Uncertainty Relation on a World Crystal and Its Applications to Micro Black Holes

PHYSICAL REVIEW D 81, 084030 (2010) Uncertainty relation on a world crystal and its applications to micro black holes

Petr Jizba,1,2,* Hagen Kleinert,1,† and Fabio Scardigli3,‡ 1ITP, Freie Universita¨t Berlin, Arnimallee 14 D-14195 Berlin, Germany 2FNSPE, Czech Technical University in Prague, Brˇehova´ 7, 115 19 Praha 1, Czech Republic 3Leung Center for Cosmology and Particle Astrophysics (LeCosPA), Department of Physics, National Taiwan University, Taipei 106, Taiwan (Received 17 December 2009; published 16 April 2010) We formulate generalized uncertainty relations in a crystal-like universe—a ‘‘world crystal’’—whose lattice spacing is of the order of Planck length. In the particular case when energies lie near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relation for position and momenta does not pose any lower bound on involved uncertainties. We apply our results to micro black holes physics, where we derive a new mass-temperature relation for Schwarzschild micro black holes. In contrast to standard results based on Heisenberg and stringy uncertainty relations, our mass-temperature formula predicts both a ﬁnite Hawking’s temperature and a zero rest-mass remnant at the end of the micro black hole evaporation. We also brieﬂy mention some connections of the world-crystal paradigm with ’t Hooft’s quantization and double special relativity.

DOI: 10.1103/PhysRevD.81.084030 PACS numbers: 04.70.Dy, 03.65. w À

I. INTRODUCTION from numerical computations. One may, however, inves- tigate the consequences of taking the lattice no longer as a Recent advances in gravitational and quantum physics mere computational device, but as a bona fide discrete indicate that in order to reconcile the two fields with each network, whose links define the only possible propagation other, a dramatic conceptual shift is required in our under- directions for signals carrying the interactions between standing of spacetime. In particular, the notion of spacetime as a continuum may need revision at scales where fields sitting on the nodes of the network. gravitational and electroweak interactions become compa- Recently one of us proposed a model of a discrete, rable in strength [1]. For this reason there has been a recent crystal-like universe—a ‘‘world crystal’’ [21,24,25]. revival of interest in approximating the spacetime with There, the geometry of Einstein and Einstein-Cartan discrete coarse-grained structures at small, typically spaces can be considered as being a manifestation of the Planckian, length scales. Such structures are inherent in defect structure of a crystal whose lattice spacing is of the many models of quantum gravity [2], such as spacetime order of ‘p. Curvature is due to rotational defects, torsion foam [3], loop quantum gravity [4–6], noncommutative due to translational defects. The elastic deformations do geometry [7–10], black hole physics [11], or cosmic cel- not alter the defect structure, i.e., the geometry is invariant lular automata [12–16]. under elastic deformations. If one assumes these to be controlled by a second-gradient elastic action, the forces Despite a vast gap between the Planck length (‘p 35 between local rotational defects, i.e., between curvature 1:6 10À m) and smallest length scales that can be Â 18 singularities, are the same as in Einstein’s theory [26]. probed with particle accelerators ( 10À m), the issue of Planckian physics might not be so speculative as it Moreover, the elastic fluctuations of the displacement seems. In fact, probes such as Planck Surveyor [17] or fields possess logarithmic correlation functions at long the related IceCube [18]—which just started or are planned distances, so that the memory of the crystalline structure to start in the near future—are supposed to set various is lost over large distances. In other words, the Bragg peaks important limits on prospective models of the Planckian of the world crystal are not -function-like, but display the world. typical behavior of a quasi-long-range order, similar to the One of the simplest toy-model systems for Planckian order in a Kosterlitz-Thousless transition in two- physics is undoubtedly a discrete lattice. Discrete lattices dimensional superfluids [24]. are routinely used, for instance, in computational quantum The purpose of this paper is to study the generalized field theory [19–21], but with a few notable exceptions uncertainty principle (GUP) associated with the quantum [22–24], they mainly serve as numerical regulators of physics on the world crystal and to derive physical con- ultraviolet divergences. Indeed, a major point of renormal- sequences related to micro black hole physics. In view of ized theories is precisely to extract lattice-independent data the fact that micro black holes might be formed at energies as low as the TeV range [27–29]—which will be shortly *[email protected] available in particle accelerators such as the LHC—it is †[email protected] hoped that the presented results may be more than of a ‡[email protected] mere academic interest.

