∆ n ∆ n . (28)The ∓ sign will be chosen depending on whether we consider an expanding or a contracting branch of the universe.In order to study the properties of the multiverse in the semiclassical approximation, it is useful to write down theleading order of the asymptotic behavior of Ω n ( a ) for large values of the scale factor a :Ω n ∼ Ω ≡ (cid:18) H (1 ∓ ϕ ) ± i H e mϕ (cid:19) / σa . (29)In the third quantization formalism, Eq. (27) can formally be seen as the equation of an harmonic oscillator placedin a dispersive medium with time dependent mass and frequency, M ≡ M ( a ) and Ω n ≡ Ω n ( a, ϕ ), where the scalefactor plays the role of the intrinsic time variable and the scalar field that of a spatial variable. Nevertheless, thefrequency Ω n = | Ω n | e iθ n is, in general, complex. Using Eq. (29), we obtain the polar angle θ n does not depend on a at the leading order: ˙ θ n ≈ ˙ θ = 0.We would like to have a pure real frequency in order to ease the analogy with a proper harmonic oscillator anddefine, afterwards, the associated creation and annihilation operators. The WKB solution to the Wheeler-DeWittequation takes the form: φ scn ( a, ϕ ) = 1 p M ( a )Ω n ( a, ϕ ) e ± iS ( n ) ( a,ϕ ) , (30)with ˙ S ( n ) ( a, ϕ ) ≡ Ω n . This expression can be written in the following equivalent way: φ scn ( a, ϕ ) = 1 p N n ( a ) ω n ( a, ϕ ) e ± iS ( n ) ω ( a,ϕ ) , (31)with ˙ S ( n ) ω ( a, ϕ ) ≡ ω n , and ω n = | Ω n | cos θ n , N n ( a ) = M ( a ) | cos θ n | e ± S ( n ) I , where S ( n ) I ≡ ℑ ( S ( n ) ) = R | Ω n | sin θ n da , is the imaginary part of the action S ( n ) . Now both ω n ∈ R and N n ( a ) ∈ R .Besides, the solution (31) satisfies the following dumped harmonic oscillator equation:¨ φ scn + ˙ N n ( a ) N n ( a ) ˙ φ scn + ω n ( a, ϕ ) φ scn = 0 , (32)that can be derived from the Hamiltonian H n = 12 N n ( a, ϕ ) p φ n + N n ( a, ϕ ) ω n φ n . (33)The total Hamiltonian of the i -type of universes, H ( i ) in Eq. (4), corresponds to the sum of all the contributions ofthe modes, i.e., H ( i ) ≡ P n H ( i ) n , where the label i of the type of universe has now been reintroduced for explicitness.In the following sections, we will concentrate in one of the modes (of just one type of universe) and the Hamiltonian(33) will be used to determine the evolution of the semiclassical state of a single universe. III. BOUNDARY CONDITIONS AND INVARIANT STATES OF THE MULTIVERSE
For definiteness, we shall consider from now on just one single n -mode of the i -type of universes in the multiverse.However, as it has been pointed out previously, the same procedure that we will use in this section can be applied toother modes of the i -universe as well as to the rest of types of universes of the multiverse. It is worth noticing thatthe multiverse turns out to be thus an enormously rich scenario interpretable in terms of a ‘particle-soup’ of universes,whose complete phenomenology is still to be uncovered.Following the new point of view offered at the end of the previous section we shall consider that our n -mode of the i -type of universe (the index i will be omitted throughout the rest of the section) is described by an harmonic oscillatorwhose mass N n and frequency ω n depend on the scale factor a which is, as it has been commented previously, thetime-like coordinate of the minisuperspace. The equation of motion and the Hamiltonian of the system are given byEqs. (32-33), respectively. On the basis of the ‘third quantization’ formalism, the quantum state of each universe ofthe multiverse will be given in terms of the states of an harmonic oscillator, for any kind of potential of the field V ( ϕ )(the potential V ( ϕ ) is related to the frequency ω ).There are two representations of the Hamiltonian (33) that can naturally be chosen with a sensible physical inter-pretation. On the one hand we will consider the usual creation and annihilation operators of the harmonic oscillator c n ( a ) , c † n ( a ) that diagonalize the Hamiltonian. On the other hand, we construct other kind of creation and annihilationoperators b n ( a ) , b † n ( a ) in such a way that the number operator constructed with them is constant under the changeof the scale factor. Notice that in the case where the frequency and mass of the harmonic oscillator do not dependon the time, these two representations coincide. Nevertheless, that is not our case.Let us start with the “diagonal representation”, i.e., the representation that diagonalizes the Hamiltonian at a givenvalue a of the scale factor for each value n of the normal mode, ˆ c ,n ≡ ˆ c n ( a ) and ˆ c † ,n ≡ ˆ c † n ( a ), withˆ c n = r N n ω n (cid:18) ˆ φ n + i N n ω n ˆ p φ n (cid:19) , (34)ˆ c † n = r N n ω n (cid:18) ˆ φ n − i N n ω n ˆ p φ n (cid:19) . (35)In the third quantization formalism, the wave function of the universe, φ ( a, ϕ ), is promoted to an operator that canbe written as ˆ φ ( a, ϕ ) = X n A n ( a, ϕ )ˆ c ,n + A ∗ n ( a, ϕ )ˆ c † ,n , (36)where