Uncertainty Relation on World Crystal and Applications to Micro Black Holes

1,2, 1, 3,4, Petr Jizba, ∗ , † and Fabio Scardigli ‡ 1ITP, Freie Universit¨at Berlin, Arnimallee 14 D-14195 Berlin, Germany 2FNSPE, Czech Technical University in Prague, B˘rehov´a7, 115 19 Praha 1, Czech Republic 3Leung Center for Cosmology and Particle Astrophysics (LeCosPA), Department of Physics, National Taiwan University, Taipei 106, Taiwan 4Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Abstract still to be written

PACS numbers:

I. INTRODUCTION black hole remnants. Such a decay behavior has the ad- vantage that is is more likely to be observed experimen- The geometry of Einstein and Einstein-Cartan spaces tally in the LHC when it will restarts operations in the can be considered as being a manifestation of the defect near future. structure of a world crystal whose lattice spacing ilon is of the order of the Planck length ℓp [1–3]. is II. DIFFERENTIAL CALCULUS ON A due to rotational defects, torsion to translational defects. LATTICE The elastic deformations do not alter the defect structure, i.e., the geometry is invariant under elastic deformations. If we assume these to be controlled by a second-gradient On a lattice of spacing ǫ in one dimension, the lattice elastic action, the forces between local rotational defects, sites lie at xn = nǫ where n runs through all integer i.e., between curvature singularities, are the same as in numbers. There are two fundamental derivatives of a Einstein’s theory [4]. Moreover, the elastic fluctuations function f(x): of the displacement fields possess logarithmic correlation 1 functions at long distances, so that the memory of the ( f)(x) = [f(x + ǫ) f(x)] , crystalline structure is lost over large distances. In other ∇ ǫ − 1 words, the Bragg peaks of the world crystal are not δ- ( ¯ f)(x) = [f(x) f(x ǫ)] . (1) function-like, but display the typical behavior of a quasi- ∇ ǫ − − long-range order, similar to the order in a Kosterlitz- They obey the generalized Leibnitz rule Thousless transition in two-dimensional superfluids [5]. The purpose of this note is to study the generalied un- ( fg)(x) = ( f)(x)g(x)+ f(x + ǫ)( g)(x) , ∇ ∇ ∇ certainty principle (GUP) associated with the quantum ( ¯ fg)(x) = ( ¯ f)(x)g(x)+ f(x ǫ)( ¯ g)(x) . (2) field theory on the world crystal and to derive physical ∇ ∇ − ∇ consequences from it. We shall find that the GUP implies On a lattice, integration is performed as a summation: that quantum physics tends to the classical limit at the scale of the lattice spacing. Our results have interesting dxf(x)= ǫ f(x), (3) implications for the physics of micro black holes to be Z Xx discussed in Section V.

