WKB approximation for a deformed Schrodinger-like equation and its applications to quasinormal modes of black holes and quantum cosmology

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.1 (1-31)

Available online at www.sciencedirect.com

ScienceDirect 1

1 2

2

Nuclear Physics B ••• (••••) •••–•••

3

www.elsevier.com/locate/nuclphysb

3 4

5

5

RO OF

4

6 7 8 9 10 11 12

WKB approximation for a deformed Schrodinger-like equation and its applications to quasinormal modes of black holes and quantum cosmology

13

Fenghua Lu, Bochen Lv, Peng Wang, Haitang Yang

14 15

Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University, Chengdu, 610064, China

DP

16 17

Received 28 November 2017; received in revised form 25 March 2018; accepted 6 June 2018

18 19

Editor: Stephan Stieberger

20

TE

21 22

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

EC

26

RR

25

In this paper, we use theWKB approximation method to approximately solve a deformed Schrodinger  like differential equation: −h¯ 2 ∂ξ2 g 2 −i h¯ α∂ξ − p 2 (ξ ) ψ (ξ ) = 0, which are frequently dealt with in various effective models of quantum gravity, where the parameter α characterizes effects of quantum gravity. For an arbitrary function g (x) satisfying several properties proposed in the paper, we find the WKB solutions, the WKB connection formulas through a turning point, the deformed Bohr–Sommerfeld quantization rule, and the deformed tunneling rate formula through a potential barrier. Several examples of applying the WKB approximation to the deformed quantum mechanics are investigated. In particular, we calculate the bound states of the Pöschl–Teller potential and estimate the effects of quantum gravity on the quasinormal modes of a Schwarzschild black hole. Moreover, the area quantum of the black hole is considered via Bohr’s correspondence principle. Finally, the WKB solutions of the deformed Wheeler–DeWitt equation for a closed Friedmann universe with a scalar field are obtained, and the effects of quantum gravity on the probability of sufficient inflation is discussed in the context of the tunneling proposal. © 2018 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

CO

24

Abstract

UN

23

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

E-mail addresses: [email protected] (F. Lu), [email protected] (B. Lv), [email protected] (P. Wang), [email protected] (H. Yang).

43

https://doi.org/10.1016/j.nuclphysb.2018.06.002 0550-3213/© 2018 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

45

44

46 47

JID:NUPHB AID:14369 /FLA

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.2 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

2

1. Introduction

1 2

2

4 5 6 7 8 9 10

The time-independent Schrodinger equation   h¯ 2 2 ∂ + V (ξ ) ψ (ξ ) = Eψ (ξ ) , − 2m ξ

3 4

RO OF

3

can be rewritten as   2 h¯ 2 ∂ξ − p (ξ ) ψ (ξ ) = 0, i

11

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

[X, P ] = i hf ¯ (P ) ,

37 38 39 40 41 42 43 44 45 46 47

The GUP has been extensively studied recently, see for example [6–14]. For a review of the GUP, see [15,16]. To study 1D quantum mechanics with the deformed commutators (4), one can exploit the following representation for X and P in the position representation:  h¯ ∂ X = X0 and P = P , (6) i ∂x where the function P (x) is the solution of the differential equation could write P (x) in terms of the function g (x) as

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

CO

36

(4)

where f (P ) is some function. This model is inspired by the prediction of the existence of a minimal length in various theories of quantum gravity, such as string theory, loop quantum gravity and quantum geometry [3–5]. For example, if f (P ) = 1 + βP 2 , the minimal measurable length is  lmin = h¯ β. (5)

UN

35

10 11

The properties of the function g (x) will be discussed in section 2. Note that the parameter α characterizes effects of quantum gravity. For example, the deformed Schrodinger-like equation (3) could appear in two effective models, namely the Generalized Uncertainty Principle (GUP) and the modified dispersion relation (MDR). The first one is the GUP [1,2], derived from the modified fundamental commutation relation:

33 34

9

(2)

DP

15

8

TE

14

7

EC

13

5 6

where V (ξ ) is the potential, and p2 (ξ ) = 2m [E − V (ξ )]. The real function p 2 (ξ ) can be either positively or negatively valued. The WKB approximation, named after Wentzel, Kramers, and Brillouin, is a method for obtaining an approximate solution to a one-dimensional Schrodingerlike differential equation (2). The WKB approximation has a wide range of applications. Its principal applications are in calculating bound-state energies and tunneling rates through potential barriers. On the other hand, the construction of a quantum theory for gravity has posed one of the most challenging problems of the theoretical physics. Although there are various proposals for quantum gravity, a comprehensive theory is not available yet. Rather than considering a full quantum theory of gravity, we can instead study effective theories of quantum gravity. In various effective models of quantum gravity, one always deals with a deformed Schrodinger-like equation:

  2 h¯ h¯ 2 ∂ξ − p (ξ ) ψ (ξ ) = 0. (3) g α ∂ξ i i

RR

12

(1)

dP (x) dx

= f (P ). Usually, we

34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.3 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

3 4 5 6 7 8 9 10 11 12 13 14 15

P (x) = xg (αx) ,

(7)

1

where α is a parameter and can be related to the minimal length: lmin = hα. ¯ In the f (P ) = 1 + βP 2 case, one finds that

2

tan (x) g (x) = , (8) x √ with α = β. The second is the MDR. It is believed that the trans-Planckian physics manifests itself in certain modifications of the existing models. Thus, even though a complete theory of quantum gravity is not yet available, we can use a “bottom-to-top approach” to probe the possible effects of quantum gravity on our current theories and experiments [17]. One possible way of how such an approach works is via Planck-scale modifications of the usual energy-momentum dispersion relation

5

RO OF

2

p =E −m , 2

2

2

DP

1

16

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

TE

21

E = p g (αp) + m , 2

2 2

2

P = P0 g (αP0 ) ,

3 4

6 7 8 9 10 11 12 13 14 15 16

(10)

where α = −1 , and  is the cut off scale which characterizes the new physics in Planck scale. To obtain the deformed wave equations, we can define the modified differential operator by

EC

20

(11)

∂ where P0 = hi¯ ∂x , and replace P0 with P [29,30]. Another way to obtain eqn. (11) is using effective field theories (EFT). In fact, one has to break or modify the global Lorentz symmetry in the classical limit of the quantum gravity to have a MDR. There are several possibilities for breaking or modifying the Lorentz symmetry, one of which is that Lorentz invariance is spontaneously broken by extra tensor fields taking on vacuum expectation values [24]. The most conservative approach for a framework in which to describe MDR is considering an EFT, in which modifications to the dispersion relation can be described by the higher dimensional operators. In [31], we constructed a such EFT for a scalar field, which only contained the kinetic terms and the usual minimal gravitational couplings. It showed for the MDR (10) that the deformed Klein–Gordon equation could be obtained via making the replacement (11). The WKB approximation in deformed space and its applications have been considered in effective models of quantum gravity. For example, in the framework of GUP, the WKB wave functions were obtained in [32]. Moreover, the deformed Bohr–Sommerfeld quantization rule and tunneling rate formula were used to calculate bound states of Harmonic oscillators and Hydrogen atoms [32], α-decay [33,34], quantum cosmogenesis [33], the volume of a phase cell [35], and electron emissions [34], for some specific function g (x). In the context of both GUP and MDR, the deformed Bohr–Sommerfeld quantization rule was used to compute the number of quantum states to find the entanglement entropy of black holes in the brick wall model [36–38]. In [34,35], we found the WKB connection formulas and proved√ the deformed Bohr–Sommerfeld quantization rule and tunneling rate formula for the g (x) = 1 + x 2 case. In this paper, we

RR

19

CO

18

(9)

whose possibility has been considered in the quantum-gravity literature [18–21]. The modified dispersion relation (MDR) has been reviewed in the framework of Lorentz violating theories in [22,23]. It has also been shown that the MDR might play a role in astronomical and cosmological observations, such as the threshold anomalies of ultra high energy cosmic rays and TeV photons [18,24–28]. In most cases, the MDR could take the form

UN

17

3

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

2 3 4 5 6 7 8 9 10

will consider the case with an arbitrary function g (x), for which the WKB connection formulas, Bohr–Sommerfeld quantization rule and tunneling rate formula are obtained. The organization of this paper is as follows. In section 2, the deformed Schrodinger-like differential equation (3) are first approximately solved by the WKB method. After the asymptotic behavior of exact solutions of eqn. (3) around turning points are found, we obtain the WKB connection formulas through a turning point by matching these two solutions in the overlap regions. Accordingly, the Bohr–Sommerfeld quantization rule and tunneling rate formula are also given. In section 3, the formulas obtained in section 2 are used to investigate several examples, namely harmonic oscillators, the Schwinger effect, the Pöschl–Teller potential, and quantum cosmology. Section 4 is devoted to our discussion and conclusion.

RO OF

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.4 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

4

11 12

2. WKB method

13

15 16 17 18 19

2.1. Usual WKB method

TE

21 22

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

ψ (ξ ) = e

  S0 (ξ )+ hi¯ S1 (ξ )+···

.

EC

26

i h¯

  Plugging the expansion (12) into eqn. (2) and solving for Si (ξ ) to O h¯ 2 , one has      i i 1 k ψW exp exp − C = p dξ + C p dx √ (ξ ) (ξ ) (ξ ) 1 2 KB h¯ h¯ |p (ξ )| 2 for p (ξ ) > 0,      1 1 1 k |p (ξ )| dξ + C2 exp − |p (ξ )| dx C1 exp ψW KB (ξ ) = √ h¯ h¯ |p (ξ )| for p 2 (ξ ) < 0.

RR

25

6 7 8 9 10

14 15 16 17 18 19

21 22 23 24 25

(12)

26 27 28 29 30 31 32 33 34

(13)

The above WKB solutions are reliable in the allowed regions where the wavelength or the decay length changes only slightly over a distance of one wavelength or decay length, respectively. However, the allowed regions are often separated by “breakdown regions,” in which the WKB solutions diverge unphysically due to the vanishing of p (ξ ) = 0. To establish the connection between the WKB solutions in the allowed regions on either side of the breakdown region, an accurate solution of the equation (12) is needed within the breakdown region, where the potential p 2 (ξ ) can be approximated as a linear potential: p 2 (ξ ) ≈ −F ξ.

5

20

CO

24

4

13

To apply the WKB method to approximate solutions to the equation (2), we can express ψ (ξ ) in terms of the perturbation expansion of h¯ :

UN

23

3

12

The WKB method is a technique for obtaining approximate solutions to a one-dimensional Schrodinger-like differential equation (2). In this section, we will generalize this method to the deformed Schrodinger-like equation (3). First, we will briefly review the usual WKB method for the differential equation (2) as a warm-up. For later use, we will then summarize the results from solving the deformed Schrodinger-like equation (3) using the WKB method. In the rest of this section, we will show how to obtain these results.

