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Some consequences of spacetime fuzziness
Chaos, Solitons and Fractals 34 (2007) 224–234 www.elsevier.com/locate/chaos
Some consequences of spacetime fuzziness Kourosh Nozari *, Behnaz Fazlpour Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47416-1467 Babolsar, Iran Accepted 16 March 2006
Abstract Finite resolution of spacetime at Planck scale gives it a fuzzy structure (the so-called foamy or fractal spacetime). This fuzzy structure of spacetime is a consequence of quantum fluctuation of geometry itself and can be described within non-commutative geometry and some alternative approaches to quantum gravity. In this paper, some consequences of spacetime fuzziness are studied. Due to this fuzzy structure, some basic notions of ordinary quantum mechanics such as position space representation, wave packet broadening during its propagation and coherent states of quantum mechanical systems should be re-examined. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The problem of reconciling quantum mechanics with general relativity is one of the task of modern theoretical physics which until now has not yet found a consistent and satisfactory solution. The difficulty arises since general relativity deals with the events which define the world-lines of particles, while quantum mechanics does not allow the definition of trajectory; in fact the determination of the position of a quantum particle involves a measurement which introduces an uncertainty into its momentum. These conceptual difficulties have their origin in the violation, at quantum level, of the weak principle of equivalence (universality of free fall) on which general relativity is based [1,2]. Such a problem becomes more involved in the formulation of quantum theory of gravity, owing to the non-renormalizability of general relativity when one quantizes it as a local quantum field theory [3]. Nevertheless, one of the most interesting consequences of this unification is that in quantum gravity there exists a non-vanishing minimal observable length on the qffiffiffiffi h order of the Planck length, lp ¼ G 1033 cm. One cannot set up a position measurement with uncertainty less than c3 this minimal value. The existence of such a fundamental length is a dynamical phenomenon due to the fact that, at Planck scale, there are fluctuations of the background metric, i.e. a limit of the order of Planck length appears when quantum fluctuations of the gravitational field are taken into account. Existence of minimal length scale has been motivated by several promising candidates of quantum gravity [4–7] and its consequences have been studied extensively [7– 17]. This natural cut-off guarantees the renormalizability of underlying quantum field theory [18–20]. Also existence of this minimal cut off results in the modification of usual Heisenberg algebra to incorporate gravitational uncertainty from very beginning [21–24].
*
Corresponding author. E-mail address: [email protected] (K. Nozari).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.066
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Non-commutativity of spacetime [25–27] at quantum level led to spacetime fuzziness [28–35]. Existence of non-vanishing minimal uncertainty in position cause undeterminability of spacetime points. In another words, locality is no longer possible and all events are uncertain at least up to Planck length. Spacetime in this viewpoint has a fractal or foamy structure. Recently a new point of view to this issue has been developed by El Naschie. In this novel scenario, fractal structure of spacetime in quantum gravity era has been formulated within Cantorian (E1 ) space [36–39]. It seems that this scenario has the strong potential to be ultimate framework for unification of fundamental interactions [40,41] and solves some mysteries of physics such as spacetime dimensionality [42,43], the nature of time in quantum gravity [41] and some other key problems of modern physics [44–46]. The fuzzy structure of spacetime has very novel implications for the rest of quantum theory. We will see that the picture of position space representation of usual quantum mechanics breaks down. Due to non-commutativity of spacetime, dispersion relations will generalize and as a consequence, waves profile will encounter anomalous dispersions. In addition, the notion of coherency and coherent states should be re-examined within this framework. In this paper, after an introduction to spacetime fuzziness, we discuss some of its consequences in the spirit of quantum theory. Position space representation, wave profile anomalous dispersion and the notion of coherency will be reexamined within fuzzy spacetime framework.
2. Fuzzy spacetime A key characteristic of quantum theory is the emergence of uncertainties, and one might expect that the distance observable would also be affected by uncertainties. Actually various heuristic arguments suggest that for such a distance observable the uncertainties might be more pervasive: in ordinary quantum theory one is still able to measure sharply any given observable, though at the cost of renouncing all information on a conjugate observable, but it appears plausible that a quantum gravity distance observable would be affected by irreducible uncertainties. Quantum gravity suggests that in the Planck-scale regime there should be some absolute limitations on the measurability of distances. This restricted resolution of spacetime structure is referred as spacetime fuzziness (‘‘foamy or fractal spacetime ’’). Most authors consider a dD P Lp relation, meaning that the uncertainty in the measurement of distances could not be reduced below the Planck-length level, but measurability bounds