Bekenstein Bound from the Pauli Principle∗

Bekenstein bound from the Pauli principle∗

G. Acquaviva,1, † A. Iorio,1, ‡ and L. Smaldone1, § 1Faculty of Mathematics and Physics, Charles University, V Holeˇsoviˇckách2, 18000 Praha 8, Czech Republic. 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 dissipative-like fermionic system.

I. INTRODUCTION identification with the standard formulae (see [7], where the bound is rigorously defined in quantum field theory This paper moves from the results of previous research (QFT) and, e.g., the review [8]) one needs to associate ge- [1], but reversing the point of view. There, Bekenstein’s ometrical concepts to the Xons. For instance, one could argument that a black hole (BH) reaches the maximal make each dof correspond to one elementary Planck cell. entropy at disposal of a physical system (i.e., that it sat- Nonetheless, in our picture we do not need the exact ex- urates the Bekenstein bound [2]), leads to two main pro- pression of the bound. What is crucial is that the Xons posals: i) the degrees of freedom (dof) responsible for are taken to be finite in number and fermionic, other- the BH entropy have to take into account both matter wise the entanglement entropy would just indefinitely and spacetime and hence must be of a new, more funda- grow without reaching a maximal value. It is sugges- mental nature than the dof we know (with Feynman [3], tive, though, that taking on board the geometric picture here we call such dof “Xons”, see also [4] and [5]); ii) the of Xons as quanta of area (Planck cells), the horizon of Hilbert space H of the Xons of a given BH is necessarily the BH is of nonzero size as an effect of a Pauli exclusion finite dimensional principle. Before entering the details of what just discussed, let us now briefly put our work into the context dimH = eSBH , (1) of current literature. Bekenstein entropy [9, 10] is traditionally regarded as a with SBH the Bekenstein entropy. With these, in [1] it measure of our ignorance about the dof which formed the was shown that the (average) loss of information is an un- BH [10–13] and as a consequence of the no-hair theorem avoidable consequence of the non-vanishing relic entan- [14]. However, other interpretations have been proposed glement between the evaporated matter and the space- in literature, as in Loop Quantum Gravity (LQG), where time. BH entropy is a counting of microstates corresponding In search of a unifying view of the various types of en- to a given macroscopic horizon area A [15, 16]. Along tropies involved in the BH evaporation (i.e., Bekenstein, these lines, Bekenstein proposed a universal upper bound thermodynamical, and entanglement entropies, see, e.g., for the entropy of any physical system contained in a [6]), here we reverse that logic. Namely, we start off by finite region [2], which is saturated by BHs. This implies supposing that in a BH only free Xons exist (hence there [17] that the entropy of every system in a finite region is can only be one kind of entropy at that level), and we bounded from above by the Bekenstein entropy of a BH, suppose that they are finite in number and fermionic in whose horizon coincides with the area of the boundary of arXiv:2005.13973v2 [hep-th] 5 Nov 2020 nature. This amounts to have a finite dimensional H. that region (see [7], and also [8]). With these assumptions, here we show that the evapo- Using the approach of QFT in curved spacetimes, ration is a dynamical mechanism producing a maximal Hawking discovered the black body spectrum of BH radi- entanglement entropy, equal to the initial entropy of the ation [18]. In the meantime, Umezawa and Takahashi de- BH. veloped their Thermofield dynamics (TFD) [19] (see also This is an instance of the Bekenstein bound, obtained Ref. [20]), that immediately appeared to be a fruitful tool here with arguments that do not assume pre-existing ge- to describe BH evaporation [21]. In [22], with the help of ometrical (spatiotemporal) concepts. In fact, for a full an entropy operator, whose structure is natural in TFD, the BH-radiation entropy is viewed as an entanglement entropy of radiation modes with their “TFD-double” (the modes beyond the horizon). ∗To our friend, colleague and inspirer Martin Scholtz (1984 - 2019). †Electronic address: [email protected]ff.cuni.cz Although the relation between QFT in curved space- ‡Electronic address: [email protected]ff.cuni.cz time and TFD was studied already in Refs. [23, 24], the §Electronic address: [email protected]ff.cuni.cz renewed interest comes in connection with the AdS/CFT 2 correspondence [25], where in a two-sided Anti-de Sit- requirement that the number of slots available were finite ter (AdS) BH, the specular asymptotic region is mapped would not have been sufficient to guarantee the finiteness into two copies of a conformal field theory (CFT). The of dim H. Let us recall now that, in the picture of [1], it thermal nature of the BH is then naturally seen through is only at energy scales below those of quantum gravity TFD. Extensions to incorporate dissipative effects are in (e.g., at the energy scales of ordinary matter) that the the recent [26, 27]. field modes are distinguishable from those “making” the Since a BH, initially described as a pure state, could spacetime, hence we can write end up in a mixed state (this is actually the view of [1]), H ⊗ H ⊆ H . (2) questions arise on the unitary evolution, as first noticed F G by Hawking [28] and then extensively discussed, from Here F and G stand for “fields” and “geometry”, re- different points of view, see e.g. Refs. [29–39]. In par- spectively. In other words, at low energy, the F -modes ticular, in Refs. [31, 40] Page studied the bipartite sys- will form quantum