20

2

11

DP

14

1

35 36 37 38 39 40 41 42 43

(14)

For a reasonably smooth potential, such linear approximation could hold beyond the breakdown region. So there exist overlap regions where both the linear approximation and WKB approximation are reliable. On the overlap regions, one can match the WKB solutions with the accurate

44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.5 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

3 4 5 6 7 8

solution for the linear potential to obtain the WKB connection formulas, which link WKB solutions across a classical turning point. There are two famous applications of the WKB connection formulas: Bohr–Sommerfeld quantization and tunneling rates. For bound states in a potential well V (ξ ), the uniqueness of the wave function in the classically allowed region leads to the Bohr–Sommerfeld quantization condition B

9 10 11 12 13 14 15



1 p (ξ ) dξ = n + π h, ¯ 2

A

T =e

− h2¯

B A

|p(ξ )|dξ

,

where A and B are two turning points for the potential V (ξ ). 2.2. Summary of deformed WKB results

18

22 23 24 25 26 27 28 29 30 31 32

35 36 37 38 39 40 41

sg (−ias) = e and satisfy λk (0) = e

44 45 46 47

iπk/2

.

, with k = 0, 1, 2, 3,

1 h¯

B A

 1 |p (ξ ) λ1 (α |p (ξ )|)| dξ = n + π + O (h¯ ) , 2

5 6

8 9

11 12

14

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

(18)

34 35 36

(19)

Note that when α = 0, the WKB solutions (17) reduce to eqns. (13) as expected. After the WKB connection formulas are obtained, we find that the Bohr–Sommerfeld quantization condition for bound states is given by

42 43

iπk/2

CO

34

4

15

where λk (a) are the solutions to the equations

UN

33

3

13

(16)

TE

21

2

10

EC

20

(15)

The remainder of this section is devoted to using WKB method to find approximated solutions to the deformed Schrodinger-like differential equation (3). Before giving the detailed calculations of the WKB solutions, we first present our key results for later use and convenience. To obtain the WKB solutions to the equation (3), we need to impose some conditions on g (x), which will be carefully discussed in section 2.3. For such g (x), the WKB solutions are given by    1  Ck exp h¯ |p (ξ )| λk (α |p (ξ )|) dξ 2 ψW KB (ξ ) =   for p (ξ ) > 0,     k=1,3  x 2 g 2 (αx) |x=−i|p(ξ )|λk (α|p(ξ )|)     1  Ck exp h¯ |p (ξ )| λk (α |p (ξ )|) dξ 2 (17) ψW KB (ξ ) =   for p (ξ ) < 0,     2 2 k=0,2  x g (αx) |x=−i|p(ξ )|λk (α|p(ξ )|) 

RR

19

1

7

where A and B are two turning points for the potential V (ξ ), and n is an integer. For tunneling through a potential barrier V (ξ ), the WKB transmission probability is given by

16 17

RO OF

2

DP

1

5

37 38 39 40 41 42

(20)

where n is a nonnegative integer. For tunneling through a potential barrier, the WKB transmission probability is given by

43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.6 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

6 2

1 2 3 4

T = e− h¯

B A

|p(ξ )|λ0 (α|p(ξ )|)dξ

(21)

,

where A and B are two turning points. Note that more conditions on g (x) are required to obtain eqns. (20) and (21), which are given in section 2.6.

RO OF

2.3. Deformed WKB solutions

7

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

• In the complex plane, g (z) is assumed to be analytic except for possible poles. We assume that g (0) = 1. • For a positive real number a > 0, each of the equations sg (−ias) = e

iπk/2

, with k = 0, 1, 2, 3,

DP

13

(22)

possess only one regular solution λk (a), which is regular as a → 0 and becomes λk (0) = eiπk/2 , and the possible runaway solutions (i)

ηk (a) ∼

(i) ηk

(i) ηk (a),

,

TE

12

11 12 13 14 15 16 17 18

21

23 24

(24)

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

(27)

for some function S (ξ ), which can be expanded in power series over h¯ h¯ S (ξ ) = S0 (ξ ) + S1 (ξ ) + · · · . i Plugging eqn. (28) into eqn. (3) gives

10

22

which becomes

For example, g (x) = 1 ± x 2 satisfies the above conditions with ∃0 < c1 < 1 and 0 < c2 ≤ √ 1/ 2. The function g (x) = tanx x also satisfies the above conditions with ∃0 < c1 < π and 0 < c2 ≤ arctan (3/2). To find an approximate solution via the WKB method, we could make the change of variable ψ (ξ ) = e

9

20

(23)

a when a  1. We also assume that for small enough value of a, there exists a c1 > 0 such that for all possible i and k, c1  (i)  < ηk (a) . (25) a If there is no runaway solution, we simply set c1 = ∞. • Finally, we assume that there exists a c2 > 0 such that   1  2  (26) g (−is) − 1 ≤ for |s| ≤ c2 . 2

iS(ξ ) h¯

8

19

EC

11

RR

10

4

7

CO

9

3

6

We now apply the WKB method to approximately solve the deformed Schrodinger-like differential equation (3). In what follows, we choose that arg p (ξ ) = 0 for p 2 (ξ ) > 0 and arg p (ξ ) = π2 for p 2 (ξ ) < 0. For later use, we define P (x) = xg (αx). To study the WKB solutions of eqn. (3) and the connection formulas through a turning point, we shall impose the following conditions on the function g (x):

UN

8

2

5

5 6

1

42 43 44

(28)

45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.7 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

P2    P 2 S0 (ξ )

1 2 3 4

6

S0 (ξ ) = −i |p (ξ )| λk (α |p (ξ )|) , with k = 1 and 3,

8

10

and when p 2 (ξ ) < 0, S0 (ξ ) = −i |p (ξ )| λk (α |p (ξ )|) ,

11 12 13 14

 S0 (ξ ) = p 2 (ξ ) ,  h¯ i h¯  2   S1 (ξ ) = P S0 (ξ ) S0 (ξ ) , i 2

with k = 0 and 2,

21 22 23 24 25 26

1 ψW KB (ξ ) = C1 ψW KB

28

3 (ξ ) + C3 ψW KB

29

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

(ξ ) for p (ξ ) > 0

RR

32

0 2 2 ψW KB (ξ ) = C0 ψW KB (ξ ) + C2 ψW KB (ξ ) for p (ξ ) < 0,

(31)

11 12 13 14 15

(32)

16 17 18 19 20 21 22 23 24 25 26 27

(34)

28 29 30 31

(35)

32 33

where Ci are constants, and we define

  1 1 k ψW KB (ξ ) =   exp h |p (ξ )| λk (α |p (ξ )|) dξ .  ¯  2 2   x g (αx) |x=−i|p(ξ )|λk (α|p(ξ )|) 

CO

31

and

34 35 36 37 38

(36)

These WKB solutions are valid if the RHS of the second equation in eqn. (29) is much less than that of the first one. Specifically, they are valid when        h¯  2    2  (37) p (ξ )   P S0 (ξ ) S0 (ξ ) . 2

UN

30

2

8

10

The expression for the WKB solutions are

27

6

9

TE

20

5

7

(30)

These solutions are called “runaways” solutions since they do not exist in the limit of α → 0. In [39], it was argued that these “runaways” solutions were not physical and hence should be discarded. A similar argument was also given in the framework of the GUP [40]. Therefore, we will discard the “runaways” solutions and keep only the solutions (30) and (31) in this paper. Solving the second equation in eqn. (29) gives     1  S1 (x) = − ln  P 2 (x) |x=S0 (ξ )  . (33) 2

EC

19

3 4

DP

S0 (ξ ) = −i |p (ξ )| ηk (α |p (ξ )|) .

16

18

2

(29)

where λk (a) are regular solutions of eqn. (22). It is noteworthy that there are other possible solutions, namely

15

17

1

where the prime denotes derivative with respect to the argument of the corresponding function. The first equation in eqn. (29) can be solved for S0 (ξ ). In particular, when p 2 (ξ ) > 0,

7

9



RO OF

5

7

However, the condition (37) fails near a turning point where p (ξ ) = 0. In the following of this section, we will derive WKB connection formulas through the turning points.

39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.8 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

8

2.4. Analytical solutions for linearized potential

1 2

2

4 5 6 7 8 9 10 11 12

Suppose there is a turning point at ξ = 0, where p 2 (0) = 2m [E − V (0)] = 0. The equation (37) shows that the WKB solutions fail in the neighborhood of ξ = 0. To connect the WKB solutions separated by ξ = 0, we need to find analytical solutions straddling the turning point. In the neighborhood of ξ = 0, the potential can be approximated as a straight line:

RO OF

3

V (ξ ) E + V  (0) ξ .

The deformed Schrodinger-like equation (3) for this linearized V (ξ ) becomes

  2 h¯ h¯ ∂ξ + F ξ ψ (ξ ) = 0, g α ∂ξ i i

13

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

41 42 43 44 45 46 47

12 13

(40)

17

15 16

18 19 20 21 22 23 24

where the contour C in the complex plane will be discussed below. The equation for ψ˜ (t) in terms of the complex variable t reads

25

d ψ˜ (t) + t 2 g 2 (−i αt) ˜ ψ˜ (t) = 0, (42) dt where we use the integration by parts to obtain the second term. Note the integration by parts used in eqn. (42) requires that eρt ψ˜ (t) vanishes at endpoints of C. Up to an irrelevant pre-factor, its solution is ⎛ ⎞ t   ˜  dt  ⎠ . ψ˜ (t) = exp ⎝− t 2 g 2 −i αt (43)

28

0

0

  s 2 g 2 −ias  ds  ,

29 30 31 32 33 34 35

38 39 40 41 42 43

where we define a = α˜ |ρ|1/2 and f± (s) = ±s −

27

37

1

C

s

26

36

To apply the saddle point method, we make the change of variables t = |ρ| 2 s and rewrite the Laplace transformation in eqn. (41) as   3  1 2 ψ (ρ) = |ρ| exp |ρ| 2 f± (s) ds, (44)

UN

40

11

14

C

37

39

(39)

In what follows, we will find the integral representations of the solutions to eqn. (40). Using the saddle point method, we then investigate the asymptotic behavior of these solutions in the overlap region. To solve eqn. (40), it is useful to Laplace transform it via  ψ (ρ) = eρt ψ˜ (t) dt, (41)

36

38

8

and we assume that F > 0. If we make change of variables = h¯ 2/3 F −1/3 , eqn. (39) becomes

DP

18

  ˜ ρ ψ − ρψ = 0. ∂ρ2 g 2 −i α∂

6

10

TE

17

5

9

EC

16

where F ξ = F ρ

4

7

(38)

RR

15

2 ≡ − dpdξ(ξ ) |ξ =0 = 2mV  (0), and α = F α˜ h¯ −1 , where F

CO

14

3

44 45

(45)

46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.9 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

9

1

2

2

3

3

4

4

5

5

RO OF

1

6 7 8 9 10 11 12 13 14

16

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

11 12 13 14 15 16

with + for ρ > 0 and − for ρ < 0. The contour C in eqn. (44) is chosen so that the integrand vanishes at endpoints of C. Now consider a large circle CR of radius R = ac , where c = min {c1 , c2 }. The saddle points (i) of f+ (s) (f− (s)) are λk (a) and ηk (a) with k = 0 and 2 (1 and 3). Thus, all the saddle points except λk (a) are outside the circle CR . To discuss the properties of the steepest descent contours passing through λk (a), we first prove two propositions. In the following, let Cλk denote the steepest descent contours passing through λk (a). To study the properties of the contours, we need to introduce two propositions. The proofs of these propositions are given in appendixes A and B.