the probability amplitudes, A n ( a, ϕ ) and A ∗ n ( a, ϕ ) in Eq. (36), satisfy the Wheeler-DeWitt equation (27), orequivalently Eq. (32).However, the operators ˆ c † ,n ( a ) and ˆ c ,n ( a ) in Eq. (36) cannot properly be interpreted as the creation and annihila-tion operators of universes because, in such representation, the number operator given by ˆ N c ,n ≡ ˆ c † ,n ( a )ˆ c ,n ( a ), is notan invariant operator, i.e., d ˆ N c ,n da = i [ ˆ H n , ˆ N c ,n ] = 0, where ˆ H n , given by Eq. (33), is the third quantized Hamiltonianthat determines the evolution of the state of the n -mode. It means that the excitation number of universes withinthe multiverse, which is given by the eigenvalues of the number operator, would depend on the value of the scalefactor in a particular single universe. This is not expected in the quantum multiverse and a different representationhas to be chosen in order to have a parallel notion to that of the creation and annihilation of particles. Notice, thatwe might have considered the creation and annihilation operators of universal states with the proper frequency ofthe Hamiltonian, given directly by Eqs. (34-35) for any value of the scale factor, for which ˆ H n = ω n ( a )( ˆ N c,n + )and [ ˆ H c,n , ˆ N c,n ] = 0. Nevertheless, the number of universes would be scale factor dependent too, because in this case ∂ a ˆ N c = 0.An invariant representation, ˆ b n ( a ) and ˆ b † n ( a ), is constructed such that the eigenvalues of the associated numberoperator, ˆ N b,n ( a ) ≡ ˆ b † n ( a )ˆ b n ( a ), are invariant under the evolution of the scale factor a . That is, dda ˆ N b,n ( a ) = ∂∂a ˆ N b,n ( a ) + i [ ˆ H n ( a ) , ˆ N b,n ( a )] = 0 . The boundary condition we are imposing on the state of the multiverse (the global properties of the multiverse mustnot depend on the value of the scale factor) fixes the representation that has to be chosen. We look for a representationfor which the number operator is invariant. This can be given by the so called Lewis representation [46] (see also,Refs. [49, 50, 52, 53]), for which the creation and annihilation operators are defined for each single mode n as [25]ˆ b n ( a ) ≡ r ˆ φ n R + i ( R ˆ p φ n − N n ˙ R ˆ φ n ) ! , (37)ˆ b † n ( a ) ≡ r ˆ φ n R − i ( R ˆ p φ n − N n ˙ R ˆ φ n ) ! , (38)with R ≡ R n ( a ) = p φ ( a ) + φ ( a ), where φ and φ are two linearly independent solutions of the Wheeler-DeWittequation (32) that make real the value of the function R ( a ) , ( R ( a ) ∈ R ). The Lewis representation given by the This is to ensure the Hermitian condition of the invariant operator ˆ I n ≡ ˆ N n + . b † n ˆ b n | N, a i = N n | N, a i , with N n = N n ( a ). Then, the operatorsˆ b † n and ˆ b n can properly be interpreted as the ladder operators of the n -mode of the i -type of universes in the multiverse.Let us notice that the mode is expected to remain at the value n because we are not considering interactions betweendifferent modes of the wave function of the i -universe, i.e., there are no interaction terms in the Hamiltonian of the i -universe, H ( i ) = P n H ( i ) n .However, in terms of the creation and annihilation operators of the Lewis representation, the Hamiltonian turnsout to be ˆ H ( a ) = X n ˆ H n = X n (cid:18) β − ˆ b n ˆ b − n + β + ˆ b † n ˆ b †− n + β ( a ) (cid:18) ˆ b † n ( a )ˆ b n ( a ) + 12 (cid:19)(cid:19) , (39)where, β ± and β are two non-trivial functions of R [17, 25]. The structure of the Hamiltonian (39) is formallythe same to that used in quantum optics to represent the creation and annihilation of entangled pairs of photons[9, 54–56]. This analogy, together with the isotropy of the minisuperspace that suggests that the universes are createdin pairs with opposite momenta n and − n , allows us to interpret the multiverse as made up of entangled pairs ofuniverses whose properties of entanglement can be analyzed. Notice also that for the case n = 0 we still have twodifferent entangled universes (ˆ b ( a ) = ˆ b − ( a )).All the representations of the harmonic oscillator corresponding to different frequencies are always related by a socalled squeezing relation (see Ref. [50]). The invariant and the diagonal representations are related by a Bogoliubovtransformation, that is, for each single mode there is a relation such thatˆ b n = µ ∗ n ˆ c n + ν n ˆ c †− n , ˆ b † n = µ n ˆ c † n + ν ∗ n ˆ c − n , (40)where, from Eqs. (34-35) and (37-38), µ ∗ n = 12 R r N n ω n + R p N n ω n − i ˙ R r N n ω n ! , (41) ν n = 12 R r N n ω n − R p N n ω n − i ˙ R r N n ω n ! , (42)with | µ n | − | ν n | = 1. The ground state of the invariant representation can then be written as [57] | n, − n i ( b ) = 1 | µ n | ∞ X k =0 (cid:18) ν n µ n (cid:19) k | k n , k − n i ( c ) . (43)It can be checked that the Lewis representation turns out to be the diagonal representation in the limit of large valuesof the scale factor. In that limit, µ n → ν n →
