Bekenstein Bound from the Pauli Principle

PHYSICAL REVIEW D 102, 106002 (2020)

Bekenstein bound from the Pauli principle

† ‡ G. Acquaviva ,* A. Iorio , and L. Smaldone Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Praha 8, Czech Republic

(Received 8 June 2020; accepted 1 October 2020; published 2 November 2020)

Assuming that the degrees of freedom of a black hole are finite in number and of fermionic nature, we naturally obtain, within a second-quantized toy model of the evaporation, that the Bekenstein bound is a consequence of the Pauli exclusion principle for these fundamental degrees of freedom. We show that entanglement, Bekenstein, and thermodynamic entropies of the black hole all stem from the same approach, based on the entropy operator whose structure is the one typical of Takahashi and Umezawa’s thermofield dynamics. We then evaluate the von Neumann black hole-environment entropy and noticeably obtain a Page-like evolution. We finally show that this is a consequence of a duality between our model and a quantum dissipativelike fermionic system.

DOI: 10.1103/PhysRevD.102.106002

I. INTRODUCTION supposing that in a BH only free Xons exist (hence, there can only be one kind of entropy at that level), and we This paper moves from the results of previous research suppose that they are finite in number and fermionic in [1] but reversing the point of view. There, Bekenstein’s nature. This amounts to having a finite dimensional H. argument that a black hole (BH) reaches the maximal With these assumptions, here we show that the evaporation entropy at disposal of a physical system (i.e., that it is a dynamical mechanism producing a maximal entangle- saturates the Bekenstein bound [2]), leads to two main ment entropy, equal to the initial entropy of the BH. proposals: i) the degrees of freedom (d.o.f) responsible for This is an instance of the Bekenstein bound, obtained the BH entropy have to take into account both matter and here with arguments that do not assume preexisting geo- spacetime and hence must be of a new, more fundamental metrical (spatiotemporal) concepts. In fact, for a full nature than the d.o.f we know (with Feynman [3], here we identification with the standard formulae (see [7], where call such d.o.f “Xons”, see also [4] and [5]); ii) the Hilbert the bound is rigorously defined in quantum field theory space H of the Xons of a given BH is necessarily finite (QFT) and, e.g., the review [8]) one needs to associate dimensional geometrical concepts to the Xons. For instance, one could make each d.o.f correspond to one elementary Planck cell. S dim H ¼ e BH ; ð1Þ Nonetheless, in our picture we do not need the exact expression of the bound. What is crucial is that the Xons are S with BH the Bekenstein entropy. With these, in [1] it was taken to be finite in number and fermionic, otherwise the shown that the (average) loss of information is an unavoid- entanglement entropy would just indefinitely grow without able consequence of the nonvanishing relic entanglement reaching a maximal value. It is suggestive, though, that between the evaporated matter and the spacetime. taking on board the geometric picture of Xons as quanta of In search of a unifying view of the various types of area (Planck cells), the horizon of the BH is of nonzero size entropies involved in the BH evaporation (i.e., Bekenstein, as an effect of a Pauli exclusion principle. Before entering thermodynamical, and entanglement entropies, see, e.g., the details of what we just discussed, let us now briefly put [6]), here we reverse that logic. Namely, we start off by our work into the context of current literature. Bekenstein entropy [9,10] is traditionally regarded as a * measure of our ignorance about the d.o.f, which formed the [email protected] † – [email protected] BH [10 13], and as a consequence of the no-hair theorem ‡ [email protected] [14]. However, other interpretations have been proposed in literature, as in Loop Quantum Gravity (LQG), where BH Published by the American Physical Society under the terms of entropy is a counting of microstates corresponding to a the Creative Commons Attribution 4.0 International license. A Further distribution of this work must maintain attribution to given macroscopic horizon area [15,16]. Along these the author(s) and the published article’s title, journal citation, lines, Bekenstein proposed a universal upper bound for and DOI. Funded by SCOAP3. the entropy of any physical system contained in a finite