TE

21

10

EC

20

9

Proposition 1. For small a, if Cλk intersects CR at Reiθ∗ , then there exists an n ∈ {0, 1, 2} such that    ∗ 2nπ  π ≤ θ − + O (a) . (46)  3  18

RR

19

8

Fig. 1. The saddle points (red dots) and steepest descent contours (blue thick lines) of f± (s) when a = 0. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

    −2  c3 for θ − Proposition 2. On the circle CR , Re f± Reiθ ≤ − 12a 3 +O a where n ∈ {0, 1, 2}.

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Moreover, one finds     c3 Re f± Reiθ∗  − 3 + O a −2 . 6a

CO

18

UN

17

7

DP

15

6

36

(47) 

2nπ  π 3 ≤ 18

37 38 39

+ O (a),

For a = 0, we plot saddle points (red dots) of f± (s) and the steepest descent contours (blue thick lines) passing through them in Fig. 1. When a > 0, more possible saddle points and poles could appear and these steepest descent contours could change dramatically around them, e.g. a contour that goes to infinity in the case a = 0 could change to the one that ends at a new saddle point or a pole. However within CR , there are no new saddle points or poles, and hence

40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.10 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

10

1

2

2

3

3

4

4

5

5

RO OF

1

6 7 8 9 10 11 12 13 14

DP

15 16 17 18 19 20

TE

21 22 23

Fig. 2. Contours (blue thick lines) and saddle points (red dots) of ψ1 (ρ) and ψ2 (ρ) in the ρ > 0 and ρ < 0 cases.

24

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

EC

28

RR

27

Cλk would change continuously as a is varied away from 0. Fig. 1 shows that for a = 0, Cλ2 4π approaches θ = 2π of |s|. So when a > 0, Cλ2 will intersect CR twice, 3 and 3 for a large value    ∗ 4π  π   −  ≤ π + O (a), and the intersections Reiθ∗ are within θ ∗ − 2π 3 ≤ 18 + O (a) and θ 3  18  π  π respectively. Similarly, Cλ1 intersects CR within |θ ∗ | ≤ 18 + O (a) and θ ∗ − 2π 3 ≤ 18 + O (a),   π  π Cλ3 intersects CR within |θ ∗ | ≤ 18 + O (a) and θ ∗ − 4π 3 ≤ 18 + O (a), and Cλ0 ends at λ2 (a) π ∗ and intersects CR within |θ | ≤ 18 + O (a). Since the saddle points and hence the solutions in eqn. (44) depend on the sign of ρ, it is convenient to choose different contours in the complex plane for ρ > 0 and ρ < 0, making sure that they are deformable to each other. In Fig. 2(a), the contour considered in the ρ > 0 case is the steepest descent contour Cλ2 through λ2 (a). For simplicity, we assume arg λk (a) = kπ/2 to illustrate the contours in Fig. 2. In Fig. 2, Cλi k denotes the part of Cλk inside the circle CR while

CO

26

denotes the parts of Cλk outside CR . Note that f+ (s) → −∞ when one moves away from Cλ1,2 k the saddle point, and hence the corresponding integrand in eqn. (44) vanishes at endpoints of Cλ2 . Since Cλ2 is a steepest descent contour, when 1  ρ  α˜ −2 , the dominant contribution to the integral over Cλ2 in eqn. (44) comes from the neighborhood of the saddle point λ2 (a). Thus by the method of steepest descent, one has for 1  ρ  α˜ −2 that   3  1 2 ψ1 (ρ) = |ρ| exp |ρ| 2 f+ (s) ds ∼ Iλ2 (a) , (48)

UN

25

C>

where C> = Cλ2 , and Iλk (a) is the contribution from the saddle point λk (a). Using Watson’s lemma, we find

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.11 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

√

1

Iλk (a) ∼

2

 3   π exp |ρ| 2 f± (λk (a)) 1

7 8

where k = 0 and 2 is for +, and k = 1 and 3 for −. To study the asymptotic behavior of ψ1 (ρ) when −1  ρ  −α˜ −2 , we consider   3  1 ψ1 (ρ) = |ρ| 2 exp |ρ| 2 f− (s) ds, (50) C<

9

11 12 13 14 15 16 17 18 19 20 21 22 23

30 31 32 33 34 35 36 37 38

ψ1 (ρ) ∼ Iλ1 (a) + Iλ3 (a) .

C>

42 43 44 45 46 47

is

12 13 14 15 16 17 18 19

21 22 23 24

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ψ2 (ρ) ∼ Iλ0 (a) .

(54)

42 43

When −1  ρ  −α˜ −2 , the asymptotic behavior of ψ2 (ρ) is

44

ψ2 (ρ) ∼ Iλ3 (a) .

To better illustrate the contours, we plot these contours for g (x) =

11

25

UN

41

(52)

Since there is no singularity inside CR , the contour C< used in the ρ < 0 case can be deformed to C> in the ρ > 0 case. Similarly in Fig. 2(b), we consider the contour C> = Cλ22 + Cλi12 + Cλi 0 + Cλ10 in the ρ > 0 case and C< = Cλ22 + CR2 + Cλi 3 + CR4 + Cλ10 in the ρ < 0 case. It is noteworthy that the contours C> and C< are deformable to each other. As argued before, the contributions from CR2 and CR4 can be neglected. Since the leading contribution to the integral over a steepest descent contour comes from a small vicinity of the saddle point, the contributions from Cλ10 and Cλ22 can also be neglected.   λ2(a) is on the steepest descent contour Cλ0 passing through λ0 (a), and  Moreover, hence Iλ2 (a)   Iλ0 (a) . So Iλ2 (a) can be neglected for the integral over C> . Therefore when 1  ρ  α˜ −2 , the asymptotic behavior of the solution   3  1 ψ2 (ρ) = |ρ| 2 exp |ρ| 2 f− (s) ds, (53)

39 40

8

20

EC

29

7

where i = 1, 2, 3. Since |f− (s)| ∼ O (1) at s = λ1 (a) and λ3 (a), the contributions from CRi can also be neglected. Thus considering the contributions from Cλi 1 and Cλi 3 around the saddle points λ1 (a) and λ3 (a), we find that when −1  ρ  −α˜ −2 ,

RR

28

6

10

25

27

5

where the contour C< consists of Fig. 2(a). Since the contributions from Cλ12 and neglected in eqn. (50). The contours CRi connect two adjacent steepest descent contour along the circle CR . Thus, Propositions 1 and 2 implies that the contributions from CRi is        3   3   πc |ρ| 12  1 c3 |ρ| 2 |ρ| 2 exp |ρ| 2 f− (s) ds  ∼ exp − , (51)   9a 12a 3     CRi

24

26

4

9

Cλ22 , Cλi 1 , Cλi 3 , CR1 , CR2 , and CR3 , as shown in Cλ22 are neglected in eqn. (48), they can also be

Cλ12 ,

DP

10

RO OF

6

2 3

TE

5

(49)

,

CO

4

1



s 2 g 2 (−ias) |s=λk (a)

|ρ| 4

3

2



11

(55)

45 46

tan(x) x

in the appendix C.

47

JID:NUPHB AID:14369 /FLA

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.12 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

12

2.5. Connection formulas

1 2

2

5 6 7 8 9 10 11 12 13 14 15 16

RO OF

4

After the asymptotic behavior of analytical solutions straddling the turning point ξ = 0 for the linearized potential is obtained, we can use it to splice the two WKB solutions together. The key observation is that there exist overlap regions, where the linear approximation and WKB approximation are both reliable. As shown below, the overlap regions are α˜ −2  |ρ|  1, where we already have the asymptotic behavior of analytical solutions and the WKB solutions. On the overlap regions, we can match the coefficients Ck of the WKB solutions with the analytical solutions. Since the analytical solutions straddle the turning point, we can link the WKB solutions across ξ = 0. As shown above, a linear approximation to the potential p 2 (ξ ) near the turning point ξ = 0 is p 2 (ξ ) ≈ −F ξ.

|p (ξ )| λk (α |p (ξ )|) dξ 0

21 22 23 24

=−

3F

27 28

⎡

−ρ

  |p| λk (α |p|) d p2

0

⎡

30

= |ρ| 2 ⎣sgn (ρ) λk (a) − 3

32

34 35 36 37 38 39

3

λk (a) = |λk (a)| ei(πk/2+αk (a))

43 44 45 46 47

9 10 11

15 16

18 19 20 21 22 23 24

⎤

25 26

(57)

27 28 29

⎤

30 31 32 33 34

where√we use eqn. (56) for in the second line, u = |p| λk (α |p|) in the third line, and s = u/ h¯F |ρ| in the fourth line. Defining αk (a) and θk (a) as in p2

8

17

s 2 g 2 (−ias) ds ⎦

= |ρ| 2 f± (λk (a)) ,

7

14

⎥ u2 g 2 (−iαu) du⎥ ⎦

  and θk (a) = arg 1 − iaλ2k (a) e−iπk/2 g  (−iaλk (a)) ,

35 36 37 38 39

(58)

40

one obtains for the linear approximation of p 2 (ξ ) that        2 2   x g (αx) |x=−i|p(ξ )|λ (α|p(ξ )|)  = h¯ |ρ| s 2 g 2 (−ias) |s=λ (a) e−i(πk/2+θk (a)−αk (a)) , k k    F (59)

41

UN

42

 0

λk (a)

0

40 41

√ |ρ|λk (a)

CO

33

h¯ F

√ 3 ⎢ h¯ 3 ρ |ρ| − λk (a) + = − F3 ⎢ h¯ ⎣ 3F

29

31



h¯ 3

25 26

h¯ F

√

TE

20

1 h¯

6

13

EC

19

ξ

5

α˜ h−1 .

RR

18

4

12

Moreover, eqn. (39) becomes eqn. (40) through changing of variables ξ = F ρ and α = F ¯ Thus in the region where eqn. (56) holds, we conclude that ψ1 (ρ) and ψ2 (ρ) are solutions of eqn. (3). On the other hand, we find for the linear approximation of p 2 (ξ ), that

17

3

(56)

DP

3

F where s = i h√ x. From eqns. (57) and (59), we have |ρ| ¯

42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.13 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

1 2 3

    exp h1¯ |p (ξ )| λk (α |p (ξ )|) dξ F ∼ Ck ei(πk/4+θk (a)/2−αk (a)/2) Iλk (a) .    2π h¯  2 2   x g (αx) |x=−i|p(ξ )|λk (α|p(ξ )|) 

13

1

(60)

3 4

4

6 7 8 9 10 11

When |ξ |  F −1 α −2 (α |p (ξ )|  1), the condition (37) for validity of the WKB approximation becomes |ξ |  F .