0, and the ground states that correspond to both representationsturn out to be the same, i.e., lim a →∞ | n, − n i ( b ) = | n, − n i ( c ) . (44)Let us summarize this section by discussing the interpretation of both representations we used. The most naturalrepresentation (in the sense that is more direct and standard) is the diagonal representation given by the ˆ c operators.However, the very fact that we are dealing with a system analogous to an harmonic oscillator in a dispersive medium,avoids the possibility that the number operator constructed with the diagonal representation be invariant. Takinginto account that we are working under certain sensible boundary conditions, given by the independence of the globalproperties of the multiverse with respect to the scale factor of a single universe, we were forced to look for anotherrepresentation, the Lewis representation, whose number operator remains invariant.Thus, the Lewis representation is constructed in order to deal with the boundary conditions of the whole multiverse(as a collection of universes), so we can associate to it a notion of external observer. In the limit of large values ofthe scale factor, the Hamiltonian written in terms of the Lewis representations acquires a diagonal form (the Lewisand the diagonal representations coincide in this limit). Indeed, we expect that in this limit the universes becomeasymptotically independent. In this way, it is reasonable to relate the diagonal representation with an observer insidea given universe (that could be us). To conclude, notice that for each value of the scale factor we obtain bothrepresentations (Lewis and diagonal), but it is possible to relate all of them with a Bogoliubov transformation.1 IV. THERMODYNAMICS OF ENTANGLEMENT IN THE MULTIVERSE
We have described a multiverse scenario with a collection of universes created in pairs whose quantum states aregiven by the equation of an harmonic oscillator. In this section we compute the thermodynamic properties derivedfrom the entanglement between the universes.If we consider the Lewis representation, that is, the representation of an external observer, the operator ˆ N b,n hasthe interpretation of the excitation number of the multiverse. Given that we are describing the whole multiverse,where by definition there is no external force, we expect the multiverse to stay in the ground state, at least as a firstapproximation. Then, in order to study the thermodynamics of entanglement, it is sensible to consider the groundstate of two universes whose quantum states are given in the b -representation, | n, − n i ( b ) , for a particular value of thefield mode | n | . Nevertheless, the procedure followed here could be easily extended to more complicated states, oreven general states, although the equations would be more intricate and it would be more difficult to obtain relevantresults. In any case, we do not expect qualitative changes in the behavior of the thermodynamics for the excited states.Regarding the mode n of the scalar field, we will see that the asymptotic limit for large values of a is independent ofthe mode considered for the scalar field, although the subsequent corrections depend on it.As we have shown in the preceding sections, the invariant and the diagonal representations are related by theBogoliubov transformation (43). The density matrix ρ that represents the quantum state of the entangled pair canbe expressed as ρ ≡ | n, − n i ( b ) ( b ) h n, − n | . (45)It represents, in the invariant representation, the ground state of the entangled pair of universes that were born fromthe Euclidean instanton represented in Fig. 1. Following the same procedure of that used in the context of a quantumfield theory in a curved background (see appendix A), the reduced density matrix ρ n (that represents the quantumstate of one single universe of the entangled pair in the diagonal representation, more concretely, the universe with apositive value of the mode n ) can be obtained by tracing out from ρ the degrees of freedom of the partner universe.Using Eq. (43), it yields ρ n ≡ Tr − n ρ = ∞ X j =0 h j − n | ρ | j − n i = 1 | µ n | X j (cid:18) | ν n || µ n | (cid:19) j | j n ih j n | (46)or, written in the Gibbs form ρ n ( r ) = 1 Z n ( r ) ∞ X j =0 e − ωnT ( r ) ( j n + ) | j n ih j n | , (47)where r is the parameter of squeezing, | µ n | ≡ cosh r , | ν n | ≡ sinh r , and Z − n = 2 sinh ω n T , with ω n the energy thatcorresponds to the ground state of the positive modes in the diagonal representation, | n i ( c ) . The two universes ofthe entangled pair evolve then in thermal equilibrium with respect to each other, with a temperature T ( r ) = ω n r ) (48)that depends on the value of the parameter of squeezing r (that in turn depends on the value of the scale factor). Itis worth noticing that the thermal state represented by the density matrix (47) is indistinguishable from a classicalmixture [24]. Thus, at first sight, the observers inside a single universe see their respective