There exists also a close contact with two interesting where x runs over all xn. mainstreams of contemporary research. One is ’t Hooft’s For periodic functions on the lattice or for functions approach to deterministic quantum mechanics, the other vanishing at the boundary of the world crystal, the lat- deformed (or double) special relativity. tice derivatives can be subjected to the lattice version of On a more phenomenological side, our version of GUP integration by parts: provides a nice way out of two long standing puzzling situations: the final Hawking temperature of a decaying f(x) g(x)= g(x) ¯ f(x) , (4) ∇ − ∇ micro black hole remains finite, in contrast to the infi- Xx Xx nite temperature of the standard result where the Heisen- f(x) ¯ g(x)= g(x) f(x) . (5) berg’s uncertainty principle operates. On the world crys- ∇ − ∇ Xx Xx tal, the final mass of the evaporation process is zero, thus avoiding the problems caused by the existence of massive One can also define the lattice Laplacian as 1 ¯ f(x)= ¯ f(x)= [f(x+ǫ) 2f(x)+f(x ǫ)] (6) ∇∇ ∇∇ ǫ2 − − ∗Electronic address: [email protected] †Electronic address: [email protected] which reduces in the continuum limit to an ordinary ‡Electronic address: [email protected] Laplace operator ∂2. Note that the lattice Laplacian can 2 also be expressed in terms of the difference of the two The generator Mˆ corresponds to a rotation, while Pˆ1 lattice derivatives: and Pˆ represent two translations. In the limit ǫ 0, 2 → 1 the Lie algebra of E(2) contracts to the standard Weyl– ¯ f(x)= f(x) ¯ f(x) . (7) ˆ ˆ ˆ ∇∇ ǫ ∇ − ∇ Heisenberg algebra: Xǫ xˆ, Pǫ pˆ, Iǫ 1. Thus   mathematically ordinary QM→ is obtained→ from→ lattice QM The calculus can be easily be extended to any number by a contraction of the E(2) algebra via the limit ǫ 0 D of dimensions [6]. of the deformation parameter ǫ. → All functions on the lattice can be Fourier-decomposed III. POSITION AND MOMENTUM with wave numbers in the Brillouin zone: OPERATORS π/ǫ dk f(x)= f˜(k)eikx , (14) Z π/ǫ 2π Consider now the quantum mechanics on a 1D lattice − in a Schr¨odinger-like picture. Wave function are square- with the coefficients integrable complex functions on the lattice, where “inte- ˜ ikx gration” means here summation, and scalar products are f(k)= ǫ f(x)e− . (15) defined by Xx This implies the good-old de Brogllie relation ψ ψ = ǫ ψ∗(x)ψ (x). (8) h 1| 2i 1 2 Xx (ˆpf˜)(k)= ~kf˜(k), (16) It follows from Eq. (4) that and its lattice version f g = ¯ f g , (9) h |∇ i −h∇ | i ( i f˜)(k)= Kf˜(k), ( i ¯ f˜)(k)= K¯ f˜(k) (17) − ∇ − ∇ so that (i )† = i ¯ , and neither i nor i ¯ are hermitian operators.∇ The lattice∇ Laplacian∇ (6), however,∇ is hermi- with the eigenvalues tian. ikǫ K (e 1)/iǫ = K¯ ∗. (18) The position operator Xˆǫ acting on wave functions of ≡ − x is defined by simple multipication with x: From (17) we find the Fourier transforms of the operators Xˆǫ, Pˆǫ, Iˆǫ: (Xˆǫf)(x)= xf(x) . (10) ˆ d Similarly we define the lattice momentum operator Pǫ. (Xˆǫf˜)(k)= i f˜(k) , (19) In order to assure hermiticity we shall relate it to the dk 1 symmetric lattice derivative [8–10]. Using (9) we have (Pˆ f˜)(k)= sin(kǫ)f˜(k) , (20) ǫ ǫ 1 ˆ ˜ ˜ (Pˆ f)(x) = [( f)(x) + ( ¯ f)(x)] (Iǫf)(k) = cos(kǫ)f(k) , (21) ǫ 2i ∇ ∇ 1 With the help of (16) and (21) we can rewrite the com- = [f(x + ǫ) f(x ǫ)] . (11) 2iǫ − − mutation relation (13) as For small ǫ, this reduces to the ordinary momentum op- ˆ ˆ ~ ~ eratorp ˆ i~∂ : [Xǫ, Pǫ]f (x) = i cos(ǫp/ˆ ) f(x) . (22) ≡− x   Pˆǫ =p ˆ + O(ǫ) . (12) IV. UNCERTAINTY RELATIONS ON LATTICE The “canonical” commutator between Xˆǫ and Pˆǫ on the lattice reads We are now prepared to derive the generalized uncer- i tainty relation implied by the previous commutators. We [Xˆǫ, Pˆǫ]f (x) = [f(x + ǫ)+ f(x ǫ)]   2 − shall define the uncertainty of an observable A in a state i(Iˆ f)(x) . (13) ψ by the standard deviation ≡ ǫ The last line defines a lattice-version of the unit operator 2 (∆A)ψ ψ (Aˆ ψ Aˆ ψ ) ψ . (23) as the average over the two neighboring sites. All three ≡ qh | − h | | i | i ˆ ˆ ˆ operators Xǫ, Pǫ, and Iǫ are hermitian under the scalar Following the conventional procedure (see, e.g., Ref. [11– product (8). 13]), we derive the inequality It was noted in [9] that the operators Xˆǫ, Pˆǫ and Iˆǫ generate the Euclidean algebra E(2) in 2D. Indeed, set- 1 1 ∆Xǫ∆Pǫ ψ [Xˆǫ, Pˆǫ] ψ = ψ Iˆǫ ψ ting Mˆ = ǫXˆǫ, Pˆ1 = ǫPˆǫ and Pˆ2 = Iˆǫ we obtain ≥ 2 h | | i 2 h | | i ~ ~ [Mˆ ; Pˆ ] = iPˆ , [Mˆ ; Pˆ ] = iPˆ , [Pˆ ; Pˆ ] = 0 . = ψ cos(ǫp/ˆ ) ψ . (24) 1 2 2 − 1 1 2 2 |h | | i| 3