2470-0010=2020=102(10)=106002(10) 106002-1 Published by the American Physical Society G. ACQUAVIVA, A. IORIO, and L. SMALDONE PHYS. REV. D 102, 106002 (2020) region [2], which is saturated by BHs. This implies [17] that bound itself descends from the Pauli principle. In Sec. IV the entropy of every system in a finite region is bounded we explain the relation with TFD by mapping our model to from above by the Bekenstein entropy of a BH, whose an equivalent description as a dissipativelike system. The horizon coincides with the area of the boundary of that last section is left to our conclusions, while in the Appendix region (see [7] and also [8]). we show the connection between TFD and von Neumann Using the approach of QFT in curved spacetimes, entropies in the present context. Hawking discovered the black body spectrum of BH radia- Throughout the paper we adopt units in which tion [18]. In the meantime, Umezawa and Takahashi c ¼ ℏ ¼ 1. developed their thermofield dynamics (TFD) [19] (see also Ref. [20]), that immediately appeared to be a fruitful tool to II. BASIC ASSUMPTIONS AND MODEL describe BH evaporation [21].In[22], with the help of an OF BH EVAPORATION entropy operator, whose structure is natural in TFD, the BH-radiation entropy is viewed as an entanglement entropy We assume that the fundamental d.o.f are fermionic (BH of radiation modes with their “TFD-double” (the modes models based on fermions are available in literature, see, beyond the horizon). e.g., the Sachdev-Ye-Kitaev model [43,44]). As a conse- Although the relation between QFT in curved spacetime quence, each quantum level can be filled by no more than one fermion. This assures that the Hilbert space H of and TFD was studied already in Refs. [23,24], the renewed physical states with a finite number of levels is finite interest comes in connection with the AdS=CFT corre- dimensional. In fact, if the fundamental modes were spondence [25], where in a two-sided anti-de Sitter BH, the bosons, then the requirement that the number of slots specular asymptotic region is mapped into two copies of a available were finite would not have been sufficient to conformal field theory. The thermal nature of the BH is then guarantee the finiteness of dim H. Let us recall now that, in naturally seen through TFD. Extensions to incorporate the picture of [1], it is only at energy scales below those of dissipative effects are in the recent [26,27]. quantum gravity (e.g., at the energy scales of ordinary Since a BH, initially described as a pure state, could end matter) that the field modes are distinguishable from those up in a mixed state (this is actually the view of [1]), “making” the spacetime, hence we can write questions arise on the unitary evolution, as first noticed by Hawking [28] and then extensively discussed, from differ- H ⊗ H ⊆ H: ð2Þ ent points of view, see e.g., Refs. [29–39]. In particular, F G in Refs. [31,40] Page studied the bipartite system BH- Here F and G stand for “fields” and “geometry”, respec- radiation, in a random (Haar distributed) pure state, tively. In other words, at low energy, the F-modes will form computing the radiation entanglement entropy as function quantum fields excitations, that is, the quasiparticles (from of the associated thermodynamical entropy. He found a the Xons point of view) immersed into the spacetime symmetric curve (Page curve) which goes back to zero formed by the G-modes. when the BH is completely evaporated. In Ref. [36] he Now, say N is the total number of quantum levels (slots) postulated that entanglement entropy, as a function of time, available to the BH. The evaporation consists of the follows the minimum between Bekenstein and radiation following steady process: N → ðN − 1Þ → ðN − 2Þ → . thermodynamic entropy (Conjectured Anorexic Triangular That is, the number of free Xons steadily decreases, in favor Hypothesis). Recently [41], a Page curve was also derived of the Xons that, having evaporated, are arranged into from holographic computations [42]. quasiparticles and the spacetime they live in. One might As said, in this paper we reverse the line of reasoning of think of a counter that only sees free Xons and hence keeps Ref. [1] and present a simple, purely quantum toy model clicking in one direction as the BH evaporates till its of the dynamics of BH evaporation, focusing on the complete stop. fundamental d.o.f. In Sec. II the basic assumptions are In this picture: i) there is no preexisting time because the the finiteness of slots (quantum levels) available for the natural evolution parameter is the average number of free system and the fermionic nature of such d.o.f. The finite- Xons; and ii) there is no preexisting space to define the ness of the Hilbert space of states follows from the Pauli regions inside and outside the BH because a distinction of exclusion principle. In Sec. III we compute the von the total system into two systems, say environment (I) and Neumann entropy of the subsystems during their evolution. BH (II), naturally emerges in the way just depicted. With This is remarkably given by the expectation value of the this in mind, in what follows we shall nonetheless refer to TFD entropy operator [22], and it has the same qualitative system I as outside and to system II as inside. It is a worthy behavior of a Page curve: it starts from zero and ends in remark that other authors do use the geometric notions of zero while its maximum is reached at half of the evapo- exterior and interior of BHs even at fundamental levels ration process. That maximum is identified here with the [29]. Even though this can be justified, see, e.g., [4], and Bekenstein entropy of the BH at the beginning of the permits one to produce meaningful models, see, e.g., [39], evaporation. We can, therefore, argue that Bekenstein our approach does not require it. The Hilbert space of

106002-2 BEKENSTEIN BOUND FROM THE PAULI PRINCIPLE PHYS. REV. D 102, 106002 (2020) physical states is then built as a subspace of a larger tensor where on σ we shall comment soon. Equations (10) and product (kinematical) Hilbert space (11) define a canonical transformation