17 18 19 20 21 22 23 24 25

ψ1 (ρ)

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

9 10 11

13 14 15 16 17 18 19 20 21

23

(64)

RR

32

8

22

In the usual quantum mechanics, the Bohr–Sommerfeld quantization condition and tunneling rates through potential barriers can be derived from the WKB connection formulas. However, more conditions are needed to be imposed on the function g (x) to obtain the Bohr– Sommerfeld quantization condition and tunneling rates in our case. In fact, we further require that g (z) = g (−z) for z ∈ C, and both g (x) and g˜ (x) ≡ g (ix) are real functions when x ∈ R. Note that g (x) = 1 ± x 2 and tan x/x satisfy the above requirements. Under these requirements, the solutions λk (a) of eqn. (22) satisfies the properties:

CO

31

6

12

(62)

2.6. Bohr–Sommerfeld quantization and tunneling rates

1. λ0 (a) = −λ2 (a) and λ1 (a) = −λ3 (a), 2. For small enough value of a, one has that λk (a) = |λk (a)| eikπ/2 , where the second property comes from the fact that for any real function f (x), the equation xf (ax) = 1 always has a real solution around x = 1 if a is small enough. Furthermore, these 0 properties imply that θk (a) = αk (a) = 0. In the region ξ > 0 where p 2 (ξ ) < 0, ψW KB (ξ ) is 2 exponentially increasing away from the turning point ξ = 0 while ψW KB (ξ ) is exponentially 3 1 decreasing. In the region ξ < 0 where p 2 (ξ ) > 0, ψW KB (ξ ) and ψW KB (ξ ) are oscillatory solutions and propagate toward and away from the turning point, respectively. Now suppose that p 2 (ξ ) has two simple points at ξ = A and ξ = B with A < B. We also assume that p 2 (ξ ) < 0 if ξ > B or ξ < A, and that p 2 (ξ ) > 0 if A < ξ < B. To study the

UN

30

Iλ2 (a) ,

Similarly for ψ2 (ρ), we find that another connection formula is

0 Cei[α0 (a)/2−θ0 (a)/2] ψW KB (ξ ) for ξ > 0 . ψW KB (ξ ) = 3 i[−3π/4+α (a)/2−θ (a)/2] 3 3 ψW Ce KB (ξ ) for ξ < 0

28 29

∼

we use eqn. (60) to match WKB solutions (34) and (35) with ψ1 (ρ) over the overlap region α˜ −2  |ρ|  1 and find that one connection formula around the turning point is ⎧ 2 Cei[α2 (a)/2−θ2 (a)/2] ψW ⎨ KB (ξ ) for ξ > 0 i[−π/4+α3 (a)/2−θ3 (a)/2] ψ 3 . ψW KB (ξ ) = Cei[π/4+α1 (a)/2−θ1 (a)/2] ψ 1 + Ce (ξ ) W KB W KB (ξ ) ⎩ for ξ < 0 (63)

26 27

1ρα −2

DP

16

∼

TE

15

−1ρ−α˜ −2

EC

14

Iλ1 (a) + Iλ3 (a)

5

7

(61)

In terms of ρ, the WKB solutions (34) and (35) for the linear approximation of p2 (ξ ) are valid in the overlap region α˜ −2  |ρ|  1. However, we can approximate ψ1 (ρ) and ψ2 (ρ) by their leading asymptotic behaviors for α˜ −2  |ρ|  1. Since

12 13

RO OF

5

2

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

2 3 4 5 6 7 8

boundary-value problem with ψ (±∞) = 0, we consider the two-turning-point solutions by matching two one-turning-point solutions: the first one is from +∞ through B and down to near A; the second is −∞ through A and down to near B. We can use the WKB connection formula (63) to show that the first one-turning-point solution that decays like           ξ C exp h1¯ B p ξ   λ2 α p ξ   dξ  2 (65) CψW  KB (ξ ) =  ,   2 2   x g (αx) |x=i|p(ξ )|λ2 (α|p(ξ )|) 

RO OF

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.14 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

14

9 10

as ξ → +∞ behaves like

⎛

11

13 14 15 16 17 18 19 20

26 27 28 29 30 31 32 33 34 35 36

39 40 41 42 43 44 45 46 47

7 8

(66)

12 13 14 15 16 17 18 19 20

as ξ → −∞ behaves like

27

⎛

ξ

1 h¯

A

1 |p (ξ ) λ1 (α |p (ξ )|)| dξ = n + π + O (h¯ ) , 2

23 24 25 26

29

(68)

30 31 32

in the region between A and B. In order that the two solutions in eqns. (66) and (68) match over the region between A and B, we require that the expression in the curly bracket of eqn. (66) is an integral multiple of π . Therefore, we derive the Bohr–Sommerfeld quantization condition: B

22

28

⎞

       ⎝ 1 p ξ  λ1 α p ξ    dξ  + π ⎠ ,   sin h   4 ¯   2 2   x g (αx) |x=|p(ξ )λ1 (α|p(ξ )|)|  A 2C 

37 38

6

11

TE

25

EC

24

5

21

RR

23

4

in the region between A and B. Similarly, the second one-turning-point solution that decays like           A C  exp h1¯ ξ p ξ   λ2 α p ξ   dξ  (67)   ,   2 2   x g (αx) |x=i|p(ξ )|λ2 (α|p(ξ )|) 

CO

22



33 34 35 36 37

(69)

where n is a nonnegative integer. We now consider WKB description of tunneling, in which p (−∞) = p (+∞) > 0 and p 2 (ξ ) vanishes at two turning points x = A and x = B. Moreover, there are two classical allowed regions p 2 (ξ ) > 0, Region I with ξ < A and Region III with ξ > B, and one forbidden region p 2 (ξ ) < 0, Region II with A < ξ < B. To describe tunneling, we need to choose appropriate boundary conditions in the classical allowed regions. We postulate that there is only a transmitted wave in Region III:

UN

21

3

10

⎜ 1           π ⎟ p ξ λ1 α p ξ dξ + ⎠ sin ⎝    4 h¯  2 2   x g (αx) |x=|p(ξ )λ1 (α|p(ξ )|)|  ξ          (         ) ξ B 2C sin h1¯ A p ξ  λ1 α p ξ    dξ  + π4 − h1¯ A p ξ  λ1 α p ξ    dξ  + π2   =− ,   2 2   x g (αx) |x=|p(ξ )λ1 (α|p(ξ )|)|  2C

2

9

⎞

B

DP

12

1

38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.15 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

RO OF

3

Using the WKB connection formula (64), we find that the WKB solution in Region II is           B F e3iπ/4 exp h1¯ ξ p ξ   λ0 α p ξ   dξ      2 2   x g (αx) |x=i|p(ξ )|λ0 (α|p(ξ )|)            ξ F e3iπ/4 eη exp h1¯ A p ξ   λ2 α p ξ   dξ  , =     2 2   x g (αx) |x=i|p(ξ )|λ0 (α|p(ξ )|)  where

20 21 22

η=

23

25 26 27 28 29

1 h¯

B |p (ξ )| λ0 (α |p (ξ )|) dξ. A

RR

T=

|F |2 |A|2

∼e

−2η

.

37 38

3. Application

39

41 42 43 44 45 46 47

6 7 8 9 10 11 12 13 14 15

(71)

16 17 18 19 20 21

(72)

22 23

(73)

25

27 28 29 30 31 32 33 34

(74)

In this section, we use the results obtained in the previous section to discuss some interesting examples in the deformed quantum mechanics. From now on, we take Planck units c = G = h¯ = k = 1.

UN

40

5

(70)

where the first term in the square bracket is the incident wave with the amplitude A = F eη eiπ/2 . Therefore, the transmission probability is

CO

36

4

26

34 35

3

              ξ        ξ F eη eiπ/2 exp hi¯ A p ξ  λ1 α p ξ    dξ  + eiπ exp − hi¯ A p ξ  λ1 α p ξ    dξ    ,   2 2   x g (αx) |x=|p(ξ )λ1 (α|p(ξ )|)| 

31

33

2

24

30

32

1

In Region I, the WKB approximation solution includes a wave incident the barrier and a reflected wave:

EC

24

DP

2

          1 B−ξ  p B − ξ   λ3 α p B − ξ   dξ  0 h ¯ 3 F ψW  KB (ξ ) =    2 2   x g (αx) |x=|p(ξ )λ3 (α|p(ξ )|)|            ξ F exp hi¯ B p ξ  λ1 α p ξ    dξ  =   .   2 2   x g (αx) |x=|p(ξ )λ1 (α|p(ξ )|)|  F exp

TE

1

15

3.1. Harmonic oscillator

We first study a simple example, bound states of an harmonic oscillator in the potential 2 2 V (x) = mω2 x . For the harmonic oscillator, the deformed Schrodinger equation is

35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

  x0    i 1  2 p (x) arctan (α |p (x)|) dx = n + π, α |p (x)| 2 0

35 36 37 38 39 40 41 42 43 44 45 46 47

(76)

λ ∼ L

x0

ω = . 2E0,n



6

8 9

(77)

10 11

where β in [41] is our α 2 . The WKB approximation is a good approximation when the de Broglie wavelength λ of a particle is smaller than the characteristic length L of the potential. Thus, the higher order WKB corrections are suppressed by powers of Lλ relative to the leading term. For the harmonic oscillator in the nth energy level, we find that  1 2mE0,n

5

7

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

(80)

2

Therefore, the terms proportional to 2Eω0,n in eqn. (79) are the higher order corrections, and hence the WKB result (78) agrees with the leading term of the WKB expansion of the exact result (79). It is noteworthy that the momentum representation of the position operator is quadratic in the g (x) = tan(x) x case, and hence eqn. (75) can be solved exactly. However for a generic case, the WKB approximation could provide a simple way to estimate the quantum gravity’s corrections.

