Microscopic Picture of Non-Relativistic Classicalons
Felix Berkhahn, Sophia Müller, Florian Niedermann, Robert Schneider
PPrepared for submission to JCAP
Microscopic Picture ofNon-Relativistic Classicalons
Felix Berkhahn, a Sophia Müller, a,b
Florian Niedermann a,c andRobert Schneider a,c a Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, There-sienstraße 37, 80333 Munich, Germany b Max-Planck-Institute for Physics, Foehringer Ring 6, 80805 Munich, Germany c Excellence Cluster Universe, Boltzmannstraße 2, 85748 Garching, GermanyE-mail: [email protected], [email protected],fl[email protected], [email protected]
Abstract.
A theory of a non-relativistic, complex scalar field with derivatively coupled in-teraction terms is investigated. This toy model is considered as a prototype of a classicalizingtheory and in particular of general relativity, for which the black hole constitutes a prominentexample of a classicalon. Accordingly, the theory allows for a non-trivial solution of the sta-tionary Gross-Pitaevskii equation corresponding to a black hole in the case of GR. Quantumfluctuations on this classical background are investigated within the Bogoliubov approxima-tion. It turns out that the perturbative approach is invalidated by a high occupation of theBogoliubov modes. Recently, it was proposed that a black hole is a Bose-Einstein condensateof gravitons that dynamically ensures to stay at the verge of a quantum phase transition. Ourresult is understood as an indication for that claim. Furthermore, it motivates a non-linearnumerical analysis of the model. a r X i v : . [ h e p - t h ] F e b ontents Recently, Dvali and Gomez proposed a microscopic picture of black holes [1–3]. Accordingto them, black holes can be understood as Bose-Einstein condensates of gravitons. In thispicture, the Schwarzschild geometry would effectively emerge from the interaction of a testparticle with the condensate of gravitons. In [4, 12] this picture was further elaborated andthe authors concluded that the black hole is at the point of quantum phase transition.Within the Schwarzschild radius, the graviton theory is strongly coupled. This necessi-tates to sum up a large number of equally important terms in the perturbation series. Thisfact and the relativistic nature of the graviton theory makes it hard to obtain any quantitivepredictions along the lines of [1–4, 12] within the theory of general relativity. Therefore, inthis paper we propose a non-relativistic, derivatively coupled toy model that allows to quan-titatively compute properties expected for black holes according to [1–4, 12]. Our model isconstructed such that it contains a ground state corresponding to the black hole of generalrelativity, which is nothing else but a non-relativistic classicalon state. For a description ofthe concept of classicalization in the case of gravity see [5, 6] and for its generalisation toother derivatively coupled theories compare to [7–10].We perform a quantum perturbation theory around a highly occupied classical state(so called ’Bogoliubov approximation’) which is supposed to make up the classicalon. Ourresults indicate that the perturbative approach is not applicable, which is exactly what weexpect to see if the system indeed manages to stay at the point of quantum phase transition.Therefore, we see indications for the claims of [4, 12], even though only a subsequent numericaland non-linear analysis will clearly decide about the status of our model.Our paper is organized as follows: Section 2 summarizes the main ideas of [1]. Section 3contains our model and results. Future prospects of our theory are discussed in Section 4.