universes as classicaluniverses. However, the quantum state (47) is the effect of partially tracing out the degrees of freedom of a partneruniverse in a composite entangled state which is a quantum state that has no classical counterpart [9, 10].The thermodynamic magnitudes of the thermal state given by Eq. (47) can easily be computed in terms of thesqueezing parameter. For instance, the entanglement entropy [18, 58, 59] S ent = − Tr( ρ n ln ρ n ) , (49)turns out to be S ent ( a ) = cosh r ln(cosh r ) − sinh r ln(sinh r ) , (50)and the total energy associated to ρ n reads E n ( a ) = Tr( ρ n H n ) = ω n (sinh r + 12 ) , (51)2where H n ≡ ω n (ˆ c † n ˆ c n + ). The change in the heat and work, as defined in Ref. [58, 59], are respectively δW n = Tr( ρ n d H n da ) = ˙ ω n (sinh r + 12 ) , (52) δQ n = Tr( dρ n da H n ) = ω n ˙ r sinh 2 r, (53)from which it can be checked that the first principle of thermodynamics, dE n = δW n + δQ n , is directly satisfied.From Eqs. (50) and (53), it can also be checked that the production of entropy [58], ς , is zero ς ≡ dS ent da − T δQ n da = 0 , (54)satisfying, therefore, the second principle of thermodynamics for any value of the scale factor. Furthermore, Eq. (54)can be compared with the expression which is usually used to compute the energy of entanglement (see, Refs. [60–62]), dE ent = T dS ent . (55)It enhances us to establish an energy of entanglement given by dE ent = δQ n = ω sinh 2 r dr. (56)We can now compute explicitly the previous magnitudes. The first step is to compute the function R ( a, ϕ ) thatappears in the definition of ˆ b n ( a ), Eqs. (37) and (38). The function R is defined as R ( a, ϕ ) = p φ ( a ) + φ ( a ), with φ and φ two solutions of the Wheeler-DeWitt equation (32), which can be written, in the semiclassical limit, as φ = s N n ( a ) ω n ( a, ϕ ) cos S ω ( a, ϕ ) , (57) φ = s N n ( a ) ω n ( a, ϕ ) sin S ω ( a, ϕ ) , (58)so the expression for R takes the form R ≈ s N n ( a ) ω n ( a, ϕ ) . (59)Then, inserting Eq. (59) into Eqs. (41-42), we obtain for the entanglement parameters: µ ≈ i ˙ R s N n ω n ( a, ϕ ) , ν ≈ − i ˙ R s N n ω n ( a, ϕ ) . (60)Therefore, the parameter of entanglement r is given bysinh r ≡ | ν | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ R s N n ω n ( a, ϕ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Taking into account that ˙ R = − R NN + ˙ ωω ) ∼ − R | Ω | sin θ, where θ was defined at the end of section II as the polar angle of the complex frequency Ω, we can easily computethe explicit value of sinh r at the leading order:sinh r = 12 | tan θ | = 12 (cid:12)(cid:12)(cid:12)(cid:12) tan (cid:18)
12 arctan (cid:18) e mϕ H | ∓ ϕ / | (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (61)Some comments are in order at this point. First notice that, as we had anticipated, the parameter r does not dependon the mode chosen at the leading order (although it will depend on it in the next order). Nevertheless, it crucially3depends on the value of the cosmological constant Λ as well as on the potential of the field ϕ . Particularly, in ourcase with V ( ϕ ) = m ϕ , it depends on the value of ϕ and on the mass m .In order to compare our results with the case in which no scalar field is considered [25, 63], it is interesting tocompute the next two terms in the corrections of the expansion of sinh r . It is a straightforward although a lengthycomputation. If we consider the next order in the scale factor, a , in the expression for Ω (Eq. 29), that is,Ω n ∼ (cid:18) H (1 ∓ ϕ ) ± iH e mϕ (cid:19) σ a + (2 e m ∓ iH )( n + 12 ) σa, (62)we obtain the following expansion for sinh r up to order (1 /a ) :sinh r = (cid:12)(cid:12)(cid:12)(cid:12) f ( ϕ ) + g n ( ϕ ) 1 a + O (cid:18) a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (63)where f ( ϕ ) = | tan θ | , as expected from Eq. (61) and, in the limit of small values of the scalar field, f ( ϕ ) ∼ − m H ϕ , g n ( ϕ ) ∼ n − H σ + (cid:18) n H σ + 3 ˜ m ( n + )2 H (cid:19) ϕ . (64)The consideration of a scalar field in the formulation of the inter-universal entanglement has important and unex-pected consequences. Let us first consider the scale factor a and the scalar field ϕ as independent variables. In the( a, ϕ )-space, consider the limit of ϕ → n = − / ϕ → (sinh r ) (cid:12)(cid:12)(cid:12)(cid:12) n = − = 34 H σa , (65)that exactly coincides with the results found in previous works [25, 63].During the oscillatory regime, after the slow-roll approximation has failed, the scalar field and the scale factor followa classical trajectory in the ( a, ϕ )-space. In particular, at late times the scale factor behaves like a ∼ e Ht , and withthe approximations made in this paper ( m ≫ H ), the scalar field behaves like a dumped wave (see, for instance, Eq.