Bekenstein bound from the Pauli principle∗ G. Acquaviva,1, y A. Iorio,1, z and L. Smaldone1, x 1Faculty of Mathematics and Physics, Charles University, V Holeˇsoviˇck´ach2, 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 dis- cussed, 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). yElectronic address: [email protected]ff.cuni.cz Although the relation between QFT in curved space- zElectronic address: [email protected]ff.cuni.cz time and TFD was studied already in Refs. [23, 24], the xElectronic 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.