Let us study two different extremal regimes, first for long Since k lies always inside the Brillouin zone, implying wavelengths where pˆ ψ 0, and second near the bound- that pˆ π~/2ǫ, see that the long-wavelength GUP (30) ary of the Brillouinh zonei → where pˆ π~/2ǫ. To this chengesh i≤ close to the boundary of the Brillouin zone to h iψ → end we first rewrite cos(ǫp/ˆ ~) ψ as h i ~ π ǫ > ˆ 2n ∆Xǫ∆Pǫ Pǫ ψ . (34) ∞ ∞ (ǫp/~) 2 2 − ~ h i cos(ǫp/ˆ ~) = dp ̺(p) ( 1)n , (25) ∼   h iψ Z − (2n)! nX=0 0 As the momentum reaches the boundary of the Brillouin zone, the right-hand side vanishes so that the lattice 2 where ̺(p) ψ(p) . In the first case, ̺(p) is peaked quantum mechanics at short wavelengths shows classical around p ≡0, so| that| the relation (25) becomes approx- ≃ behavior. imately It is worth noting that the uncertainty relation (34) has the same form of that found, on different grounds, by ǫ2p2 cos(ǫp/ˆ ~) = 1 + (p4) . (26) Magueijo and Smolin in [17]. In particular the deforming ψ ~2 h i − 2 O term is also there linear in the momentum. It results 2 2 that in this model the world becomes ”classical” at the where p pˆ ψ. If we now apply the identity ≡ h xi Planck scale, in the sense that no blurred quantities show 2 2 2 up in the measure of fundamental dynamical variables, as Aˆ ψ = (∆A) + Aˆ , (27) h i h iψ similarly devised by ’t Hooft in models on ”deterministic” we obtain from (24) quantum mechanics [18, 19].

~ ǫ2p2 ∆X ∆P > 1 ǫ ǫ 2 − 2~2 V. IMPLICATIONS FOR PHOTONS ∼ ~ ǫ2 = 1 (∆ p)2 + pˆ 2 . (28) 2 − 2~2 h iψ The vector potential of a photon in the Lorentz gauge   in 1+1 dimensions satisfies the wave equation