1550-7998=2010=81(8)=084030(13) 084030-1 Ó 2010 The American Physical Society PETR JIZBA, HAGEN KLEINERT, AND FABIO SCARDIGLI PHYSICAL REVIEW D 81, 084030 (2010) The structure of our paper is as follows: In Sec. II we fg x f x g x f x g x ; present some fundamentals of a differential calculus on a ðr Þð Þ ¼ ðr Þð Þ ð Þþ ð þ Þðr Þð Þ (2) fg x f x g x f x g x : lattice that will be needed in the text. In Sec. III we ðr Þð Þ¼ðr Þð Þ ð Þþ ð À Þðr Þð Þ construct position and momentum operators on a 1D lattice On a lattice, integration is performed as a summation: and compute their commutator. We then demonstrate that the usual Weyl-Heisenberg algebra W1 for p^ and x^ opera- dxf x f x ; (3) tors is on a 1D lattice deformed to the Euclidean algebra ð Þ x ð Þ E 2 . By identifying the measure of uncertainty with a Z X standardð Þ deviation we derive the related GUP on a lattice. where x runs over all xn. This is done in Sec. IV. There we focus on two critical For periodic functions on the lattice or for functions regimes: long-wave regime and the regime where momenta vanishing at the boundary of the world crystal, the lattice are at the border of the first Brillouin zone. Interestingly derivatives can be subjected to the lattice version of inte- enough, our GUP implies that quantum physics of the gration by parts: world-crystal universe becomes ‘‘deterministic’’ for ener- f x g x g x f x ; (4) gies near the border of the Brillouin zone. In view of x ð Þr ð Þ¼À x ð Þr ð Þ applications to micro black hole physics, we derive in X X Sec. V the energy-position GUP for a photon. f x g x g x f x : (5) Implications for micro black holes physics are discussed x ð Þr ð Þ¼À x ð Þr ð Þ X X in Sec. VI. There we derive a mass-temperature relation for One can also define the lattice Laplacian as Schwarzschild micro black holes. On the phenomenological side, the latter provides a nice resolution of a long- 1 f x f x f x 2f x f x ; standing puzzle: the final Hawking temperature of a decay- rr ð Þ¼rr ð Þ¼2 ½ ð þ ÞÀ ð Þþ ð À Þ ing micro black hole remains finite, in contrast to the (6) infinite temperature of the standard result where which reduces in the continuum limit to an ordinary Heisenberg’s uncertainty principle operates. Besides, the 2 final mass of the evaporation process is zero, thus avoiding Laplace operator @x. Note that the lattice Laplacian can the problems caused by the existence of massive black hole also be expressed in terms of the difference of the two remnants. Entropy and heat capacity are discussed in lattice derivatives: Sec. VII. Finally, in Sec. VIII we outline a connection of 1 f x f x f x : (7) our results with ’t Hooft’s approach to deterministic quan- rr ð Þ¼ ½r ð ÞÀr ð Þ tum mechanics and with deformed (or double) special The above calculus can be easily extended to any num- relativity. Section IX is devoted to concluding remarks. ber D of dimensions [19,20,24]. For completeness, we present in the Appendix an alternative derivation of the micro black hole mass-temperature formula. III. POSITION AND MOMENTUM OPERATORS ON A LATTICE II. DIFFERENTIAL CALCULUS ON A LATTICE Consider now the quantum mechanics (QM) on a 1D lattice in a Schro¨dinger-like picture. Wave functions are In this section we quickly review some features of a square-integrable complex functions on the lattice, where differential calculus on a 1D lattice. An overview discus- ‘‘integration’’ means here summation, and scalar products sing more aspects of such a calculus can be found, e.g., in are defined by Refs. [19–21]. Independent and very elegant derivation of these can be also done in the framework of a noncommu- f g fÃ x g x : (8) tative geometry [30–32]. h j i¼ x ð Þ ð Þ X On a lattice of spacing in one dimension, the lattice It follows from Eq. (4) that sites lie at xn n where n runs through all integer numbers. There¼ are two fundamental derivatives of a func- f g f g ; (9) h jr i¼ Àhr j i tion f x : ð Þ so that i y i , and neither i nor i are Hermitian operators.ð r TheÞ ¼ latticer Laplacian (6r), however,r is Hermitian. 1 ^ f x f x f x ; The position operator X acting on wave functions of x is ðr Þð Þ¼ ½ ð þ ÞÀ ð Þ (1) defined by a simple multiplication with x: 1 f x f x f x : X^ f x xf x : (10) ðr Þð Þ¼ ½ ð ÞÀ ð À Þ ð Þð Þ¼ ð Þ Similarly we can define the lattice momentum operator P^. They obey the generalized Leibnitz rule In order to ensure Hermiticity we should relate it to the

084030-2 UNCERTAINTY RELATION ON A WORLD CRYSTAL AND ... PHYSICAL REVIEW D 81, 084030 (2010) symmetric lattice derivative [20,31,33]. Using (9) we have d X^ f~ k i f~ k ; (19) ð Þð Þ¼ dk ð Þ P^ f x f x f x ð Þð Þ¼2@i ½ðr Þð Þþðr Þð Þ ^ ~ ~ Pf k @ sin k f k ; (20) f x f x : (11) ð Þð Þ¼ ð Þ ð Þ ¼ 2@i ½ ð þ ÞÀ ð À Þ I^ f~ k cos k f~ k : (21) For small , this reduces to the ordinary momentum op- ð Þð Þ¼ ð Þ ð Þ erator p^ i @ , or more precisely À x With the help of (21) we can rewrite the commutation @ relation (13) equivalently as P^ p^ O 2 : (12) ¼ þ ð Þ X^ ; P^ f x i cos p=^ f x : (22) ^ ^ The ‘‘canonical’’ commutator between X and P on the ð½ Þð Þ¼ @ ð @Þ ð Þ ^ lattice reads The latter allows us to identify the lattice unit operator I with cos p=^ . Indeed, I^ ^ on all lattice nodes. i 1 ^ ^ ^ ð @Þ ¼ X; P f x @ f x f x i If x : ð½ Þð Þ¼ 2 ½ ð þ Þþ ð À Þ @ð Þð Þ (13) IV. UNCERTAINTY RELATIONS ON LATTICE The last line defines a lattice version of the unit operator as We are now prepared to derive the generalized uncer- the average over the two neighboring sites. Note that all tainty relation implied by the previous commutators. We ^ ^ ^ shall define the uncertainty of an observable A in a state c three operators X, P, and I are Hermitian under the scalar product (8). by the standard deviation It was noted in [33] that the operators X^ , P^ , and I^ ÁA c A^ c A^ c 2 c : (23) generate the Euclidean algebra E 2 in 2D. Indeed, setting ð Þc h jð Àh j j iÞ j i ð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M^ X^ , P^ P^ = , and P^ I^ we obtain Following the conventional Robertson-Schro¨dinger proce- ¼ 1 ¼ 2 ¼ @ dure (see, e.g., Refs. ), we derive on the spacetime lattice M;^ P^ iP^ ; M;^ P^ iP^ ; P^ ; P^ 0: ½ 1¼ 2 ½ 2¼À 1 ½ 1 2¼ the inequality The generator M^ corresponds to a rotation, while P^ and P^ 1 1 2 ÁX ÁP c X^ ; P^ c c I^ c represent two translations. In the limit 0, the Lie ð Þc ð Þc 2 jh j½ j ij ¼ 2@ jh j j ij algebra of E 2 contracts to the standard! Weyl- ð Þ Heisenberg algebra W : X^ x^, P^ p^, I^ ^ . Thus c cos p=^ c : (24) 1 1 ¼ 2@ jh j ð Þj ij ordinary QM is obtained from! lattice QM! by a contraction! @ of the E 2 algebra, with the lattice spacing playing the For brevity we will omit in the following the subscript c in ð Þ ÁA and set c c . role of the deformation parameter. ð Þc h jÁÁÁj ihÁÁÁic All functions on the lattice can be Fourier-decomposed Let us now study two critical regimes of the GUP (24): with wave numbers in the Brillouin zone: the first is the long-wavelengths regime where p^ c 0; h i ! the second regime is near the boundary of the Brillouin = dk ~ ikx zone where p^ =2. To this end we first rewrite f x f k e ; (14) h ic ! ð Þ¼ = 2 ð Þ cos p=^ as @ ZÀ c h ð @Þi with the coefficients 2n 1 1 n p= cos p=^ c dp% p 1 ð Þ ; (25) ikx h ð Þi ¼ 0 ð ÞðÀ Þ 2n@! f~ k f x eÀ : (15) @ n 0 X¼ Z ð Þ ð Þ¼ x ð Þ X where % p c p 2. This implies the good-old de Broglie relation In theð firstÞ case,j ð%Þpj is peaked around p 0, so that the relation (25) becomesð Þ approximately ’ p^ f~ k kf~ k ; (16) ð Þð Þ¼ ð Þ 2 2 @ p 4 and its lattice version cos p=^ c 1 2 O p ; (26) h ð @Þi ¼ À 2 þ ð Þ i f~ k Kf~ k ; i f~ k K f~ k ; (17) where p2 p^ 2 . We should stress@ that expansion (26) is ðÀ r Þð Þ¼ ð Þ ðÀ r Þð Þ¼ ð Þ h ic not an expansion in but rather in p= . So if we speak of with the eigenvalues % p as being peaked around p 0 we@ mean that p ik =ð.Þ ’ K e 1 =i K Ã: (18) ð À Þ ¼ @ Applying now the identity From (17) we find the Fourier transforms of the operators ^2 2 ^ 2 ^ ^ ^ A c ÁA A ; (27) X, P, I: h i ¼ð Þ þh ic