fields excitations, that is, the quasi- tem BH-radiation, in a random (Haar distributed) pure particles (from the Xons point of view) immersed into state, computing the radiation entanglement entropy as the spacetime formed by the G-modes. function of the associated thermodynamical entropy. He Now, say N is the total number of quantum levels found a symmetric curve (Page curve) which goes back (slots) available to the BH. The evaporation consists of to zero when the BH is completely evaporated. In Ref. the following, steady process: N → (N −1) → (N −2) → [36] he postulated that entanglement entropy, as function ··· . That is, the number of free Xons steadily decreases, of time, follows the minimum between Bekenstein and in favor of the Xons that, having evaporated, are ar- radiation thermodynamic entropy (Conjectured Anorexic ranged into quasi-particles and the spacetime they live Triangular Hypothesis). Recently [41], Page curve was in. One might think of a counter that only sees free also derived from holographic computations [42]. Xons, hence keeps clicking in one direction as the BH As said, in this paper we reverse the line of reasoning of evaporates, till its complete stop. Ref. [1] and present a simple, purely quantum toy-model In this picture: i) there is no pre-existing time, because of the dynamics of BH evaporation, focusing on the fun- the natural evolution parameter is the average number damental dof. In Section II the basic assumptions are of free Xons; and ii) there is no pre-existing space to the finiteness of slots (quantum levels) available for the define the regions inside and outside the BH, because a system, and the fermionic nature of such dof. The finite- distinction of the total system into two systems, say en- ness of the Hilbert space of states follows from the Pauli vironment (I) and BH (II), naturally emerges in the way exclusion principle. In Section III we compute the von just depicted. With this in mind, in what follows we shall Neumann entropy of the subsystems during their evolu- nonetheless refer to the system I as outside, and to system tion. This is remarkably given by the expectation value of II as inside. It is a worthy remark that other authors do the TFD entropy operator [22] and it has the same qual- use the geometric notion of exterior and interior of BH, itative behavior of Page curve: it starts from zero and even at fundamental level [29]. Even though this can be ends in zero, while its maximum is reached at half of the justified, see, e.g., [4], and permits to produce meaning- evaporation process. That maximum is identified here ful models, see, e.g., [39], our approach does not require with the Bekenstein entropy of the BH at the beginning it. The Hilbert space of physical states is then built as a of the evaporation. We can therefore argue that Beken- subspace of a larger tensor product (kinematical) Hilbert stein bound itself descends from the Pauli principle. In space Section IV we explain the relation with TFD by mapping H ⊆ H ⊗ H . (3) our model to an equivalent description as a dissipative- I II like system. The last Section is left to our conclusions, We now assume that such a Hilbert space can be con- while in the Appendix we show the connection between structed with the methods of second quantization. This TFD and von Neumann entropies in the present context. provides a language contiguous to the language of QFT, Throughout the paper we adopt units in which c = which should be recovered in some limit. Therefore, BH ~ = 1. and environment modes will be described by two sets of creation and annihilation operators, which satisfy the usual canonical anticommutation relations II. BASIC ASSUMPTIONS AND MODEL OF BH n † o EVAPORATION χτn, χτ 0n0 = δτ,τ 0 δnn0 , (4) with n, n0 = 1,...,N, τ = I, II, and all other anticom- We assume that the fundamental dof are fermionic (BH mutators equal to zero. Then, we introduce the simplified and models based on fermions are available in literature, notation see, e.g., the SYK model [43, 44]). As a consequence, each quantum level can be filled by no more than one χI n = an ⊗ 1III ≡ an , χII n = 1II ⊗ bn ≡ bn . (5) fermion. This assures that the Hilbert space H of phys- We suppose that the initial state is ical states with a finite number of levels is finite dimensional. In fact, if the fundamental modes were bosons, the |0,Ni ≡ |0, 0,..., 0iI ⊗ |1, 1,..., 1iII , (6) 3 where both kets, I and II, have N entries and and (8). Furthermore, the choice of phases introduced does not affect any of the results presented. This is a † † † |1, 1,..., 1iII = b1b2 . . . bN |0, 0,..., 0iII . (7) consequence of the fact that we are working with two types of modes (BH and environment). If we had more The state in Eq.(6) represents the BH at the beginning than two, we would have to deal with one or more physi- of the evaporation process, with all the slots occupied by cal phases, as is well known in quark and neutrino physics free Xons. Although the Xons, during the evaporation, [45]. We can thus safely set ϕn = 0 = ψn. are progressively arranged into less fundamental struc- With our choice of parameters, the state (13), can also tures (and hence no longer are the dof to be used for the be written as emergent description) we keep our focus on them. For us this “transmutation” only helps identifying what to call N n 1−n Y X † i † i “inside” and what “outside”, so that evaporation is the |Ψ(σ)i = Ci(σ) ai bi |0iI ⊗ |0iII , process that moves the Xons from II to I. In this way, i=1 ni=0,1 the final state (for which there are no free Xons left, as (14) ni 1−ni they all recombined to form fields and spacetime), has with Ci = (sin σ) (cos σ) . This form would suggest the form the following generalization