For mirror-symmetric states where pˆ ψ = 0 this implies h i 1 2 µ 2 µ ∂t A (x, t)= ∂x A (x, t) (35) ~ ǫ2 c2 ∆X ∆P > 1 (∆p)2 . (29) ǫ ǫ ~2 ∼ 2  − 2  A plane wave solution Aµ(x)= ǫµ exp[i(kx ω(k)t)] pos- sesses the well know linear dispersion relation− Here we have substituted ... with (...) since we assume | | that ǫ ℓp (Planckian lattice) and that ∆p is close to ω(k)= c k, (36) zero (this≃ is our original assumption). For Planckian lattices we can neglect in (29) higher where ǫµ is a polarization vector. On a one-dimensional orders in ǫ and write 2 lattice, the operator ∂x is replaced by the lattice Lapla- ¯ ~ ǫ2 cian , and the spectrum becomes, on account of ∆X ∆P > 1 (∆P )2 . (30) ∇∇ ǫ ǫ ~2 ǫ Eq. (6) and (17), ∼ 2  − 2  ω(k) [2 2 cos(kǫ)] 2 kǫ In the second case, where pˆ ψ ~π/2ǫ, we are going ¯ h i → = KK = − = sin , (37) to examine the behavior of (24) at the border of the first c p p ǫ ǫ  2  Brillouin zone, where the averaged momentum takes its which reduces to (36) for ǫ 0. Denoting the energy on maximum value → the lattice ~ω by Eǫ, we obtain the dispersion relation π~ pˆ ψ = . (31) 2ǫ Eǫ 2 pǫ h i = sin (38) ~ c ǫ 2 ~ we expand on the right-hand side of (24):   For small momenta p, this has the expansion cos[π/2 + (ǫp/ˆ ~ π/2)] = sin(π/2 ǫp/ˆ ~) h − iψ h − iψ 2n+1 2 3 ~ ǫ p 4 ∞ ∞ n (π/2 ǫp/ˆ ) E = cp c + O(ǫ ) . (39) = dp ̺(p) ( 1) − , (32) ǫ ~2 Z − (2n + 1)! − 24 nX=0 0 The correction to the continuum dispersion relation E = Under the assumption that ̺(p) is centered around p 2 ≃ pc + O(ǫ ), so that we have ∆Pǫc ∆Eǫ, which allows π~/2ǫ, the first term in the expansion (32) is dominant, us to rephrase (30) as ≈ and the uncertainty relation becomes ~c ǫ2 ~ π ǫ ∆X ∆E 1 (∆E )2 . (40) ǫ ǫ ~2 2 ǫ ∆Xǫ∆Pǫ ~ pˆ . (33) ≥ 2  − 2 c  ≥ 2 2 − h i

4

VI. APPLICATIONS TO MICRO BLACK as a reduced temperature Θ = T/Tp, we can rewrite Eq. HOLES PHYSICS (44) as

1 Let us see which consequences the modified uncertainty 2m = β 2πΘ . (46) relations (30) and (40) have for black holes. In particu- 2πΘ − lar we want to study how mass-temperature relations is In the continuum limit ǫ 0 where β 0 and (43) modified at short distances near the lattice spacing ǫ. For reflects the ordinary uncertainty→ principle,→ this reduces simplicity, we follow here the treatment of Refs. [20–25]. to Small distances can be explored by high-momentum collisions. The size of the smallest detectable details by 1 ~ m = (47) photons of energy E is is δx = c/2E. The lattice version 4πΘ of this is, from Eq. (40): which is the dimensionless version of Hawking’s formula ~c ǫ2 for large black holes: δXǫ 1 2 2 Eǫ (41) ≃ 2Eǫ  − 2 ~ c  ~c We now suppose that the lattice spacing is not precisely T = . (48) 4πkB RS eqaul to the Planck length, but merely prortional to it by a factor a> 0 with some nonextreme value of a. Let It is instructive to compare our mass-temperature re- us denote the Planck energy by lation (46) with that suggested by the so-called stringy version of the GUP : here ~ p = c/2ℓp . (42) refer- E 1 2m = + β 2πΘ . (49) ences Then we can write (41) as 2πΘ ~c E δX βℓ ǫ (43) The physical consequences of our relation (46) are ǫ ≃ 2E − p ǫ Ep quite different from rhose of the stringy result (49), due to the opposite signs of the deformation term. In Fig. 1 where β = a2/8. we compare the two results, and add also the curve for the ordinary Hawking relation (47). Considering m and We now imagine having found a black hole on the lat- tice as a Schwarzschild solution. If the Schwarzschild m radius is much larger than the lattcie spacing ǫ, this will 8 not look much different from the well-known continuum solution. We must avoid too small black holes, for other- 6 wise, completely new physics will set in near the center, due to a pileup of defects. These will cause the melting 4 of the world crystal at a crtical defect density [26], and the emerging general relativity on the world crystal will 2 look completely different from Einstein’s theory. Consider now, as in Refs.[20–25], a photon of Hawking Q radiation just outside the event horizon. The position 0.05 0.1 0.15 0.2 0.25 0.3 uncertainty of such a photon is of the order of the in the Schwarzschild radius R , i.e., δX 2R . This, in S ǫ ≃ S turn, is equal to 2 ℓp m, where m is the reduced mass FIG. 1: Diagrams for the three Mass - Temperature relations, of the black hole, m = M/Mp. According to the above ours (red), Hawking’s (green), and stringy UPS result (blue), arguments, this must be assumed to be much larger than with β = 2, as an example. As a consequence of lattice uncer- unity, in order to avoid the melting transition. In this tainty principle the evaporation ends at a finite temperature. regime we can rephrase (43) as