H ⊆ H ⊗ H † † I II: ð3Þ fcnðσÞ;cmðσÞg ¼ fdnðσÞ;dmðσÞg ¼ δnm: ð12Þ

We now assume that such a Hilbert space can be con- We thus get the evolution of the initial state (6) as structed with the methods of second quantization. This provides a language contiguous to the language of QFT, YN Ψ σ ≡ † σ 0 ⊗ 0 which should be recovered in some limit. Therefore, BH j ð Þi cnð Þj iI j iII and environment modes will be described by two sets of n¼1 YN creation and annihilation operators, which satisfy the usual − ψ † † φ i n σ i n σ 0 ⊗ 0 canonical anticommutation relations ¼ e ðbn cos þ an e sin Þj iI j iII: n¼1 χ χ† δ δ ð13Þ f τn; τ0n0 g¼ τ;τ0 nn0 ; ð4Þ with n; n0 ¼ 1; …;N, τ ¼ I; II, and all other anticommu- Strictly speaking, σ should be regarded as a discrete tators equal to zero. Then, we introduce the simplified parameter, counting the free Xons that leave the BH, notation according to the picture described above (see also the discussion in the next section). Nonetheless, in order to χ ⊗ 1 ≡ χ 1 ⊗ ≡ simplify computations, and with no real loss of generality, I n ¼ an III an; II n ¼ II bn bn: ð5Þ we use the continuous approximation. Given our initial We suppose that the initial state is state [Eq. (6)] and final state [Eq. (8)], σ can be seen as an interpolating parameter describing the evolution of the 0 ≡ 0 0 … 0 ⊗ 1 1 … 1 system from σ ¼ 0 corresponding to the beginning of j ;Ni j ; ; ; iI j ; ; ; iII; ð6Þ the evaporation till σ ¼ π=2, corresponding to complete where both kets, I and II, have N entries and evaporation. Let us also notice that the linear canonical transformation † † † j1; 1; …; 1i ¼ b b …b j0; 0; …; 0i : ð7Þ defined in Eqs. (10) and (11) is very general, given the II 1 2 N II requests. In fact, if we mix creation and annihilation opera- σ ∼ † The state in Eq. (6) represents the BH at the beginning of tors, cnð Þ ðan þ bnÞ, one cannot interpolate Eqs. (6) the evaporation process with all the slots occupied by free and (8). Furthermore, the choice of phases introduced does Xons. Although the Xons, during the evaporation, are not affect any of the results presented. This is a conse- progressively arranged into less fundamental structures quence of the fact that we are working with two types of (and hence no longer are the d.o.f to be used for the modes (BH and environment). If we had more than two, we emergent description), we keep our focus on them. For us would have to deal with one or more physical phases, as is “ ” well known in quark and neutrino physics [45]. We can thus this transmutation only helps in identifying what to call φ 0 ψ “inside” and what to call “outside” so that evaporation is safely set n ¼ ¼ n. the process that moves the Xons from II to I. In this way, the With our choice of parameters, the state (13) can also be final state (for which there are no free Xons left, as they all written as recombined to form fields and spacetime), has the form YN X † † ni 1−ni jΨðσÞi ¼ CiðσÞða Þ ðb Þ j0i ⊗ j0i ; ð14Þ jN;0i ≡ j1; 1; …; 1i ⊗ j0; 0; …; 0i ; ð8Þ i i I II I II i¼1 ni¼0;1

ni 1−ni where with Ci ¼ðsin σÞ ðcos σÞ . This form would suggest the following generalization 1 1 … 1 † †… † 0 0 … 0 j ; ; ; iI ¼ a1a2 aNj ; ; ; iI: ð9Þ YN X † † Φ σ σ ni mi 0 ⊗ 0 In order to construct a state of the system compatible j ð Þi ¼ Dið Þðai Þ ðbi Þ j iI j iII; ð15Þ 1 0 1 with the previous assumptions, let us consider the evolved i¼ ni;mi¼ ; operators as ni mi with Di ¼ðsin σÞ ðcos σÞ . However, we easily compute ψ − φ c ðσÞ¼ei n ðb cos σ þ a e i n sin σÞ; ð10Þ n n n Φ 0 0 … 0 ⊗ 1 … 1 j ð Þi¼j 1; ; NiII j 1; ; NiII ψ − φ σ i n i n σ − σ 0 … 0 ⊗ 0 … 0 dnð Þ¼e ðe an cos bn sin Þ; ð11Þ þj 1; ; NiII j 1; ; NiII; ð16Þ

106002-3 G. ACQUAVIVA, A. IORIO, and L. SMALDONE PHYS. REV. D 102, 106002 (2020) which is incompatible with our boundary condition (6). In order to enforce the latter, we need to impose the 700 constraint mi ¼ 1 − ni. 600