of other forms, generically of the type dD P f(D, Lp) (with f some function such that f(D, 0) = 0) are also being considered. The presence of such an irreducible measurement uncertainty could be significant in various contexts. For example, one can characterize operatively this spacetime fuzziness as an irreducible (fundamental) Planck-scale contribution to the noise levels in the readout of interferometers [30,34,47]. In classical physics, interferometer noise can in principle be reduced to zero. Ordinary quantum properties of matter already introduce an irreducible noise contribution. Spacetime fuzziness would introduce an additional irreducible source of noise, reflecting the fact that the distances involved in the experiment would be inherently unsharp in a foamy spacetime picture. This inherent unsharp distance measurement is the essence of spacetime fuzziness. In fact all approaches to the quantum gravity predict some limitations on the accuracy of localization, and therefore predict some spacetime fuzziness. One is usually unable to rigorously derive from first principles a detailed description of the physical consequences of this fuzziness, such as the mentioned interferometric noise. Nevertheless, some phenomenological arguments can be perform in this direction. The key input needed for this phenomenology turns out to be the power spectrum q(f) of the Planck-scale-induced strain noise in terms of frequency f [35]. Combining some intuition about the stochastic-like features of spacetime fuzziness and the dependence on the Planck length, one can easily reach a model of q(f). For example if the effects depend linearly on the Planck length Lp 1/Ep and the underlying phenomena are of random-walk type, one is inevitably led to q Lp f 2 K2 fLp f 2 L2 :
ð1Þ
The proportionality to the square of the inverse of the frequency is a direct result of the assumption of random-walktype processes. The length scale K is needed on the basis of the dimensional analysis of the equation, and is to be treated as a free parameter to be constrained experimentally. One may choose to make reference to a dimensionless parameter, f, which may be used to express the ratio of K with the length L of the arms of an interferometer or of an optical resonator. Other hypothesis about the stochastic-like features of the underlying Planck-scale processes led to other forms of f dependence (and Lp dependence) of the strain noise power spectrum. In general one should find (a > 1) [35] 2b 2ab q fab L1þa L : p f
ð2Þ
There is therefore some interest in attempting to improve limits on the parameters fab [48]. The idea of spacetime fuzziness also motivates some research work on Planck-scale-induced decoherence. It is in fact rather plausible that
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spacetime fuzziness might affect the time evolution of quantum mechanical states in such a way that, for example, a pure state might evolve into a mixed state. A key attribute of spacetime fuzziness is that in certain non-commutative spacetimes, departures from Lorentz symmetry will be encountered [49]. There, the Planck-scale structure of spacetime can introduce a systematic dependence of the speed of photons on their wavelength. After a time duration T, the difference between the expected position of the photon and the Planck-scale-corrected position could take the form Dx Tdv cTLp/k, where k is the photon wavelength. If we instead focus on how distance fuzziness could affect the propagation of photons, it is natural to expect that a group of photons would all travel the same average distance in a given time T (and this average distance is still given by cT), but for each individual photon the distance traveled might be slightly different from the average, as a result of distance fuzziness. Just to be more specific let us imagine that distance fuzziness effectively introduces a Planck-length uncertainty pffiffiffiffiffiffiffiffiffiffiin position per each Planck time of travel. Then the final position uncertainty would be of the type Dx cTLp . The square root here (assuming a random-walk-type description) is the result of the fact that non-systematic effects do not add linearly, but rather according to rules familiar in the analysis of stochastic processes. After a brief introduction to spacetime fuzziness, in forthcoming sections we consider some consequences of this fuzzy structure.
3. Some consequences of spacetime fuzziness 3.1. Failure of position space representation In ordinary quantum mechanics position operator, x could be represented as multiplication or differentiation operator acting on square integrable position- or momentum-space wave functions w(x) := hxjwi, where the jxi is position eigenstate. Strictly speaking the jxi is not physical state since it is not normalizable and thus not in the Hilbert space. However, the operator x is essentially self-adjoint and the eigenstates can be approximated to arbitrary precision by sequences jwni of physical states of increasing localization in position space: lim Dxjwn i ¼ 0:
n!1
This situation changes drastically with the introduction of minimal uncertainties Dx0 P 0. A non-zero minimal uncertainty in position ðDxÞ2jwi ¼ hwjðx hwjxjwiÞ2 jwi P Dx0
8jwi
ð3Þ
