Modified Wave Particle Duality and Black Hole Physics

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1 Commun. Theor. Phys. (Beijing, China) 48 (2007) pp c International Academic Publishers Vol. 48, No. 1, July 15, 2007 Modified Wave Particle Duality and Black Hole Physics LI Xiang Department of Physics, Nanchang University, Nanchang , China (Received August 8, 2006) Abstract de Broglie relation is revisited, in consideration of a generalization of canonical commuting relation. The possible effects on particle s localization and black hole physics are also discussed, in a heuristic manner. PACS numbers: m, Dy Key words: wave-particle duality, dispersion relation, localization limit, black hole entropy 1 Introduction Measurement is one of fundamental concepts of quantum mechanics, where position and momentum are two fundamental dynamical variables. The position and momentum of a particle cannot be measured simultaneously, with arbitrarily high accuracy. The uncertainties of position and its conjugate momentum are related by Heisenberg s uncertainty relation x p h, i.e., one of fundamental principles of quantum mechanics. However, this principle only meets the quantum effect of matter, and it does not directly describe the quantum fluctuation of spacetime. Based on some theoretical considerations and gedanken experiments, quantum effects of gravity may influence Heisenberg s uncertainty principle. [1 8] The first appearance of generalized uncertainty principle (GUP) is closely related to string theory, and it reads x 1 p + α2 ( p), (1) which should be regarded as an incorporation of gravity and quantum theory, and it means that there is a minimal length that we can measure, α. The second term on the right-hand side of Eq. (1) means a new duality, x p, which is different from ordinary quantum mechanics. Certainly, α is supposed to be of order of Planck length, new effects of this term are too slight to be detected. Corresponding to Eq. (1), a possible generalization of canonical commuting relation is given by [6] [ˆx, ˆp] = i(1 + α 2 ˆp 2 ), (2) which induces the relation (1). A more general form of the commuting relation is given by [3,7] [ˆx, ˆp] = i 1 + α 2 (ˆp 2 + m 2 ), (3) which returns to Eq. (2) in the regime of low-energy far away from Planck scale. The modification to the relativistic dispersion relation (MDR) is another possible effect of quantum gravity. MDR can be motivated by loop quantum gravity (MDR), [9 12] it changes the relativistic dispersion relation into g 2 (E) = m 2 + p 2, where g(e) is a function of the energy E, and its concrete formula is model-dependent. For standard dispersion relation, g(e) = E. Some works [12 15] imply that there are some similarities between GUP and MDR, based on the observations on their applications to black holes physics. The effects of GUP and MDR were taken into account respectively, both gave the logarithmic correction to black hole entropy. In those applications, GUP and MDR are independent: uncertainty principle maintains invariant when MDR is considered, or dispersion relation is changeless while uncertainty principle is generalized. Indeed, GUP and MDR are different concepts and come from two different physical considerations: the former is the generalization of one of fundamental principles of quantum mechanics, and the latter changes one of important conclusions of relativistic theory. The dynamics of a system is determined by Hamiltonian, and then determined by energy-momentum relation. One may choose different Hamiltonian, based on different considerations. For example, reference [7] considered two kinds of Hamiltonian, H = p 2 + m 2, sinh (αh) = α p 2 + m 2. (4) They have different signatures for relativity, when incorporating them with the generalized commuting relation (3). The first Hamiltonian seems to be a natural choice, because it corresponds to the standard relativistic energymomentum relation. However, it makes the speed of light variable and dependent on the energy of photon. [7] The second Hamiltonian (4) deforms the dispersion relation, but it keeps the speed of light invariant. The aim of this The project supported by National Natural Science Foundation of China under Grant No , Natural Science Foundation of Jiangxi Province under Grant No xiang.lee@163.com In this paper, Planck constant h, speed of light c, and Newtonian constant G are set to be unity, unless we need to stress them