p Eǫ Θ as functions of time, we can follow the evolution of 2m = E β . (44) a micro black hole from the curves in Fig. 1. For the E − ǫ Ep stringy GUP, the blue line predicts a maximum temper- The energy Eǫ, associated with this limiting photon ature via (41) sets the scale for the thermal energy, so that we 1 obtain the temperature T from the relation Θmax = , (50a) 2π√β Eǫ = πkB T . (45) and a minimum mass Defining the Planck temperature T from = 1 k T , p Ep 2 B p and measuring the temperature in terms of Planck units mmin = β . (50b) p 5

The end of the evaporation process is reached after a long-wavlength relation Eqs. (45). The reason is sim- finite time, the final temperature is finite, and there is a ply that for small Θ, the term proportional to β in (46) remnant of finite rest mass (see Refs.[20–25]). becomes irrelevant. From the standard Heisenberg uncertainty principle we find the green curve, representing the usual Hawking for- mula. Here the evaporation process ends, after a finite VII. COMPARISON WITH ’T HOOFT’S MODEL time, with a zero mass and a worrisome infinite temper- ature. The quantum properties of the model proposed by In contrast to these results, our lattice GUP predicts ’t Hooft’s to derive quantum physics from a determin- the red curve. This yields a finite end temperature istic system [18, 19] displays quite similar properties to 1 our lattice quantum mechanics. Θmax = (51) 2π√β Let us first recapitulate the relevant points of ’t Hooft’s scenario. He starts with the deterministic system consist- with a zero-mass remnant. ing of a finite set of states, (0), (1), ...(N 1) equidis- In the literature, the undesirable infinite final temper- tantly distributed on a circle.{ Time evolution− } is imple- ature prediction of Hawking’s formula has so far been mented by means of discrete clockwise time jumps of cured only with the help of the stringy GUP, which length τ): brings the final temperature to a finite value. This result is, however, also questionable since implies the t t + τ : (ν) (ν + 1modN) (54) existence of finite-mass remnants in the universe. Some → → people like them as candidates for dark matter [27], On a basis spanned by the states (ν), the evolution op- But it has also been pointed out that their existence erator introduced by ’t Hooft is would create other difficult problems in context with the entropy/information problem [cite...], the issue of their 0 1 detectability, and their possibly too large abundance in 0 1 π   the early universe [cite...]. iHτ i N . . U(∆t = τ)=e− =e− . . . (55)  . .    Our result (46) coming from the lattice GUP formula  0 1    (30) solves all these problems my predicting an end to  1 0  the evaporation process at a finite final temperature With the hindsight of quantum mechanics, ’t Hooft in- with zero-mass remnants, which thus seems to be a troduced the phase factor π/N which provides the 1/2- desiderable result. contribution to the energy− spectrum of H. Indeed, if we diagonalize the previous matrix we obtain that the di- A comment is useful concerning the relation (34) at agonal entries are ei2πn/N with n = 0,...,N 1. By the largest momentum near the boundary of the Brillouin denoting the corresponding eigenstates as n we− obtain zone. The relation (45) relating the temperature to the the energy spectrum: | i energy of the emitted photons is deduces from the semi- classical Hawking argument, which is