500 III. ENTROPY OPERATORS, PAGE CURVE, AND THE BEKENSTEIN BOUND 400 The Hilbert space of physical states has the dimension 300 200 Σ ≡ dim H ¼ 2N: ð17Þ 100 The state defined in Eq. (13) is an entangled state. This is † 0.5 1.0 1.5 due to the fact that cnðσÞ cannot be factorized as an and bn † ⊗ in Eq. (5), i.e., it cannot be written as cn ¼ AI BII, where FIG. 1. The von Neumann entropy as a function of σ in the case H H 1000 AI (BII) acts only on I ( II). N ¼ . To quantify such entanglement, we define the entropy operator for environment modes as in TFD [19,20,22] entanglement (von Neumann) entropy of the environment XN with the BH (and, of course, vice versa). This happens † 2 † 2 S ðσÞ¼− ðana ln sin σ þ a an ln cos σÞ: ð18Þ exactly when the modes have half probability to be inside I n n 1 n¼1 and half probability to be outside the BH, and then a large amount of bits are necessary to describe the system. Thus, We also define the entropy operator for BH modes in a the system has an intrinsic way to know how big is the rather unconventional way: physical Hilbert space, hence to know how big is the BH at the beginning of the evaporation: when the maximal XN † 2 † 2 entanglement is reached, that value of the entropy S S ðσÞ¼− ðb b ln cos σ þ b b ln sin σÞ: ð19Þ max II n n n n tells how big was the original BH. Hence, S must be n¼1 max some function of M0, Q0, J 0, i.e., the initial mass, charge, The reason for such unconventional definition will be clear or angular momentum of the BH. in the next section. For the moment, notice that we have This maximal entropy bound is obtained here as a mere two different operators for I and for II, but we see that since consequence of the finiteness of the fermionic fundamental d.o.f, hence of a Pauli principle. No geometric notions † 2 † hananiσ ¼ sin σ ¼ 1 − hbnbniσ; ð20Þ (distance, area, Planck length, etc.) are employed. When such notions are eventually introduced, this bound must then correspond to the Bekenstein bound. In other words, the necessary dynamical map connecting the Xons to fields S σ ≡ σ Ið Þ hSIð Þiσ and geometry will be introduced in such a way that this ¼ −Nðsin2 σ ln sin2 σ þ cos2 σ ln cos2 σÞ fundamental nongeometrical bound becomes the emergent geometrical Bekenstein bound. A brief discussion on the σ ≡ S σ ¼hSIIð Þiσ IIð Þ; ð21Þ dynamical map is offered in the last section. S S … ≡ Ψ σ … Ψ σ Therefore, for a full identification of max with BH we where h iσ h ð Þj j ð Þi. Therefore, the averages of need more than what we have here. In particular, we need the operators coincide, as it must be for a bipartite system. the concept of area that, somehow, is what has been evoked This entropy is the entanglement entropy between envi- in LQG [15,16] when in (22) one identifies ronment and BH when the system evolves. Remarkably, it has a behavior in many respects similar to that of the Page A curve [31], as shown in Fig. 1. N ≡ pffiffiffi ; ð24Þ 4πγl2 3 The maximum value is P where γ is the Immirzi parameter. We shall comment more S ¼ N ln 2 ¼ ln Σ; ð22Þ max on this later. so that We want now to bring into the picture the two missing pieces: how the entropy of the BH, that should always S Σ ¼ e max : ð23Þ decrease, and the entropy of the environment, that should

As we see here through Eq. (23) [that is the analogue of 1Recall that we have an intrinsic nongeometric notion of the Eq. (1)], in our model dim H is related to the maximal partition into inside/outside.

