Advances in High Energy Physics

Volume 2015, Article ID 627264, 15 pages

http://dx.doi.org/10.1155/2015/627264

## Maximally Localized States and Quantum Corrections of Black Hole Thermodynamics in the Framework of a New Generalized Uncertainty Principle

^{1}School of Physics, Nankai University, Tianjin 300071, China^{2}State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China^{3}Bethe Center for Theoretical Physics and Institute of Physics, University of Bonn, Nussallee 12, 53115 Bonn, Germany^{4}Center of Astronomy and Astrophysics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 25 May 2015; Accepted 21 September 2015

Academic Editor: Piero Nicolini

Copyright © 2015 Yan-Gang Miao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

As a generalized uncertainty principle (GUP) leads to the effects of the minimal length of the order of the Planck scale and UV/IR mixing, some significant physical concepts and quantities are modified or corrected correspondingly. On the one hand, we derive the maximally localized states—the physical states displaying the minimal length uncertainty associated with a new GUP proposed in our previous work. On the other hand, in the framework of this new GUP we calculate quantum corrections to the thermodynamic quantities of the Schwardzschild black hole, such as the Hawking temperature, the entropy, and the heat capacity, and give a remnant mass of the black hole at the end of the evaporation process. Moreover, we compare our results with that obtained in the frameworks of several other GUPs. In particular, we observe a significant difference between the situations with and without the consideration of the UV/IR mixing effect in the quantum corrections to the evaporation rate and the decay time. That is, the decay time can greatly be prolonged in the former case, which implies that the quantum correction from the UV/IR mixing effect may give rise to a radical rather than a tiny influence to the Hawking radiation.

#### 1. Introduction

To unify general relativity and quantum mechanics is one of the most difficult tasks because the existing quantum gravity theories are ultraviolet divergent and thus nonrenormalizable. Various candidates of quantum gravity, including string theory [1–6], loop quantum gravity [7], and quantum geometry [8], have pointed out that it is essential to introduce a fundamental length scale of the order of the Planck length and then the corresponding momentum provides a natural UV cutoff. Furthermore, Gedanken experiments of black holes [9] tend to support the existence of a minimal length. One of the approaches to introduce a fundamental length scale is to modify the Heisenberg uncertainty principle (HUP) and then to obtain the so-called generalized uncertainty principle (GUP) [10–14] whose commutation relations between position and momentum operators on a Hilbert space are no longer constants but depend in general on position and momentum operators. In the HUP framework, the restriction upon the position measurement precision does not exist. On the contrary, in the GUP framework that can be regarded as a phenomenological description of quantum gravity effects, a minimal position measurement precision is predicted with the order of the Planck length cm below which the spacetime cannot be probed effectively [12–15]. In other words, a finite resolution appears in the spacetime.

The idea of GUP has been utilized to modify fundamental physical concepts and to analyze the gravity effects on fundamental physical quantities, such as maximally localized states [10, 16] and energy spectra and wavefunctions of some interesting quantum systems [17–24], where the physical states displaying the minimal length uncertainty and the quantum corrections to energy spectra and wavefunctions have been calculated.

On the other hand, the recent applications of GUP to the investigation of quantum black holes have attracted much attention and several achievements have been made [25–34]. For instance, according to Hawking’s black hole thermodynamics [35, 36], a small black hole can radiate continuously and the black hole temperature can rise infinitely during the whole evaporation process until the black hole mass decreases to zero. However, in the framework of GUP the minimal length scale provides a natural restriction that the mass of a black hole cannot be less than the scale of the Planck mass at the end of the evaporation process, and the black hole remnant at the final stage of evaporation has zero entropy, zero heat capacity, and a finite temperature. Moreover, the entropy of a black hole does not strictly obey the area theorem but contains an additional logarithmic correction.

There exist some typical forms of GUP that give rise to modifications of basic concepts in quantum mechanics and to quantum gravity effects on black holes. Here we merely mention two of them that are intimately related to the present paper. One is called the quadratic form [10] (noted by in the figures, figure captions, and tables for the sake of a concise presentation) in which the commutators of position and momentum operators contain an additional quadratic term of momentum operator. The other is called the exponential form [37] (noted by for the same purpose as ) in which the commutators depend on an exponential function of the square of momentum operator. In fact, the former is just the first-order approximation of the Taylor expansion of the latter in the Planck length. Based on the quadratic GUP, the maximally localized states are derived [10] and then developed [16] for a class of quite general GUPs. In the framework of the exponential GUP, the quantum corrections to the thermodynamic quantities of the Schwardzschild black hole are computed and some interesting results related to the black hole evaporation process are obtained, such as the faster evaporation and larger remnant mass than that deduced in the framework of the quadratic GUP.