a long-wavelength H 1 2π argument. Here we are confronted with the opposite limit n = (n + 2 ) n , ω . (56) ω | i | i ≡ Nτ of short wavelengths. In fact, Eq. (38) tell us that in this limit Because of this spectrum, H seems at first sight equal to the Hamiltonian of the 1D harmonic oscillator. However, √2 E ~c (52) there are two major differences. First, the eigenvalues of ≃ ǫ ’t Hooft’s H have an upper bound implied by the finite value N value. Only in the continuum limit, i.e., for which for a world lattice of Planckian spacing ǫ = aℓp is N , one can expect a complete correspondence with of the order of the Planck energy, E = (2√2/a) p . → ∞ Considering the associations of the previous paragraph,E the harmonic oscillator. However, even that is not true since in a the harmonic oscillator, the contribution 1/2 to ∆Xǫ m, ∆Pǫ Eǫ T (where m and T are mass and temperature∼ of the∼ micro∼ black hole), we see that close to the ground state energy is a consequence of a “geometric phase” (recall that in the semiclassical Bohr–Sommerfeld the border of the Brillouin zone, where Pǫ π~/(2ǫ), the energy is E , and h i ≃ quantization it originates directly from the Morse index ǫ ≃Ep [29]) and as such it reflects a nontrivial global structure of m T 0 . (53) the theory. So the ground state cannot be decided by any · ≃ ad hoc choice of a phase factor in an evolution operator. Since the temperature in this regime approaches the It is only at the moment when ’t Hooft’s model is able Planck temperature, T Tp,we conclude that the mass to predict in the large N limit the same geometric phase of the micro black hole≃ must go to zero. This is con- as the usual 1D harmonic oscillator when both theories sistent with the previous result (46) deduced from the match at N . → ∞ 6

From the group theoretical reasonings ’t Hooft was able obtained by considering that the above matrix (includ- to find an explicit realization for quantum mechanical ing the phase) should satisfy the condition U N = 1I, we analogues of position and momenta operators. His new get the eigenstates of the form q l = 1 eiπ(2l+1)/N h l| i √N operatorsx ˆ andp ˆ fulfilled the “deformed” commutation with l =0, 1, ..., N. rules Applying the definition of geometric phase, we obtain τ (the first line is the definition for the continuous case): [ˆx, pˆ ] = i 1 H , (57)  − π  T d and the Hamiltonian could be recast into the form Γ = i φ(t) φ(t) Z0 h |dt | i 2 1 1 τ ω → 2 2 2 2 = i ǫ l ql ∂ ql l H = ω xˆ + pˆ + + H . (58) h | i h | i 2 2 2π  4  Xl Xl

1 iπ(2l+1)/N → iπ(2l+1)/N In the continuous limit the Hamiltonian goes to the one = i ǫ e− ∂ e N of the harmonic oscillator, and thex ˆ andp ˆ commuta- Xl Xl tor goes to the canonical one. In that limit the original state space (finite N) changes becoming the infinite di- with x = lǫ mensional state space.

To show the consistency of ’t Hooft’s model we try to VIII. CONCLUSIONS re-derive it in the framework of the non–commutative differential calculus with a particular emphasis on the ********************* geometric phase. ***** We now want to investigate what is the geometric phase (if any) for the above model: we will then have IX. ACKNOWLEDGEMENTS an unambiguous definition of the ground state energy for the harmonic oscillator. Let us consider the eigen- states of the unitary operator Eq. (55). This is easily *********************

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