(7.17) of Ref. [57]), ϕ k ( t ) ∼ e − H t e ± imt . (66)Then, the relation ϕ ( a ) turns out to give | ϕ ( a ) | ∼ ( a /a ) , where we have introduced a constant initial value for thescale factor, a , in order to be consistent with the dimensional analysis. This is a very interesting result because theleading order term is modified by the presence of the scalar field, and is given bysinh r ∼ (cid:12)(cid:12)(cid:12)(cid:12) − n + 2 a ˜ mσ )8 H σa (cid:12)(cid:12)(cid:12)(cid:12) . (67)This is a result coming from the consideration of the scalar field which introduces differences in the rate of entanglementbetween universes that depend on the values of the mode n (see fig. 3) and on the specific potential considered forthe field, that is, on the value of ˜ m . Notice also that the quantum fluctuations of the vacuum state of the scalar fielddo contribute to the entanglement (see fig. 4), as we have a different contribution to the entanglement with ϕ → n = − / n = 0 (zeroth mode of the field).These results will have important consequences in the behavior of the thermodynamic magnitudes of entanglementof each single universe of the entangled pair, that we discuss in the following. A. Thermodynamic magnitudes of entanglement
In this section we compute the thermodynamic magnitudes of entanglement associated to the state (47), whichrepresents the quantum state of each single universe of an entangled pair. The temperature (48) turns out to be, atthe leading order, T ( ϕ, a ) = ξ ( ϕ ) cos θ (cid:18) √ ( r )sinh( r ) (cid:19) a , (68)4
15 20 25 30 - PSfrag replacements sinh r a n = 10 n = 50 n =100 FIG. 3: We observe in this plot how the entanglement parameter (more concretely sinh r ) decreases for large values of the scalefactor, a , for three different values of the mode n . Indeed, it decreases for all possible values of n . PSfrag replacements sinh r (cid:12)(cid:12) ϕ → an = − n = 0 FIG. 4: In this plot we show the non zero contribution to the entanglement of the zeroth mode ( n = 0) of the scalar field, incomparison with the case where there is no field at all ( n + 1 / where | Ω | = ξ ( ϕ ) a , with ξ ( ϕ ) ≡ (cid:16) ( H ∓ H ϕ ) + 9 H e m ϕ (cid:17) . It grows, essentially, because the frequency growsdue to the expansion of the universe. However, the specific temperature per frequency T /ω , is a measure of the rateof entanglement and it decreases as the universes expand, becoming asymptotically independent universes.The standard measure of the entanglement between the parts of a composite system is the entropy of entanglement[18, 59]. Nevertheless, both sinh r and the entropy S behave in the same way, due to the fact that the entropy is amonotonic function of sinh r (given that sinh r ≥ r , which in the oscillatory regime is given by Eq.(67). The entropy of entanglement turns out to be a monotonic decreasing function whose variation with respect tothe scale factor behaves like ˙ S ent ∼ − a − log a. (69)It may still provide us with an arrow of time for each single universe of the entangled pair [30] because of itsmonotonicity. Let us also notice that the second principle of thermodynamics, which can be formulated as [58] therequirement of a non-negativity value of the production of entropy (54), is satisfied because the entanglement betweenuniverses is not an adiabatic process, in the quantum information sense, and the production of entropy (54) is zerofor any change rate of the scale factor. Thus, the classical formulation of the second principle of thermodynamicsapplied to the entropy of inter-universal entanglement would provide us with no direction of time because ˙ ς = 0 bothfor a decreasing and a increasing value of the scale factor. However, the quantum information formulation that states[64] that the entanglement rate between the parts of a composite system cannot be increased by any local operationand classical communication alone, does imply an arrow of time in each single universe of the entangled multiverse.By local operations we mean, in the context of the multiverse, processes that happen within each single universe ofthe multiverse. Therefore, anything that happens in a single real universe would make the entanglement rate betweenuniverses decrease as the universes expand. Besides, the entanglement arrow of time in the multiverse might be a5testable property of each single universe provided that the entropy of entanglement is eventually related to the totalentropy of the universe.Regarding the energy, there are two contributions to the total energy given by the quantum information work andheat, W and Q , respectively, in Eqs. (51-53). By inspecting these equations, we can easily see that the work W is dueto the variation of the frequency that is caused by the expansion of the universe. It is therefore the change of energycaused by the change of the proper volume of the universe. On the other hand, the variation of heat, Eq. (53), is dueto the change of the rate of entanglement, r . It is therefore an energy purely associated to the entanglement betweenuniverses and it supplies a correction term to the total energy of an unentangled universe. Thus, entangled andunentangled