implies that there cannot be any physical state which is a position eigenstate since an eigenstate would of course have zero uncertainty in position. All of these arguments show the failure of position space representation in quantum mechanics. Is there any way to retain such representation? Actually, this failure does not exclude the existence of unphysical, ‘formal position eigenvectors’ which lie in the domain of x alone but not in Dx;x2 ;p;p2 . In fact, such formal x-eigenvectors do exist and are of infinite energy. Most importantly however, unlike in ordinary quantum mechanics, it is no longer possible to approximate formal eigenvectors through a sequence of physical states of uncertainty in positions decreasing to zero. This is because now all physical states have at least a finite minimal uncertainty in position. A minimal uncertainty in position means that the position operator is no longer essentially self-adjoint but only symmetric. Since there are no more position eigenstates jxi in the representation of the Heisenberg algebra, the Heisenberg algebra will no longer find a Hilbert space representation on position wave functions hxjwi. Where there is no minimal uncertainty in momentum, one can work with the convenient representation of the commutation relations on momentum space wave functions and the states of maximal localization will be proper physical states [17]. We can use states of maximal localization to define a ‘quasi-position’ representation. This representation has a direct interpretation in terms of position measurements, although it does of course not diagonalize x. The modified commutation relations proposed by Kempf et al. [17] are of the form ½^xi ; ^pj ¼ ihdij ð1 þ a^x2 þ b^p2 Þ:
ð4Þ
To be more specific, let us consider the case a = 0. The other commutators read ½^pi ; ^pj ¼ 0; ½^xi ; ^xj ¼ 2ihbð^ pi ^xj ^pj^xi Þ:
ð5Þ
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The last relation means that we have a non-commutative geometry since translations along different directions do not commute anymore. This non-commutativity of space coordinate is the signature of fuzzy structure of space in quantum level. The presence of ~ p2 ;~ x2 in Eq. (4) implies that the rotational symmetry, whose generators are ^ij ¼ L
1 ð^xi ^pj ^xj ^pi Þ 1 þ b~ p2
ð6Þ
is still preserved. As has been mentioned, one of the remarkable features of this theory is the existence of a minimal length uncertainty pffiffiffi h b because of the new Heisenberg inequalities ! n X h 2 2 Dxi Dpj P dij 1 þ b : ð7Þ ðD~ pk Þ þ h~ pk i 2 k¼1 The momentum representation is found to be given by the operators ^pi :wð~ pÞ ¼ pi wð~ pÞ
ð8Þ
^xi wð~ pÞ ¼ ihð1 þ b~ p2 Þopi wðpÞ
ð9Þ
and
acting on a Hilbert space in which the scalar product is given by Z p d3~ / ðpÞwðpÞ: h/jwi ¼ ð1 þ b~ p2 Þ
ð10Þ
As we have argued, the presence of a minimal length uncertainty implies that no position representation exists. The concept which proves to be the closest to it is the quasi-position representation in which the operators are non-local pffiffiffi pffiffiffi ^xi ¼ ni þ b tan h boni ð11Þ and ^pi ¼
qffiffiffiffiffiffiffi pffiffiffi b1 tan ih boni ;
ð12Þ
where ni is the mean value of the position operator. The formalism provided in this subsection can be used to retain position space representation in ordinary quantum mechanics. 3.2. Waves in fuzzy spacetime Existence of minimal observable length which restricts the accuracy of position measurements and complete resolution of spacetime points, will affect wave packet propagation in a background fuzzy spacetime. Here we are going to study the effect of spacetime fuzziness on the wave packet propagation. First we give an outline to the ordinary quantum mechanical considerations. For simplicity, consider the following plane wave profile: f ðx; tÞ / eikxixt :
ð13Þ
and m ¼ kc, this relation can be written as f(x, t) / eik(xct). Now the superposition of these plane Since x = 2pm, k ¼ 2p k waves with amplitude g(k) can be written as, Z 1 f ðx; tÞ ¼ dkgðkÞeikðxctÞ ¼ f ðx ctÞ ð14Þ 1
where g(k) can have a Gaussian profile. This wave packet is localized at x ct = 0. In the absence of dispersive properties of the medium, wave packet will suffer no broadening with time. In this case the relation x = kc holds. In general, the medium has dispersive behavior and therefore x becomes a function of wave number, that is, x = x(k). In this situation, Eq. (14) becomes Z f ðx; tÞ ¼ dkgðkÞeikxixðkÞt : ð15Þ
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K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 34 (2007) 224–234 2
Suppose that gðkÞ ¼ eaðkk0 Þ . With expansion of x(k) around k = k0, one find 2 dx 1 dx þ ðk k 0 Þ2 ; xðkÞ xðk 0 Þ þ ðk k 0 Þ dk k0 2 dk 2 k 0
ð16Þ
where using the definitions, dx 1 d2 x ¼ vg ; ¼ g; k k 0 ¼ k 0 : dk k0 2 dk 2 k0 Eq. (15) can be written as Z f ðx; tÞ ¼ eik0 xixðk0 Þt
1
02
0
dk 0 eak eik ðxvg tÞ eik 1
02
gt
ð17Þ
¼ eik0 xixðk0 Þt
Z
1
0
02
dk 0 eik ðxvg tÞ eðaþigtÞk :
ð18Þ
1
Now completing the square root in exponent and integration gives 12 hðxvg tÞ2 i p i½k 0 xxðk 0 Þt e 4ðaþigtÞ : f ðx; tÞ ¼ e a þ igt Therefore one finds, 12 h aðxvg tÞ2 i p2 2 e 2ða2 þg2 t2 Þ ; jf ðx; tÞj ¼ a2 þ g2 t2
ð19Þ
ð20Þ
which is the profile of the wave packet (density of probability) in position space (note that regarding the failure of position space representation we should interpret the last equation within maximally localized or quasi-position represeng2 t2 tation). The quantity 1 which in t = 0 was a, now has became a þ a . This means that broadening has increased by a g2 t2 2 factor of 1 þ a2 . This argument shows that a wave packet with width (Dx)0 in t = 0, after propagation will have the following width: 1 g2 t2 2 ð21Þ ðDxÞt ¼ ðDxÞ0 1 þ 2 : a In the presence of quantum gravitational effects, usual uncertainty principle of Heisenberg should be generalized due to fuzzy structure of spacetime. From string theoretical considerations (see for example [5]), this generalization can be written as DxDp P h½1 þ bðDpÞ2 ; where b Suppose that p Dp and Dx x k. Then we find 1 kp hð1 þ bp2 Þ ) 1 kp hð1 þ bp2 Þ :
ð22Þ
l2P .
ð23Þ
Since quantum gravitational effects are very small, one can write (1 + bp2)1 ’ 1 bp2. Therefore we find 1 p 2 k ’ h ð1 bp Þ:
ð24Þ