N n m |N, 0i ≡ |1, 1,..., 1iI ⊗ |0, 0,..., 0iII , (8) Y X † i † i |Φ(σ)i = Di(σ) ai bi |0iI ⊗|0iII , where i=1 ni,mi=0,1 (15) † † † ni mi |1, 1,..., 1iI = a1a2 . . . aN |0, 0,..., 0iI . (9) with Di = (sin σ) (cos σ) . However, we easily compute In order to construct a state of the system, compat- ible with the previous assumptions, let us consider the |Φ(0)i = |01,..., 0N iII ⊗ |11,..., 1N iII evolved operators as

+ |01,..., 0N iII ⊗ |01,..., 0N iII . (16) iψn −iϕn cn(σ) = e bn cos σ + an e sin σ , (10) which is incompatible with our boundary condition (6). iψn −iϕn dn(σ) = e e an cos σ − bn sin σ , (11) In order to enforce the latter, we need to impose the constraint mi = 1 − ni. where on σ we shall comment soon. Eqs.(10) and (11) deﬁne a canonical transformation

† † III. ENTROPY OPERATORS, PAGE CURVE cn(σ) , cm(σ) = dn(σ) , dm(σ) = δnm . (12) AND THE BEKENSTEIN BOUND We thus get the evolved of the initial state (6) as The Hilbert space of physical states has dimension N Y † Σ ≡ dim H = 2N . (17) |Ψ(σ)i ≡ cn(σ) |0iI ⊗ |0iII (13) n=1 The state defined in Eq.(13) is an entangled state. This is N † Y −iψ † † iϕ due to the fact that cn(σ) cannot be factorized as an and n n † = e bn cos σ + an e sin σ |0iI ⊗ |0iII . bn in Eq.(5), i.e. it cannot be written as cn = AI ⊗ BII , n=1 where AI (BII ) acts only on HI (HII ). Strictly speaking, σ should be regarded as a discrete pa- To quantify such entanglement we define the entropy rameter, counting the free Xons that leave the BH, ac- operator for environment modes as in TFD [19, 20, 22] cording to the picture above described (see also the dis- N cussion in the next Section). Nonetheless, in order to X S (σ) = − a† a ln sin2 σ + a a† ln cos2 σ , simplify computations, and with no real loss of gener- I n n n n n=1 ality, we use the continuous approximation. Given our (18) initial state (Eq.(6)) and final state (Eq.(8)), σ can be We also define the entropy operator for BH modes, in a seen as an interpolating parameter, describing the evo- rather unconventional way lution of the system, from σ = 0, corresponding to the beginning of the evaporation, till σ = π/2, corresponding N to complete evaporation. X † 2 † 2 SII (σ) = − bn bn ln cos σ + bn bn ln sin σ . Let us also notice that the linear canonical transfor- n=1 mation defined in Eqs.(10),(11) is very general, given the (19) requests. In fact, if we mix creation and annihilation op- The reason for such unconventional definition will be † erators, cn(σ) ∼ (an +bn), one cannot interpolate Eqs.(6) clear in the next Section. For the moment, notice that 4 we have two different operators, for I and for II, but we maximal entanglement is reached, that value of the en- see that, since tropy, Smax, tells how big was the original BH. Hence Smax must be some function of M0, Q0, J0, i.e., the ini- † 2 † han aniσ = sin σ = 1 − hbn bniσ , (20) tial mass, charge or angular momentum of the BH. This maximal entropy bound is obtained here as a mere then consequence of the finiteness of the fermionic fundamental dof, hence of a Pauli principle. No geometric notions S (σ) ≡ hS (σ)i I I σ (distance, area, Planck length, etc) are employed. When such notions are eventually introduced, this bound must = −N sin2 σ ln sin2 σ + cos2 σ ln cos2 σ correspond to the Bekenstein bound. In other words, the necessary dynamical map connecting the Xons to fields = hSII (σ)iσ ≡ SII (σ) , (21) and geometry will be introduced in such a way that this where h...iσ ≡ hΨ(σ)| ... |Ψ(σ)i. Therefore the averages fundamental, non-geometrical bound becomes the emer- of the operators coincide, as it must be for a bipartite sys- gent, geometrical Bekenstein bound. A brief discussion tem. This entropy is the entanglement entropy between on the dynamical map is offered in the last section. environment and BH, when the system evolves. Remark- Therefore, for a full identification of Smax with SBH ably, it has a behavior in many respect similar to that of we need more than what we have here. In particular, we the Page curve [31], as shown in Fig. 1. need the concept of area, that somehow is what has been evoked in LQG [15, 16] when in (22) one identifies S ⅈ A 700 N ≡ √ , (24) 2 4πγlP 3 600 where γ is the Immirzi parameter. We shall comment 500 more on this later. We want now to bring into the picture the two missing 400 pieces: how the entropy of the BH, that should always de- 300 crease, and the entropy of the environment, that should always increase (hence, can be related to a standard ther- 200 modynamical entropy), actually evolve in our model. To 100 this end, let us introduce the following number operators