CO

34

4

which gives the energy levels of bound states  α 2 mE0,n En = E0,n 1 + , (78) 2   for n = 0, 1, 2 · · · , where E0,n = n + 12 ω. In [41], the differential equation (75) with g (x) = tan(x) was solved exactly in the momentum x space, and the exact energy levels were given by ⎡ ⎤    2 m2 ω2 1 β 1 βmω ⎦ En = ω ⎣ n + 1+ + n2 + n + 2 4 2 2 ⎡ ⎤   2 2 2 mE 2 mE α α ω ω 0,n 0,n 2 ⎦, = E0,n ⎣ 1 + α 4 m2 E0,n + + (79) 2E0,n 2 2 2E0,n

32 33

RO OF

6

i λ1 (a) = arctan a. a In this case, the Bohr–Sommerfeld quantization condition becomes

3

one then has

DP

5

1 2

tan(x) x ,

TE

4

(75)

EC

3

RR

2

  −∂x2 g 2 (−iα∂x ) − p 2 (x) ψ (x) = 0, *   2E where p 2 (x) = m2 ω2 x02 − x 2 and x0 = mω 2 . Considering g (x) =

UN

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.16 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

16

3.2. Schwinger effect

The Schwinger effect [42] is an example of creation of particles by external fields, which consists in the creation of electron–positron pairs by a strong electric field. The WKB approximation and the Dirac sea picture can be used to illustrate its main physical features in a heuristic way. Suppose the electrostatic potential is

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.17 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

17

1

2

2

3

3

4

4

5

5

RO OF

1

6 7 8 9 10 11 12 13

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

DP

19

8 9 10 11 12 13 14 15

(81)

16 17 18

If eEL > 2m, there exist states in the Dirac sea with x > L having the same energy as some positive energy states in the region x < 0. The electrons with energy m ≤ E ≤ eEL − m in the Dirac sea could tunnel through the classically forbidden region leaving a hole behind, which can be described as the production of an electron–positron pair out of the vacuum by the effect of the electric field. For simplicity, we assume that electrons are described by a wave function  (t, x) satisfying the 1 + 1 dimensional deformed Klein–Gordon equation ( ) (82) [i∂t + eV (x)]2 − (i∂x )2 g 2 (−iα∂x ) − m2  (t, x) = 0.

19

Since the potential only depends on x, we could the following ansatz for  (t, x)

28

TE

18

 (t, x) = e−iEt ψ (x) .

EC

17

−∂x2 g 2 (−iα∂x ) ψ (x) − p 2 (ξ ) ψ (x) = 0,

23 24 25 26 27

33

35

(85) p 2 (x)

30

34

in Fig. 3(a). Then, the WKB

36 37 38 39 40 41

(86)

42 43

x_

44

The number of pairs produced per unit time with energies between E and E + dE is dN dE = 2T , dt 2π

22

32

(84)

where p 2 (x) = [E + eV (x)]2 − m2 . The function p 2 (x) has two turning points:

where 0 < x_ < x+ < L since m ≤ E ≤ eEL − m. We plot transmission coefficient is given by ⎡ ⎤ x+ ⎢ ⎥ T = exp ⎣−2 |p (x)| λ0 (α |p (x)|) dx ⎦ .

21

31

Substituting this expression in eqn. (82) results in a deformed Schrodinger-like equation

1 x± = (E ± m) , eE

20

29

(83)

RR

16

⎧ x<0 ⎨0 V (x) = −Ex 0 < x < L . ⎩ −EL L < x

CO

15

7

Fig. 3. The plot of p2 (x) and p2 (a).

UN

14

6

45 46

(87)

47

JID:NUPHB AID:14369 /FLA

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

eE T. π Considering the g (x) = tan (x) /x case in which λ0 (a) = arctanh (a) /a, we find    2π  2 α2 T = exp − 1 − m 1 − . eEα 2 W=

Thus, the pair production rate per unit length is    eE 2π  2 α2 W= 1 − m exp − 1 − . π eEα 2

2K 2 α 2

45 46 47

5

(88)

6 7 8 9

(89)

10 11 12 13

(90)

14 15

a . (92) √ 2π 1 − 1 − 4K 2 α 2 For a Schwarzschild black hole of the mass M, the event horizon is at rh = 2M. Since the gravitational acceleration at the event horizon is given by Tu ∼

M 1 a= 2 = , 4M rh

Th ∼ T 0

2Th2 α 2

* , 1 − 1 − 4Th2 α 2

T0 , 1 + α 2 T02

18 19 20 21 22 23

25 26 27 28 29 30 31

33 34 35 36

(94)

37 38 39

1 where T0 = 8πM , and we estimate that the energy of radiated particles K ∼ Th . Solving the above equation for Th gives

Th =

17

32

(93)

it follows from eqn. (92) that the Hawking temperature is

UN

44

4

24

42 43

3

where we identify the reduced mass m2 = K as the energy associated with the pair production process. The Unruh temperature reads

39

41

2

16

38

40

1

The Schwinger mechanism can explain the Unruh effect which predicts that an accelerating observer will observe a thermal spectrum of photons and particle–antiparticle pairs at temperature a T = 2π , where a is the acceleration [43]. In fact, considering a free particle of charge e and mass m moving in a static electric field E , one finds that the acceleration of the particle is a = em E . It follows that the pair production rate per unit length is   √ K 1 − 1 − 4K 2 α 2 W∼ − , (91) a/2π 2K 2 α 2

36 37

RO OF

6

DP

5

TE

4

EC

3

where the factor of 2 takes into account the two polarizations of the electron. Note that the turning points x± are the positions at which the two particles of the pair are produced. Therefore, shifting the energy by dE results in a change in the positions of the particles by dx = dE eE . It follows from eqn. (87) that the pair production rate per unit length is

RR

2

CO

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.18 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

18

40 41 42

(95)

which shows that in the g (x) = tan (x) /x case, the quantum gravity effects always lower the Hawking temperature. Using the first law of the black hole thermodynamics, we find that the black hole’s entropy is

43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.19 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

 1 2 3 4

S=

19

 dM A α2 A ∼ + ln , Th 4 8π 16π

(96)

2 3

where A = 4πrh2 = 16πM 2 is the area of the horizon.

4 5

RO OF

5 6

3.3. Pöschl–Teller potential and quasinormal modes of a black hole

7

9

VP T (x, b) = −

12 13 14 15 16

V0 cosh2 bx

,

   1 2 1 V0 2 1 2 En (b) = −b − n + , + + 2 4 b2

20 21

28

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

−x0

where x = ±x0 ≡ ±b−1 arccosh



1/2 V √0



are the turning points, and p (x) =

−E (x) = 1 ± x 2 case,

Now consider the g± are     2 λ± + O a4 . = i 1 ± a (a) 1

CO

29

16 17 18 19 20 21 22

(99)

23 24 25

1 |p (x) λ1 (α |p (x)|)| dx ≈ n + π, 2 

EC

x0

RR

27

15

(98)

The Bohr–Sommerfeld quantization condition (69) then leads to

26

(100)

27 28 29

√

30

E − VP T (x, b).

in which we find that the solutions of sg± (−ias) = i

31 32 33 34

(101)

35 36 37

Solving eqn. (100) for E, we find that the bound states are

38

   1 +  1 ,2  V0 2 1 2 1 1 3 V0 2 ± 2 2 2 En (b) ≈ −b − n + ∓α b n+ , + n+ − 2 b2 2 2 2 b2

UN

26

14

−∂x2 g 2 (−iα∂x )  (x, b) + [VP T (x, b) − E (b)]  (x, b) = 0.

24 25

2

11

13

for n = 0, 1, 2, · · · , N −1, where N + 12 > 14 + Vb20 . We now use the WKB method to solve the deformed Schrodinger equation for the bound states in the Pöschl–Teller potential. The deformed Schrodinger equation is given by

22 23

1

9

12

TE

19



8

10

(97)

is exactly solvable. For a particle of the mass m = 1/2, the exact bound states are given by

17 18

7

It has been long known that the usual Schrodinger equation for the Pöschl–Teller potential of the form

10 11

6

DP

8

1

(102)

where n = 0, 1, 2, · · · such that the sum of terms in the square bracket is non-negative. If α = 0, from comparing the exact result (98) with the WKB one (102), it follows that the higher WKB corrections are suppressed by powers of b2 V0−1 . Therefore, combining results from eqns. (98) and (102), one obtains the bound states:

39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

2

   1 1 V0 2 1 + + − n+ 2 4 b2 ,  +  1      2 1 2 1 3 V0 2 −1/2 2 2 4 4 ∓α b n + + O bV0 . +O α b n+ − 2 2 2 b2

3 4 5 6

9 10 11

ds 2 = h (r) dr 2 −

12

14 15 16 17 18 19 20 21 22 23 24 25

d 2 R (r∗ ) − + V (r∗ ) R = ω2 R (r∗ ) , dr∗2 where dr∗ = dr/ h (r) is the tortoise coordinate, and  l (l + 1) 1 dh V (r∗ ) = h (r) + . r2 r dr

30 31 32 33 34 35 36 37 38

(r∗ ) |r∗ =r∗,0 . dr∗2

8 9

11 12 13 14 15 16 17

(105)

18 19 20 21

(106)

22 23 24 25 26

(107)

The quasinormal modes can be estimated by using a simpler potential −VP T (x, b) that approximates (106) closely, especially near its maximum [45]. The quantities V0 and b are given by the height and curvature of the potential V (r) at its maximum r∗ = r∗,0 :   1 V0 = V r∗,0 and b2 = − 2V0

7

10

(104)

In deformed quantum mechanics, the Schrodinger-like equation (105) could be changed to   d2 d − 2 g2 α + V (r∗ ) − ω2 R (r∗ ) = 0. (108) idr∗ dr∗

d 2V

5 6

RR

29

R ∼ e±iωr∗ , r∗ → ±∞.

CO

28

4

(103)

For asymptotically flat black holes, quasinormal modes are solutions of the wave equation (105), satisfying specific boundary conditions [44]

26 27

3

where d is the solid angle. The wave equation for the scalar particles is the Klein–Gordon equation. After the wave function  (t, r, ) is decomposed into eigenmodes of normal frequency ω and angular momentum l,  (t, r, ) = e−iωt Ylm () R (r) /r, the Klein–Gordon equation gives a Schrodinger-like equation for R (r) in stationary backgrounds:

TE

13

dr 2 − r 2 d, h (r)

DP

8

2

To study quasinormal modes of a static and spherically symmetric, we consider the propagation of the massless and minimally coupled scalar particles in a general Schwarzschild-like metric:

EC

7

1

En± (b) = −b2

RO OF

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.20 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

20

27 28 29 30 31 32 33 34 35 36

(109)

37 38

47

where

47

40 41 42 43 44 45

UN

39

46

For a Schwarzschild black hole with h (r) = 1 − we find   4l 3 (l + 1)3 al − 3 − l − l 2 [1 + 3l (l + 1) + al ] V0 = , (110) [3l (l + 1) − 3 + al ]4 M 2 16l 2 (l + 1)2 {9 (−3 + al ) + l (l + 1) [−33 + 4al + l (l + 1) (−13 + 9l (l + 1) + 3al )]} b2 = , [−3 + 3l (l + 1) + al ]4 [1 + 3l (l + 1) + al ] M 2 (111)

39

2M r ,

40 41 42 43 44 45 46

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.21 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••



9 + l (l + 1) (14 + 9l (l + 1)).

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

−1/2

Note that the ratio bV0 −1/2

bV0

(112)

∼ l −1 .

In this approximation, eqn. (108) becomes   d2 2 d 2 − 2g α − VP T (r∗ , b) − ω (b) R (r∗ , b) = 0. dr∗ idr∗

x → −ir∗ and b → ib such that VP T (x, b) = VP T (−ir∗ , ib). Let us define  (x, b) = R (−ix, ib) , E (b) = −ω2 (ib) .