Now the generalized De Broglie principle can be written as p = hkf(k). Using this generalization, Eq. (24) can be written as 1 2 3 3 ð25Þ k ¼ kf ðkÞ bh k f ðkÞ: Anomalous dispersion of wave profile within fuzzy spacetime will be more manifest if we write x ¼ c= k or equivalently x ¼ c½kf ðkÞ bh2 k 3 f 3 ðkÞ:
ð26Þ
Note that we have considered only the first order effects. This relation shows that in quantum gravity regime the very basic notion of spacetime itself induces dispersion on the wave packets profile. This fact has origin on the quantum fluctuation of spacetime which can be described as fuzzy (fractal or foamy) spacetime. Now look at the group velocity. Within generalized De Broglie framework, we find " #! dx df ðkÞ df ðkÞ 2 2 3 3 2 vg ¼ ¼ c f ðk 0 Þ þ k 0 3bh k 0 f ðk 0 Þ þ k 0 f ðk 0 Þ : ð27Þ dk k0 dk k0 dk k0
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229
The functional form of f(k) is important in the rest of above calculations. The actual form of f(k) should be quantum gravity model dependent via modified dispersion relations [50]. Suppose that f(k) can be written as f ðkÞ ¼ 1 l2P k 2 ;
ð28Þ
which has been motivated by Adler and Santiago [51]. Then we have velocity up to second order of Planck length becomes
df ðkÞ dk
¼
2l2P k
and
d2 f ðkÞ dk 2
¼
2l2P
and therefore group
vg ¼ cð1 3bh2 k 20 3l2p k 20 þ 15bh2 l2P k 40 Þ:
ð29Þ
In the same manner, we find for g ! 2 1 d2 x c df ðkÞ df ðkÞ df ðkÞ d2 f ðkÞ d2 f ðkÞ 2 2 2 2 3 2 3 2 3 2 2 18bh k f ðkÞ 6b h k f ðkÞ g¼ ¼ 6bh kf ðkÞ þ k 3bh k f ðkÞ 2 dk 2 k0 2 dk dk dk dk 2 dk 2 k¼k
0
ð30Þ
and using (28), this quantity becomes c g¼ h2 k 3 ð1 l2P k 2 Þð2l2P kÞ2 6b h2 kð1 l2P k 2 Þ3 þ kð2l2P Þ 2ð2l2P kÞ 18bh2 k 2 ð1 l2P k 2 Þ2 ð2l2P kÞ 6b 2 3bh2 k 3 ð1 l2P k 2 Þ2 ð2l2P Þ
:
k¼k 0
ð31Þ
Up to second order in Planck length this equation gives c g ¼ ð6l2P k 0 6bh2 k 0 þ 60bh2 l2P k 30 Þ: 2
ð32Þ
Using these generalized quantities for vg and g in Eq. (20), we find the following wave profile in spacetime background with quantum gravitational effects 0 1 !12 2 aðx cð1 3b h2 k 20 3l2p k 20 þ 15bh2 l2P k 40 ÞtÞ p2 2 @ jf ðx; tÞj ¼ exp h 2 iA: a2 þ ½2c ð6l2P k 0 6bh2 k 0 þ 60bh2 l2P k 30 Þ2 t2 2 a2 þ 2c ð6l2P k 0 6b h2 k 0 þ 60b h2 l2P k 30 Þ t2 ð33Þ
The terms containing b and lP are the direct consequences of spacetime fuzziness and fractal nature of spacetime in Planck scale. It is evident that in ordinary picture, these extra terms will be disappeared since in usual quantum mechanics we should set b ! 0 and lP ! 0. Now Eq. (21) generalizes to c 2 !12 ð6l2P k 0 6bh2 k 0 þ 60bh2 l2P k 30 Þ t2 2 ðDxÞt ¼ ðDxÞ0 1 þ : ð34Þ a2 Note that when we set b ! 0 and lP ! 0 in the last two equations, we find the case corresponding to x = kc which contains no broadening for wave profile(corresponding to Eq. (14)). This is a restricted situation. To find more reliable picture which contains usual quantum mechanical broadening in the absence of quantum gravitational effects (i.e. corresponding to Eqs. (20) and (21)), we proceed as follows: Suppose that 1 k ¼ kgðkÞ;
ð35Þ
where is a generalization of (25) to separate quantum mechanical and quantum gravitational contributions from each other. In this situation we can write gðkÞ ¼ gq ðkÞ þ gqg ðkÞ; therefore we have c x ¼ ¼ ckgðkÞ ¼ ckgq ðkÞ þ ckgqg ðkÞ: k One can decompose this equation to the following two pieces: xq ¼ ckgq ðkÞ and
xqg ¼ ckgqg ðkÞ
ð36Þ
ð37Þ
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for quantum mechanical and quantum gravitational contributions respectively. In the same manner one can define dxq 1 d2 xq ¼ vg and ¼ g: 2 2 dk k
dk k
Now suppose that gqg ðkÞ ¼ a1 l2p k 2 þ a2 l4p k 4 þ Oðl6p k 6 Þ:
ð38Þ
With this form of gqg(k), we find
dgqg ðkÞ dxqg v0g ¼ ¼ c gqg ðkÞ þ k ; dk dk k
k
ð39Þ
which leads to v0g ¼ c½3a1 l2p k 2 þ 5a2 l4p k 4 :
ð40Þ
On the other hand, for g 0 we have 1 d2 xqg c 0 ¼ ½6a1 l2p k þ 20a2 l4p k 3 ¼ c½3a1 l2p k þ 10a2 l4p k 3 : g ¼ 2 dk 2 k 2
ð41Þ
Now Eqs. (20) and (21) will generalize to ! !12 a½x ðvg þ v0g Þt2 p2 2 exp jf ðx; tÞj ¼ a2 þ ðg þ g0 Þ2 t2 2ða2 þ ðg þ g0 Þ2 t2 Þ
ð42Þ
and ðDxÞt ¼ ðDxÞ
ðg þ g0 Þ2 t2 1þ a2
!12 ð43Þ
respectively. Since ðg þ g0 Þ2 ¼ g2 þ 2gg0 þ g02 ¼ g2 þ 6ga1 ck l2p þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p ; we find for profile of the wave packet jf ðx; tÞj2 ¼
p2 2 2 2 a þ ½g þ 6ga1 ck lp þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p t2
!12 exp
aðx ½vg þ c½3a1 l2p k 2 þ 5a2 l4p k 4 tÞ2
!
2ða2 þ ½g2 þ 6ga1 ck l2p þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p t2 Þ
:
ð44Þ
Finally, wave packet broadening is given by ðDxÞt ¼ ðDxÞ
½g2 þ 6ga1 ck l2p þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p t2 1þ a2
!12 :
ð45Þ
It is obvious that in this situation in the limit of b ! 0 and lP ! 0, we find the usual quantum mechanical results consisting ordinary dispersion of wave profile(or wave broadening). Fig. 1 shows the phenomenon of wave packet broadening in three candidate models. 3.3. Coherent states in fuzzy spacetime Within ordinary quantum mechanics, the coherent state was introduced by Schro¨dinger as the quantum state of the harmonic oscillator which minimizes the uncertainty equally distributed in both position x and the momentum p. By definition, coherent state is the normalized state jki 2 H, which is the eigenstate of annihilation operator and satisfies the following equation: ajki ¼ kjki where hkjki ¼ 1 and
ð46Þ
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231
3 a 2.5
2
y
1.5
b
1
0.5
–20
–10
c
10
20 x
Fig. 1. Wave packet broadening. The profile of wave packet (a) in a non-dispersive medium, (b) in usual quantum mechanical case and (c) in quantum gravity regime. Vertical axis measures y = jf(x, t)j2 in a given time. Obviously, wave packet has more broadening within fuzzy spacetime.
