GUP parameter from quantum corrections to the Newtonian potential

We propose a technique to compute the deformation parameter of the generalized uncertainty principle by using the leading quantum corrections to the Newtonian potential. We just assume General Relativity as theory of Gravitation, and the thermal nature of the GUP corrections to the Hawking spectrum. With these minimal assumptions our calculation gives, to first order, a specific numerical result. The physical meaning of this value is discussed, and compared with the previously obtained bounds on the generalized uncertainty principle deformation parameter.


I. INTRODUCTION
Research on generalizations of the uncertainty principle (GUP) of quantum mechanics has nowadays a long history [1]. One of the main lines of investigation focuses on understanding how the Heisenberg Uncertainty Principle (HUP) should be modified once gravity is taken into account. Given the pivotal role played by gravitation in these arguments, it is not surprising that the most relevant modifications to the HUP have been proposed in string theory, loop quantum gravity, deformed special relativity, and studies of black hole physics [2][3][4][5][6][7].
As it is well know, the dimensionless deforming parameter of the GUP, henceforth denoted by β, is not (in principle) fixed by the theory, although it is generally assumed to be of order one (this happens, in particular, in some models of string theory, see for instance Ref. [2]).
There have been many studies that aim at setting bounds on β, for instance Refs. [8], [9], [10]. In these works, a specific (in general, non linear) representation of the operators in the deformed fundamental commutator is utilized 1 in order to compute corrections to quantum mechanical quantities, such as energy shifts in the spectrum of the hydrogen atom, or to the Lamb shift, the Landau levels, Scanning Tunneling Microscope, charmonium levels, etc. The bounds so obtained on β are quite stringent, ranging from β < 10 21 to β < 10 50 . A further group of bounds can be found in Refs. [11] and [12], where a deformation of classical Newtonian mechanics is introduced by modifying the standard Poisson brackets in a way that resembles the quantum commutator where β 0 = β/m 2 p . However, in the limit β → 0, Ref. [11] recovers only the Newtonian mechanics but not GR, and GR corrections must be added as an extra structure. Clearly, the physical relevance of this approach and the bound that follows for β remains therefore questionable.
Finally, in Refs. [13] and [14], the authors consider the gravitational interaction when evaluating bounds on β. They use a covariant formalism which firstly is defined in Minkowski space, with the metric η µν = diag(1, −1, −1, −1), which can be easily generalized to curved space-times via the standard procedure η µν → g µν . In addition, these papers, as the previous ones, start from a deformation of classical Poisson brackets, although posited in covariant form. From the deformed covariant Poisson brackets, they obtain interesting consequences, like a β-deformed geodesic equation, which leads to a violation of the Equivalence Principle. This formalism remains covariant when β → 0 and it reproduces the standard GR results in the limit β → 0 (unlike papers as Ref. [11]).
Among the papers which consider the gravitational interaction when evaluating bounds on β, it is Ref. [15]. This approach differs from the previous ones because Poisson brackets and classical Newtonian mechanics remain untouched. Additionally, GR and standard quantum mechanics are recovered, when β → 0. Therefore, the Equivalence Principle is preserved, and the equation of motion of a test particle is still given by the standard geodesic equation. The bounds on β proposed by papers which take into account gravity range from β < 10 19 , for those papers admitting a violation of equivalence principle, to β < 10 69 for the papers preserving the aforesaid principle.
In the present paper, we exhibit a computation of the value of β obtained by comparing two different low energy (first order in ) corrections for the expression of the Hawking temperature. The first is due to the GUP, and therefore involves β. The second correction, instead, is obtained by including the deformation of the metric due to quantum corrections to the Newtonian potential. Then we demand the two corrections to be equal (at the first order), and this yields a specific numerical value for β. It results to be of order of unity, in agreement with the general belief and with some particular models of string theory.