084030-3 PETR JIZBA, HAGEN KLEINERT, AND FABIO SCARDIGLI PHYSICAL REVIEW D 81, 084030 (2010) we obtain from (24) Let us remark that the scenario in which the universe at Planckian energies is deterministic rather than being domi- 2p2 ÁX ÁP * 1 nated by tumultuous quantum ﬂuctuations is a recurrent @ 2 2 À 2 theme in ’t Hooft’s deterministic quantum mechanics [38–

2@ 44]. 1 Áp 2 p^ 2 : (28) @ 2 c It is straightforward to generalize the above formulas to ¼ 2 À 2 ½ð Þ þh i higher dimensions. In this context, a useful inequality is @ For mirror-symmetric states where p^ c 0 this implies h i ¼ i î ^ i i ÁXÁ P @ c P= P cos p^ = c 2 j j2 jh jð j jÞ ð @Þj ij ÁX ÁP * 1 Áp 2 : (29) @ 2 2 À 2 ð Þ c " p^ i cos ip^ i= c ; (34) @ ¼ 2@ jh j ð Þ ð Þj ij Here we have substituted ... by ... since we assume @ that ‘ (Planckian lattice)j andj thatð ÁÞ p is close to zero which will be needed in the following. Here " ... is the ’ p sign function, and ð Þ (this is our original assumption). Therefore 2 Áp 2=2 2 1. ð Þ D sin jp^ j= 2 For Planckian@ lattices with the relation (12), we can P^ ð Þ : (35) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij @ neglect higher powers of in (29) and write j j¼@uj 1 uX¼ tu 2 Inequality (34) should be contrasted with inequality (24) ÁX ÁP * 1 ÁP 2 : (30) @ 2 where the momentum is without an absolute value. 2 À 2 ð Þ @ In a particular case when states c are a combination of In the second case, where p^ =2, i.e. near the h ic ! only positive or only negative momentum eigenstates (e.g., border of the Brillouin zone, we use the@ expansion: incident or reflected particle states) we can simply write cos =2 p=^ =2 c ÁXi Á P c cos ip^ i= c h ½ þð @ À Þi @ sin =2 p=^ j j2 jh j ð @Þj ij ¼h ð À Þic @ 2n 1 2 i i =2 p= 1 2 sin p^ =2 c : (36) 1 1 n þ ¼ 2@ ½ À h ð Þi dp% p 1 ð À @Þ : (31) @ ¼ n 0 0 ð ÞðÀ Þ 2n 1 ! X¼ Z ð þ Þ Under the assumption that % p is peaked near the border V. IMPLICATIONS FOR PHOTONS of the Brillouin zone, the firstð Þ term in the expansion is We may now use the inequality (36) to derive the GUP dominant, and the uncertainty relation reduces to for photons. The vector potential of a photon in the Lorentz gauge in ÁX ÁP p^ : (32) 2@ 2 À h ic 1 1 dimensions satisfies the wave equation þ @ 1 Since k p= lies always inside the Brillouin zone, we @2A x; t @2A x; t : (37) 2 t x have p^ ¼ @ =2 and can therefore in (32) substitute c ð Þ¼ ð Þ h ic ... by ... . Finally,@ using again (12), we can write for A plane wave solution A x exp i kx ! k t thej GUPj ð closeÞ to the boundary of the Brillouin zone possesses the well-known linearð Þ¼ dispersion½ relationð À ð Þ Þ ! k c k ; (38) ^ ð Þ¼ j j ÁXÁP * @ P c : (33) 2 2 À h i with being a polarization vector. On a one-dimensional @ 2 As the momentum reaches the boundary of the Brillouin lattice, the operator @x is replaced by the lattice Laplacian , and the spectrum becomes, on account of Eq. (6) and zone, the right-hand sides of (32) and (33) vanish, so that rr lattice quantum mechanics at short wavelengths is permit- (17), ted to exhibit classical behavior—no irreducible lower ! k 2 1 cos k 2 k bound for uncertainties of two complementary observables ð Þ KK ½ À ð Þ sin ; c ¼ ¼ ¼ 2 appears. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi It is worth noting that the uncertainty relation (33) leads (39) to the same physical conclusions as those found, on a which reduces to (38) for 0. Denoting the energy on different ground, by Magueijo and Smolin in Ref. [37]. ! the lattice ! by E, we obtain the dispersion relation In particular, the world-crystal universe can become deter- @ E 2 p ministic for energies near the border of the Brillouin zone, sin : (40) c ¼ 2 i.e., for Planckian energies. @ @

084030-4 UNCERTAINTY RELATION ON A WORLD CRYSTAL AND ... PHYSICAL REVIEW D 81, 084030 (2010)