jki ¼ ejkj
2
=2
1 X 2 kn y pffiffiffiffi jni ¼ ejkj =2 eka j0i: n! n¼0
ð47Þ
Actually k can be complex because a (annihilation operator) is not Hermitian. Now we are going to consider the situation in a fuzzy spacetime. There are two approaches for definition of generalized coherent states: non-commutative geometry considerations and fuzzy sphere approach. The non-commutative plane is defined by the relations [25,26] ½^xj ; ^xk ¼ ihj;k ;
h > 0; k; j ¼ 1; 2:
ð48Þ
The coherent states are defined as eigenvalues of a destruction operator and saturate the Heisenberg uncertainty Dx1Dx2 = h/2. The situation of the fuzzy sphere is more complicated. It is a matrix model defined by the following relations [52] iR ½^xk ; ^xl ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi klm^xm jðj þ 1Þ
and
dlk ^xl^xk ¼ R2
ð49Þ
with j integer or half-integer and k, l, m = 1, 2, 3. Here we use re-scaled variables such that R = 1. There is no state saturating simultaneously all the Heisenberg uncertainties Dx1 Dx2 ¼
1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jh^x3 ij; 2 jðj þ 1Þ
Dx2 Dx3 ¼
1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jh^x1 ij; 2 jðj þ 1Þ
Dx3 Dx1 ¼
1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jh^x2 ij: 2 jðj þ 1Þ
ð50Þ
As a consequence, one has to resort to other criteria to define coherent states in this context. One way to tackle this problem is based on the construction of a deformation of the creation-destruction operators. In this approach, one uses the stereographic projection to define the operators z ¼ ð^x1 i^x2 Þð1 ^x3 Þ1 ;
zþ ¼ ð1 ^x3 Þ1 ð^x1 þ i^x2 Þ;
ð51Þ
which obey the commutation relation ½z; zþ ¼ F ðzzþ Þ; where
ð52Þ
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1 a F ðzz Þ ¼ av 1 þ jzj2 v 1 þ jzj2 ; 2 2 þ
n ¼ 1 þ ajzj2 ;
and
2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 24 a 1 a 5 ; 1þ þ v¼ a 2n n 2n
1 a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : jðj þ 1Þ
ð53Þ
Now, there exists a function f which relates the operators z and z+ to the usual creation and destruction operators aˆ+ and aˆ respectively z ¼ f ð^aþ ^a þ 1Þ^a:
ð54Þ
Now the generalized coherent states jfi are taken to be the eigenstates of the destruction operator z and read as follows: jfi ¼ N ðjfj2 Þ1=2 exp½ff 1 ð^aþ ^aÞ^aþ f 1 ð^aþ ^aÞj0i;
ð55Þ
2 1/2
the function N(jfj ) enforcing the normalization of the wave function. The vacuum j0i is annihilated by aˆ. This approach to definition of coherent states opens the way for more investigation of the issue specially comparison between alternative scenarios of quantum gravity.
4. Summary and conclusion Today, loop quantum gravity, string theory and non-commutative geometry are three main approaches to quantum gravity. A common feature of these scenarios is the existence of a minimal length scale on the order of Planck length which restricts the accuracy of position measurements. In another words, one cannot probe distances smaller than Planck length. In this framework, in the Planck-scale regime spacetime points have a finite resolution. This finite resolution of spacetime points can be described within foamy or fractal picture of underlying geometry and usually is referred to spacetime fuzziness. There are signatures of this spacetime fuzziness from non-standard features of underlying Planck scale quantum optics which have been supported in recent observations of gamma-ray bursts spectra. These effects could led us to observable cosmological predictions of the discrete nature of quantum spacetime. In this paper, we have studied the effects of fuzzy structure of spacetime in the spirit of quantum theory. Due to the fuzzy nature of spacetime structure, the notion of locality breaks down and therefore the usual picture of position space representation is no longer suitable for description of quantum states. In this situation, a quasi-position representation of quantum states seems to be helpful in order to retain the role of position space representation via a maximally localized picture. We have used this maximally localized states to investigate the effects of spacetime fuzziness on the wave packets propagation. As a result of the fuzzy structure of spacetime, wave packets suffer more broadening relative to standard situation during their propagation. This extra broadening can be detected in principle and this feature provides a possible experimental test of underlying quantum gravity proposal. Interferometric noise experiments are suitable candidates of such test experiments. Finally, coherent states of quantum mechanical systems have been re-examined within non-commutative geometry and stereographic projection algorithm. Obviously, fuzzy nature of spacetime requires a redefinition of coherent states and generalization of these states to more realistic situation in quantum gravity era. Finally, it should be noted that spacetime fuzziness has several other important consequences such as break down of local Lorentz invariance and modification of dispersion relations which recently have attracted considerable attentions.
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Some consequences of spacetime fuzziness Kourosh Nozari *, Behnaz Fazlpour Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47416-1467 Babolsar, Iran Accepted 16 March 2006
Abstract Finite resolution of spacetime at Planck scale gives it a fuzzy structure (the so-called foamy or fractal spacetime). This fuzzy structure of spacetime is a consequence of quantum fluctuation of geometry itself and can be described within non-commutative geometry and some alternative approaches to quantum gravity. In this paper, some consequences of spacetime fuzziness are studied. Due to this fuzzy structure, some basic notions of ordinary quantum mechanics such as position space representation, wave packet broadening during its propagation and coherent states of quantum mechanical systems should be re-examined. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The problem of reconciling quantum mechanics with general relativity is one of the task of modern theoretical physics which until now has not yet found a consistent and satisfactory solution. The difficulty arises since general relativity deals with the events which define the world-lines of particles, while quantum mechanics does not allow the definition of trajectory; in fact the determination of the position of a quantum particle involves a measurement which introduces an uncertainty into its momentum. These conceptual difficulties have their origin in the violation, at quantum level, of the weak principle of equivalence (universality of free fall) on which general relativity is based [1,2]. Such a problem becomes more involved in the formulation of quantum theory of gravity, owing to the non-renormalizability of general relativity when one quantizes it as a local quantum field theory [3]. Nevertheless, one of the most interesting consequences of this unification is that in quantum gravity there exists a non-vanishing minimal observable length on the qffiffiffiffi h order of the Planck length, lp ¼ G 1033 cm. One cannot set up a position measurement with uncertainty less than c3 this minimal value. The existence of such a fundamental length is a dynamical phenomenon due to the fact that, at Planck scale, there are fluctuations of the background metric, i.e. a limit of the order of Planck length appears when quantum fluctuations of the gravitational field are taken into account. Existence of minimal length scale has been motivated by several promising candidates of quantum gravity [4–7] and its consequences have been studied extensively [7– 17]. This natural cut-off guarantees the renormalizability of underlying quantum field theory [18–20]. Also existence of this minimal cut off results in the modification of usual Heisenberg algebra to incorporate gravitational uncertainty from very beginning [21–24].
