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

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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 space- time 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  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 phenomenologi- cal 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 alterna- tive 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 fà 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.
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