106002-4 BEKENSTEIN BOUND FROM THE PAULI PRINCIPLE PHYS. REV. D 102, 106002 (2020) always increase (hence, can be related to a standard thermodynamical entropy), actually evolve in our model. 700 To this end, let us introduce the following number 600 operators: 500 XN XN ˆ † ˆ † 400 NI ¼ an an; NII ¼ bn bn; ð25Þ n¼1 n¼1 300 that count the number of modes of the radiation and the 200 number of modes of the BH, respectively. Although it 100 should be clear from the above example, it is nonetheless important to stress now again that, in our formalism, the full kinematical Hilbert spaces associated to both sides have 0.5 1.0 1.5 H H 2N S fixed dimensions (dim I ¼ dim II ¼ ), while only a FIG. 2. Here we plot: BH in black, monotonically decreasing; H ⊆ H ⊗ H H 2N S S subspace I II such that dim ¼ is the one env in green, monotonically increasing; I in red with a Page- S of physical states. Note that H cannot be factorized, and like behavior. Note that the maximal value of I, Smax coincides this is the origin of BH/environment entanglement. with the initial BH entropy, as well as with the final environment 1000 Nonetheless, one could think that the physical Hilbert entropy, as inferred in the text. The plots are done for N ¼ . spaces of the two subsystems have to take into account only the number of modes truly occupied at any given stage of S S S the evaporation. Hence, the actual dimensions would be The plots of the three entropies I, BH, and env are σ σ 2NIð Þ and 2NIIð Þ, where one easily finds that shown in Fig. 2 and must be compared with similar results of Ref. [36]. There are, though, two main differences worth σ ≡ ˆ 2σ NIð Þ hNIiσ ¼ Nsin ; ð26Þ stressing. First, we have a common single origin behind all involved entropies, as explained. Second, since the overall and system here is based on the most fundamental entities, the curve for SI cannot be always below the other two, as σ ≡ ˆ S NIIð Þ hNIIiσ happens in [36], but its maximum max must reach the starting point of S (and the ending point of S ). In our ¼ N − N ðσÞ¼Ncos2σ: ð27Þ BH env I case, the inequality Recall that σ is, in fact, a discrete parameter essentially S ≤ S þ S ¼ S ; ð31Þ counting the diminishing number of free Xons (as said I BH env max earlier and shown in greater detail later). is always satisfied. Note also that the dynamics of our In other words, when we take this view, the partition system is unitary because we keep our focus on the into I and II becomes in all respects similar to the one of evaporated Xons and not on the emerging structures, as Page [31], that is was done in [1]. Hence, we are not in the position here to σ σ spot the relic entanglement between fields and spacetime 2N 2NIIð Þ 2NIð Þ ≡ ¼ × n × m; ð28Þ S S that would make the curves for BH and I end at a nonzero −1 −1 value. The latter is precisely the source of the information with n ¼ 2N; 2N ; …; 1 and m ¼ 1; …; 2N ; 2N, while σ lost in the quasiparticle picture of [1]. Whether or not this runs in discrete steps in the interval ½0; π=2. Number formally unitary evolution is physically tenable at the fluctuations, which makes it necessary to invoke the entire emergent level, and the impact of this on the validity of Hilbert space H at each stage, represent a measure of the Stone–von Neumann uniqueness theorem [20,46,47] in entanglement of these modes, as we shall see below. It is a quantum system with a finite-dimensional Hilbert space, then natural to define the Bekenstein entropy as is under scrutiny in ongoing research [48]. Recently, the relation between unitarity and the existence of a maximal S ≡ ln n ¼ N ln 2 cos2 σ; ð29Þ BH entropy has been also investigated in Ref. [49]. 2 It is worth noticing that the total entropy S þ S in and the environment entropy as BH env Eq. (31) is a constant along the evolution parameter σ:itis tempting to recognize this as a generalized second law S ≡ ln m ¼ N ln 2 sin2 σ: ð30Þ env (GSL), saturated in this case, due to the fact that we are probing the fundamental non-coarse-grained level of the 2We could also call it thermodynamical entropy in comparison Xons. However, we stress again here that σ cannot be with the nomenclature of Ref. [31]. directly related to time evolution yet and hence a

106002-5 G. ACQUAVIVA, A. IORIO, and L. SMALDONE PHYS. REV. D 102, 106002 (2020)

N comparison with a GSL can be premature. Instead, one could expect that a proper GSL should arise from the 8 coarse-grained description given by a dynamical map, something that was already hinted to in [1] due to the presence of the final residual entropy. 6 It is perhaps worthwhile to stress that Eq. (21) represents exactly a von Neumann entropy. We can write the density 4 matrix ρðσÞ¼jΨðσÞihΨðσÞj and evaluate the reduced ρ ρ ρ ρ density matrices I ¼ TrII and II ¼ TrI [50]. We can then easily check that (see Appendix) 2

S σ − ρ σ ρ σ Ið Þ¼ TrIð Ið Þ ln Ið ÞÞ 2 2 2 2 ¼ −Nðcos σ ln cos σ þ sin σ ln sin σÞ 0.5 1.0 1.5 − ρ σ ρ σ S σ ¼ TrII ð IIð Þ ln IIð ÞÞ ¼ IIð Þ: ð32Þ FIG. 3. ΔNj as a function of σ in the case N ¼ 1000.