We can also define the associated energy operator E^c by tainty relation. From this one can deduce nontrivial replacing p by p^. phenomenological consequences. The passage from the For states c with % p sharply peaked around small p, energy-position uncertainty relation to the micro black we can use a spectral expansionð Þ analog of (25) to obtain hole mass-temperature relation has been intensively studied in recent years. For definiteness we shall follow ÁE cÁ p cÁ P : (41) ’ j j’ j j here the treatment of Refs. [45–52]. An alternative deriva- Here we have neglected higher powers of momentum and tion based on the so-called Landauer principle will be used the fact that we deal with a Planckian lattice. In presented in the Appendix. deriving we have also applied the cumulant expansion: We start with an assumption that the lattice spacing is roughly of order Planck length, i.e., a‘ , where a>0 ^ ¼ p E c 2 c= sin p^=2 c is of order of unity. Let us now imagine that we have found h i ¼ð @ Þhj ð @Þji 2p3 a black hole on the lattice as a discretized version of a cp c O p5 : (42) Schwarzschild solution. It is a pile up of disclinations. If ¼ À 24 2 þ ð Þ the Schwarzschild radius is much larger than the lattice @ With the help of (36 ), (40), and (41) we can write in the spacing , this will not look much different from the well- long-wavelength regime known continuum solution. We must avoid too small black holes, for otherwise, completely new physics will set in c 2 ÁX ÁE 1 E2 : (43) near the center, due to the high concentration of defects. @ 2 2 c 2 À 2 c h i These will cause the ‘‘melting’’ of the world crystal at a @ Here E2 is the average quadrat of the photon energy, critical defect density [53], and the emerging trans-horizon h ic and thus the square root of it can be formally identified general relativity would look completely different from Einstein’s theory. with the energy change in the detector, i.e. ÁE. From this follows that if the uncertainty of a photon position in a state Following the classical argument of the Heisenberg microscope [54], we know that the smallest resolvable c is ÁX, then the energy of a detector changes at least by amount detail x of an object goes roughly as the wavelength of the employed photons. If E is the (average) energy of the c 2 1 ÁE 1 E2 ; (44) photons used in the microscope, then @ 2 2 c ’ 2 À 2 c h i ÁX @ c per particle. Remembering the Einstein relation ÁE x @ : (48) ¼ ’ 2E 2 c=, we can interpret 4ÁX as being a lattice equivalent@ of photon’s wavelength . Conversely, with the relation (48) one can compute the It is interesting to observe that in the short-wavelength energy E of a photon with a given (average) wavelength case we can deduce from the exact GUP x. As a consequence of Eq. (43), we can write the lattice’ version of this standard Heisenberg formula as 2 ÁX Á P 1 E2 ; (45) @ 2 2 c c 2 j j2 À 2 c h i X 1 E 2 ; (49) @ 2@E 2 2c2 that near the border of the Brillouin zone ÁX takes the ’ À ð Þ @ approximate form [cf. Eq. (40)] which links the (average) wavelength of a photon to its 2 energy E. Since the lattice spacing is a‘p and the 2 ^ ¼ ÁX 1 E c P c : (46) Planck energy c=2‘ , Eq. (49) can be rewritten as ’ À 2 2c2 h i ’ 2 À h i Ep p ¼ @ @ @ 2 In the derivation we have used that fact that c a ‘pE X @ : (50) ’ 2E À 8Ep Á P P^2 : (47) 2@ j jqffiffiffiffiffiffiffiffiffih i ’ Let us now loosely follow the argument of Refs. [45–52] Relation (46) represents the smallest attainable positional and consider an ensemble of unpolarized photons of uncertainty near the border of the Brillouin zone. It will be Hawking radiation just outside the event horizon. From a useful in the following two sections. geometrical point of view, it is easy to see that the position uncertainty of such photons is of the order of the Schwarzschild radius RS of the hole. An equivalent argu- VI. APPLICATIONS TO MICRO BLACK HOLES ment comes from considering the average wavelength of An interesting playground where one can apply the the Hawking radiation, which is of the order of the geo- above lattice GUP’s is the hypothetical physics of micro metrical size of the hole (see, e.g., Ref. [50], chapter 5). By black holes. Their mass-temperature relation depends sen- recalling that R ‘ m, where m M=M is the black S ¼ p ¼ p sitively on the actual form of the energy-position uncer- hole mass in Planck units (M E =c2), we can estimate p ¼ p

084030-5 PETR JIZBA, HAGEN KLEINERT, AND FABIO SCARDIGLI PHYSICAL REVIEW D 81, 084030 (2010) the photon positional uncertainty as The phenomenological consequences of the relation (55) are quite different from those of the stringy result X 2R 2‘ m: (51) ’ S ¼ p (57). In Fig. 1 we compare the two results, and add also the The proportionality constant is of order unity and will be curve for the ordinary Hawking relation (56). Considering ﬁxed shortly. According to the above arguments, m must be m and Â as functions of time, we can follow the evolution assumed to be much larger than unity, in order to avoid the of a micro black hole from the curves in Fig. 1. For the melting transition. With (51) we can rephrase Eq. (50) as stringy GUP, the blue line predicts a maximum temperature

2 Ep a E 1 2m : (52) Âmax ; (58a) ’ E À 8 Ep ¼ 2

According to the equipartition principle the average energy and a minimum rest mass E of unpolarized photons of the Hawking radiation is linked with their temperature T as m : (58b) min ¼ E kBT: (53) ¼ The end of the evaporation process is reached in a finite In order to fix , we go to the continuum lattice limit time, the final temperature is finite, and there is a remnant 0 (a 0), and require that formula (52) predicts the! ! of a finite rest mass (cf. Refs. [45–49,51,52,57]). standard semiclassical Hawking temperature: From the standard Heisenberg uncertainty principle we c3 c find the green curve, representing the usual Hawking for- TH : (54) mula. Here the evaporation process ends, after a finite time, ¼ 8Gk@ M ¼ 4@k R B B S with a zero mass and a worrisome infinite temperature. In This fixes . the literature, the undesired infinite final temperature pre- Defining the¼ Planck temperature T so that E dicted by Hawking’s formula has so far been cured only p p ¼ kBTp=2 and measuring all temperatures in Planck units with the help of the stringy GUP, which brings the final as Â T=Tp, we can finally cast formula (52) in the form temperature to a finite value. This result is, however, also ¼ questionable since it implies the existence of finite-mass 1 2m 22Â; (55) remnants in the universe. Though by some authors such ¼ 2Â À remnants are greeted as relevant candidates for dark matter where we have defined the deformation parameter [58], others point out that their existence would create ¼ further complications such as the entropy/information a= 2p2 . problem [59], detectability issue, or their (excessive) pro- Asð alreadyÞ mentioned, in the continuum limit both and ffiffiffi duction in the early universe [28,60]. a tend to zero and (50) reduces to the ordinary Heisenberg In contrast to these results, our lattice GUP predicts the uncertainty principle. In this case Eq. (55) boils down to red curve. This yields a finite end temperature 1 m : (56) ¼ 4Â m This is the dimensionless version of Hawking’s formula 8 (54) for large black holes. Historically, the validity of (54) was also postulated for 6 micro black holes on the assumption that the black hole thermodynamics is universally valid for any black hole, be 4 it formed via star collapse, or primordially via quantum fluctuations. Such an assumption is by no means warranted without some further input about mesoscopic and/or mi- 2 croscopic energy scales (much like in ordinary thermodynamics) and, in fact, we have seen that corrections should 0.05 0.1 0.15 0.2 0.25 0.3 be expected at short world-crystal scales. It is instructive to compare our mass-temperature relation (55) with the one suggested by the so-called stringy FIG. 1 (color online). Diagrams for the three mass- uncertainty relation [55,56]. There the sign of the correc- temperature relations, ours (lower line), Hawking’s (middle tion term in (55) is positive: line), and stringy GUP result (upper line), with p2, as an ¼ 1 example. As a consequence of the lattice uncertainty principleffiffiffi 2m 22Â: (57) the evaporation ends at a finite temperature with a zero rest-mass ¼ 2Â þ remnant.