σ N N 0.5 1.0 1.5 ˆ X † ˆ X † NI = an an , NII = bn bn , (25) FIG. 1: The von Neumann entropy as a function of σ, in the n=1 n=1 case N = 1000. that count the number of modes of the radiation and the number of modes of the BH, respectively. Although it The maximum value is should be clear from the above, it is nonetheless important to stress now again that, in our formalism, the full Smax = N ln 2 = ln Σ , (22) kinematical Hilbert spaces associated to both sides have N ﬁxed dimension (dim HI = dim HII = 2 ), while only a so that N subspace H ⊆ HI ⊗ HII such that dim H = 2 is the one of physical states. Note that H cannot be factorized and Smax Σ = e . (23) this is the origin of BH/environment entanglement. Nonetheless, one could think that the physical Hilbert As we see here through Eq. (23) (that is the analogue of spaces of the two subsystems have to take into account Eq.(1)), in our model dim H is related to the maximal en- only the number of modes truly occupied, at any given tanglement (von Neumann) entropy of the environment stage of the evaporation. Hence, the actual dimensions with the BH (and, of course, viceversa). This happens N (σ) N (σ) would be 2 I , and 2 II , where one easily ﬁnds that exactly when the modes have half probability to be inside and half probability to be outside the BH1, and then ˆ 2 NI (σ) ≡ hNI iσ = N sin σ , (26) a large amount of bits are necessary to describe the system. Thus, the system has an intrinsic way to know how and big is the physical Hilbert space, hence to know how big N (σ) ≡ hNˆ i is the BH at the beginning of the evaporation: when the II II σ 2 = N − NI (σ) = N cos σ . (27) Recall that σ is, in fact, a discrete parameter, essentially 1 Recall that we have an intrinsic, non-geometric notion of the counting the diminishing number of free Xons (as said partition into inside/outside. earlier, and shown in greater detail later). 5

In other words, when we take this view, the partition system is unitary, because we keep our focus on the evap- into I and II becomes in all respects similar to the one of orated Xons, and not on the emerging structures, as was Page [31], that is done in [1]. Hence we are not in the position here to spot

N N (σ) N (σ) the relic entanglement between fields and spacetime, that 2 = 2 II × 2 I ≡ n × m , (28) would make the curves for SBH and SI end at a nonzero with n = 2N , 2N−1,..., 1, and m = 1,..., 2N−1, 2N , value. The latter is precisely the source of the informa- while σ runs in discrete steps in the interval [0, π/2]. tion loss in the quasi-particle picture of [1]. Whether Number fluctuations, which makes it necessary to invoke or not this formally unitary evolution is physically ten- the entire Hilbert space H at each stage, represent a mea- able at the emergent level, and the impact of this on the sure of entanglement of these modes, as we shall see be- validity of the Stone-von Neumann uniqueness theorem low. It is then natural to define the Bekenstein entropy [20, 46, 47] in a quantum system with a finite-dimensional as Hilbert space, is under scrutiny in ongoing research [48]. Recently, the relation between unitarity and the existence 2 SBH ≡ ln n = N ln 2 cos σ , (29) of a maximal entropy has been also investigated in Ref. [49]. and the environment entropy2 as It is worth noticing that the total entropy SBH + Senv 2 Senv ≡ ln m = N ln 2 sin σ . (30) in Eq.(31) is a constant along the evolution parameter σ: it is tempting to recognize this as a generalized second The plots of the three entropies, SI , SBH , Senv are law (GSL), saturated in this case, due to the fact that we shown in Fig. 2, and must be compared with similar are probing the fundamental, non-coarse-grained level of results of Ref. [36]. There are, though, two main differ- the Xons. However we stress again here that σ cannot be ences worth stressing. First, we have a common single directly related to time evolution yet and hence a com- origin behind all involved entropies, as explained. Sec- parison with a GSL can be premature. Instead, one could ond, since the overall system here is based on the most expect that a proper GSL should arise from the coarse- fundamental entities, the curve for SI cannot be always grained description given by a dynamical map, something below the other two, as happens in [36], but its maxi- that was already hinted to in [1] due to the presence of mum Smax must reach the starting point of SBH (and the final residual entropy. the ending point of Senv). In our case, the inequality It is perhaps worthwhile to stress that Eq.(21) represents exactly a von Neumann entropy. We can write the SI ≤ SBH + Senv = Smax , (31) density matrix ρ(σ) = |Ψ(σ)ihΨ(σ)| and evaluate the is always satisfied. Note also that the dynamics of our reduced density matrices ρI = TrII ρ and ρII = TrI ρ [50]. We can then easily check that (see Appendix A) S

700 SI (σ) = −TrI (ρI (σ) ln ρI (σ)) 2 2 2 2 600 = −N cos σ ln cos σ + sin σ ln sin σ

500 = −TrII (ρII (σ) ln ρII (σ)) = SII (σ) . (32)