2 94 LI Xiang Vol. 48 paper is to investigate some problems related to quantum mechanics and black holes physics, based on the generalized commuting relation (3) and the modified dispersion relation (4). The contents are arranged as follows. The second section is devoted to the modified wave-particle duality. [6] The third section discusses the particle localization limit, an alternative to Compton wavelength is suggested by revisiting the measurement process. Since the speed of light is a constant, equations (3) and (4) can be applied to black hole physics in a logically consistent manner. In Sec. 4, Bekenstein s argument about black hole entropy is reexamined, based on the discussion in Sec. 3. It is convenient that a classical background is fixed, in the following discussions. The quantum effect of spacetime is transferred to matter, and the motion of matter is assumed to be described by GUP and MDR. The similar philosophy was also adopted in Refs. [8] and [12] [15]. 2 Modified Wave-Particle Duality de Broglie relation is the quantitative description of wave-particle duality. As a conjecture, it appeared before Heisenberg s uncertainty principle. However, de Broglie relation can be obtained within the framework of quantum mechanics, if Heisenberg s uncertainty principle is treated as a fundamental principle. Concretely speaking, starting from the fundamental commuting relation [ˆx, ˆp] = i, [ f(ˆp) = (1 + α 2 m 2 + α 2 ˆp 2 ) 1/2 d ˆp = α 1 arcsinh αˆp ] (1 + α 2 m 2 ) 1/2 a momentum eigenstate in coordinate representation is given by [16] ϕ p (x) = exp(ipx). (5) Comparing it with a plane wave amplitude, exp(ikx), we obtain the relation between the wave-length and the momentum, λ = 2π/k = 2π/p. Once the canonical commuting relation is changed, wave-particle duality is expected to suffer a modification. [5,6] We want to know how the de Broglie relation is generalized, in the context of the commuting relation (3). In the following discussion, we will see that the momentum eigenstate of a free particle is associated with the plane wave approximately. To obtain the modified de Broglie relation, we need to know the momentum eigenstate in the coordinate representation and compare it with the plane wave amplitude. Obviously, the generalized commuting relation (3) is complicated. For convenience, we introduce a function f(ˆp) and make it satisfy a simple commuting relation as follows: [ˆx, f(ˆp)] = i, (6) According to the operator algebra, we have [ˆx, ˆp] = [ˆx, f(ˆp)] ˆp f = i ˆp f, (7) where p is supposed to be expanded as a series of f. Comparing the above equation with Eq. (3), we obtain [ = α 1 ln αˆp (1 + α 2 m 2 ) 1/2 + ] 1 + α2 ˆp α 2 m 2, (8) which satisfies [ˆp, f(ˆp)] = 0, and the common eigenstate ψ p reads ˆp ψ p = p ψ p, f(ˆp) ψ p = f(p) ψ p. (9) To obtain the coordinate representation of ψ p, we first introduce a real small parameter ε and define an operator and then Ŝ = exp[iεf(ˆp)] 1 + iεf(ˆp), (10) x Ŝ ψ p ψ p (x) + iε x f(ˆp) ψ p = [1 + iεf(p)] ψ p (x), (11) where ψ p (x) = x ψ p is the momentum eigenstate in coordinate representation. On the other hand, we have and then [ˆx, Ŝ] = [ˆx, f(ˆp)] Ŝ f = εŝ, (ˆx + ε)ŝ = Ŝˆx, (12) x Ŝ(x + ε) = x (ˆx + ε)ŝ = x Ŝˆx, Equation (11) becomes x Ŝ = x + ε. (13) ψ p (x + ε) = [1 + iεf(p)]ψ p (x). (14) Setting ε = x, we obtain the following equation: f(p)ψ p (x) = i ψ p x i d dx ψ p(x). (15) The momentum eigenstate in coordinate representation is given by ψ p exp[ixf(p)], (16) which describes a plane wave. Comparing the wave amplitude exp(ikx), the wave-number reads k = f(p). The wave s spatial period, i.e., the modified de Broglie wavelength, is given by λ db = 2π k = As α 0, 2π f(p) = 2πα arcsinh [ αp/(1 + α 2 m 2 ) 1/2]. (17) f(p) p, λdb 2π p, (18) which is just the standard de Broglie formula. It is noted that the position eigenstate x is unobservable, because the error x in this state is required to be zero and GUP is violated. We expect that the concept of coordinate representation is still applicable to the