universes behave differently, so this effect makes inter-universal entanglement to be a falsifiable propertyof the universe in the multiverse scenario.The energy of entanglement is usually defined [60–62] as it is defined in Eq. (55). Therefore, we can compute it byintegrating Eq. (56). First, we can check that the limiting value of a vanishing scalar field with n = − / E ent ≡ Q = Z da ω n sinh 2 r ˙ r ∼ H σa . (70)The consideration of the scalar field introduces a different behavior of the energy of entanglement. During the oscil-latory phase of the scalar field, | ϕ ( a ) | ∝ ( a /a ) , so using Eq. (67) the energy of entanglement can be approximatedfor large values of the scale factor as E ent ∼ Q ∝ − n + 2 a ˜ mσ ) H σa . (71)In both cases the energy of entanglement asymptotically vanish for an infinite value of the scale factor as it does theentanglement rate between the universes. Particularly, Eq. (71) becomes Eq. (70) for the limiting values n → − and ˜ m →
0, as it was expected. However, Eqs. (70) and (71) give different contributions to the total energy ofthe universe. That should have, in principle, observable consequences provided that the energy of inter-universalentanglement gives a contribution to the total energy of each single universe.Let us also notice two other consequences of Eq. (71). First, as we already pointed out in the case of the sinh r ,there is a non-vanishing contribution of the n = 0 mode of the scalar field, that is, the quantum fluctuation of thevacuum state of the scalar field contribute. Second, the energy of entanglement, Eq. (71), can be written as theenergy of entanglement (70) with an effective value of the Hubble parameter given by H eff ≡ H (1 − n + 2 a ˜ mσ ) . (72)We observe that for most of the values of the field mode (except those such that n ∼ a ˜ mσ ), the effective Hubbleparameter satisfy H eff ≪ H . Therefore, we can conclude that, in general (for all the values of the field mode exceptthe highly implausible case pointed out before), the effect of the scalar field in the value of the inter-universal energyof entanglement is equivalent to reduce the value of the effective cosmological constant, Λ eff ≡ H eff ) ≪ Λ . V. CONCLUSIONS
The multiverse has become during the last years an interesting and fruitful scenario where exploring customaryproblems of classical and quantum cosmology. The third quantization procedure discussed here allows us to mimic, inthe context of the entangled multiverse, well known techniques used in quantum field theories in curved space-timeswhich uncover fundamental phenomena of the space-times with event horizons, like those involved in the formulationof Hawking radiation or in the particle creation in a de-Sitter space-time. In both cases it is possible to relate differentrepresentations of the time-dependent harmonic oscillator by Bogoliubov transformations. We dealt here, however,with a new scenario in which two or more universes are entangled and we computed some of their thermodynamicproperties of entanglement.More precisely, our model of the multiverse consists in a collection of universes filled with a scalar field with aquadratic potential. The addition of such a scalar field provides a realistic framework that allows the study of boththe inflationary stage of the universe and the subsequent oscillatory regime of the scalar field that would give rise tothe current universe at large values of the scale factor. It is worth noticing that a similar formalism can be followedusing other potentials, including those representing interacting matter fields, that would lead to the same equation6(22) with another corresponding Hamiltonian. Therefore, the developments made in this paper are rather general andcan straightforwardly be applied to multiple cosmological scenarios.In order to work out the model we have imposed certain boundary conditions. It is reasonable to assume thatthe global properties of the multiverse do not depend on the value of the scale factor of a specific universe. Thisis analogous to imposing space-time invariance to the vacuum state of a specific curved space-time, like it happensfor instance in the choice of the vacuum state in a de-Sitter space-time [23, 57, 65]. The parallel reasoning in theminisuperspace is to consider an invariant representation with respect to the value of the scale factor, which is formallythe time-like variable. That makes the vacuum state of the multiverse to be invariant under reparametrizations ofthe scale factor and consequently under time reparametrizations in a particular single universe, as it is expected.However, as it also happens in a de-Sitter space-time, in which an observer is better described by static coordinates,or in the case of the black hole radiation where the particles are meaningfully defined in the asymptotic flat region ofthe space-time, the representation of universes from the point of view of internal (actual) observers is preferably