The starting point in the approach of [1–4] is the observation that the graviton interactionstrength α gr is momentum dependent due to the derivatively coupled nature of interaction– 1 –erms of the metric fluctuation field with itself: α gr = hG N λ − , (2.1)where G N is Newtons constant and λ is the typical graviton wavelength involved in a givenscattering process. For the case of black holes, the characteristic wavelength is set by theSchwarzschild radius r g = 2 G N M ∼ λ , where M is the mass of the black hole. Accord-ingly, each graviton contributes an energy ∼ h/ (2 G N M ) . The total number N of gravitonsconstituting a black hole is thus N = 2 G N M h ∼ λ L P , (2.2)where we have introduced the planck length L P = √ hG N . Equation (2.2) is also true for thenumber of gravitons contained in the gravitational field of other objects such as planets sinceit can be obtained from summing up the Fourier modes of any Newtonian gravitational field φ = − r g /r . Inserting (2.2) in (2.1) yields the dependence of the coupling with Nα gr = 1 N . (2.3)The occupation number N can be understood as the parameter measuring the classical-ity of a given object composed out of gravitons, in this case black holes. Intrinsic quan-tum processes such as the decay into a two particle state are exponentially suppressed (cid:104) Out | exp ( − S ) | In (cid:105) ∼ exp ( − N ) . Additionally, the number of gravitons produced in the grav-itational field of any elementary particle is negligibly small, for example for an electron weget N = 2 G n m e /h ≈ − . This shows why elementary particles cannot be considered as aclassical gravitating object (even though they contribute a standard Newton law at large dis-tances), and in particular it becomes clear why a single elementary particle does not collapseinto a black hole.Let us contrast black holes with the gravitational field of other objects such as planets.Assuming that the characteristic wavelength of the gravitons is in any case given by thecharacteristic size R of the object, we obtain as the gravitational part of the energy E grav ∼ N hR ∼ M r g R . (2.4)This shows that for objects not being a black hole (i.e., for
R > r g ) a substantial part ofthe energy is carried by other constituents than gravitons. This is why the gravitationalfield of other objects than black holes cannot exist without an external source, for examplea planet. However, once the extension of the gravitational object reaches R = r g , the wholeenergy M of our object is stored in the gravitational field, so that an external source is notrequired to balance the energy budget. It is exactly at this point where the interaction ofan individual graviton with the collective potential generated by the other gravitons becomessignificant. This can most easily be seen by appreciating that the classical perturbationseries in the metric fluctuation field h about a Minkowski background breaks down at thehorizon r g . However, the interaction of two individual gravitons is still small as long as weconsider regions r > L P . Given that the dominant interaction is gravity itself, the authorsof [1] concluded that black holes are self-sustained bound states of gravitons. Moreover, blackholes are maximally packed in the sense that the only characteristic of a black hole in the– 2 –emi-classical limit is the number of gravitons N composing it, and any further increase of thisnumber results inevitably in an increase of the size and mass of the black hole. This becomesclear since by default the extension of the black hole is no free parameter but given by r g ,and accordingly all physical black hole quantities (mass, size, entropy, etc.) can be quantifiedby N . This is nothing else but the famous no-hair theorem translated in the language ofgravitons. An important consequence of this picture is that black holes always balance on theverge of self-sustainability, since the kinetic energy h/r g of a single graviton is just as largeas the collective binding potential − α gr N h/r g produced by the remaining N − gravitons.Thus, if you give a graviton just a slight amount of extra energy, its