3 No. 1 Modified Wave Particle Duality and Black Hole Physics 95 case that we consider in this paper. After all, most cases that we deal with is far away from Planck scale. The position eigenstate in usual quantum mechanics is actually unobservable too, since the momentum uncertainty is explosive. However, this does not throw serious obstacles in the way of introducing the coordinate representations. 3 Particle Localization: Alternative to Compton Wavelength Compton wavelength of a particle with mass m is defined as λ c = 2π h mc, (19) which can be regarded as the length of the particle. In other words, it imposes a constraint on the scenario that the particle can be localized. According to Heisenberg s uncertainty principle x p h, the position of particle can be localized/measured with arbitrarily high accuracy, if the momentum uncertainty is allowed to be arbitrarily large. If the coordinate of the particle is detected by a photon, the photon s wavelength is required to be arbitrarily short, or the energy is required to be arbitrarily large. This is the justification for applying Schrödinger equation of non-relativistic quantum mechanics. However, when the photon s energy is comparable to the rest mass of the particle, the momentum uncertainty of particle is p mc and relativistic effect cannot be ignored. Schrödinger equation is no longer valid and it is replaced by Klein Gordon equation or Dirac equation in relativistic quantum mechanics. The relativistic wave function has the negative frequency modes, which means the appearance of antiparticles in the process that the particle is localized in a small spatial region. The production of new particle changes the physical significance of wave function essentially. It becomes meaningless to talk about the probability density that the particle can be found in a particular spatial region. Corresponding to the ultra relativistic momentum mc, a characteristic wavelength is naturally defined as x h/ p h/mc λ c. It is a scale that the production of particle cannot be ignored and the relativistic wave function does not play the role of probability density. Compton wavelength imposes a constraint on the localization description of a particle. The speed of light c and Planck constant h are introduced in the definition of Compton wavelength. Obviously, this is an incorporation of quantum and relativistic effects. However, the absence of Newtonian constant means that (quantum) gravity is not considered when identifying Compton wavelength as the particle localization limit. We want to know how quantum gravity influences the localization description of the particle, in consideration of the modified dispersion relation (4) and the modified de Broglie relation (17). The following is devoted to a gedanken experiment measuring the position of a rest particle with mass m. The probe is a photon with wavelength λ. From Eq. (4), the energy of a massless particle is given by E γ = α 1 arcsinh(αp γ ). (20) If the particle is measured with the precision δ, one should use a photon with wavelength λ δ. According to the modified de Broglie relation (17), the momentum of the photon, p γ satisfies λ = 2πα arcsinh(αp γ ) = 2π δ, (21) E γ where E γ is the photon s energy, which satisfies Eq. (20). Considering Eq. (4), the static energy of the particle reads E = α 1 arcsinh (αm). (22) In the measuring process, the energy of the photon is transferred to the particle and increases the energy uncertainty of the latter. It is necessary to avoid the production of new copy of the particle, otherwise the measurement becomes meaningless. This actually means that the photon s energy is not allowed to be too large, i.e., E E γ E. Combining Eq. (21) with Eq. (22), we have δ 2πα arcsinh (αm) = 2πα ln[αm + (αm) 2 + 1], (23) which is the gravitational induced correction to the particle localization limit. α appears from the generalization of uncertainty principle and denotes the quantum effect of gravity. At a first sight, the above expression is very different from Compton wavelength. However, it can be expanded as a series δ 2π (1 αm m 2 + α2 m 2 ) +, (24) 6 where the leading term is just Compton wavelength. Here we suppose that the mass of particle is less than Planck mass, i.e., m < α 1. 4 Correction to Black Hole Entropy: a Heuristic Method Black hole thermodynamics is an important probe of quantum gravity. About thirty years ago, Beknestein introduced the concept of black hole entropy, [17] based on some heuristic arguments. Starting from information theory, quantum theory and general relativity, Bekenstein argued that the entropy should be proportional to the area of black hole. This conclusion is qualitatively valid, except that a coefficient is corrected by Hawking s rigid works. [18] The probability density is not always positive, if it is still defined by the wave function in Klein Gordon equation. However, in the lower-energy limit, the relativistic equation returns to Schrödinger equation and the wave function is well defined.