In the present paper we revisit the maximally localized states and the quantum corrections to the thermodynamic quantities of the Schwardzschild black hole in the framework of our newly proposed GUP [38], the so-called improved exponential GUP (noted by for the same purpose as and ). The motivation emerges directly from our recent interpretation that the origin of the cosmological constant problem may arise from the GUP issue. Through choosing a suitable index introduced in our GUP and considering the UV/IR mixing effect, we can give the cosmological constant that coincides exactly with the experimental value provided by the most recent Planck 2013 results [39]. We are curious about how the maximally localized states are modified and how the thermodynamic quantities of the Schwardzschild black hole are corrected in the framework of our specific GUP. Following the scenario proposed in [16], we obtain for our GUP the maximally localized states in terms of special functions. On the other hand, besides the expected outcomes that the corrected Hawking temperature, entropy, and heat capacity are distinct from that in the frameworks of other GUPs, our significant consequences lie on the two observations: one is that the evaporation rate is extremely small; in other words, the lifetime of black holes is remarkably prolonged, when the UV/IR mixing effect is particularly considered, and the other observation is that the larger the index is, the less radiation the Schwardzschild black hole emits.

The paper is arranged as follows. In the next section, we briefly review our improved exponential GUP with a particularly introduced positive integer , give its minimal length and corresponding momentum, and then derive the maximally localized states. Based on our GUP, we work out in Section 3 the corrected Hawking temperature, entropy, and heat capacity of the Schwarzschild black hole and compare our results with that computed in the frameworks of the Hawking proposal, the quadratic GUP, and the exponential GUP. We then turn to the Hawking evaporation process of the Schwarzschild black hole and calculate the quantum corrections to the evaporation rate and the decay time in Section 4, where we focus on the significant difference between the situations without and with the consideration of the UV/IR mixing effect. Finally, we make a brief conclusion in Section 5.

#### 2. The Improved Exponential GUP and Its Corresponding Maximally Localized States

##### 2.1. Representation of Operators and the Minimal Length

In [38] we propose our improved exponential GUP as follows:where is a dimensionless parameter with the order of unity that describes the strength of gravitational effects and is a positive integer. Note that the parameter introduced in our original form (see [38]) has been set to be in order for us to make a direct comparison with the exponential GUP [37] which is only our special case for . We point out that is very small due to when is taken to be the order of unity in our discussion of micro-black holes. Therefore, the deviation of our GUP from the HUP is kept small because the momentum of a particle is less than the Planck scale even if it is relatively large in some sense, which can be seen obviously from the Taylor expansion of our GUP. For phenomena at the other energy scales much less than the Planck one, such as those analyzed in [20, 40], can have a large upper bound. As being unity corresponds to the phenomena with momenta less than the Planck scale but much larger than that of those phenomena investigated, for instance, in [20, 40], our setup of has no conflict with the present experimental data.

In the momentum space the position and momentum operators can be represented as and the symmetric condition [10], gives rise to the following scalar product of wavefunctions and the orthogonality and completeness of eigenstates: where and mean momentum eigenstates and stands for a wavefunction in the momentum space.

From (1) we get the uncertainty relation In light of the properties and , we reduce the uncertainty relation to be

For simplicity, we take . By using the definition of the Lambert function [41], we write the saturate uncertainty relation as follows: where we have set up and . The Lambert function remains single-valued when it is restricted to be not less than in the range . As a result, it is straightforward to give the minimal length from (7), and its corresponding momentum measurement precision, which can also be regarded as the critical value to distinguish the sub- and trans-Planckian modes [42].

We make two comments on the minimal length and the critical momentum. The first is that is certainly in the order of the Planck momentum, , when is in the order of the Planck length, . The second comment that further demonstrates the minimal length and the critical momentum is that the minimal length never goes to a macroscopic order of magnitude even for a quite great , like ; see [38]. That is, the minimal length is always around the Planck length and the critical momentum is always around the Planck momentum for any , which gives a good property for our improved exponential GUP.

At the end of this subsection we solve (7) and give the momentum measurement precision in terms of the position measurement precision for our use in Section 3:

##### 2.2. Functional Analysis of the Position Operator

The eigenvalue equation for the position operator in the momentum space in the framework of is given by and the wavefunctions, that is, the position eigenfunctions, can be obtained by solving the above equation: where is the gamma function defined as and is the generalized exponential integral function defined as . Note that this piecewise-defined function is continuous at the point .

The scalar product of wavefunctions can be calculated: Note that, according to KMM’s result [10], because of the existence of the minimal length there are no exact eigenvalues for the position operator and the formal eigenfunctions attained by solving the eigenvalue equation are in fact unphysical. For this reason we have to recover information on position by using the maximally localized states which will be analyzed in the following.