kinetic energy will beabove the escape energy of the bound state. In [1] it was therefore concluded that black holesare leaky condensates .The above reasoning strictly applies only in the (semi-)classical limit N → ∞ . This isimportant, because we might wonder how a quantum effect like Hawking radiation can beunderstood in our picture of highly occupied graviton states, since usually we expect quantumeffects to be exponentially suppressed. Actually, to explain this, the authors in [4] conjecturedthat the black hole is at a point of quantum phase transition . Thus, quantum excitations arealways significant and cannot be ignored. In particular, given that black holes are leakycondensates, every quantum excitation will lead to the escape of the corresponding graviton.These escaped particles are interpreted as the Hawking radiation of the black hole.Moreover, due to the quantum phase transition, the leading corrections to the above(semi-)classical ( N → ∞ ) picture are not exponentially but only /N suppressed. Thismakes it possible for any finite N to retrieve information from the black hole (for instancethe Hawking spectrum contains /N corrections, making it for example possible to read outthe amount of Baryons originally stored in the black hole). The famous information paradoxis thus just a relict of working in the strict (semi-)classical N → ∞ approach in which thehair of the black hole is negligible compared to the N graviton state.In the next section we discuss the well known physics of quantum phase transition forthe example of a non-relativistic condensed matter system. Assuming that black holes behavesimilar to this model, we will qualitatively discuss the implications for black hole physics, asit was done in [4]. The discussion of this section closely follows [11], where the properties of a quantum phasetransition are studied. We want to describe a system of N bosons of mass m with an at-tractive interaction in one dimension of size V at zero temperature. The second quantizedfield ˆΨ( x, t ) in the Heisenberg representation is measuring the particle density at position x .The corresponding Hamiltonian reads ˆ H = (cid:126) m (cid:90) V d x ( ∂ x ˆΨ) † ( ∂ x ˆΨ) − U (cid:90) V d x ˆΨ † ˆΨ † ˆΨ ˆΨ , (2.5)where U is a positive parameter of dimension [energy] × [length] controlling the interactionstrength. The dynamics of ˆΨ( x, t ) are given by the Heisenberg equation i (cid:126) ∂∂t ˆΨ = (cid:104) ˆΨ , ˆ H (cid:105) (2.6) = (cid:18) − (cid:126) m ∂ x − U ( ˆΨ † ˆΨ) (cid:19) ˆΨ (2.7)– 3 –here the equal time commutation relations (cid:104) ˆΨ( x, t ) , ˆΨ † ( x (cid:48) , t ) (cid:105) = δ ( x − x (cid:48) ) (cid:104) ˆΨ( x, t ) , ˆΨ( x (cid:48) , t ) (cid:105) = 0 (2.8)have been used. Applying the mean-field approximation amounts to replacing the opera-tor ˆΨ( x, t ) by a classical field Ψ ( x, t ) . This replacement is justified when the quantumground state is highly occupied. In this case the non-commutativity of the field operatoris a negligible effect. Since we are looking for stationary solutions, the time dependence isseparated in the usual way Ψ ( x, t ) = Ψ ( x ) exp (cid:18) − iµt (cid:126) (cid:19) , (2.9)where µ is the chemical potential. Inserting this ansatz in (2.6), yields the stationary Gross-Pitaevskii equation. A trivial solution that fulfils the periodic boundary conditions Ψ (0) =Ψ ( V ) is given by Ψ (BE)0 ( x ) = (cid:114) NV = const. (2.10)This solution corresponds to the homogenous Bose-Einstein condensate. However, this so-lution is the minimal energy configuration only for U < U c . The critical value has been bederived in [11] to be: U c = (cid:126) π / ( mV N ) . For U > U c the ground state is given by aninhomogenous solution Ψ (sol)0 ( x ) describing a soliton. By increasing the parameter U , i.e. theinteraction strength, the ground state of the system undergoes a phase transition from theBose-Einstein phase to the soliton phase once the critical point U c is reached. As the authorsin [11] have shown, this point of phase transition is characterized by a cusp in the chemicalpotential µ ( U ) , the kinetic energy (cid:15) kin ( U ) and the interaction energy (cid:15) int ( U ) per particle asfunctions of U .The main result of [11] was to show that at the point of phase transition quantumcorrections to Ψ become important and a purely classical description is no longer possible,therefrom the name ’quantum phase transition’. A suitable way to investigate this effect isprovided by the Bogoliubov approximation in which the classical field Ψ is furnished withsmall quantum corrections δ ˆΨ . A proper quantum mechanical treatment, of which the detailsare given in the next section, allows to derive the famous energy spectrum of the Bogoliubovexcitations (cid:15) ( k ) = (cid:32)(cid:18) (cid:126) δk m (cid:19) − (cid:126) U NmV δk (cid:33) / (2.11) = (cid:32)(cid:18) π (cid:126) mV (cid:19) δk (cid:34)(cid:18) V π (cid:19) δk − UU c (cid:35)(cid:33) / . Due to the periodic boundary conditions, the momentum δk of the Bogoliubov modes is quan-tized in steps of π/V . From (2.11) it is clear that once the interaction strength approachesthe value U c , the energy of the first Bogoliubov mode ( δk = 2 π/V ) vanishes. Consequently,the excitation of the first mode becomes energetically favourable and the condensate is deplet-ing very efficiently. This is the characteristic property of a quantum phase transition. This– 4 –icture is further substantiated by calculating the occupation number of excited Bogoliubovmodes n ( δk ) = (cid:126) δk / m − U N/V (cid:15) ( δk ) − , (2.12)which shows that the vanishing of (cid:15) ( δk ) is accompanied by an extensive occupation of thecorresponding quantum states. This means that the Bogoliubov approximation is no longerapplicable and quantum corrections are significant. For values U > U c the energy becomesimaginary, which signals the formation of a new ground state that is given by the solitonsolution Ψ (sol)0 ( x ) , compare to the discussion in [11]. Moreover, the work of [13, 14] showsthat the system becomes drastically quantum entangled at the critical point, which is yetanother characterization of quantum phase transition.By making the N dependence of U c explicit and introducing the new dimensionlesscoupling parameter α = U mV / ( (cid:126) π ) , the condition for the breakdown of the Bogoliubovapproximation becomes α = 1 N . (2.13)This is exactly the condition for self-sustainability in the case of a black hole (2.3). Theseconsiderations closely follow [4], where the authors wanted to illustrate the relation betweenblack hole physics and Bose-Einstein condensation at the critical point. Of course, in thistoy model the relation (2.13) is not generically realized, but has to be imposed by adjustingthe model parameters by hand. (For a given value of N , the interaction strength U hasto be chosen appropriately.) In the case of GR the left hand side of equation (2.13) is k -dependant which in principal could allow for a generic cancelation between the two terms inthe squared bracket in the last line of (2.11). This cancelation is assumed to take place upto /N –corrections.The aim of our work is to present a non-relativistic scalar model that is in principle able toaccount for this cancelation and thus generically stays at the point of quantum phase transitionindependent of the chosen parameters. It is not possible to derive this result within theBogoliubov approximation since a high occupation of quantum states is the defining propertyof a quantum phase transition. However, the breakdown of the perturbative approach is anecessary condition and therefore provides an indication for it. Non-relativistic classicalizing theories have the advantage of being computable without aresummation of infinitely many equally important terms as it would be the case for examplein GR. In the following, we will consider a special non-relativistic, classicalizing theory thatwas constructed to mimic general relativity. As in [11], we choose to confine our theory in a1-dimensional box of size V . To be concrete, we consider the following Hamiltonian for thesecond quantized field ˆΨ( x ) measuring the particle density at position x : ˆ H = (cid:126) m (cid:90) V d x : ( ∂ x ˆΨ) † ( ∂ x ˆΨ) : + λ (cid:90) V d x : (cid:16) ( ∂ x ˆΨ) † ( ∂ x ˆΨ) (cid:17) : + κ (cid:90) V d x : (cid:16) ( ∂ x ˆΨ) † ( ∂ x ˆΨ) (cid:17) : , (3.1)– 5 –here : : denotes the normal ordering. We are looking for homogenous solutions of theHeisenberg equation i (cid:126) ∂∂t ˆΨ = (cid:104) ˆΨ , ˆ H (cid:105) , (3.2)in which the field operator is again replaced by a classical field Ψ ( x ) . (The subscript willbe suppressed throughout the rest of this work.) We try to generalize the known homogenousBEC solution (2.10). We can separate the time dependence as in (2.9). Since Ψ( x ) is acomplex field, (3.2) has in general the following class of solutions Ψ k ( x ) = (cid:114) NV exp ( ikx ) , (3.3)where the momentum k is quantized in steps of π/V by implementing periodic boundaryconditions. The number of particles is denoted by N . Inserting (3.3) in the Hamiltonian (3.1)results in the polynomial H (0) V = (cid:126) m z + λz + κz (3.4)where z = NV k .However, not every solution (3.3) is a local minimum of the energy (3.4). For sure, oneminimum is given by k = 0 (since the kinetic energy contributes positively), which wouldexactly correspond to the Minkowski vacuum in the case of general relativity given that thisis the global energetic minimum of the theory (3.1). Moreover, by appropriately choosing thecoefficients λ and κ , we can construct a second minimum of (3.4) at z = N k /V with positiveenergy, denoted with Ψ k , where k > . It is easy to show that the corresponding solutionnot only minimizes (3.4) (that is, minimizing the energy within the sub-class of homogenoussolutions (3.3)) but is also given as a minimum in complete field space (that is, it is a minimumfor general fluctuations Ψ = Ψ k + δ Ψ ). It is this solution that will turn into the classicalonwhich corresponds to the black hole solution of general relativity. Furthermore, it should benoted that the chemical potential is zero due to the relation µ ∝ ∂H (0) /∂z | z . We will study the leading quantum perturbations δ ˆΨ( x ) about the classical condensate Ψ k ( x ) .To this end, we write ˆΨ( x ) = 1 √ V (cid:88) k ˆ a ( k ) e ikx = 1 √ V ˆ a ( k ) e ik x + 1 √ V (cid:88) k (cid:54) = k ˆ a ( k ) e ikx , (3.5)where ˆ a ( k ) is the annihilation operator of the momentum mode k . The Bogoliubov approx-imation consists in treating the first term in (3.5) classically due to the large occupation ofthe state with momentum k . The second term presents a small quantum correction. Onaccount of this, the replacement ˆ a ( k ) → (cid:112) N (3.6)is introduced, which allows to identify Ψ k ( x ) with the first term in (3.5). The second term issimply the Fourier representation of the quantum perturbation δ ˆΨ( x ) . We want to calculatethe perturbation series up to second order in δ ˆΨ( x ) or ˆ a ( k (cid:54) = k ) . Note that once we allowfor an occupation of the momentum states with k (cid:54) = k , we have to distinguish between N ,– 6 –he number of particles in the ground state, and N , the total number of particles. Since wewant to express everything in terms of N , the normalisation condition ˆ a † ( k )ˆ a ( k ) = N − (cid:88) k (cid:54) = k ˆ a † ( k )ˆ a ( k ) (3.7)has to be employed. This means that the zeroth order H (0) terms contribute to the secondorder H (2) when we express N in terms of N . Inserting (3.5) and (3.7) into the Hamiltonian(3.1), results in the following quadratic order expression: H (2) = (cid:88) δk (cid:54) =0 (cid:104) (cid:15) (1)0 ˆ a † ˆ a + (cid:15) (2)0 ˆ b † ˆ b + (cid:15) (ˆ a † ˆ b † + ˆ b ˆ a ) (cid:105) , (3.8)where the decomposition k = k + δk has been used and the (re-)definitions ˆ a ( δk ) ≡ ˆ a ( k + δk ) , (3.9) ˆ b ( δk ) ≡ ˆ a ( k − δk ) , (3.10)as well as (cid:15) (1)0 = ( k + δk ) P + Λ , (3.11) (cid:15) (2)0 = ( k − δk ) P + Λ , (3.12) (cid:15) = ( k − δk ) P , (3.13)apply. Here, the polynomials P , P and Λ are functions of the combination z and thecoefficients m , λ and κ : P = (cid:126) m + 2 λz + 92 κz (3.14) Λ = − k (cid:18) (cid:126) m + λz + 32 κz (cid:19) (3.15) P = λz + 3 κz (3.16)Note that when using the minimal energy condition ∂H (0) /∂z | z = 0 , see equation (3.4), weobtain P = P and Λ = 0 due to the relations V ( P − P ) = ∂H (0) /∂z | z and V Λ /k = − ∂H (0) /∂z | z , respectively. Furthermore, it can be checked that P > if z corresponds tothe minimum of (3.4) because V P = z ∂ H (0) /∂z | z . The Hamiltonian (3.8) is almost ofthe Bogoliubov form and can be diagonolised by means of the transformation ˆ α = u ˆ a + v ˆ b † and ˆ β = u ˆ b + v ˆ a † , (3.17)where u, v ∈ R . Setting the off-diagonal terms to zero and requiring standard commutationrelations for ˆ α and ˆ β implies (cid:15) (cid:0) u + v (cid:1) − u v (cid:15) (1)0 + (cid:15) (2)0 , (3.18)as well as u − v = 1 . (3.19)– 7 –hese two equations are solved by u = ± √ (cid:32) (cid:15) (1)0 + (cid:15) (2)0 (cid:15) + 1 (cid:33) / , v = ± √ (cid:32) (cid:15) (1)0 + (cid:15) (2)0 (cid:15) − (cid:33) / , (3.20)where (cid:15) = (cid:114) (cid:16) (cid:15) (1)0 + (cid:15) (2)0 (cid:17) − (cid:15) . (3.21)Note that (cid:15) (1)0 and (cid:15) (2)0 are strictly positive, whereas the sign of (cid:15) depends on the value of δk .Thus in order to fulfill (3.18), we have to choose u and v in (3.20) both positive when δk < k and one of both has to be chosen negative when δk > k . In both cases the diagonalizedversion of (3.8) reads H (2) = (cid:88) δk (cid:54) =0 (cid:20)(cid:18) (cid:15) + 12 ( (cid:15) (1)0 − (cid:15) (2)0 ) (cid:19) ˆ α † ˆ α + (cid:18) (cid:15) −
12 ( (cid:15) (1)0 − (cid:15) (2)0 ) (cid:19) ˆ β † ˆ β + (cid:15) −
12 ( (cid:15) (1)0 + (cid:15) (2)0 ) (cid:21) . (3.22)Using the definitions (3.11), (3.12) and (3.13), we find (cid:15) = 2 P k | δk | and ( (cid:15) (1)0 − (cid:15) (2)0 ) / P k δk . Note that (cid:15) is strictly positive. By employing the relation ˆ α ( δk ) = ˆ β ( − δk ) we find H (2) = (cid:88) δk (cid:54) =0 (cid:20) (cid:18) (cid:15) + 12 ( (cid:15) (1)0 − (cid:15) (2)0 ) (cid:19) ˆ α † ˆ α + (cid:15) −
12 ( (cid:15) (1)0 + (cid:15) (2)0 ) (cid:21) . (3.23)Accordingly, the vacuum | (cid:105) of the Fock space is defined as ˆ α | (cid:105) = 0 . (3.24)It follows from the Hamiltonian (3.23) that the combination e ( δk ) ≡ (cid:18) (cid:15) + 12 ( (cid:15) (1)0 − (cid:15) (2)0 ) (cid:19) (3.25)is the energy of the quasi particles created by ˆ α † ( δk ) with momentum k + δk . Since thevacuum of our theory is defined with respect to ˆ α , it contains a non-vanishing amount ofexcited real particles associated with ˆ a (and ˆ b equivalently). This effect goes under the namequantum depletion and occurs physically due to the interactions amongst the particles whichnecessarily pushes some of them to excited states. Their precise number is given by (cid:104) | ˆ a † ( δk )ˆ a ( δk ) | (cid:105) = v ( δk ) . (3.26)This allows to rewrite the energy of the quasi particles associated with ˆ α as e ( δk ) = (cid:40) P k δk for δk > for δk ≤ (3.27)and the number of depleted real particles with momentum k + δk as v ( δk ) = 12 (cid:18) k + δk k | δk | − (cid:19) . (3.28)– 8 –he above results can easily be generalized to a derivatively coupled theory with an arbitrarynumber of higher order terms H = r max (cid:88) r =1 c r (cid:90) V d x : ( ∂ x Ψ † ∂ x Ψ) r : . (3.29)Note that the coefficients c r have dimension [energy][length] − . The standard kinetic termcorresponds to r = 1 for which the coefficient is c = (cid:126) / (2 m ) . The energy of the quasiparticles and the number of depleted particles are given by (3.27) and (3.28) where P nowis given by the generalized expression P = r max (cid:88) r =1 c r r (cid:18) NV (cid:19) r − (cid:0) k (cid:1) r − , (3.30)and k is determined as a minimum of the generalized version of (3.4) H (0) V = r max (cid:88) r =1 c r (cid:0) k (cid:1) r (cid:18) NV (cid:19) r . (3.31)The coefficients c r have again to be chosen such that there is a non trivial minimum. Our results incorporate the vanishing of the energy gap for δk < . This (at least partly)vanishing energy gap can be considered as an indication for the occurrence of a quantumphase transition, as we discussed in section 2.2. Moreover, we see that the Bogoliubov modesbecome highly occupied for δk (cid:29) k . This in fact signals a breakdown of the