4 96 LI Xiang Vol. 48 The black hole entropy, i.e., so-called Bekenstein Hawking entropy is given by S = A 4lp 2. (25) The appearance of Planck length means that black hole entropy is associated with the degrees of freedom of gravitational field. However, quantum effect of gravity does not enter directly in the derivation of Bekenstein Hawking formula. In fact, equation (25) is attributed to the contribution of classical gravitational action and quantum fluctuation of spacetime is not taken into account. [19] Bekenstein s argument is heuristic, but it is insightful. The following relation A mδ, (26) is the basis for his original work. Following it, the minimal increase in horizon area is associated with the mass and length of a particle, when the particle is lowed down to the black hole by an adiabatic process. The particle localization limit is crucial to Bekenstein s argument. Substituting Compton wavelength for δ, one obtains that a nonzero value of the minimum increases in horizon area, i.e., A lp. 2 Two consequences deserve to be mentioned: one is the discrete spectrum of black hole area, [20] A nlp, 2 where n is an integer; another is the linear relation between black hole entropy and horizon area. What we care about is related to the second point. Starting from information theory, the information of one bit is lost when a particle is absorbed by black hole, and the increase in black hole entropy reads S = ln 2. So one has ds da S A ln 2 lp 2, (27) which results in the linear relation between entropy and horizon area. This section deals with the correction to this linear relation, in consideration of Eqs. (3) and (4). The discussion will be performed within the framework of general relativity. It means that the speed of light is a constant and the general covariance need to be maintained. Are these two principles compatible with Eqs. (3) and (4)? Firstly, the velocity of a particle is given by [7] ẋ = [ˆx, Ĥ] = [ˆx, ˆp] H p = p p2 + m. (28) 2 It is evident that the speed of light is invariant when the mass is set to be zero. On the other hand, the dispersion relation (4) is rewritten as where we define p µ p µ = η µν p µ p ν = m 2, (29) p 0 = α 1 sinh(αe), (30) which is different from the usual definition p 0 = E. However, this difference ensures the covariant form of the modified energy-momentum relation. In curved spacetime, η µν g µν, we have p µ p µ = g µν p µ p ν = m 2. (31) We see that the generalized commuting relation (3) and the modified dispersion relation (4) is compatible to general relativity. We are going to revisit Beknestein s argument, on the basis of the covariant relativistic relation (31). Considering a rest particle in a static gravitational field with spherical symmetry, p 1 = p 2 = p 3 = 0, we have p 0 = α 1 sinh(αe) = m g 00. (32) For the particle just outside the horizon, the proper distance between it (located by r ) and the location of the horizon reads r r dr r dr d = g11 dr = r 0 r 0 g00 r 0 2κ(r r 0 ) 2(r r 0 ) =. (33) κ Equation (32) can be deduced to α 1 sinh(αe) m 2κ(r r 0 ) = mκd mκδ. (34) When the particle with energy is absorbed by the black hole, the minimal increase in the black hole mass is given by the right-hand side of the above inequality, i.e., M = mκδ. In terms of the first law of black hole dynamics, The minimal increase in the area is given by A = 8π M = 8πmδ, (35) κ where δ is the particle localization limit, as mentioned in the last section. The minimal increase in horizon area, as given by Eq. (35), is just the same as the starting point of Bekenstein s argument. However, the minimal change in horizon area is no longer a constant, as the particle localization limit has been corrected by Eq. (23) or (24). The simple relation, A lp, 2 is no longer valid, which should be changed to ( 2πα ) A = 8πα 1 δ sinh = 16π 2( ) 1 + α 2 δ 6δ π 2( ) 1 + α 2 6r0 2 +, (36) where α = 2πα. The last inequality is due to a reasonable assumption that the size of particle should be less than the radius of black hole. With the information loss of one bit, the black hole entropy is increased by ln 2. We obtain ds da S A = Ω ln 2 ( ) 16π 2 1 α 2 6r0 2 +, (37) where a calibration factor Ω is introduced. The above equation is dependent on the radius of the black hole and leads to the corrections to black hole entropy. The entropy is obtained by integration S Ω ln 2 ( ) S da = A 16π 2 1 α 2 + da 6r 2 0