084030-6 UNCERTAINTY RELATION ON A WORLD CRYSTAL AND ... PHYSICAL REVIEW D 81, 084030 (2010) 1 Rewriting Eq. (63) with the dimensionless variables m Â ; (59) max ¼ 2 and Â we get k dm with a zero-mass remnant. The mass-temperature formula dS B : (64) (55) thus solves at once several problems by predicting the ¼ 2 Â end of the evaporation process at a finite final temperature Inserting here formula (55) we find with zero-mass remnants [61]. 2 It should be stressed that since the photon GUP (43) and kB dm kB 1 2 2 2 2 dS 3 dÂ: (65) (49) holds only for states c where p^ c = , our ¼ 2 Â ¼À 4 2Â þ Â h i reasonings are warranted only for @ By integrating dS we obtain S S Â . Just as formula ¼ ð Þ 1 (55), the relation (65) can be trusted only for Â Âmax E c= Ep=a 2 : (60) ¼ ’ ) Â 1=2. Thus, when integrating (65), we should do this @ ~ only up to a cutoff Âmax Âmax. The additive constant in This implies, in particular, that when Â is close to Âmax our S can be then be fixed by requiring that S 0 when Â long-wavelength approximation cannot be trusted. ¼ ! Â~ . This is equivalent to what is usually done when To understand the behavior of the system close to Âmax max we must turn to the short-wavelength limit, Eqs. (33) and calculating the Hawking temperature for a Schwarzschild (46). In this regime the momenta lie close to the border of black hole. There one fixes the additive constant in the the Brillouin zone p^ P^ = 2 , and Eq. (40) entropy integral to be zero for m 0, so that S m 0 h ic ’h ic ’ ð Þ ¼ ð ¼ Þ¼ implies @ S Â 0 (the minimum mass attainable in the stan- dardð ! Hawking 1Þ ¼ effect is m 0). Thus we obtain ¼ p2 ~ 2 E c; (61) k Âmax 1 2 S B dÂ ; (66) ’ ffiffiffi @ 3 0 ¼ 4 Â 2Â0 þ Â0 which for the Planckian lattice, where a‘ , gives E Z ¼ p ’ where the sign was chosen in order to have a positive E = . Considering again the uncertainty in the photon p ð Þ entropy. position as X 2‘ m [cf. Equation (51)], GUP (46) ’ p The integral (66) yields then predicts k 1 1 Â~ a a2‘2 E2 S Â B 822 log max : (67) X m 1 p p 0: (62) 16 2 ~ 2 Â 2 2 2 2 2 ð Þ¼ Â À Âmax þ / ’ 2 À 2 c ¼ @ The entropy is always positive, and S 0 for Â Â~ . We can thus conclude that the mass of the micro black hole ! ! max must go to zero. This is also consistent with our previous long-wavelength considerations. The micro black hole B. Heat Capacity therefore evaporates completely, without leaving remnants. With entropy formulas (65) and (67) at hand we can now compute the heat capacity of a (micro) black hole in the VII. ENTROPY AND HEAT CAPACITY world crystal. This will give us important insights on the final stage of the evaporation process. Again, we shall In this section we exhibit the modified thermodynamic obtain formulas valid only for Â Âmax 1= 2 . entropy and heat capacity of a black hole implied by the The heat capacity C of a black hole is¼ definedð viaÞ the new mass-temperature formula (55). relation dQ dE CdT: (68) A. Entropy ¼ ¼ From the first law of black hole thermodynamics [62] we The pressure exerted on the environment by the expanding know that the differential of the thermodynamical entropy black hole surface is zero. Hence we do not need to specify of a Schwarzschild black hole reads which C is meant. With the help of (63) and (68) we obtain dE dS ; (63) dS dS ¼ TH C T Â ; (69) ¼ dT ¼ dÂ where dE is the amount of energy swallowed by a black which yields hole with Hawking temperature TH. In Eq. (63) the increase in the internal energy is equal to the added heat kB 1 because a black hole makes no mechanical work when its C 2 : (70) ¼À 2 þ 2Â 2 entropy/surface changes (expanding surface does not exert ð Þ any pressure). From this clearly follows that C is always negative.