Bogoliubovtheory anyways, as two succeeding terms in the quantum perturbation theory compare as N ( k + δk ) k δN ∼ N / ( k + δk ) k δN / , (3.32)where δN denotes the number of excited particles in the momentum state k + δk . Equa-tion (3.32) clearly shows that the number of excited particles should at least be suppressedas δN ∼ N k /δk . The result for the number of depleted particles (3.28) is, however, com-pletely the opposite, as it is not suppressed but enhanced for large δk . Therefore, we cansafely conclude that the perturbative approximation has broken down anyways. Again, thisis in accordance with the expectation of being at the quantum critical point because at thispoint the system behaves purely quantum and cannot even approximately be described clas-sically. Therefore, the breakdown of the Bogoliubov theory was expected, since it amountsto calculate the perturbative quantum corrections around a classical ground state.Note that the breakdown is also intuitive from the viewpoint of a vanishing energy gapfor the quasi particles with δk < . Of course, neither ˆ a or ˆ b particles can directly be relatedwith the direction of ˆ α or ˆ β particles in phase space. But the vanishing of the energy gapshould somehow be transferred into the sector of physical ˆ a and ˆ b particles. Since a vanishingenergy gap means that it is indefinitely easy to excite the quasi particles, we seem to recoverthis behavior in the high momentum sector of ˆ a and ˆ b particles.We can also perform the Bogoliubov approximation around the global minimum of (3.4)at k = 0 . Due to the derivatively coupled nature of the interaction terms, the higher orderterms in (3.1) do not contribute, which in turn implies that the Hamiltonian (3.8) is alreadydiagonal. Therefore, there is no depletion of the vacuum which allows us to further extendthe GR analogy: This state would simply correspond to the Minkowski vacuum in the caseof GR. – 9 – Future Prospects
Contrary to model (2.5), where the critical point is actually reached and crossed by sufficientlyincreasing the interaction strength U , in our model there is some indication that the systemstays at the point of quantum phase transition and does not organize itself in a new classicalground state. However, this indication is only inferred from the observation of the breakdownof the Bogoliubov theory. To get some solid measures, we need to go beyond the Bogoliubovapproximation in the next step [15]. This can be achieved by a full quantum mechanicaltreatment of the theory (3.1). The diagonalization of the Hamiltonian can be performedunder the assumption that only the lowest l momentum eigenstates are significantly occupied(given that we are supposed to sit in a local minimum, this seems to be a good assumption).Therefore, it suffices to diagonalize the Hamiltonian within a Hilbert subspace containing onlya finite number of states describing N bosons occupying l different momentum eigenstates.For l chosen appropriately small the calculation is numerically feasible and has been performedin the case of the non-derivativly coupled model in [11]. By means of this calculation we wouldbe able to address quantitative questions, such as the size of the energy gap, the number andspectrum of depleted particles or the amount of quantum entanglement in the system.The generalization of our results to a relativistic classicalon theory offers another promis-ing prospect of future research. This necessitates to apply the ideas of the Bogoliubov ap-proach to a relativistic theory and would be a significant step towards a more quantitativetreatment of the black hole condensate in general relativity. Acknowledgements
The authors would like to thank Gia Dvali, Daniel Flassig, Stefan Hofmann, Michael Kopp,Florian Kühnel, Alexander Pritzel and Nico Wintergerst for inspiring discussions. The workof FB was supported by TRR 33 ’The Dark Universe’. The work of SM was supported bya research grant of the Max Planck Society. The work of FN and RS was supported by theDFG cluster of excellence ’Origin and Structure of the Universe’.
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