5 No. 1 Modified Wave Particle Duality and Black Hole Physics 97 = Ω ln 2 ( ) 16π 2 A 2πα 2 ln A +, (38) 3 the logarithmic correction appears. To obtain the correct coefficient of the leading term, the calibration factor Ω is set to equal 4π 2 / ln 2. 5 Discussions The modified dispersion relation (4) can be expanded as a series E = p 2 + m 2 α2 6 (p2 + m 2 ) 3/2 +, (39) where the linear in α term is absent. This is in agreement with the argument that the linear in l p deformation to relativistic dispersion relation should be declined, [12] since it leads to a A-type correction to black hole entropy and is inconsistent with loop quantum gravity. One difference is that present correction to dispersion relation is negative, while correction considered in Ref. [12] is positive. The present line of argument is different from Refs. [12] and [14]. The effect of generalized uncertainty principle was not considered in Ref. [12]. On the contrary, only the generalized uncertainty principle was taken into account in Ref. [14]. The present work is within the framework that the group velocity of light is a constant, and the logarithmic correction to black hole is attributed to the combined effects of the modified de Broglie relation and MDR. In Ref. [15], MDR and GUP were regarded as two independent aspects of quantum gravity and the correlation between them was not established, in which the speed of light was energy-dependent. In this paper, the modified dispersion relation takes a special form correlated with the generalized uncertainty principle, and then the speed of light is guaranteed to be independent of the photon s energy. This is the most important difference from Ref. [15]. A constant speed of light is more appropriate for the applications to the black holes in general relativity. Equation (35) is crucial to Bekenstein s argument. It also enters in Ref. [12], and becomes the starting point of the argument therein. At a first sight, the modified dispersion relation seems to violate the covariance of relativity, and Eq. (35) itself should be corrected. However, the covariance of modified dispersion relation is maintained by redefining p 0. Furthermore, the discussion from Eq. (29) to Eq. (35) has shown that the relation, A mδ is still valid. Certainly, the final result deduced from Eq. (35) suffers a modification, because the particle localization limit has been replaced by Eq. (23) and the simple relation δ 1/m is no longer valid. The quantum fluctuations of gravity may lead to some observable consequences for the high-energy particles propagating through cosmic distances. Some strong constraints are imposed on the modification of dispersion relation. [21] For example, the linear in l p correction to dispersion relation may be ruled out. [21] This constraint is supported by the argument based on black hole thermodynamics. [12] One of dramatic consequences is that vacuum Čerenkov effect occurs at a threshold momentum, [25] [ m 2 ] 1/n p th =, (40) (n 1)f n above which all the charged particles become unstable if they are governed by the following modified dispersion relation E 2 = m 2 + p 2 + f n p n. (41) However, the threshold formula (40) requires f n > 0 if n is an even number. So it cannot be applied to the modified dispersion relation as follows: E 2 = m 2 + p 2 α2 3 p4, (42) which is deduced from Eq. (39), for the ultra-high-energy particles with p m. There is no threshold for the decay of charged particles if the dispersion relation is modified by a negative extra term. [25] In my opinion, it remains open whether the speed of light is a constant. The present work adapts the invariant speed of light for the consistent incorporation of GUP and general relativity. However, this is not a unique choice. One may start with a theory with variable speed of light, such as gravity s rainbow. [26] In which, Einstein s field equation has been corrected, and classical metric of spacetime suffers a modification in the context of quantum gravity. The thermodynamics of a rainbow black hole has been investigated, [27] in a heuristic manner. A rigorous investigation is being performed. The relation between gravity s rainbow and present work will be considered. References [1] Veneziano, Europhys. Lett. 2 (1986) 199. [2] D.J. Gross and P.F. Mende, Nucl. Phys. B 303 (1986) 407. [3] M. Maggiore, Phys. Lett. B 304 (1993) 65; Phys. Rev. D 49 (1994) 5182; Phys. Lett. B 319 (1993) 83. [4] L.J. Garay, Int. J. Mod. Phys. A 10 (1995) 145. [5] D.V. Ahluwalia, Phys. Lett. A 275 (2000) 31. [6] A. Kempf, G. Mangano, and R.B. Mann, Phys. Rev. D 52 (1995) also see Refs. [22] [24] for the review of quantum gravity phenomenology

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