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

TE

25

(114)

(115)

ω (b) = −E (−ib) .

9 10 11 12 13

15 16 17

(116)

18 19 20

(117)

The quasinormal modes in the g (x) case can be found by the bound states of the Pöschl–Teller potential in the g˜ (x) case 2

8

14

where g˜ (x) = g (−ix), and the boundary conditions for the quasinormal modes are reduced to     (x, b) ∼ exp ∓ −E (b)x , as x → ±∞. (118)

EC

24

(x, b) − E (b)]  (x, b) = 0,

5

7

(119)

For the g± (x) = 1 ± x 2 case, it follows from eqn. (103) the quasinormal modes ω ≡ ωR + iωI of a Schwarzschild black hole in the deformed quantum mechanics can be estimated as 

 

    b2 3α 2 b2 1 2 −1 4 4 |ωR | = V0 − +O l 1± n+ +O α b , 4 2 2 

 

2     1 1 + O l −1 + O α 4 b4 , 1 ± α 2 b2 n + (120) ωI = −b n + 2 2

RR

23

−∂x2 g˜ 2 (−iα∂x )  (x, b) + [VP T

CO

22

Then  (x, b) satisfies

4

6

To relate the quasinormal modes of the above equation to the bound states of the Pöschl–Teller potential, we consider the formal transformations [45]

where n = 0, 1, 2, · · · such that the sum of terms in the square bracket of eqn. (103) is nonnegative. If l  1, it follows that eqn. (120) work for n < l when a corresponding bound state exists. Our WKB method gives quite accurate results for the regime of high multipole numbers l of a Schwarzschild black hole of the mass M  1, since αb ∼ αM −1 . The Pöschl–Teller approximate potential method gives best result for low overtone number. However for the higher modes, it is known that the Pöschl–Teller potential method gives higher values of ωI [46]. In fact, for l  1, eqn. (110) gives that b ≈ √1 in the Pöschl–Teller approximate potential method. On 3 3M the other hand, the asymptotic quasinormal mode of a Schwarzschild black hole is given by

UN

21

3

(113)

19 20

1 2

controlling the WKB expansion is given by

RO OF

al =

DP

1

21

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.22 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

22

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

ln 3 i − 8πM 4M

 n+

1 . 2

(121)

It appears that if b = 1/4M, we could have better approximations for ωI for the higher modes. In [47], Hod used Bohr’s correspondence principle to argue that the highly damped black-hole oscillations frequencies were transitions from an unexcited black hole to a black hole in a mode with n  1. Later, Maggiore*argued that these highly damped black-hole oscillations frequen-

2 + ω2 [48]. In high damping limit n  1, it is easy to see that cies should be interpreted as ωR I * 2 2 |ωR |  |ωI |, and hence ωR + ωI ∼ |ωI |. First consider the α = 0 case. It concludes from the above arguments that the energy absorbed in the n → n − 1 transition with n  1 is the minimum quantum that can be absorbed by the black hole. Therefore, one obtains for the minimum quantum that

1 , (122) 4M where we use b = 1/4M. Since for a Schwarzschild black hole the horizon area A is related to the mass M by A = 16πM 2 , a change M in the black hole mass produces a change M = |ωI |n − |ωI |n−1 =

A = 32πMM = 8π,

19

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

TE

24

which becomes negative or infinity as n → ∞, depending on the sign in front of α 2 . This means that contributions from higher order terms become important and have to be included for very large value of n. Despite the ignorance of higher order contributions, one may introduce an upper cutoff nc on n, when higher order contributions are important. Thus, the minimum quantum can be estimated as  1 3α 2 n2c M ∼ 1± , (125) 4M 16M 2 which gives that the area of the horizon is quantized in units  3πα 2 n2c A = 8π 1 ± . A

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

EC

23

RR

22

CO

21

(123)

which coincides with the Bekenstein result [49]. For the g± (x) = 1 ± x 2 case, it follows from eqn. (120) that for n  1, the minimum quantum absorbed by the black hole is    1 3α 2 1 2 M = |ωI |n − |ωI |n−1 ≈ 1± n+ , (124) 4M 2 16M 2

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

(126)

Since the minimum increase of entropy is ln 2 which is independent of the value of the area, one then concludes that  dS S 1 3πα 2 n2c ≈ ≈ 1∓ , (127) dA A 4 A

UN

20

1 2

RO OF

2

ω≈

DP

1

where a “calibration factor” ln 2/2π is introduced in A [50]. From this, it follows that A S ≈ ∓ 3πα 2 n2c ln A, (128) 4 where the logarithmic term is the well known correction from quantum gravity to the classical Bekenstein–Hawking entropy.

36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.23 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

1

23

3.4. Quantum cosmology

1 2

2

4 5 6 7

We now consider the case of a closed Friedmann universe with a scalar field with the potential V (). The Einstein–Hilbert action plus the Gibbons–Hawking–York boundary term is    ε 1 4 √ Sg = 2 d x −gR + 2 d 3 x |h|K, (129) 4κ 2κ M

RO OF

3

∂M

8

11 12 13 14 15 16 17

M

The ansatz for the classical line element is ds = −N dt + a 2

2

2

2

18

20 21 22 23 24 25

(t) d23 ,

where N (t) is the lapse function, and   d23 = d 2 χ + sin2 χ d 2 θ + sin2 θ dφ 2 S 3.

is the standard line element on Thus, one has for the curvature scalar: +  2 , 6 a¨ a˙ N˙ a˙ 6 R= 2 − + + + 2. Na a a N a

26

5 6 7

9 10 11 12 13 14 15 16

(131)

17 18 19

(132)

20 21 22 23 24

(133)

EC

19

DP

10

4

8

where κ 2 = 4π , hab is the induced metric on the boundary, K is the trace of the second fundamental form, ε is equal to 1 when ∂M is timelike, and ε is equal to −1 when ∂M is spacelike. The action for the single scalar field is   √ 1 Sm = d 4 x −g − g μν ∂μ ∂ν  − V () . (130) 2

TE

9

3

25 26

32

where we make rescalings

32

30

33 34 35 36 37 38 39 40 41 42

√

κ 3 κ 9π 2 a → √ a, N → √ N ,  → , and V () → 4 V () . κ κ 6π 6π

CO

29

π2 π2 V (a, ) H = − a + 2 + , 2 2 2a where   V (a, ) = a 2 a 2 V () − 1 .

43 44 45 46 47

(135)

= −∂a2 g 2 (−iα∂a )

and

2 π

29 30 31

34 35 36 37

(136)

38 39 40 41

(137)

42 43

To quantize this model, we could make the following replacements for πa and π πa2

28

33

With the choice of the gauge N = a, the Hamiltonian can be written as

UN

28

RR

27

31

After partial integration of the second term in the parentheses of eqn. (133), we find that the minisuperspace action becomes    2  ˙ 1 V () a a˙ 2  S = S g + Sm = − dtN − 2 + a + dtN a 3 , (134) 2 N 2N 2 2

27

2 2 = −∂ g (−iα∂ ) .

Then, the Wheeler–DeWitt equation, Hψ (a, ) = 0, reads

44

(138)

45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.24 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

24

 1

∂a2 g 2 (−iα∂a ) −

2

 2 g 2 (−iα∂ ) ∂  + V (a, ) ψ (a, ) = 0. a2

(139)

8 9

1 3 2 ψW KB (a, ) = C1 () ψW KB (a, ) + C3 () ψW KB (a, ) for a V () > 1

10

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

0 2 2 ψW KB (a, ) = C0 () ψW KB (a, ) + C2 () ψW KB (a, ) for a V () < 1,

where

*    *  a exp V −1 () a  a 2 V () − 1λk αa  a 2 V () − 1 da  k ψW .   KB (ξ ) =    * *    x 2 g 2 (αx)  |       2  2    

x=−ia

1 ψW KB (a, ) = C () ψW KB

38 39 40 41 42 43 44 45 46 47

6 7 8 9 10

13 14 15 16

(142)

17 18

a V ()−1

19

(143)

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

(144)

CO

37

2 ψW KB (a, ) = e3iπ/4 C () ψW KB (a, ) .

5

12

(141)

are admitted in the oscillatory region a 2 V () > 1. Note that since ∂ V (a,) |a=V −1 () > 0, we ∂a 1 −1 (), and ψ 2 have that ψW ) propagates away from the turning point a = V (a, KB W KB (ξ ) is exponentially increasing away from a = V −1 (). The WKB connection formula (64) gives the wave function ψW KB (a, ) in the classically forbidden region a 2 V () < 1:

It appears that eqn. (139) can be described as a particle of zero energy moving in a potential V (a, ). The universe can start at a = 0 and tunnel through the potential barrier to the oscillatory region. The tunneling probability is given by eqn. (74): ⎡ ⎤ V −1/2  ()     ⎢ ⎥ (145) P () ∼ exp ⎣−2 a 1 − a 2 V ()λ0 αa 1 − a 2 V () da ⎦ .

UN

36

(a, )

4

11

To specify the WKB solution of the Wheeler–DeWitt equation, we need to make a choice of boundary condition. The tunneling proposal was proposed by Vilenkin [51], which states that the universe tunnels into “existence from nothing.” The tunneling proposal of Vilenkin [52] is that the wavefunction ψ should be everywhere bounded, and at singular boundaries of superspace ψ includes only outgoing modes. In our case, the boundary a = ∞ with φ finite is the singular boundary. For the WKB solutions, it follows that the tunneling proposal demands that only the outgoing modes with

34 35

a V ()−1 λk αa

(140)

TE

12

and

RR

11

RO OF

7

DP

6

Confining ourselves to regions in which the potential can be approximated by a cosmological constant, we can drop the term involving derivatives with respect to  in eqn. (139), thereby obtaining a simple 1-dimensional problem which is amenable to the WKB analysis. In this case, eqn. (139) becomes eqn. (3) with p2 (a) = V (a, ). The function p 2 (a) is illustrated in Fig. 3(b). Thus, we find that the WKB solutions are

EC

5

2 3

3 4

1

0

P () can be interpreted as the probability distribution for the initial values of  in the ensemble of nucleated universes. For a chaotic potential V () = λ2p , there will then be a minimum value of the scalar field, s , for which sufficient inflation is obtained. The probability of sufficient inflation is given by a conditional probability:

35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.25 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••



P ( > s |1 <  < 2 ) =  s 2 1

3 4 5 6 7 8 9 10 11 12 13 14 15

P () d P () d

19 20 21 22 23 24

4

The probability of sufficient inflation is

10

  2 . s exp − 3V () d 6α 2   P ( > s |1 <  < 2 ) =  1± [F (1 ) − F (s )] , 2 2 35 1 exp − 3V () d  2

where we define  2

  2 d φ exp − 3V () V 2 ()   F (φ) =  . 2 2 exp − d φ 3V ()

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

RR

31

g (x) = 1 + x 2 /3,

7 8 9

11 12 13 14 15 16

19

(149)

CO

30

6

18

In the first part of this paper, we used the WKB approximation method to approximately solve the deformed Schrodinger-like differential equation (3) and applied the steepest descent method to find the exact solutions around turning points. Matching the two sets of solutions in the overlap regions, we obtained the WKB connection formulas through a turning point, the deformed Bohr–Sommerfeld quantization rule, and tunneling rate formula. In the second part, several examples of applying the WKB approximation to the deformed quantum mechanics were discussed. In the example of the harmonic oscillator, we used the WKB approximation to calculate bound states in the g (x) = tan x/x case. After compared with the exact solutions, our WKB ones were shown to agree with the leading term of the WKB expansion of the exact result. The pair production rate of electron–positron pairs by a strong electric field was computed in the case with g (x) = tan x/x ≈ 1 + x 2 /3 and found to be   πm2 α 2 m2 W ∼ exp − 1+ . (150) eE 4

UN

29

5

17

EC

4. Conclusion

27 28

(148)

Since F  (φ) < 0, we find that F (1 ) > F (s ), and the probability of sufficient inflation is higher/lower in the g+ (x)/g− (x) case than in the usual case.