Let us now consider some simple cases (i) If N ¼ 1, then we have We could then expect that at each step the number of BH/environment modes was fixed. What is the meaning of Ψ σ σ 0 ⊗ 1 σ 1 ⊗ 0 fluctuations of Nˆ and Nˆ on jψðσÞi? A direct computation j ð Þi ¼ cos j iI j iII þ sin j iI j iII: ð33Þ I II shows that In general, this is an entangled state whose maximal pffiffiffiffi 2σ entanglement is reached for σ ¼ π=4: Δ σ Δ σ N sinð Þ NIð Þ¼ NIIð Þ¼ 2 ; ð37Þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ψ π 4 ffiffiffi 0 ⊗ 1 1 ⊗ 0 j ð = Þi ¼ p ðj iI j iII þj iI j iIIÞ: ð34Þ Δ ≡ ˆ 2 − ˆ 2 2 where Nj hNj iσ hNjiσ is the standard deviation of Nˆ on jΨðσÞi. In agreement with the general results of (ii) For N ¼ 2 we have j Ref. [51], Fig. 3 clearly shows that this is a measure of the 2 2 entanglement. Moreover, for N ¼ 1, ðΔN ðσÞÞ is propor- jΨðσÞi ¼ cos σj0102i ⊗ j1112i j I II tional to the linear entropy or impurity 2 σ 1 1 ⊗ 0 0 þ sin j 1 2iI j 1 2iII 2 j σ σ 0 1 ⊗ 1 0 ðΔNjðσÞÞ ¼ 2S ðσÞ;j¼ I; II; ð38Þ þ cos sin j 1 2iI j 1 2iII L σ σ 1 0 ⊗ 0 1 þ cos sin j 1 2iI j 1 2iII: ð35Þ defined as [50,52]

It is then clear that the mean number (20) represents j ≡ 1 − ρ2 SL Tr j : ð39Þ the probability of the nth mode to “leave the BH ” phase (to go from II to I). Note that ΔNj can be easily discretized as explained above. As mentioned earlier, σ for us is a continuous approxi- We finally turn our attention to the generator of the mation of a discrete parameter that counts the Xons canonical transformations in (10) and (11) (with our choice transmuting from being free (in the BH, II) to being of parameters) arranged into fields and spacetime (that is what happens, −i σ G i σ G eventually, in I). Now we can formalize that statement by anðσÞ ≡ dnðσÞ¼e anð0Þe ; ð40Þ inverting Eq. (26) and getting −i σ G i σ G rffiffiffiffiffi bnðσÞ ≡ cnðσÞ¼e bnð0Þe ; ð41Þ N σðN Þ¼arcsin I: ð36Þ I N where one can easily check that

XN When N is constrained to be an integer N ¼ m, the † † I I G i a b − b a : σ N σ is discretized. Therefore, the evolution param- ¼ ð n n n nÞ ð42Þ ð IÞ¼ m 1 eter is just a way of counting how many modes jumped out. n¼ It cannot be regarded as time, which should emerge, like With the above recalled limitations, the existence of such space, at low energy from Xons dynamics. The von unitary generators is guaranteed by the Stone–von Neumann entropy as a function of σ ¼ σm is reported Neumann theorem and, in the general meaning of [16], in Fig. 1. it can be seen to enter the Wheeler–DeWitt equation

106002-6 BEKENSTEIN BOUND FROM THE PAULI PRINCIPLE PHYS. REV. D 102, 106002 (2020) P Ψ σ 0 N † † 0 ⊗ 0 Hj ð Þi ¼ ; ð43Þ with jIi¼exp ð n¼1 AnBnÞj iA j iB, and the entropy operators with H ≡ i∂σ − G. This constrains the kinematical Hilbert H ⊗ H H space I II to the physical Hilbert space as XN † 2 † 2 previously extensively commented. Let us remark that SA ¼ − ðAnAn ln sin σ þ An An ln cos σÞ; ð52Þ 1 for σ ¼ σm, Eq. (43) becomes a linear difference (recursion) n¼ equation. XN † † S ¼ − ðB B ln sin2 σ þ B B ln cos2 σÞ: ð53Þ IV. CONNECTION WITH DISSIPATIVE B n n n n n¼1 SYSTEMS Therefore, through (44),wehavenowtheusual entropy In the previous section we have shown how our toy operators of TFD [19,20,22] to be compared with the model possesses a Page-like behavior for entanglement unusual definitions of (19): S ¼ S and S ¼ S . entropy and this can be easily computed by means of the I A II B The physical picture here is that, when the system TFD entropy operator. We now ask if this is a mere evolves, a pair of A and B particles is created. The coincidence or if the connection with TFD can be made B-modes enter into the BH, annihilating BH modes, while more precise. the A-modes form the environment. This mechanism is Let us perform the canonical transformation heuristically the same as the one proposed by Hawking [18] † and, lately, formalized via the tunneling effect [56]. A ¼ a ;B¼ bn: ð44Þ n n n The A- and B-modes do not discern explicitly fields and This is not a special Bogoliubov transformation [53].In geometric d.o.f. However, taking the view of Ref. [1], some fact, this transformation can be obtained from d.o.f are indeed responsible for the reduction of the BH’s horizon area during the evaporation and annihilators of † An ¼ an cos θn − bn sin θn; ð45Þ geometric modes can be defined. In order to make this idea more precise, we can decompose An in their geometric (G) † Bn ¼ bn cos θn þ an sin θn; ð46Þ and field (F) parts as follows: X for θ ¼ 0. Then it is not connected with the identity. k ⊗ 1 1 ⊗ k n An ¼ ðgk;n AG;n IF;n þ fk;n IG;n AF;nÞ; ð54Þ However, we can still define vacua in the new representation k 0 0 0 Anj iA ¼ Bnj iB ¼ : ð47Þ where k labels the emergent modes. Equation (54) can be regarded as a dynamical (Haag) map at linear order [20]. One can check that The full dynamical map—available once the quantum theory of gravity is specified—should connect the 0 0 j iA ¼j iI; ð48Þ fundamental d.o.f to the emergent notions of geometry and fields. The coefficients of the map should then lead to 0 1 1 …1 j iB ¼j 1 2 NiII: ð49Þ the Hawking thermal behavior of the latter at the emergent † † 2 level. Note that the action of A on j0i , creates both a The second relation follows from the fact that ðb Þ ¼ 0. n A n matter mode and a geometric mode outside the horizon: Therefore, the generator (42) becomes the region of spacetime surrounding the BH and available XN to an external observer increases because the horizon area † † G ¼ i ðAnBn − BnAnÞ: ð50Þ decreases. n¼1 Let us remark that, in this picture, a relationship of the form (54) makes sense only for A-modes (or equivalently This is nothing but the (fermionic version of the) interac- a-modes) and not for B-modes (or equivalently b-modes). tion Hamiltonian of quantum dissipative systems, as intro- In fact, by definition, the former are the ones which ducedinRef.[54] (see also [55]). This operator noticeably rearrange to form matter and geometry, in contrast with coincides with the generator of a special Bogoliubov the latter which describes free Xons. As previously stated, Ψ σ transformation. Therefore, j ð Þi has a TFD-vacuum-like such a distinction is at the basis of our intrinsic notion of structure [19,20] interior/exterior of the BH. A more detailed analysis of this issue will be addressed in a forthcoming work. YN Ψ σ σ σ † † 0 ⊗ 0 It is only once we have the notion of area that we could j ð Þi ¼ ðcos þ sin AnBnÞj iA j iB try to fix N in terms of the BH parameters M0, Q0, J 0, n¼1 1 and of the Planck length l . In fact, as remarked in − S ðσÞ P ¼ e 2 A jIi; ð51Þ Refs. [57,58], a quantum of BH horizon area measures