givenby the asymptotic representation of large universes with a large value of their scale factors. Then, continuing withthe analogy with a curved space-time, the fact that a particular observer has no access to a region of the space-time(the region behind the event horizon in a de-Sitter or a Schwarzschild space-time), that is, the partner universe of theentangled pair, makes our universe to effectively stay in a thermal state. It is worth pointing out that for an internalobserver such a thermal state of the universe is indistinguishable from a classical mixture [24, 30]. However, it comesfrom a composite entangled state that has no classical counterpart whatsoever [9, 10].The former boundary condition translates into the condition that the number operator associated with a givenrepresentation has to be invariant. We achieved it by using the Lewis representation defined by the operators ˆ b andˆ b † . On the other hand, we considered the natural representation that diagonalizes the Hamiltonian and that representslarge parent universes, plausibly with observers inhabiting them. The fact that both representations coincide in theasymptotic limit and given that the Lewis representation encodes global properties of the multiverse, suggest us thatwe can associate the invariant representation to an external observer, i.e., a hypothetical observer that would live inthe multiverse.In such a framework, we considered that the universes could be created in entangled pairs. Then, if we considerthat the multiverse is in the ground state of the Lewis representation, an internal observer would see her universe ina state with a non-zero value of its excitation number. From our point of view, the consideration that the multiverseis in the ground state (in the Lewis representation) is sensible, and it is the way we followed in the present paper.Nevertheless, it would be possible to extend our results to a general quantum state, although the technical difficultieswould increase and we do not expect qualitative differences, at least at the leading order of the asymptotic limit.In this paper, we have generalized previous results on the thermodynamics of entanglement between universes byadding a scalar field. In this way, we enriched the model and considered a more realistic scenario. We obtained thethermodynamic magnitudes of entanglement for each single universe of an entangled pair of universes that generalizeprevious works on the subject [25, 63] recovering their results in the appropriate limits. The entropy of entanglementstill supplies us with an arrow of time for each single universe. The energy of entanglement, given by the quantuminformation heat, provides us as well with a correction to the total energy of the expanding universe that would not bepresent in the case of an isolated universe, becoming thus a falsifiable property of the universe. In a pair of entangleduniverses, the energy of entanglement decays at late times becoming asymptotically zero for large values of the scalefactor, where the universes become uncorrelated. The quantum fluctuations of the vacuum state of the scalar fieldcontribute to the energy of entanglement between different universes. However, the more remarkable effect of thescalar field in the energy of entanglement would be the effective reduction of the cosmological constant in each singleuniverse.The contribution of the energy of entanglement to the total energy of each single universe would thus presumablyhave observable consequences on the dynamical properties of the universe, making therefore testable the whole multi-verse proposal. However, the study of the dynamical consequences on a single universe of its entanglement propertieswith respect to other universes of the multiverse needs first of a framework that would account for the backreactionof the thermodynamic magnitudes of inter-universal entanglement on the physical properties of each single universe,which we expect to develop in future works.In that sense, it is worth pointing out that although the complete relationship between the thermodynamic mag-nitudes of entanglement and those of the customary formulation of thermodynamics is not clear yet, it is currently apromising area of study [26–29, 64]. If such an ambitious program is finally achieved in quantum optics and quantuminformation theory, it will entail an important tool for testing the entanglement among universes, the multiverseproposal and the theories that underlie it. Therefore, in our opinion, the entanglement between universes representsa new interesting way to explore possible observable consequences of the multiverse, offering physically admissiblemodels and a novel way to look at some of the main open problems in cosmology.7 Acknowledgments
This work was in part supported by the Spanish MICINN research grants FIS2009-11893 and FIS2012-34379. IGis supported by the CNPq-Brasil. SRP was supported by the Basque Government project IT-221-07.