25 26

2 3

TE

18

1

where the initial value of  lies in the range 1 <  < 2 , and the values 1 and 2 are respectively lower and upper cutoffs on the allowed values of . For g± (x) = 1 ± x 2 , we find     6α 2 2 P ± () ∼ exp − 1± + O α4 . (147) 3V () 35V ()

16 17

(146)

,

RO OF

2

 2

DP

1

25

In the GUP case with the scalar particles pair creation rate by an electric field was calculated in the context of the deformed QFT [53] and given by   πm2 4α 2 e2 E 2 W ∼ exp − 1+ . (151) eE 3π 2 m2

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

RO OF

5

Although the expressions for the quantum gravity correction are different in the above cases, the effects of the minimal length all tend to lower the pair creation rates. Using the Bohr–Sommerfeld quantization rule, we calculated the bound states of the Pöschl– Teller potential in the g (x) = 1 ± x 2 case. The quasinormal modes of a black hole could be related to the bound states of the Pöschl–Teller potential by approximating the gravitational barrier potential of the black hole with the inverted Pöschl–Teller potential. In this way, the effects of quantum gravity on quasinormal modes of a Schwarzschild black hole were estimated. Moreover, the effects of quantum gravity on the area quantum of the black hole was considered via Bohr’s correspondence principle. In the g (x) = 1 ± x 2 case, we found that the minimum increase of area was  3πα 2 n2c A = 8π 1 ± , (153) A

DP

4

where nc is some upper cutoff on n. On the other hand, authors of [55] followed the original Bekenstein argument [56] and gave that  6πα 2 A = 8π 1 ± (154) A

TE

3

The pair creation rate was also calculated by using Bogoliubov transformations [54] and given by    m2 π α 2 m2 e2 E 2 W ∼ exp − 1+ 1− 4 . (152) eE 4 m

in the MDR scenario in which g (x) = 1 ± x 2 , and  4πα 2 A = 8π 1 + A

EC

2

Acknowledgements

37 38 39

UN

43

45

f± (s) = ±s −

46 47

where

|s|3 ρ (s) ei[3θ+σ (s)] , 3

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

27

29 30 31 32 33 34

36 37 38 39 40 41 42

Proof. For f± (s), we have

44

4

35

Appendix A. Proof of Proposition 1

42

3

28

We are grateful to Houwen Wu and Zheng Sun for useful discussions. This work is supported in part by NSFC (Grant No. 11375121 11005016 and 11175039).

40 41

(155)

CO

36

2

26

in the GUP scenario in which g (x) = 1 + x 2 . Finally, we used the WKB approximation method to find the WKB solutions of the deformed Wheeler–DeWitt equation for a closed Friedmann universe with a scalar field. In the context of the tunneling proposal, the effects of quantum gravity on the probability of sufficient inflation was also discussed.

35

1

25

RR

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.26 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

26

43 44

(A1)

45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.27 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

s = |s| eiθ ,

1 2

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

RO OF

11

|as|    3 1  2  2  f (s) ≤ x  g (−ix) − 1 d |x| ≤ , 3 2 |as| 0

and hence 1 3 ≤ 1 − f (s) ≤ ρ (s) ≤ 1 + f (s) ≤ , 2 2

1 π |sin σ (s)| ≤ f (s) ⇒ |σ (s)| ≤ arcsin f (s) ≤ arcsin = . 2 6

4

6 7 8 9 10

(A3)

Im f± (s) = Im f± (λk (a)) .

At s = s∗ , this equation becomes  ∗  ∗ 2 ∗ 3 ρ (s) sin 3θ + σ (s ) ±ca sin θ − c = a 3 Im f± (λk (a)) , 3

13 14 15 16

(A4)

(A5)

σ (s ∗ ) 2nπ (2n + 1) π θ + = + O (a) or + O (a) , 3 3 3 where n ∈ {0, 1, 2}. However for θ ∗ +

  σ s∗ 3

=

(2n+1)π 3

18 19 20 21 22

24 25

(A6)

26 27 28 29

(A7)

30 31 32

+ O (a), we find at s = s∗ that

ρ (s∗ ) c3 Re f± (s∗ ) ∼  Re f± (λk (a)) ∼ O (a) , 3a 3

17

23

where we use R = ac . For small a, one has Im f± (λk (a)) ∼ O (a) and hence ∗

11 12

Suppose that the contour Cλk intersects CR at s∗ = Reiθ∗ . Since Cλk is also a constant-phase contour, Cλk is determined by

EC

10

3

5

ρ (s) eiσ (s) = 1 + f (s) eiα(s) .   Since g 2 (−ix) − 1 ≤ 12 for |x| ≤ aR, one finds for |s| ≤ R that

RR

9

(A2)

CO

8

2



x 2 g 2 (−ix) − 1 dx,

33 34

(A8)

35 36

which contradicts Cλk being the steepest descent contour. Thus, for the steepest descent contour Cλk , eqn. (A7) gives for some n ∈ {0, 1, 2} that    ∗ 2nπ  |σ (s ∗ )| π = θ − + O (a) ≤ + O (a) .   3 3 18

37

It can be easily shown that

42

UN

7



0

5 6

as

DP

4

3 f (s) eiα(s) = 3 3 a s

1

TE

3

27

    c3 ρ (s∗ ) c3 −2 −2 Re f± (s∗ ) = + O a + O a ≤ − , 3a 3 6a 3 where we use ρ (s∗ ) ≥ 12 .

2

38 39 40 41

43 44

(A9)

45 46 47

JID:NUPHB AID:14369 /FLA

28

[m1+; v1.285; Prn:8/06/2018; 10:32] P.28 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

1

Appendix B. Proof of Proposition 2

2

6

 Proof. Since |σ (s)| ≤ π6 on the CR , if θ −     θ + σ (s) − 2nπ  ≤ π + O (a) ,  3 3  9

7

which leads to

8

1 cos [3θ + σ (s)] ≥ cos = + O (a) . (B2) 3 2  π   Thus, it shows that on θ − 2nπ 3 ≤ 18 + O (a),            R 3 ρ Reiθ cos 3θ + σ Reiθ c3 iθ Re f± Re + O a −2 , + O a −1 ≤ − =− 3 3 12a (B3)  iθ  1 where we use ρ Re ≥ 2 . 2

5

9 10 11 12 13 14 15 16

π 

Appendix C. Contours in g(x) =

19

21 22 23 24

case

In this appendix, we consider the contours C> and C< in the g(x) = we have four regular saddle points:   arctanh eiπk/2 a λk (a) = , for k = 0, 1, 2, 3. a

TE

20

tan(x) x

26 27 28 29

31 32 33 34

38 39 40 41 42 43 44 45 46 47

UN

37

7 8 9 10 11 12 13 14 15 16 17 18 19

tan(x) x

case. In this case,

20 21 22

(C1)

CO

35

5 6

RR

30

36

(B1)

EC

25

3 4

17 18

2

+ O (a), we have

RO OF

4



2nπ  π 3 ≤ 18

DP

3

1

Fig. 4. Contours (blue lines) and saddle points (red dots) of ψ1 (ρ) and ψ2 (ρ) in the g(x) = tan(x) case. The green dots x are poles where the contours end.

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.29 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

2 3 4 5 6 7

Note that f± (s) both have poles at s = ± iπ 2a . The endpoints of a contour are at either infinity case, the contours C> and C< both start or singularities. It turns out that in the g(x) = tan(x) x . Following the conventions adopted in section 2, we plot from and terminate at poles s = ± iπ 2a the contours C> and C< in Fig. 4, where the green dots are poles. The circle CR is also plotted in Fig. 4.