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Δ α 2 A ¼ lP; ð55Þ Through such an operator, we have evaluated the von Neumann BH–environment entropy and noticeably where α is a constant to be fixed. Therefore, the BH entropy 3 obtained a Page-like evolution. assumes the form We finally have shown that the latter is a consequence of α a duality between our model and a dissipativelike fermionic S N BH ¼ 4 ; ð56Þ quantum system, and hence it has a natural TFD-like description. where N is the number of Planck cells that, as remarked in Many directions for further research need be thoroughly the Introduction, would correspond to the number of our explored, the most important being a reliable dynamical quantum levels/slots. map from the fundamental modes to the emergent fields/ The value of α was fixed to α ¼ 4 ln 3 in Ref. [57],by spacetime structures. This would pave the road for the means of arguments based on Bohr’s correspondence investigation of important aspects of the emergent picture, principle, and to α ¼ 8π in Ref. [58], by means of argu- such as the Hawking thermality of the field sector. Moreover, ments based on the BH quasinormal modes. In our case, a a geometric interpretation of the results presented here would comparison between (56) and (22) gives allow one to investigate the discreteness of the spectrum of quasinormal modes, which is known to be related to the α ¼ 4 ln 2: ð57Þ quantization of the BH horizon’s area (see Ref. [58]). Anyway, even in the absence of a full dynamical map, α 4 This value agrees with the condition ¼ ln k with k we believe that our simple, although nontrivial, consider- integer, which was proposed in Ref. [57],inordertoconstrain ations are necessary to fully take into account the fascinating Σ the number of microstates to an integer [see Eq. (23)]. and far-reaching consequences of the Bekenstein bound. Therefore, our model gives k ¼ 2. A comparison with the Q 0 J standard Bekenstein formula, when 0 ¼ ¼ 0,gives ACKNOWLEDGMENTS 2 4πM0 The authors would like to thank G. Vitiello, P. Lukeš,D. N ¼ 2 2 ; ð58Þ lP ln Lanteri, and L. Buoninfante for enlightening discussions. The authors profoundly thank Martin Scholtz, who sadly which, of course, is again Eq. (24) when the area is expressed passed away, for his continuous inspiration. A. I. and L. S. in terms of M0 and the Immirzi parameter is fixed to pffiffiffi are partially supported by UNiversity CEntre (UNCE) ln 2=ð 3πÞ. In fact, both derivations—ours and LQG’s— of Charles University (Prague, Czech Republic) Grant rely on the identification of the entropy with SBH.Clearly,a No. UNCE/SCI/013. more detailed analysis, beyond these qualitative arguments, will be possible only once a complete dynamical map will be APPENDIX: EQUIVALENCE OF TFD ENTROPY available. AND VON NEUMANN ENTROPY