400 Let us now consider some simple cases • If N = 1 we have 300

|Ψ(σ)i = cos σ|0iI ⊗ |1iII + sin σ|1iI ⊗ |0iII . (33) 200 In general, this is an entangled state, whose maxi- 100 mal entanglement is reached for σ = π/4: 1 σ |Ψ(π/4)i = √ (|0i ⊗ |1i + |1i ⊗ |0i ) . (34) 0.5 1.0 1.5 2 I II I II • For N = 2 we have FIG. 2: Here we plot: SBH in black, monotonically decreas- |Ψ(σ)i = cos2 σ |0 0 i ⊗ |1 1 i ing; Senv in green, monotonically increasing; SI in red, with a 1 2 I 1 2 II Page-like behavior. Note that the maximal value of SI , Smax, 2 coincides with the initial BH entropy, as well as with the ﬁnal + sin σ |11 12iI ⊗ |01 02iII environment entropy, as inferred in the text. The plots are done for N = 1000. + cos σ sin σ |01 12iI ⊗ |11 02iII

+ cos σ sin σ |11 02iI ⊗ |01 12iII . (35) It is then clear that the mean number (20) repre- 2 We could also call it thermodynamical entropy, in comparison sents the probability of the n-th mode to “leave the with the nomenclature of Ref. [31]. BH phase” (to go from II to I). 6

As mentioned earlier, σ for us is a continuous approx- ΔNj imation of a discrete parameter, that counts the Xons 8 transmuting from being free (in the BH, II) to being arranged into ﬁelds and spacetime (that is what happens, eventually, in I). Now we can formalize that statement, 6 by inverting Eq. (26) and getting r NI σ(N ) = arcsin . (36) 4 I N

When NI is constrained to be an integer NI = m, the

σ(NI ) = σm is discretized. Therefore, the evolution 2 parameter is just a way of counting how many modes jumped out. It cannot be regarded as time, which should emerge, like space, at low energy from Xons dynamics. σ 0.5 1.0 1.5 The von Neumann entropy as a function of σ = σm is reported in Fig.1. We could then expect that at each step the number of BH/environment modes was fixed. What is the mean- FIG. 3: ∆Nj as a function of σ, in the case N = 1000. ˆ ˆ ing of fluctuations of NI and NII on |ψ(σ)i? A direct computation shows that √ IV. CONNECTION WITH DISSIPATIVE N sin(2σ) SYSTEMS ∆N (σ) = ∆N (σ) = , (37) I II 2 q In the previous Section we have shown how our toy ˆ 2 ˆ 2 where ∆Nj ≡ hNj iσ − hNjiσ is the standard deviation model possesses a Page-like behavior for entanglement entropy and this can be easily computed by means of of Nˆj on |Ψ(σ)i. In agreement with the general results of Ref. [51], Fig. 3 clearly shows that this is a measure the TFD entropy operator. We now ask if this is a mere 2 coincidence or if the connection with TFD can be made of the entanglement. Moreover, for N = 1, (∆N (σ)) is j more precise. proportional to the linear entropy or impurity Let us perform the canonical transformation 2 j (∆Nj(σ)) = 2 S (σ) , j = I, II , (38) L † An = an ,Bn = bn . (44) defined as [50, 52]

j 2 This is not a Special Bogoliubov transformation [53]. In SL ≡ 1 − Trρj . (39) fact, this transformation can be obtained from

Note that ∆Nj can be easily discretized as explained † above. We finally turn our attention to the generator An = an cos θn − bn sin θn , (45) of the canonical transformations in (10) and (11) (with B = b† cos θ + a sin θ , (46) our choice of parameters) n n n n n −i σ G i σ G for θ = 0. Then it is not connected with the identity. an(σ) ≡ dn(σ) = e an(0) e , (40) n However, we can still define vacua in the new represen- −i σ G i σ G bn(σ) ≡ cn(σ) = e bn(0) e , (41) tation where one can easily check that An |0iA = Bn |0iB = 0 . (47) N X † † One can check that G = i an bn − bn an . (42) n=1 |0iA = |0iI , (48) With the above recalled limitations, the existence of such unitary generator is guaranteed by the Stone–von Neu- |0iB = |11 12 ... 1N iII . (49) mann theorem and, in the general meaning of [16], it can † 2 be seen to enter the Wheeler–DeWitt equation The second relation follows from the fact that (bn) = 0. Therefore, the generator (42) becomes H |Ψ(σ)i = 0 , (43) N with H ≡ i ∂σ − G. This constrains the kinematical X † † G = i An Bn − Bn An . (50) Hilbert space HI ⊗HII to the physical Hilbert space H as n=1 previously extensively commented. Let us remark that for σ = σm, Eq. (43) becomes a linear difference (recur- This is nothing but the (fermionic version of the) inter- sion) equation. action Hamiltonian of quantum dissipative systems, as 7 introduced in Ref. [54] (see also [55]). This operator stated, such a distinction is at the basis of our intrinsic noticeably coincides with the generator of a Special Bo- notion of interior/exterior of the BH. A more detailed goliubov transformation. Therefore |Ψ(σ)i has a TFD- analysis of this issue will be addressed in a forthcoming vacuum-like structure [19, 20] work. It is only once we have the notion of area that we could N Y try to fix N in terms of the BH parameters, M0, Q0, J0, |Ψ(σ)i = cos σ + sin σ A† B† |0i ⊗ |0i n n A B and of the Planck length lP . In fact, as remarked in Refs. n=1 [57, 58], a quantum of BH horizon area measures − 1 S (σ) = e 2 A |Ii , (51) 2 ∆A = α lP , (55)