*
Corresponding author. E-mail address: [email protected] (K. Nozari).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.066
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Non-commutativity of spacetime [25–27] at quantum level led to spacetime fuzziness [28–35]. Existence of non-vanishing minimal uncertainty in position cause undeterminability of spacetime points. In another words, locality is no longer possible and all events are uncertain at least up to Planck length. Spacetime in this viewpoint has a fractal or foamy structure. Recently a new point of view to this issue has been developed by El Naschie. In this novel scenario, fractal structure of spacetime in quantum gravity era has been formulated within Cantorian (E1 ) space [36–39]. It seems that this scenario has the strong potential to be ultimate framework for unification of fundamental interactions [40,41] and solves some mysteries of physics such as spacetime dimensionality [42,43], the nature of time in quantum gravity [41] and some other key problems of modern physics [44–46]. The fuzzy structure of spacetime has very novel implications for the rest of quantum theory. We will see that the picture of position space representation of usual quantum mechanics breaks down. Due to non-commutativity of spacetime, dispersion relations will generalize and as a consequence, waves profile will encounter anomalous dispersions. In addition, the notion of coherency and coherent states should be re-examined within this framework. In this paper, after an introduction to spacetime fuzziness, we discuss some of its consequences in the spirit of quantum theory. Position space representation, wave profile anomalous dispersion and the notion of coherency will be reexamined within fuzzy spacetime framework.
2. Fuzzy spacetime A key characteristic of quantum theory is the emergence of uncertainties, and one might expect that the distance observable would also be affected by uncertainties. Actually various heuristic arguments suggest that for such a distance observable the uncertainties might be more pervasive: in ordinary quantum theory one is still able to measure sharply any given observable, though at the cost of renouncing all information on a conjugate observable, but it appears plausible that a quantum gravity distance observable would be affected by irreducible uncertainties. Quantum gravity suggests that in the Planck-scale regime there should be some absolute limitations on the measurability of distances. This restricted resolution of spacetime structure is referred as spacetime fuzziness (‘‘foamy or fractal spacetime ’’). Most authors consider a dD P Lp relation, meaning that the uncertainty in the measurement of distances could not be reduced below the Planck-length level, but measurability bounds of other forms, generically of the type dD P f(D, Lp) (with f some function such that f(D, 0) = 0) are also being considered. The presence of such an irreducible measurement uncertainty could be significant in various contexts. For example, one can characterize operatively this spacetime fuzziness as an irreducible (fundamental) Planck-scale contribution to the noise levels in the readout of interferometers [30,34,47]. In classical physics, interferometer noise can in principle be reduced to zero. Ordinary quantum properties of matter already introduce an irreducible noise contribution. Spacetime fuzziness would introduce an additional irreducible source of noise, reflecting the fact that the distances involved in the experiment would be inherently unsharp in a foamy spacetime picture. This inherent unsharp distance measurement is the essence of spacetime fuzziness. In fact all approaches to the quantum gravity predict some limitations on the accuracy of localization, and therefore predict some spacetime fuzziness. One is usually unable to rigorously derive from first principles a detailed description of the physical consequences of this fuzziness, such as the mentioned interferometric noise. Nevertheless, some phenomenological arguments can be perform in this direction. The key input needed for this phenomenology turns out to be the power spectrum q(f) of the Planck-scale-induced strain noise in terms of frequency f [35]. Combining some intuition about the stochastic-like features of spacetime fuzziness and the dependence on the Planck length, one can easily reach a model of q(f). For example if the effects depend linearly on the Planck length Lp 1/Ep and the underlying phenomena are of random-walk type, one is inevitably led to q Lp f 2 K2 fLp f 2 L2 :
ð1Þ
The proportionality to the square of the inverse of the frequency is a direct result of the assumption of random-walktype processes. The length scale K is needed on the basis of the dimensional analysis of the equation, and is to be treated as a free parameter to be constrained experimentally. One may choose to make reference to a dimensionless parameter, f, which may be used to express the ratio of K with the length L of the arms of an interferometer or of an optical resonator. Other hypothesis about the stochastic-like features of the underlying Planck-scale processes led to other forms of f dependence (and Lp dependence) of the strain noise power spectrum. In general one should find (a > 1) [35] 2b 2ab q fab L1þa L : p f
ð2Þ
There is therefore some interest in attempting to improve limits on the parameters fab [48]. The idea of spacetime fuzziness also motivates some research work on Planck-scale-induced decoherence. It is in fact rather plausible that
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spacetime fuzziness might affect the time evolution of quantum mechanical states in such a way that, for example, a pure state might evolve into a mixed state. A key attribute of spacetime fuzziness is that in certain non-commutative spacetimes, departures from Lorentz symmetry will be encountered [49]. There, the Planck-scale structure of spacetime can introduce a systematic dependence of the speed of photons on their wavelength. After a time duration T, the difference between the expected position of the photon and the Planck-scale-corrected position could take the form Dx Tdv cTLp/k, where k is the photon wavelength. If we instead focus on how distance fuzziness could affect the propagation of photons, it is natural to expect that a group of photons would all travel the same average distance in a given time T (and this average distance is still given by cT), but for each individual photon the distance traveled might be slightly different from the average, as a result of distance fuzziness. Just to be more specific let us imagine that distance fuzziness effectively introduces a Planck-length uncertainty pffiffiffiffiffiffiffiffiffiffiin position per each Planck time of travel. Then the final position uncertainty would be of the type Dx cTLp . The square root here (assuming a random-walk-type description) is the result of the fact that non-systematic effects do not add linearly, but rather according to rules familiar in the analysis of stochastic processes. After a brief introduction to spacetime fuzziness, in forthcoming sections we consider some consequences of this fuzzy structure.