084030-7 PETR JIZBA, HAGEN KLEINERT, AND FABIO SCARDIGLI PHYSICAL REVIEW D 81, 084030 (2010) Most condensed-matter systems have C>0. However, VIII. FURTHER APPLICATIONS because of instabilities induced by gravity this is generally So far we have studied the consequence of the GUP on not the case in astrophysics [63,64], especially in black the micro black holes. Let us briefly mention two further hole physics. A Schwarzschild black hole has C<0 which applications. indicates that the black hole becomes hotter by radiating. The result (70) implies that this scenario holds also for micro black holes in the world crystal. A. ’t Hooft’s proposal In case of stringy GUP, we have to use Eq. (57) as the The first application relates to ’t Hooft’s proposal which mass-temperature formula. The expression for the heat purports to justify that our quantum world is merely a low- capacity then reads energy limit of a deterministic system operating at a deeper, perhaps Planckian, level of dynamics [38,39]. As k 1 a deterministic substrate ’t Hooft has proposed various C B 2 : (71) ¼ 2 À 2Â 2 cellular automata (CA) models. ð Þ In general, a CA is an array of cells forming a discrete lattice. All cells are typically equivalent and can take one Since also here 0 < Â < Â 1=2, black holes have max of a finite number of possible discrete states. Like space, negative specific heat also according¼ to the stringy GUP. time is discrete as well. At each time step every cell However, stringy GUP displays a striking difference with updates its state according to a transition rule which takes respect to lattice GUP. In fact, since in principle we can into account the previous states of cells in the neighbor- trust Eq. (57) also when Â Â 1=2, then from max hood, including its own state. In this sense the evolution is (71) we have, in such limit, C’ 0. This¼ means that for the deterministic. stringy GUP the specific heat¼ vanishes at the end point of One of the simplest CA considered by ’t Hooft is the the evaporation process in a finite time, so that the black one-dimensional periodic CA with 4-state cells, and with hole at the end of its evolution cannot exchange energy the nearest neighbor (N-N) transition rule; see Fig. 2. This with the surrounding space. In other words, the black hole ‘‘clocklike’’ CA can be generally described with 2N 1 stops to interact thermodynamically with the environment. cells i N;...;N each with two possible statesþ The final stage of the Hawking evaporation, according to i 0; 1 (cellð ¼À is, e.g., whiteÞ or black). The discrete time the stringy GUP scenario, contains a Planck-size remnant evolutionf g with the elementary time step t is described with a maximal temperature Â Â , but thermody- max by the N-N transition rule namically inert. The remnant behaves¼ like an elementary particle—there are no internal degrees of freedom to excite i 1;i;i 1 i0 i t t : in order to produce a heat absorption or emission. ð À þ Þ! ¼ ð þ Þ To understand the heat exchange in the last live stage of 0; 0; 0 0; 0; 0; 1 0; 0; 1; 0 0; ð Þ! ð Þ! ð Þ! (74) the world-crystal black hole we cannot use the long- 1; 0; 0 1; 1; 1; 0 0; 0; 1; 1 0; wavelength formula (70). Instead, we must turn to Eq. ð Þ! ð Þ! ð Þ! (46). Since in our scenario the micro black hole disappears 1; 0; 1 0; 1; 1; 1 0; ð Þ! ð Þ! at the critical temperature Âmax, it is more appropriate to write the mass-temperature formula (46) in the form with the Born-von Karman periodicity condition i ¼ i 2N 1. þTheþ cells can be algebraically represented by orthonor- p 2 2 a 2 m Âmax Â 1 Â ; (72) mal vectors ’ ð À Þ ffiffiffi À 8 where t is the Heaviside step function. For the specific 4 heat thisð impliesÞ t 1 C Â Â p2k 3Â: (73) 3 ’À ð max À Þ B ffiffiffi t So, in contrast to the stringy result, a world-crystal black hole exchanges heat with its environment by radiation until 2 the last moment of its existence, and unlike the x Schwarzschild black hole, the heat exchange with the environment increases in the final stage of its evaporation. FIG. 2 (color online). CA with discrete time evolution de- In addition, because of the function in (73), the transition scribed by Eq. (74) and with the periodicity condition i ¼ from the universe with the world-crystal black hole to the i 4. The right-hand panel shows an equivalent graphical rep- one without it is of first order. resentation.þ

084030-8 UNCERTAINTY RELATION ON A WORLD CRYSTAL AND ... PHYSICAL REVIEW D 81, 084030 (2010) 0 1 0 mutator . 0 0 . 01 0 0 1 N 0 . 1; N 1 0 . 1;...N 0 1; À ¼ . À þ ¼ . ¼ 1 10 1 0 ÁÁÁ 0 B C B C B C B 1 C B 0 C B 0 C 0 01 0 1 ÁÁÁ 0 1 B C B C B C ^ ^ i @ A @ A @ A X; P B . . ÁÁÁ . C: (79) ½ ¼2 B . . . . C N N 1. On the basis spanned by i the elementary B C timeÀ step¼ evolutionþ operator is B 00 01C B C B 10ÁÁÁ 10C iH^ t B C U^ t eÀ @ ÁÁÁ A ð ¼ Þ¼ In deriving (79) we have used the periodicity condition 01 1 2N. By comparing this with the evolution operator 0 10 1 (75À) we¼ have i = 2N 1 eÀ ð ð þ ÞÞ . . ; (75) ¼ B . . C ^ ^ ^ B . . C X; P i cos H !N=2 t ; (80) B C ½ ¼ ½ð À Þ B 10C which for small t gives @B AC which, among others, defines the Hamiltonian H^ . The t2 i = 2N 1 X^ ; P^ i 1 H^ ! =2 2 : (81) prefactor eÀ ð ð þ ÞÞ is known as ’t Hooft’s phase con- ½ ’ À 2 ð À N Þ 2N 1 vention. Because U^ þ 1^ one can diagonalize U^ as ¼À In addition, we deduce from (75) and (78) that i = 2N 1 i 2N = 2N 1 U^ diag eÀ ð ð þ ÞÞdiag eÀ ðð Þ ð þ ÞÞ; ...; 1; ...; ¼ ð sin H^ ! =2 t P^ ; (82) i 2N = 2N 1 N e ðð Þ ð þ ÞÞ : ½ð À Þ ¼ Â Þ which is compatible with the result (20). From (82) follows If we denote the eigenstates of H^ as n , we find that that H^ depends only on P^ but not on X^ . So P^ and H^ are j i N simultaneously diagonalizable. By defining the operators 1 2n ^ n exp i ‘ ‘; (76) K as p 2N 1 Æ j i¼ 2N 1 ‘ N À þ ¼ÀX þ iX^ iX^ K^ eÀ P^; K^ Pê ; (83) with n N;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi...;N and the ensuing ‘‘energy’’ spectrum þ ¼ À ¼ ¼À reads (K^ K^ y ), so that À ¼ þ ^ 1 2 2N 1 2 H n !N n n ;!N : (77) K^ n þ sin n n 1 ; j i¼ þ 2 j i 2N 1 t þj i¼ 2 2N 1 j þ i ð þ Þ þ (84) 2N 1 2 The energy values En resemble the spectrum of the har- K^ n þ sin n 1 n 1 ; monic oscillator, except that the n’s are bounded and can Àj i¼ 2 2N 1 ð À Þj À i attain negative values. We shall be coming back to this þ one can persuade itself that H^ and K^ close the deformed issue shortly. algebra which in the large N limit (i.e.,Æ in the small or t Position and momentum can be represented by operators limit) reduces to with the matrix representations 2H^ N 1 00 0 H;^ K^ !K^ ; K^ ; K^ ; (85) ðÀ þ Þ ÁÁÁ ½ Æ¼Æ Æ ½ þ À¼À ! 0 0 N 2 001 ðÀ þ Þ with ! ! . ^ . . . X B . . . . C; Note¼ that for1 large N one can identify (77), (84), and (85) ¼ B C B 0 N 0 C with the representation of SU 1; 1 known as the discrete B ÁÁÁ C series D D (cf. Ref. [65ð ]). Generally,Þ the Lie alge- B C 1þ=2 È 1À=2 B 0 0 NC @ ÁÁÁ À A bra Dkþ DkÀ is defined through the relations 0100 1 È ÁÁÁ À L^ k; m m k k; m ; 10 1 0 0 3j i¼ð þ Þj i 0 À ÁÁÁ 1 ^ 1 B 0 10 1 0 C L k; m m 2k m 1 k; m 1 ; P^ B À ÁÁÁ C; (78) þj i¼ ð þ Þð þ Þj þ i (86) 2i B . . . . C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ B . . . C L^ k; m n 2k 1 m k; m 1 ; B C Àj i¼ ð þ À Þ j À i B C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 00 01C L^3; L^ L^ ; L^ ; L^ 2L^3: B ÁÁÁ C ½ Æ¼Æ Æ ½ þ À¼À B 10 10C B ÁÁÁ À C Here m k k; k 1 ; k 2 ; ... and k @ A þ ¼Æ Æð þ Þ Æð þ Þ ¼ where 2= 2N 1 . With these we obtain the com- 1 ; 1; 3 ; 2; ...is the so-called Bargmann index which labels ¼ ð þ Þ 2 2