PN † † with |Ii = exp n=1 An Bn |0iA ⊗ |0iB, and the where α is a constant to be ﬁxed. Therefore, the BH entropy operators entropy assumes the form3

N α N X † 2 † 2 SBH = , (56) SA = − An An ln sin σ + An An ln cos σ (52), 4 n=1 where N is the number of Planck cells that, as remarked N in the Introduction, would correspond to the number of X † 2 † 2 SB = − Bn Bn ln sin σ + Bn Bn ln cos σ (53). our quantum levels/slots. n=1 The value of α was fixed to α = 4 ln 3, in Ref. [57], by means of arguments based on Bohr’s correspondence Therefore, through (44), we have now the usual entropy principle, and to α = 8π in Ref. [58], by means of ar- operators of TFD [19, 20, 22], to be compared with the guments based on the BH quasi-normal modes. In our unusual definitions of (19): S = S and S = S . I A II B case, a comparison between (56) and (22) gives The physical picture here is that, when the system evolves, a pair of A and B particles is created. The B- α = 4 ln 2 . (57) modes enter into the BH, annihilating BH modes, while the A-modes form the environment. This mechanism is This value agrees with the condition α = 4 ln k, with heuristically the same as the one proposed by Hawking k integer, which was proposed in Ref. [57], in order [18], and lately formalized via the tunneling effect [56]. to constrain the number of microstates Σ to an integer The A− and B−modes do not discern explicitly fields (see Eq. (23)). Therefore, our model gives k = 2. A and geometric dof. However, taking the view of Ref. [1], comparison with the standard Bekenstein formula, when some dof are indeed responsible for the reduction of the Q0 = 0 = J0, gives BH’s horizon area during the evaporation and annihila- 2 tors of geometric modes can be defined. In order to make 4 π M0 N = 2 , (58) this idea more precise we can decompose An in their ge- lP ln 2 ometric (G) and field (F ) parts as follows which, of course, is again Eq.(24) when the area is ex- X k k pressed in terms of M and the Immirzi parameter is An = gk,n AG,n ⊗ 1IF,n + fk,n 1IG,n ⊗ AF,n ,(54) √ 0 k fixed to ln 2/( 3π). In fact, both derivations – ours and LQG’s – rely on the identification of the entropy with where k labels the emergent modes. Eqs.(54) can be re- SBH . Clearly, a more detailed analysis, beyond these garded as a dynamical (Haag) map at linear order [20]. qualitative arguments, will be possible only once a com- The full dynamical map – available once the quantum plete dynamical map will be available. theory of gravity is specified – should connect the fundamental dof to the emergent notions of geometry and fields. The coefficients of the map should then lead to V. CONCLUSIONS the Hawking thermal behavior of the latter, at emergent † level. Note that the action of An on |0iA, creates both We assumed here that the dof of a BH are finite in a matter mode and a geometric mode, outside the hori- number and fermionic in nature, and hence obey a Pauli zon: the region of spacetime surrounding the BH and principle. Then, within the approach of second quan- available to an external observer increases, because the tization, we naturally obtained that the BH evapora- horizon area decreases. tion is a dynamical mechanism producing a maximal en- Let us remark that, in this picture, a relationship of tanglement entropy, equal to the initial entropy of the the form (54) makes sense only for A-modes (or, equivalently a-modes) and not for B-modes (or, equivalently b-modes). In fact, by definition, the former are the ones 3 which rearrange to form matter and geometry, in contrast This is equal to SBH , defined in Eq.(29), only before the evapo- with the latter which describe free Xon. As previously ration, when SBH = Smax. 8