3. Some consequences of spacetime fuzziness 3.1. Failure of position space representation In ordinary quantum mechanics position operator, x could be represented as multiplication or differentiation operator acting on square integrable position- or momentum-space wave functions w(x) := hxjwi, where the jxi is position eigenstate. Strictly speaking the jxi is not physical state since it is not normalizable and thus not in the Hilbert space. However, the operator x is essentially self-adjoint and the eigenstates can be approximated to arbitrary precision by sequences jwni of physical states of increasing localization in position space: lim Dxjwn i ¼ 0:
n!1
This situation changes drastically with the introduction of minimal uncertainties Dx0 P 0. A non-zero minimal uncertainty in position ðDxÞ2jwi ¼ hwjðx hwjxjwiÞ2 jwi P Dx0
8jwi
ð3Þ
implies that there cannot be any physical state which is a position eigenstate since an eigenstate would of course have zero uncertainty in position. All of these arguments show the failure of position space representation in quantum mechanics. Is there any way to retain such representation? Actually, this failure does not exclude the existence of unphysical, ‘formal position eigenvectors’ which lie in the domain of x alone but not in Dx;x2 ;p;p2 . In fact, such formal x-eigenvectors do exist and are of infinite energy. Most importantly however, unlike in ordinary quantum mechanics, it is no longer possible to approximate formal eigenvectors through a sequence of physical states of uncertainty in positions decreasing to zero. This is because now all physical states have at least a finite minimal uncertainty in position. A minimal uncertainty in position means that the position operator is no longer essentially self-adjoint but only symmetric. Since there are no more position eigenstates jxi in the representation of the Heisenberg algebra, the Heisenberg algebra will no longer find a Hilbert space representation on position wave functions hxjwi. Where there is no minimal uncertainty in momentum, one can work with the convenient representation of the commutation relations on momentum space wave functions and the states of maximal localization will be proper physical states [17]. We can use states of maximal localization to define a ‘quasi-position’ representation. This representation has a direct interpretation in terms of position measurements, although it does of course not diagonalize x. The modified commutation relations proposed by Kempf et al. [17] are of the form ½^xi ; ^pj ¼ ihdij ð1 þ a^x2 þ b^p2 Þ:
ð4Þ
To be more specific, let us consider the case a = 0. The other commutators read ½^pi ; ^pj ¼ 0; ½^xi ; ^xj ¼ 2ihbð^ pi ^xj ^pj^xi Þ:
ð5Þ
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The last relation means that we have a non-commutative geometry since translations along different directions do not commute anymore. This non-commutativity of space coordinate is the signature of fuzzy structure of space in quantum level. The presence of ~ p2 ;~ x2 in Eq. (4) implies that the rotational symmetry, whose generators are ^ij ¼ L
1 ð^xi ^pj ^xj ^pi Þ 1 þ b~ p2
ð6Þ
is still preserved. As has been mentioned, one of the remarkable features of this theory is the existence of a minimal length uncertainty pffiffiffi h b because of the new Heisenberg inequalities ! n X h 2 2 Dxi Dpj P dij 1 þ b : ð7Þ ðD~ pk Þ þ h~ pk i 2 k¼1 The momentum representation is found to be given by the operators ^pi :wð~ pÞ ¼ pi wð~ pÞ
ð8Þ
^xi wð~ pÞ ¼ ihð1 þ b~ p2 Þopi wðpÞ
ð9Þ
and
acting on a Hilbert space in which the scalar product is given by Z p d3~ / ðpÞwðpÞ: h/jwi ¼ ð1 þ b~ p2 Þ
ð10Þ
As we have argued, the presence of a minimal length uncertainty implies that no position representation exists. The concept which proves to be the closest to it is the quasi-position representation in which the operators are non-local pffiffiffi pffiffiffi ^xi ¼ ni þ b tan h boni ð11Þ and ^pi ¼
qffiffiffiffiffiffiffi pffiffiffi b1 tan ih boni ;
ð12Þ
where ni is the mean value of the position operator. The formalism provided in this subsection can be used to retain position space representation in ordinary quantum mechanics. 3.2. Waves in fuzzy spacetime Existence of minimal observable length which restricts the accuracy of position measurements and complete resolution of spacetime points, will affect wave packet propagation in a background fuzzy spacetime. Here we are going to study the effect of spacetime fuzziness on the wave packet propagation. First we give an outline to the ordinary quantum mechanical considerations. For simplicity, consider the following plane wave profile: f ðx; tÞ / eikxixt :