##### 2.3. Maximally Localized States

In order to recover information on position the maximally localized states are introduced and used to calculate the average values of the position operator instead of the ordinary position eigenvalues. In [10] the maximally localized states are constructed from the squeezed states satisfying However, it is pointed out [16] that only for a very special GUP, like the quadratic form , can the maximally localized states be obtained in terms of squeezed states. In general, a constrained variational principle should be applied in order to find out maximally localized states. The states are solutions of the following Euler-Lagrange equation in the momentum space [16]: where and are Lagrange multipliers, the function depends on the commutator , and the other parameters emerge from the following relations:Note that is such an operator that its representing function in the momentum space diverges and is not integrable but cannot diverge faster than with when goes to infinity. However, it is not necessary to determine the concrete form of as the maximally localized states appear under the condition . For the details of relevant analysis, see [16].

Furthermore, according to the proposal in [16], when , if defined as has finite limits, one can solve the Euler-Lagrange equation (15) for and give the maximally localized states as follows:whereCorrespondingly, the minimal spread in position for equals

Now we turn to our case in which for the improved exponential GUP, has the form We thus calculate and the finite parameters and , respectively: As a result, we deduce the maximally localized states in the momentum space which can be written as a piecewise-defined function. For , it can be expressed asand for as The minimal length that corresponds to the maximally localized states then reads as

In the remaining contexts of this subsection, we give some interesting properties of the maximally localized states.

First of all, we point out that any two maximally localized states with different positions ’s (see (16)) are no longer mutually orthogonal because of the fuzziness of position space:

Next, we project an arbitrary state onto one maximally localized state and calculate the probability amplitude for the particle being maximally localized around the position . To this end, we write the transformation of a wavefunction from the momentum space to the quasiposition space:For instance, the quasiposition wavefunction (it can be obtained by simply substituting the momentum eigenfunction into (29)) of the momentum eigenfunction with the eigenvalue is always a plane wave but has a specific wavelength,which reveals that the physical states with wavelengths shorter than are naturally discarded in the process of the generalized Fourier decomposition of the quasiposition wavefunction of physical states. However, we mention that in the ordinary quantum mechanics the position wavefunction describes a physical state and thus no physical states with short wavelengths are discarded in the Fourier decomposition. By using (29), we can calculate the probability amplitude for the particle being maximally localized around the position , which can be read out from the scalar product of quasiposition wavefunctions discussed in the following.

At last, we give the scalar product of quasiposition wavefunctions by first deriving the inverse transformation of (29),and then computing the following integration:

#### 3. Black Hole Thermodynamics

In this section we calculate the quantum corrections to the Hawking temperature, the entropy, and the heat capacity of the Schwarzschild black hole in the framework of our GUP and compare our results with that obtained previously in the frameworks of the Hawking proposal, the quadratic GUP, and the exponential GUP.

In the following contexts of the present paper we adopt the units . As a result, the Planck length, the Planck mass, the Planck temperature, and the gravitational constant satisfy the relations: .

##### 3.1. Temperature

The metric of a four-dimensional Schwarzschild black hole can be written as where is the black hole mass. The Schwarzschild horizon radius is defined as . According to the near-horizon geometry, the position measurement precision is of the order of the horizon radius (we assume as done in other works; see, for instance, [37]. This is a physical estimation and we think it is reasonable. For a static observer outside the horizon, one cannot fix the position of a particle around a black hole because of the horizon, so the coordinate-uncertainty of the particle can be estimated to be the radius of the horizon. This estimation (also including others) is from physical intuition and does not depend on the explicit form of a generalized uncertainty principle. Its validity can be confirmed from its successful inducing of the standard Hawking temperature of the Schwarzschild black hole), . Therefore, we can deduce that the minimal length corresponds to the minimal mass of the Schwarzschild black hole in such a way , which gives the minimal mass, sometimes called the black hole remnant (BHR), as follows: Note that as is in the order of unity the black hole remnant is of the order of the Planck mass for any .

Following the method [25, 43–49] which connects directly the uncertainty relation with the black hole mass-temperature relation and using (10), we obtain the corrected temperature which is expressed in terms of the ratio of the minimal mass and the mass of the black hole: In order to compare the above result with others, we expand it in : It is obvious that the first term of the above result coincides with Hawking’s result and the case covers the result given in the framework of the exponential GUP [37]. Moreover, we provide the new temperature-mass relation for in the framework of our improved exponential GUP [38]. We note that when increases, the new temperature-mass relation is close to the Hawking result in the process before the end of evaporation, but quite different from Hawking’s at the end of evaporation in the aspects that the new relation leads to a finite maximal temperature and a nonvanishing minimal mass.

Using the numerical method, we plot the curves of the Hawking temperature versus the black hole mass in Figures 1 and 2 in which the curves from the Hawking proposal, the quadratic GUP (noted by ), and the exponential GUP (noted by ) are shown together for comparisons. Note that we use the Planck units (here we list some Planck units related to this work and their values in the SI units: m, kg, K, and s, the Planck unit of entropy: J/K, the Planck unit of heat capacity: J/K, and the Planck unit of power of radiation: J/s) in all figures and tables of the present paper.