BH. This phenomenon is an instance of the Bekenstein • In the case N = 1, the density matrix reads bound, obtained here with arguments that do not as- 2 2 sume pre-existing spatiotemporal concepts. Of course, ρ(σ) = cos σ |0iI |1iIIIIh1|Ih0| + sin σ |1iI |0iIIIIh0|Ih1| for a full identification with the standard formulae (see, sin(2σ) e.g., [7, 8]), one needs to link geometrical concepts, such + (|0i |1i h0| h1| + |1i |0i h1| h0|) , (A1) as elementary Planck cells, to such fundamental dof. 2 I IIII I I IIII I We then showed that entanglement, Bekenstein and where we omitted tensor product symbols. The environment (thermodynamic) entropies here are all nat- reduced density matrices have the following form: urally obtained in the same approach, based on an entropy operator whose structure is the one typical of TFD. 2 2 ρI (σ) = cos σ |0iIIh0| + sin σ |1iIIh1| , (A2) Through such operator, we have evaluated the von Neu- 2 2 mann BH–environment entropy and noticeably obtained ρII (σ) = cos σ |1iIIIIh1| + sin σ |0iIIIIh0| . (A3) a Page-like evolution. We finally have shown that the latter is a consequence Eq.(32) follows immediately from these expres- of a duality between our model and a dissipative-like sions. fermionic quantum system, and hence it has a natural In the case N = 2, we directly report the reduced TFD-like description. • density matrices: Many directions for further research need be thor- oughly explored, the most important being a reliable dy- 4 4 ρI (σ) = cos σ |00iIIh00| + sin σ |11iIIh11| namical map from the fundamental modes to the emergent fields/spacetime structures. This would pave the sin2 2σ + (|01i h01| + |10i h10|) , (A4) road for the investigation of important aspects of the 4 II II emergent picture, such as the Hawking thermality of the 4 4 field sector. Moreover, a geometric interpretation of the ρII (σ) = cos σ |11iIIIIh11| + sin σ |00iIIIIh00| results presented here would allow to investigate the dis- sin2(2σ) creteness of the spectrum of quasinormal modes, which is + (|01i h01| + |10i h10|) . (A5) known to be related to the quantization of the BH hori- 4 IIII IIII zon’s area (see Ref.[58]). Anyway, even in the absence It follows that of a full dynamical map, we believe that our simple, although nontrivial, considerations are necessary to fully S (σ) = − sin4 σ ln sin4 σ − cos4 σ ln cos4 σ take into account the fascinating and far-reaching conse- I quences of the Bekenstein bound. − 2 sin2 σ cos2 σ ln sin2 σ cos2 σ

= SII (σ) . (A6)

By using the relations ln(ab) = ln a + ln b and Acknowledgments cos2 σ + sin2 σ = 1, we get

2 2 2 2 The authors would like to thank G. Vitiello, P. Lukeˇs, SI (σ) = −2 sin σ ln sin σ + cos σ ln cos σ D. Lanteri and L. Buoninfante for enlightening discus- sions. A. I. and L. S. are partially supported by UNi- = SII (σ) , (A7) versity CEntre (UNCE) of Charles University in Prague (Czech Republic) Grant No. UNCE/SCI/013. which is equal to Eq.(32) for N = 2. One could repeat similar computations for all N. How- ever it is simpler to use the correspondence of our model with a TFD/dissipative system via Eq.(44). As already Appendix A: Equivalence of TFD entropy and von known in TFD, the “thermal vacuum” can be rewritten Neumann entropy in the form X p In this Appendix we explicitly show the equivalence |Ψ(σ)i = wn(σ) |niA|niB , (A8) of the TFD entropy of Eq.(21) with the von Neumann n=0,1 entropy of Eq.(32). We ﬁrst present the computation in the simplest cases of Eqs.(33) and (35) and then we shall where |nAiA, |nBi are eigenstates of the number opera- focus on the computation of the von Neumann entropy. tors Let us note that the expectation value of the TFD en- N N tropy operators (see Eq.(21)) immediately follows from ˆ X † ˆ X † NA = An An , NB = Bn Bn . (A9) Eq.(20). n=1 n=1 9

Moreover, the coeﬃcients wn are given by The von Neumann entropy reads [19, 22, 27]

N Y 2 wn(σ) = Cj (σ) . (A10) j=1 and Cj were ﬁrstly introduced in Eq.(14). The density matrix thus reads X X SA(σ) = SB(σ) = − wn(σ) ln wn(σ) . (A14) ρ(σ) = wn(σ) |niA|niBBhn|Ahn| , (A11) n=0,1 n=0,1 from which the reduced density matrices are easily derived: X ρA(σ) = wn(σ) |niAAhn| (A12) n=0,1 Finally, Eq.(32) follows from substituting the explicit X ρB(σ) = wn(σ) |niBBhn| . (A13) form of wn (cf. Eq.(A10)) into the last expression. n=0,1