084030-9 PETR JIZBA, HAGEN KLEINERT, AND FABIO SCARDIGLI PHYSICAL REVIEW D 81, 084030 (2010) the representations. From this we have that D1þ=2 D1À=2 i E i È x^ ; p^ j i 1 j; (89) corresponds to ½ ¼ @ À Ep 1 1 L^ ;m m 1=2 ;m ; where E is the energy scale of the particle to which the 3 2 ¼ð þ Þ 2 deformed Lorenz boost is to be applied, while E is the p Planck energy. This suggests that they have an energy- 1 1 L^ ;m m 1 ;m 1 ; (87) dependent Planck ‘‘constant’’ E 1 E= . Their þ 2 ¼ð þ Þ 2 þ Ep @ð Þ¼@ð À Þ model also implies that E 0 for E Ep. For energies 1 1 ð Þ! ! L^ ;m m ;m 1 : much below that Planck@ regime, the usual Heisenberg À 2 ¼ 2 À commutators are recovered, but when E Ep one has ’ Identification with the large- N limit of (77) and (84) is Ep 0. So the Planck energy is not only an invariant ^ ^ ^ ð Þ’ established when we identify H in (87) with !L3, K with in@ this model, but the world looks also apparently classical L^ , and set m n. Æ at the Planck scale, similarly as in ’t Hooft’s proposal. ÆAt this stage¼ one can invoke ’t Hooft’s loss of informa- The connections of the DSR model with our proposal are tion condition [38,39], and project out the negative part of at this point self-evident. Our GUP (22) and (24) implies the spectra. A plausible rationale for this step can be found, that, at the boundary of the Brillouin zone, when p^ c h i ! e.g., in irreversibility of computational process due to a =2, i.e. for Planck energies E 2p2=a E , the fun- ’ð Þ p finite storage capacity [66]. @damental commutator vanishes ffiffiffi After the negative energy spectrum is removed (erased), X^ ; P^ 0; (90) we obtain only the positive discrete series D1þ=2 (i.e., rep- ½ ’ resentation where m 0; 1; 2; ...), and the Hamiltonian and since ¼ morphs into a non-negative spectrum Hamiltonian Hþ. The usual W 1 algebra of the quantum harmonic oscil- ÁXÁP * 0; (91) ð Þ lator emerges after we introduce the following mapping in lattice quantum mechanics at short wavelengths allows for the universal enveloping algebra of SU 1; 1 : classical behavior, that is uncertainties of two complemen- ð Þ 1 1 tary observables can be simultaneously zero. a^ L^ ; a^y L^ : (88) However, if we express the fundamental commutator ¼ À ¼ þ L^ 1=2 L^ 1=2 (22) of our model in terms of energy, using the exact 3 þ 3 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relation (40), we find (for a‘p) The latter gives a one-to-one (nonlinear) mapping between ¼ the deterministic cellular automaton system (with informa- a2 E^2 X^ ; P^ f x i 1 f x : (92) tion loss) and the quantum harmonic oscillator. The reader 2 ½ ð Þ¼ @ À 8 Ep ð Þ will recognize the mapping Eq. (88) as the noncompact analog [67] of the well-known Holstein-Primakoff repre- This means that the deforming term in our model is qua- sentation for SU 2 spin systems [68]. dratic in the energy, instead of the linear dependence in the ð^ Þ ^ ^ energy of the DSR model (89). Our operators X, P, and I may be viewed as the same cellular automaton E 2 algebra as discussed in Sec. III. ð Þ Thus we can conclude that in the Planckian scale the IX. DISCUSSION AND SUMMARY system must behave deterministically—which is one of the defining property of cellular automata. It is only at It should be noted that the present lattice generalization low energies when the loss of information leads to the of the uncertainty principle is not an approximate descrip- emergent degrees of freedom resulting in the usual quan- tion, but it is an exact formula necessarily implied by our tum mechanical description. model of lattice space time. The great majority of the GUP research has always borrowed the deformed commutator x;^ p^ i 1 p^ 2 either from string theory, or from B. Double special relativity ½heuristic¼ arguments@ð þ Þ about black holes [55,56]. To be pre- The second application relates to the idea of double (or cise, even in string theory [55] the formula expressing the doubly or deformed) special relativity (DSR) (see, e.g., GUP is not derived from the basic features of the model, Refs. [37,69]). The general idea is that if the Planck length but instead it is deduced from high-energy gedanken ex- is a truly universal quantity, then it should look the same to periments of string scatterings. In contrast to this we have any inertial observer. This demands a modification (defor- derived all results from a simple lattice model of space- mation) of the Lorenz transformations, to accommodate an time, and from the analytic structure of the basic commu- invariant length scale. In Ref. [37] the nonlinearity of the tator (22). deformed Lorenz transformations lead the authors to novel We have calculated the uncertainties on a crystal-like commutators between spacetime coordinates and mo- universe whose lattice spacing is of the order of Planck menta, depending on the energy length—the so-called world crystal. When the energies lie