II. GUP-DEFORMED HAWKING TEMPERATURE
One of the most common forms of deformation of the HUP (as well as the form of GUP that we are going to study in this paper) is which, for mirror-symmetric states (with p 2 = 0), can be equivalently written in terms of commutators as As is well known from the argument of the Heisenberg microscope [18], the size δx of the smallest detail of an object, theoretically detectable with a beam of photons of energy E, is roughly given by since larger and larger energies are required to explore smaller and smaller details. From the uncertainty relation (3), we see that the GUP version of the standard Heisenberg formula (5) is which relates the (average) wavelength of a photon to its energy E 2 . Conversely, using relation (6), one can compute the energy E of a photon with a given (average) wavelength λ ≃ δx. To compute the thermal GUP corrections to the Hawking spectrum, we follow the arguments of Refs. [19][20][21][22][23][24][25], and we consider an ensemble of unpolarized photons of Hawking radiation just outside the event horizon of a Schwarzschild black hole. From a geometrical point of view, it is easy to see that the position uncertainty of such photons is of the order of the unmodified Schwarzschild radius, i.e., r H = 2 G M . An equivalent argument comes from considering the average wavelength of the Hawking radiation, which is of the order of the geometrical size of the hole. We can estimate the uncertainty in photon position to be δx ≃ 2 µ r H , where the proportionality constant µ is of order unity and will be fixed soon. According to the equipartition principle, the average energy E of unpolarized photons of the Hawking radiation is simply related with their temperature by E = T . Inserting the aforesaid expressions for the uncertainty in the photon position and for the average energy into formula (6), we obtain In order to fix µ, we consider the semiclassical limit β → 0, and require formula (7) to predict the standard semiclassical Hawking temperature, namely T (β → 0) = T H , This fixes µ = π, thus we have This is the mass-temperature relation predicted by the GUP for a Schwarzschild black hole. Of course this relation can be easily inverted, to get However, since the term proportional to β is small, especially for solar mass black holes with M ≫ m p , we can expand in powers of β, namely and it is evident that to zero order in β, we recover the usual Hawking formula (8).
Once again we stress that we are assuming that the correction induced by the GUP has a thermal character, and, therefore, it can be cast in the form of a shift of the Hawking temperature. Of course, there are also different approaches, where the corrections do not respect the exact thermality of the spectrum, and thus need not be reducible to a simple shift of the temperature. An example is the corpuscular model of a black hole of Ref. [26].
In this model, the emission is expected to gain a correction of order 1/N , where N ∼ (M/m p ) 2 is the number of constituents, and it becomes important when the mass M approaches the Planck mass.

III. TEMPERATURE FROM A DEFORMED SCHWARZSCHILD METRIC
A. Leading quantum correction to the Newtonian potential After early results by Duff [27], the leading quantum correction to the Newtonian potential has been computed by Donoghue, by assuming General Relativity as fundamental theory of Gravity. In a series of beautiful papers (see for instance Ref. [28]) he reformulated General Relativity as an effective field theory, and, in particular, he considered two heavy bodies close to rest. The leading quantum correction derived from this model shows a long-distance quantum effect. More recently, Donoghue and other authors found that the gravitational interaction between the two objects can be described by the potential energy [29] U The first correction term does not contain any power of , so it is a classical effect, due to the non-linear nature of General Relativity. However, the second correction term, i.e., the last term of (12), is a true quantum effect, linear in . The potential generated by the mass M reads B. Effective potential from the metric Now we consider the effective potential produced by a metric of the very general class where r = |x| = (x 2 1 + x 2 2 + x 2 3 ) 1/2 , and x 1 , x 2 , x 3 are the standard Cartesian coordinates. Particular cases of the metric (14) is the Schwarzschild metric, in the standard form as well as in harmonic coordinates with R = r − GM .
It can be easily seen 3 that any general metric of the form can be put in the form (14). In fact, Eq. (15) is equivalent to Once the metric is in the form of (14), in Cartesian coordinates, then, with well known procedures [30], it is easy to show that the effective Newtonian potential 4 is of the form or, equivalently, C. Metric mimicking the quantum corrected Newtonian potential At this point, we can write down the metric which is able to mimic the quantum corrected Newtonian potential proposed by Donoghue. Recalling (13), we have Let us now define Therefore, F (r) will now be of the form and it is evident that when r is large, then |ǫ(r)| ≪ 2GM/r.