V. CONCLUSIONS In this Appendix we explicitly show the equivalence of the TFD entropy of Eq. (21) with the von Neumann entropy We assumed here that the d.o.f of a BH are finite in of Eq. (32). We first present the computation in the simplest number and fermionic in nature and, hence obey a Pauli cases of Eqs. (33) and (35), and then we shall focus on the principle. Then, within the approach of second quantiza- computation of the von Neumann entropy. Let us note that tion, we naturally obtained that the BH evaporation is a the expectation value of the TFD entropy operators [see dynamical mechanism producing a maximal entanglement Eq. (21)] immediately follows from Eq. (20). entropy equal to the initial entropy of the BH. This (i) In the case N ¼ 1, the density matrix reads phenomenon is an instance of the Bekenstein bound, obtained here with arguments that do not assume preexist- ρ σ 2σ 0 1 1 0 2σ 1 0 0 1 ð Þ¼cos j iIj iIIIIh jIh jþsin j iIj iIIIIh jIh j ing spatiotemporal concepts. Of course, for a full identi- sinð2σÞ fication with the standard formulae (see, e.g., [7,8]), one 0 1 0 1 1 0 1 0 þ 2 ðj iIj iIIIIh jIh jþj iIj iIIIIh jIh jÞ; needs to link geometrical concepts, such as elementary Planck cells, to such fundamental d.o.f. ðA1Þ We then showed that entanglement, Bekenstein, and environment (thermodynamic) entropies here are all natu- where we omitted tensor product symbols. The rally obtained in the same approach, based on an entropy reduced density matrices have the following form: operator whose structure is the one typical of TFD. ρ σ 2σ 0 0 2σ 1 1 Ið Þ¼cos j iIIh jþsin j iIIh j; ðA2Þ 3 S This is equal to BH, defined in Eq. (29), only before the ρ σ 2σ 1 1 2σ 0 0 S S IIð Þ¼cos j i h jþsin j i h j: ðA3Þ evaporation when BH ¼ max. IIII IIII

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Eq. (32) follows immediately from these ex- where jnAiA, jnBi are eigenstates of the number operators pressions. 2 (ii) In the case N ¼ , we directly report the reduced XN XN ˆ † ˆ † density matrices: NA ¼ AnAn; NB ¼ BnBn: ðA9Þ n¼1 n¼1 ρ σ 4σ 00 00 4σ 11 11 Ið Þ¼cos j iIIh jþsin j iIIh j 22σ Moreover, the coefficients w are given by sin 01 01 10 10 n þ 4 ðj iIIh jþj iIIh jÞ; ðA4Þ YN ρ σ 4σ 11 11 4σ 00 00 σ 2 σ IIð Þ¼cos j iIIIIh jþsin j iIIIIh j wnð Þ¼ Cj ð Þ; ðA10Þ 1 sin2ð2σÞ j¼ 01 01 10 10 : þ 4 ðj iIIIIh jþj iIIIIh jÞ and C were first introduced in Eq. (14). The density matrix ðA5Þ j thus reads It follows that X ρðσÞ¼ w ðσÞjni jni hnj hnj; ðA11Þ S σ − 4 σ 4 σ − 4 σ 4 σ n A BB A Ið Þ¼ sin ln sin cos ln cos n¼0;1 − 2 sin2 σ cos2 σ ln ðsin2 σ cos2 σÞ S σ from which the reduced density matrices are easily derived: ¼ IIð Þ: ðA6Þ X ρ σ σ By using the relations lnðabÞ¼ln a þ ln b and Að Þ¼ wnð ÞjniAAhnjðA12Þ cos2σ þ sin2σ ¼ 1, we get n¼0;1

S σ −2 2 σ 2 σ 2 σ 2 σ X Ið Þ¼ ðsin ln sin þ cos ln cos Þ ρ ðσÞ¼ w ðσÞjni hnj: ðA13Þ S σ B n BB ¼ IIð Þ; ðA7Þ n¼0;1

which is equal to Eq. (32) for N ¼ 2. The von Neumann entropy reads [19,22,27] One could repeat similar computations for all N. However, it is simpler to use the correspondence of our model with a X TFD/dissipative system via Eq. (44). As already known in SAðσÞ¼SBðσÞ¼− wnðσÞ ln wnðσÞ: ðA14Þ TFD, the “thermal vacuum” can be rewritten in the form n¼0;1 X pffiffiffiffiffiffiffiffiffiffiffiffi Ψ σ σ j ð Þi ¼ wnð Þ jniAjniB; ðA8Þ Finally, Eq. (32) follows from substituting the explicit form n¼0;1 of wn [cf. Eq. (A10)] into the last expression.

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