References Amsterdam, 1982); H. Umezawa, Advanced field theory: Micro, Macro, and Thermal Physics, (AIP, New York, [1] G. Acquaviva, A. Iorio and M. Scholtz, Annals Phys. 1993). 387, 317 (2017). [21] W. Israel, Phys. Lett. A 57, 107 (1976). [2] J. D. Bekenstein, Phys. Rev. D 23, 287 (1981); Found. [22] A. Iorio, G. Lambiase and G. Vitiello, Ann. Phys. 309, Phys. 35, 1805 (2005). 151 (2004). [3] R. Feynman, R. Leighton, M Sands and M Gottlieb, 2006 [23] M. Martellini, P. Sodano and G. Vitiello, Nuovo Cim., The Feynman Lectures on Physics (Pearson/Addison- 48 341 (1978). Wesley); A. Iorio, J. Phys.: Conf. Series 1275, 012013 [24] C. E. Laciana, Gen. Rel. Grav. 26, 363 (1994). (2019). [25] J. M. Maldacena, JHEP 0304, 021 (2003). [4] N. Bao, S. M. Carroll and A. Singh, Int. J. Mod. Phys. [26] M. Botta Cantcheff, A. L. Gadelha, D. F. Z. Marchioro D 26, no. 12, 1743013 (2017). and D. L. Nedel, Eur. Phys. J. C 78, no. 2, 105 (2018). [5] J. D. Bekenstein, Sci. Am. 289, 58 (2003). [27] M. Dias, D. L. Nedel and C. R. Senise, arXiv:1910.11427 [6] D. Harlow, Rev. Mod. Phys. 88, 015002 (2016). [hep-th]. [7] H. Casini, Class. Quant. Grav. 25, 205021 (2008). [28] S.W. Hawking, Phys. Rev. D 14, 2460 (1976). [8] R. Bousso, Rev. Mod. Phys. 74, 825 (2002). [29] D. N. Page, Phys. Rev. Lett. 44, 301 (1980). [9] J. D. Bekenstein, Lett. Nuovo Cim. 4, 737 (1972). [30] S. B. Giddings and W. M. Nelson, Phys. Rev. D 46, 2486 [10] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973). (1992). [11] S. W. Hawking, Phys. Rev. D 13, 191 (1976). [31] D. N. Page, Phys. Rev. Lett. 71, 3743 (1993). [12] W. H. Zurek, Phys. Rev. Lett. 49, 1683 (1982). [32] G. ’t Hooft, Nucl. Phys. Proc. Suppl. 43, 1 (1995). [13] D. N. Page, Phys. Rev. Lett. 50, 1013 (1983). [33] S. W. Hawking, Phys. Rev. D 72, 084013 (2005). [14] W. Israel, Phys. Rev. 164, 1776 (1967); Commun. Math. [34] P. Hayden and J. Preskill, JHEP 0709, 120 (2007). Phys. 8, 245 (1968); B. Carter, Phys. Rev. Lett. 26, 331 [35] S. D. Mathur, Class. Quant. Grav. 26, 224001 (2009). (1971). [36] D. N. Page, JCAP 1309, 028 (2013). [15] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. [37] J. Polchinski, arXiv:1609.04036 [hep-th]. Rev. Lett. 80, 904 (1998). [38] M. Al Alvi, M. Majumdar, M. A. Matin, M. H. Rahat [16] C. Rovelli, Quantum gravity, (Cambridge University and A. Roy, Phys. Lett. B 797, 134881 (2019). Press, Cambridge, 2004); C. Rovelli and F. Vidotto, [39] L. Piroli, C. Sünderhaufand X. L. Qi, JHEP 04, 063 Covariant Loop Quantum Gravity : An Elementary In- (2020). troduction to Quantum Gravity and Spinfoam Theory, [40] D. N. Page, Phys. Rev. Lett. 71, 1291 (1993). (Cambridge University Press, Cambridge, 2014). [41] A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, [17] P.F. González-D´ıazPhys. Rev. D 27, 3042 (1983). JHEP 03, 149 (2020). [18] S.W. Hawking, Nature 248, 30 (1974); Comm. Math. [42] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 Phys. 43 199 (1975). (2006). [19] Y. Takahashi and H. Umezawa, Collective Phenomena [43] S. Sachdev, Phys. Rev. X 5, no. 4, 041025 (2015) 2, 55 (1975), reprinted in Int. J. Mod. Phys. B 10, 1755 [44] J. Maldacena and D. Stanford, Phys. Rev. D 94, no. 10, (1996). 106002 (2016). [20] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo [45] M. Kobayashi and T. Maskawa, Progr. Theor. Phys. Field Dynamics And Condensed States, (North-Holland, 49, 652 (1973); T.P. Cheng and L. Li, Gauge Theory of Elementary Particle Physics (Clarendon Press, 1982). 10

[46] K.O. Friedrichs, Mathematical aspects o! Quantum The- ory and its Macroscopic Manifestations, (World Scien- ory of Fields, (Interscience Publishers Inc., New York, tiﬁc, London, 2011). 1953); H. Umezawa and G. Vitiello, Quantum Mechan- [53] J.P. Blaizot and G. Ripka, Quantum theory of ﬁnite sys- ics, (Bibliopolis, Napoli, 1985); M. Blasone, P. Jizba and tems (MIT Press, Cambridge, 1986). L. Smaldone, Ann. Phys. 383, 207 (2017). [54] E. Celeghini, M. Rasetti and G. Vitiello, Ann. Phys. 215, [47] A. Iorio, G. Vitiello, Mod. Phys. Lett. B 8 (1994) 269. 156 (1992). [48] G. Acquaviva, A. Iorio and L. Smaldone, in preparation. [55] A. Iorio, G. Vitiello, Ann. Phys. 241, 496 (1995). [49] G. Dvali, [arXiv:2003.05546 [hep-th]]. [56] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042 [50] M.A. Nielsen and I.L. Chuang, Quantum computation (2000); L. Vanzo, G. Acquaviva and R. Di Criscienzo, and quantum information, (Cambridge University Press, Class. Quant. Grav. 28, 183001 (2011). Cambridge, 2000). [57] S. Hod, Phys. Rev. Lett. 81, 4293 (1998). [51] A.A Klyachko, B. Oztop¨ and A.S. Shumovsky, Phys. Rev. [58] M. Maggiore, Phys. Rev. Lett. 100, 141301 (2008). A 75, 032315 (2007). [52] M. Blasone, P. Jizba and G. Vitiello, Quantum Field The-