084030-10 UNCERTAINTY RELATION ON A WORLD CRYSTAL AND ... PHYSICAL REVIEW D 81, 084030 (2010) near the border of the Brillouin zone, i.e., for Planckian GUP goes to zero at Planck energy. For both models, the energies, the uncertainty relations for position and mo- world should therefore be manifestly ‘‘classical’’ in the menta do not pose any lower bound on the associated Planck regime, a feature very different from the common uncertainties. Hence the world-crystal universe can beco- believe. This is a striking prediction supported by both medeterministic at Planckian energies. In this high-energy models, although the lines of thought followed in the two regime, our lattice uncertainty relations resemble the research paths are completely different and independent. double special relativity result of Magueijo and Smolin. Moreover, this aspect presents also a strong resemblance The scenario in which the universe at Planckian energies with the results obtained in the research line of determi- is deterministic rather than being dominated by quantum nistic quantum mechanics. fluctuations is a starting point in ’t Hooft’s deterministic quantum mechanics. ACKNOWLEDGMENTS With the generalized uncertainty relation at hand we One of us (P.J.) is grateful to G. Vitiello for instigating have been able to derive a new mass-temperature relation discussions. This work was partially supported by the for Schwarzschild micro black holes. In contrast to stan- Ministry of Education of the Czech Republic (Research dard results based on Heisenberg or stringy uncertainty Plan No. MSM 6840770039), and by the Deutsche relations, our mass-temperature formula predicts both fi- Forschungsgemeinschaft under Grant No. Kl256/47. F. S. nite Hawking’s temperature and a zero rest-mass remnant acknowledges financial support by the National Taiwan at the end of the evaporation process. Especially the ab- University under the Contract No. NSC 98-2811-M-002- sence of remnants is a welcome bonus which allows to 086, and thanks ITP Freie Universita¨t Berlin for warm avoid such conceptual difficulties as entropy/information hospitality. problem or why we do not experimentally observe the remnants that must have been prodigiously produced in the early universe. APPENDIX: LANDAUER PRINCIPLE Apart from the mass-temperature relation we have also Here we wish to provide an alternative derivation of the computed two relevant thermodynamic characteristics, mass-temperature formula (55) based on Landauer’s prin- namely, entropy and heat capacity. Particularly the heat ciple. To this end we consider an ensemble of unpolarized capacity provided an important insight into the last life photons that are going to deliver to a micro black hole one stage of the world-crystal micro black holes. In contrast to single bit of information per particle. In order to be sure the stringy result, our result indicates that world-crystal that each photon delivers only one bit of information— micro black hole exchanges heat with its environment namely, the information that it is there, somewhere inside (radiate) till the last moment of its existence, and unlike in the black hole—its position uncertainty must be ‘‘maxi- the Schwarzschild micro black hole, the heat exchange mal’’, i.e., it should not be smaller than Schwarzschild’s with the environment increases in the final stage of the radius RS as otherwise the photon would deliver to black evaporation. In addition, the transition from the universe hole also extra bits of information concerning its entry with the world-crystal micro black hole to the one without point (or better sector) on the horizon. At the same time is of the first order. its wavelength should not be bigger than RS, as otherwise Since the world-crystal physics allows for deterministic the photon would bounced off the black hole without description of the physics at Planckian energies, we have getting trapped. In this view, the position uncertainty of a included in this paper a discussion of ’t Hooft’s periodic photon in the ensemble must be of order of Schwarzschild’s radius R i.e., ÁX R . Factor is automaton model which gives at low-energy scales rise to a S ’ S genuine quantum harmonic oscillator. In addition, such an Bekenstein’s deficit coefficient which ensures a correct automaton has a close connection with our world-crystal Hawking’s formula in a continuum limit. paradigm. Here we have rederived ’t Hooft’s result in a new An extra bit of information added to the micro black way. In contrast to ’t Hooft derivation [39] we have hole will increase its energy at least by amount ÁE so that matched the algebra of the automaton variables with the [cf. (43)] SU 1; 1 algebra, and in contrast to Ref. [65] we have c 2 ð Þ ÁX ÁE 1 ÁE 2 : (A1) worked with a different set of dynamical variables. In the @ 2 2 ’ 2 À 2 c ð Þ contraction limit (i.e., in the limit of many lattice @ spacings—low-energy limit) we have recovered the ca- In the following we denote ÁE simply as E to stress that nonical W 1 algebra and were able to identify the ‘‘emer- ÁE is an energy increase due to one photon. With the gent’’ harmonicð Þ oscillator variables. explicit form for Planck’s energy There are several aspects of the double special relativity c 19 that are worth noting in the connection with our general- E p 0:61 10 GeV; (A2) ¼ 2@‘ Â ized uncertainty relation. Essentially, we have seen that the p fundamental commutator in DSR as well as in our lattice the relation (A1) can be cast to

084030-11 PETR JIZBA, HAGEN KLEINERT, AND FABIO SCARDIGLI PHYSICAL REVIEW D 81, 084030 (2010) c a2‘ E photon we have that ÁX p : (A3) 2@E 8 ’ À Ep E k T: (A5) ’ B If we use further the fact that, R ‘ m, where m is the S ¼ p relative mass of the black hole in Planck units, i.e., m Relation (A5) basically expresses equipartition law for an 2 ¼ unpolarized photon in the outgoing Hawking radiation. M=Mp (Mp Ep=c ), we can rewrite (A3) as ¼ Defining the Planck temperature Tp 2Ep=kB 2 32 ¼ Ep a E 3 10 K, and measuring the temperature in terms of 2m : (A4) Â E 8 Planck units as a relative temperature Â T=Tp, we can ’ À Ep ¼ rewrite Eq. (A4) in the form According to the Landauer principle [70], when a single bit of information is erased (like in the black hole) the 1 2m 22Â; (A6) amount of energy dissipated into environment is at least ¼ 2Â À kBT ln2, where kB is Boltzmann’s constant and T is the temperature of the erasing environment (in our case the where we identify a= 2p2 and set , in order ¼ ð Þ ¼ micro black hole). Since the liberated energy per bit of lost to agree with (55) and with Hawking’sffiffiffi formula (56) in the information can not be grater the energy E of the carrier continuum limit.

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