RO OF

1

References

8

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

DP

14

TE

13

EC

12

RR

11

[1] M. Maggiore, The algebraic structure of the generalized uncertainty principle, Phys. Lett. B 319 (1993) 83, https:// doi.org/10.1016/0370-2693(93)90785-G, arXiv:hep-th/9309034. [2] A. Kempf, G. Mangano, R.B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D 52 (1995) 1108, https://doi.org/10.1103/PhysRevD.52.1108, arXiv:hep-th/9412167. [3] P.K. Townsend, Small scale structure of space–time as the origin of the gravitational constant, Phys. Rev. D 15 (1977) 2795, https://doi.org/10.1103/PhysRevD.15.2795. [4] D. Amati, M. Ciafaloni, G. Veneziano, Can space–time be probed below the string size?, Phys. Lett. B 216 (1989) 41, https://doi.org/10.1016/0370-2693(89)91366-X. [5] K. Konishi, G. Paffuti, P. Provero, Minimum physical length and the generalized uncertainty principle in string theory, Phys. Lett. B 234 (1990) 276, https://doi.org/10.1016/0370-2693(90)91927-4. [6] L.N. Chang, D. Minic, N. Okamura, T. Takeuchi, The effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem, Phys. Rev. D 65 (2002) 125028, https://doi.org/10.1103/ PhysRevD.65.125028, arXiv:hep-th/0201017. [7] F. Brau, Minimal length uncertainty relation and hydrogen atom, J. Phys. A 32 (1999) 7691, https://doi.org/10.1088/ 0305-4470/32/44/308, arXiv:quant-ph/9905033. [8] S. Das, E.C. Vagenas, Universality of quantum gravity corrections, Phys. Rev. Lett. 101 (2008) 221301, https:// doi.org/10.1103/PhysRevLett.101.221301, arXiv:0810.5333 [hep-th]. [9] S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer, H. Stoecker, Collider signatures in the Planck regime, Phys. Lett. B 575 (2003) 85, https://doi.org/10.1016/j.physletb.2003.09.040, arXiv:hep-th/0305262. [10] A.F. Ali, S. Das, E.C. Vagenas, Discreteness of space from the generalized uncertainty principle, Phys. Lett. B 678 (2009) 497, https://doi.org/10.1016/j.physletb.2009.06.061, arXiv:0906.5396 [hep-th]. [11] X. Li, Black hole entropy without brick walls, Phys. Lett. B 540 (2002) 9, https://doi.org/10.1016/S0370-2693(02) 02123-8, arXiv:gr-qc/0204029. [12] F. Brau, F. Buisseret, Minimal length uncertainty relation and gravitational quantum well, Phys. Rev. D 74 (2006) 036002, https://doi.org/10.1103/PhysRevD.74.036002, arXiv:hep-th/0605183. [13] F. Scardigli, Generalized uncertainty principle in quantum gravity from micro-black hole Gedanken experiment, Phys. Lett. B 452 (1999) 39, https://doi.org/10.1016/S0370-2693(99)00167-7, arXiv:hep-th/9904025. [14] F. Scardigli, R. Casadio, Gravitational tests of the generalized uncertainty principle, Eur. Phys. J. C 75 (9) (2015) 425, https://doi.org/10.1140/epjc/s10052-015-3635-y, arXiv:1407.0113 [hep-th]. [15] D. Chen, H. Wu, H. Yang, S. Yang, Effects of quantum gravity on black holes, Int. J. Mod. Phys. A 29 (26) (2014) 1430054, https://doi.org/10.1142/S0217751X14300543, arXiv:1410.5071 [gr-qc]. [16] S. Hossenfelder, Minimal length scale scenarios for quantum gravity, Living Rev. Relativ. 16 (2013) 2, https:// doi.org/10.12942/lrr-2013-2, arXiv:1203.6191 [gr-qc]. [17] G. Amelino-Camelia, Introduction to quantum-gravity phenomenology, Lect. Notes Phys. 669 (2005) 59, arXiv: gr-qc/0412136. [18] G. Amelino-Camelia, J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos, S. Sarkar, Tests of quantum gravity from observations of gamma-ray bursts, Nature 393 (1998) 763, https://doi.org/10.1038/31647, arXiv:astro-ph/9712103. [19] L.J. Garay, Space–time foam as a quantum thermal bath, Phys. Rev. Lett. 80 (1998) 2508, https://doi.org/10.1103/ PhysRevLett.80.2508, arXiv:gr-qc/9801024. [20] G. Amelino-Camelia, Doubly special relativity, Nature 418 (2002) 34, https://doi.org/10.1038/418034a, arXiv:grqc/0207049. [21] J. Magueijo, L. Smolin, Generalized Lorentz invariance with an invariant energy scale, Phys. Rev. D 67 (2003) 044017, https://doi.org/10.1103/PhysRevD.67.044017, arXiv:gr-qc/0207085. [22] D. Mattingly, Modern tests of Lorentz invariance, Living Rev. Relativ. 8 (2005) 5, https://doi.org/10.12942/lrr-20055, arXiv:gr-qc/0502097. [23] S. Liberati, Tests of Lorentz invariance: a 2013 update, Class. Quantum Gravity 30 (2013) 133001, https://doi.org/ 10.1088/0264-9381/30/13/133001, arXiv:1304.5795 [gr-qc].

CO

10

UN

9

29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

30

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

RO OF

7

DP

6

TE

5

EC

4

RR

3

[24] D. Colladay, V.A. Kostelecky, Lorentz violating extension of the standard model, Phys. Rev. D 58 (1998) 116002, https://doi.org/10.1103/PhysRevD.58.116002, arXiv:hep-ph/9809521. [25] S.R. Coleman, S.L. Glashow, High-energy tests of Lorentz invariance, Phys. Rev. D 59 (1999) 116008, https:// doi.org/10.1103/PhysRevD.59.116008, arXiv:hep-ph/9812418. [26] G. Amelino-Camelia, T. Piran, Planck scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV gamma paradoxes, Phys. Rev. D 64 (2001) 036005, https://doi.org/10.1103/PhysRevD.64.036005, arXiv: astro-ph/0008107. [27] T. Jacobson, S. Liberati, D. Mattingly, TeV astrophysics constraints on Planck scale Lorentz violation, Phys. Rev. D 66 (2002) 081302, https://doi.org/10.1103/PhysRevD.66.081302, arXiv:hep-ph/0112207. [28] T.A. Jacobson, S. Liberati, D. Mattingly, F.W. Stecker, New limits on Planck scale Lorentz violation in QED, Phys. Rev. Lett. 93 (2004) 021101, https://doi.org/10.1103/PhysRevLett.93.021101, arXiv:astro-ph/0309681. [29] S. Corley, T. Jacobson, Hawking spectrum and high frequency dispersion, Phys. Rev. D 54 (1996) 1568, https:// doi.org/10.1103/PhysRevD.54.1568, arXiv:hep-th/9601073. [30] T.G. Pavlopoulos, Breakdown of Lorentz invariance, Phys. Rev. 159 (1967) 1106, https://doi.org/10.1103/PhysRev. 159.1106. [31] P. Wang, H. Yang, Black hole radiation with modified dispersion relation in tunneling paradigm: static frame, arXiv:1505.03045 [gr-qc]. [32] T.V. Fityo, I.O. Vakarchuk, V.M. Tkachuk, WKB approximation in deformed space with minimal length, J. Phys. A 39 (2) (2006) 379, https://doi.org/10.1088/0305-4470/39/2/0088. [33] G. Blado, T. Prescott, J. Jennings, J. Ceyanes, R. Sepulveda, Effects of the generalized uncertainty principle on quantum tunneling, Eur. J. Phys. 37 (2016) 025401, https://doi.org/10.1088/0143-0807/37/2/025401, arXiv:1509. 07359 [quant-ph]. [34] X. Guo, B. Lv, J. Tao, P. Wang, Quantum tunneling in deformed quantum mechanics with minimal length, arXiv: 1609.06944 [gr-qc]. [35] J. Tao, P. Wang, H. Yang, Homogeneous field and WKB approximation in deformed quantum mechanics with minimal length, Adv. High Energy Phys. 2015 (2015) 718359, https://doi.org/10.1155/2015/718359, arXiv:1211. 5650 [hep-th]. [36] M. Eune, W. Kim, Lifshitz scalar, brick wall method, and GUP in Horava–Lifshitz gravity, Phys. Rev. D 82 (2010) 124048, https://doi.org/10.1103/PhysRevD.82.124048, arXiv:1007.1824 [hep-th]. [37] P. Wang, H. Yang, S. Ying, Black hole radiation with modified dispersion relation in tunneling paradigm: free-fall frame, Eur. Phys. J. C 76 (1) (2016) 27, https://doi.org/10.1140/epjc/s10052-015-3858-y, arXiv:1505.04568 [gr-qc]. [38] P. Wang, H. Yang, S. Ying, Minimal length effects on entanglement entropy of spherically symmetric black holes in the brick wall model, Class. Quantum Gravity 33 (2) (2016) 025007, https://doi.org/10.1088/0264-9381/33/2/ 025007, arXiv:1502.00204 [gr-qc]. [39] J.Z. Simon, Higher derivative Lagrangians, nonlocality, problems and solutions, Phys. Rev. D 41 (1990) 3720, https://doi.org/10.1103/PhysRevD.41.3720. [40] C.L. Ching, R. Parwani, Scattering and bound states of a deformed quantum mechanics, arXiv:1207.1519 [hep-th]. [41] L.N. Chang, D. Minic, N. Okamura, T. Takeuchi, Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations, Phys. Rev. D 65 (2002) 125027, https://doi.org/10.1103/PhysRevD.65. 125027, arXiv:hep-th/0111181. [42] J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664. [43] W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870, https://doi.org/10.1103/PhysRevD.14. 870. [44] R.A. Konoplya, A. Zhidenko, Quasinormal modes of black holes: from astrophysics to string theory, Rev. Mod. Phys. 83 (2011) 793, https://doi.org/10.1103/RevModPhys.83.793, arXiv:1102.4014 [gr-qc]. [45] V. Ferrari, B. Mashhoon, New approach to the quasinormal modes of a black hole, Phys. Rev. D 30 (1984) 295, https://doi.org/10.1103/PhysRevD.30.295. [46] H.P. Nollert, Topical review: quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class. Quantum Gravity 16 (1999) R159, https://doi.org/10.1088/0264-9381/16/12/201. [47] S. Hod, Bohr’s correspondence principle and the area spectrum of quantum black holes, Phys. Rev. Lett. 81 (1998) 4293, https://doi.org/10.1103/PhysRevLett.81.4293, arXiv:gr-qc/9812002. [48] M. Maggiore, The physical interpretation of the spectrum of black hole quasinormal modes, Phys. Rev. Lett. 100 (2008) 141301, https://doi.org/10.1103/PhysRevLett.100.141301, arXiv:0711.3145 [gr-qc]. [49] J.D. Bekenstein, The quantum mass spectrum of the Kerr black hole, Lett. Nuovo Cimento 11 (1974) 467, https:// doi.org/10.1007/BF02762768. [50] P. Chen, R.J. Adler, Black hole remnants and dark matter, Nucl. Phys. Proc. Suppl. 124 (2003) 103, https://doi.org/ 10.1016/S0920-5632(03)02088-7, arXiv:gr-qc/0205106.

CO

2

UN

1

[m1+; v1.285; Prn:8/06/2018; 10:32] P.30 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–••• 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

JID:NUPHB AID:14369 /FLA

[m1+; v1.285; Prn:8/06/2018; 10:32] P.31 (1-31)

F. Lu et al. / Nuclear Physics B ••• (••••) •••–•••

2 3 4 5 6 7 8 9

[51] A. Vilenkin, Quantum creation of universes, Phys. Rev. D 30 (1984) 509, https://doi.org/10.1103/PhysRevD.30.509. [52] A. Vilenkin, Quantum cosmology and the initial state of the universe, Phys. Rev. D 37 (1988) 888, https://doi.org/ 10.1103/PhysRevD.37.888. [53] B.R. Mu, P. Wang, H.T. Yang, Minimal length effects on Schwinger mechanism, Commun. Theor. Phys. 63 (6) (2015) 715, https://doi.org/10.1088/0253-6102/63/6/715, arXiv:1501.06020 [gr-qc]. [54] S. Haouat, K. Nouicer, Influence of a minimal length on the creation of scalar particles, Phys. Rev. D 89 (10) (2014) 105030, https://doi.org/10.1103/PhysRevD.89.105030, arXiv:1310.6966 [hep-th]. [55] G. Amelino-Camelia, M. Arzano, Y. Ling, G. Mandanici, Black-hole thermodynamics with modified dispersion relations and generalized uncertainty principles, Class. Quantum Gravity 23 (2006) 2585, https://doi.org/10.1088/ 0264-9381/23/7/022, arXiv:gr-qc/0506110. [56] J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333, https://doi.org/10.1103/PhysRevD.7.2333.

RO OF

1

10 11 12 13 14

DP

15 16 17 18 19 20

TE

21 22 23 24

EC

25 26 27 28 29

RR

30 31 32 33 34

38 39 40 41 42 43 44 45 46 47

UN

37

CO

35 36

31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47