Appendix A: Quantization of a scalar field in a de-Sitter space-time
In this appendix we review the basic formalism for the quantization of a scalar field in the background of a curvedspace-time [23, 57, 65, 66]. The aim is twofold: on the one hand, we want to stress the formal resemblance between thethird quantization formalism used in this paper and the quantization of fields in a curved space-time. On the otherhand, we show that the formalism develop in section II.C naturally leads, starting from Eq. (23), to the customaryquantum field theory of a scalar field in a de-Sitter space-time.The wave equation for a scalar field with mass m that propagates in the background of a de-Sitter space-time isgiven by [57, 65] χ ′′ − ∆ χ + ( m a − a ′′ a ) χ = 0 , (A1)where χ ≡ aϕ , f ′ ≡ ∂f∂η , with η ≡ R dta being the conformal time, and ∆ is the Laplacian on the spatial sections ofthe space-time. The general solution is given by χ ( ~x, η ) = Z d ~k (2 π ) χ ~k ( η ) e i~kη , (A2)where the amplitudes χ ~k ( η ) satisfy the equation of a harmonic oscillator, χ ′′ ~k + ω ~k ( η ) χ ~k = 0 , (A3)with a time dependent frequency given by ω ~k ( η ) = | ~k | + m a − a ′′ a . (A4)Different representations can be used for expressing the solutions of the harmonic oscillator (A3). Let us considertwo of those representations, with creation and annihilation operators (ˆ a † , ˆ a ) and (ˆ b † , ˆ b ), respectively, related by aBogoliubov transformation [57] ˆ a k = µ ∗ k ˆ b k + ν k ˆ b †− k , ˆ a † k = µ k ˆ b † k + ν ∗ k ˆ b − k , (A5)with | µ k | − | ν k | = 1. Then, for a particular value k of the mode the corresponding states are related by [57] | k, − k i (ˆ b ) = 1 | µ k | ∞ X n =0 (cid:18) ν k µ k (cid:19) n | n k , n − k i (ˆ a ) . (A6)That is, because of the isotropy the particles are produced in pairs with opposite momenta ~k and − ~k . If for anyreason we would have access only to the positive modes of the field, then the reduced density matrix that representsthe quantum state of the scalar field would be given, in the a -mode representation as ρ k ≡ Tr − k | k, − k i (ˆ b ) h k, − k | = ∞ X j =0 | µ k | ∞ X n,m =0 ν nk ( ν ∗ k ) m µ nk ( µ ∗ k ) m h j − k | n k , n − k ih m k , m − k | j − k i = 1 | µ k | X n (cid:18) | ν k || µ k | (cid:19) n | n k ih n k | , (A7)which is a Gibbs state that represents a thermal state with temperature T given by T = ω k | ν k || µ k | (cid:18) ~ k B (cid:19) . (A8)From this expression for the temperature we can easily compute the customary thermodynamic magnitudes.8Let us stress two remarkable things. First, the k = 0 mode of Eq. (A3) is also obtained from the Heisenbergequations ∂ϕ∂t = i [ h , ϕ ] , ∂p ϕ ∂t = i [ h , p ϕ ] , (A9)with h being given by Eq. (23). Thus, Eq. (23) naturally leads to the customary quantization of a scalar fieldwith mass m in a de-Sitter space-time provided that we just consider a homogeneous and isotropic scalar field, ϕ ( ~x, t ) ≡ ϕ ( t ). It is straightforward to show [57] that for other modes different from zero the hamiltonian is given by h ≡ M ( t ) p ϕ + M ( t )2 (cid:18) a ( ∇ ϕ ) + m ϕ (cid:19) . (A10)Secondly, Eq. (A3) is formally similar to Eq. (5), where the scale factor formally plays the role of the time variableand it has been used the covariant generalization of the Laplacian operator in the minisuperspace, given by Eq. (2).This fact is at the heart of the parallelism made throughout this paper between the third quantization of the wavefunction of the universe and the generalization of a scalar field in a curved background. [1] B. Carr, ed., Universe or Multiverse (Cambridge University Press, Cambridge, UK, 2007).[2] M. Tegmark,
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