D. Computing β
We can legitimately wonder what kind of (deformed) metric would predict a Hawking temperature like the one inferred from the GUP in relation (11), for a given β.
Since we are interested only in small corrections to the Hawking formula, we can consider a deformation of the Schwarzschild metric of the following kind 5 where ǫ(r) is an arbitrary, small, smooth function of r. We note that the deformation (20) makes sense when |ǫ(r)| ≪ GM/r. We can also introduce a regulatory small parameter ε and, thus, we can write ǫ(r) ≡ εφ(r). At the end of the calculation, ε can go to unity. Of course, we look for the lowest order correction in the dimensionless parameter ε. The horizon's equation, i.e., F (r) = 0, now reads Such equations can be solved, in a first approximation in ε, as follows. First, we formulate (21) in a general form It is obvious that if ε is set equal to zero, then the solution will be x 0 = a. If ε is slightly different from zero, then we can try a test solution of the form x 0 = a + η(ε) where η(ε) → 0 for ε → 0. Substituting the aforesaid test solution in (22), we get x 0 = a + εf (x 0 ) which means η = εf (a + η). To first order in η, we have η = ε[f (a) + f ′ (a)η] from which we obtain η = εf (a) 1−εf ′ (a) . Therefore, to first order in ε, the general solution of (22) reads x 0 = a + εf (a) 1−εf ′ (a) . Applying this formula to (21), we get the solution where a = 2GM . The Hawking temperature is given by From Eq. (20), one gets It is useful to write the solution (23) in the compact form and, therefore, λ ∼ ε, |λ| ≪ 1. Then 5 Recently, it was argued that in the special case in which ǫ(r) ∼ 1/r 2 , the specific metric (20) could have some drawbacks in the context of GUP formalism [31]. However, none of those drawbacks appear here and, thus, there is no problem to employ (20) in our present study.
Therefore, the deformed Hawking temperature reads It is noteworthy that the only function φ(r) that annihilates the first-order in ε temperature correction term is the solution of the differential equation 2φ(r) + rφ ′ (r) = 0, namely φ(r) = A/r 2 , where A is an arbitrary constant. In particular, for the function φ(r) = G 2 M 2 /r 2 , the coefficient of ε in (27) is zero, and the coefficient of ε 2 is −1/16. It is also interesting to investigate what kind of function will eliminate the second-order in ε correction term. This function will be the solution of the differential equation r 2 φ ′′ (r) + 2rφ ′ (r) − φ(r) = 0 which is an Euler equation. Its characteristic equation is of the form So, the functions which remove the ε 2 -correction term are φ 1 (r) = r −|λ1| and φ 2 (r) = r λ2 . We are now in the position to compute the temperature generated by the metric (19), by simply employing (27). Therefore, the metric-deformed Hawking temperature is of the form 6 while the GUP-deformed Hawking temperature reads By comparing the two respective first-order correction terms in the two aforesaid expansions, we obtain Using now expression (18) for ǫ(r), we get and using (30), the parameter β will get the value

IV. CONCLUSIONS
In this work we have computed the value of the deformation parameter β of the GUP. We obtain this result by computing in two different ways the Hawking temperature for a Schwarzschild black hole. The first way consists in using the GUP (in place of the standard HUP) to compute the Hawking formula. In this way we get an expression of the temperature containing a correction term depending on β, i.e., the GUP-deformed Hawking temperature (11).
The second way involves the consideration of the quantum correction to the Newtonian potential, computed years ago by Donoghue and others. The corrections to the Newtonian potential imply naturally a quantum correction to the Schwarzschild metric. Therefore, the Hawking temperature computed through this quantum corrected Schwarzschild metric result to get corrections in respect to the standard Hawking expression, i.e., the metric-deformed Hawking temperature (28).
The request that the first-order corrections of the two different expressions of Hawking temperature must coin-cide, fixes unambiguously the numerical value of β to be 82π/5.
Finally, a couple of comments are in order here. First, this numerical value is of order one, as expected from several string theory models, and from versions of GUP derived through gedanken experiments. In particular, this is the first time, to our knowledge, that a specific value is obtained for β by starting from the minimal assumptions we made. Second, as we know, in the last years much research has focused on the experimental bounds of the size of β, and several experiments have been proposed to test GUPs in the laboratory. In fact, it has been shown that one does not need to reach the Planck energy scale to test GUP corrections. Among the more elaborated proposals, where conditions can be created in a lab, are those of the groups of Refs. [32][33][34]. However, it is also worth of note that the best bounds on β presented in the literature are still by far much larger than the value computed here. This could require, presumably, a big leap in the experimental designs and techniques in order to search this region for the parameter β.