ð13Þ
and m ¼ kc, this relation can be written as f(x, t) / eik(xct). Now the superposition of these plane Since x = 2pm, k ¼ 2p k waves with amplitude g(k) can be written as, Z 1 f ðx; tÞ ¼ dkgðkÞeikðxctÞ ¼ f ðx ctÞ ð14Þ 1
where g(k) can have a Gaussian profile. This wave packet is localized at x ct = 0. In the absence of dispersive properties of the medium, wave packet will suffer no broadening with time. In this case the relation x = kc holds. In general, the medium has dispersive behavior and therefore x becomes a function of wave number, that is, x = x(k). In this situation, Eq. (14) becomes Z f ðx; tÞ ¼ dkgðkÞeikxixðkÞt : ð15Þ
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K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 34 (2007) 224–234 2
Suppose that gðkÞ ¼ eaðkk0 Þ . With expansion of x(k) around k = k0, one find 2 dx 1 dx þ ðk k 0 Þ2 ; xðkÞ xðk 0 Þ þ ðk k 0 Þ dk k0 2 dk 2 k 0
ð16Þ
where using the definitions, dx 1 d2 x ¼ vg ; ¼ g; k k 0 ¼ k 0 : dk k0 2 dk 2 k0 Eq. (15) can be written as Z f ðx; tÞ ¼ eik0 xixðk0 Þt
1
02
0
dk 0 eak eik ðxvg tÞ eik 1
02
gt
ð17Þ
¼ eik0 xixðk0 Þt
Z
1
0
02
dk 0 eik ðxvg tÞ eðaþigtÞk :
ð18Þ
1
Now completing the square root in exponent and integration gives 12 hðxvg tÞ2 i p i½k 0 xxðk 0 Þt e 4ðaþigtÞ : f ðx; tÞ ¼ e a þ igt Therefore one finds, 12 h aðxvg tÞ2 i p2 2 e 2ða2 þg2 t2 Þ ; jf ðx; tÞj ¼ a2 þ g2 t2
ð19Þ
ð20Þ
which is the profile of the wave packet (density of probability) in position space (note that regarding the failure of position space representation we should interpret the last equation within maximally localized or quasi-position represeng2 t2 tation). The quantity 1 which in t = 0 was a, now has became a þ a . This means that broadening has increased by a g2 t2 2 factor of 1 þ a2 . This argument shows that a wave packet with width (Dx)0 in t = 0, after propagation will have the following width: 1 g2 t2 2 ð21Þ ðDxÞt ¼ ðDxÞ0 1 þ 2 : a In the presence of quantum gravitational effects, usual uncertainty principle of Heisenberg should be generalized due to fuzzy structure of spacetime. From string theoretical considerations (see for example [5]), this generalization can be written as DxDp P h½1 þ bðDpÞ2 ; where b Suppose that p Dp and Dx x k. Then we find 1 kp hð1 þ bp2 Þ ) 1 kp hð1 þ bp2 Þ :
ð22Þ
l2P .
ð23Þ
Since quantum gravitational effects are very small, one can write (1 + bp2)1 ’ 1 bp2. Therefore we find 1 p 2 k ’ h ð1 bp Þ:
ð24Þ
Now the generalized De Broglie principle can be written as p = hkf(k). Using this generalization, Eq. (24) can be written as 1 2 3 3 ð25Þ k ¼ kf ðkÞ bh k f ðkÞ: Anomalous dispersion of wave profile within fuzzy spacetime will be more manifest if we write x ¼ c= k or equivalently x ¼ c½kf ðkÞ bh2 k 3 f 3 ðkÞ:
ð26Þ
Note that we have considered only the first order effects. This relation shows that in quantum gravity regime the very basic notion of spacetime itself induces dispersion on the wave packets profile. This fact has origin on the quantum fluctuation of spacetime which can be described as fuzzy (fractal or foamy) spacetime. Now look at the group velocity. Within generalized De Broglie framework, we find " #! dx df ðkÞ df ðkÞ 2 2 3 3 2 vg ¼ ¼ c f ðk 0 Þ þ k 0 3bh k 0 f ðk 0 Þ þ k 0 f ðk 0 Þ : ð27Þ dk k0 dk k0 dk k0
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The functional form of f(k) is important in the rest of above calculations. The actual form of f(k) should be quantum gravity model dependent via modified dispersion relations [50]. Suppose that f(k) can be written as f ðkÞ ¼ 1 l2P k 2 ;
ð28Þ
which has been motivated by Adler and Santiago [51]. Then we have velocity up to second order of Planck length becomes
df ðkÞ dk
¼
2l2P k
and
d2 f ðkÞ dk 2
¼
2l2P
and therefore group
vg ¼ cð1 3bh2 k 20 3l2p k 20 þ 15bh2 l2P k 40 Þ:
ð29Þ
In the same manner, we find for g ! 2 1 d2 x c df ðkÞ df ðkÞ df ðkÞ d2 f ðkÞ d2 f ðkÞ 2 2 2 2 3 2 3 2 3 2 2 18bh k f ðkÞ 6b h k f ðkÞ g¼ ¼ 6bh kf ðkÞ þ k 3bh k f ðkÞ 2 dk 2 k0 2 dk dk dk dk 2 dk 2 k¼k
0
ð30Þ
and using (28), this quantity becomes c g¼ h2 k 3 ð1 l2P k 2 Þð2l2P kÞ2 6b h2 kð1 l2P k 2 Þ3 þ kð2l2P Þ 2ð2l2P kÞ 18bh2 k 2 ð1 l2P k 2 Þ2 ð2l2P kÞ 6b 2 3bh2 k 3 ð1 l2P k 2 Þ2 ð2l2P Þ
:
k¼k 0
ð31Þ
Up to second order in Planck length this equation gives c g ¼ ð6l2P k 0 6bh2 k 0 þ 60bh2 l2P k 30 Þ: 2
ð32Þ
Using these generalized quantities for vg and g in Eq. (20), we find the following wave profile in spacetime background with quantum gravitational effects 0 1 !12 2 aðx cð1 3b h2 k 20 3l2p k 20 þ 15bh2 l2P k 40 ÞtÞ p2 2 @ jf ðx; tÞj ¼ exp h 2 iA: a2 þ ½2c ð6l2P k 0 6bh2 k 0 þ 60bh2 l2P k 30 Þ2 t2 2 a2 þ 2c ð6l2P k 0 6b h2 k 0 þ 60b h2 l2P k 30 Þ t2 ð33Þ
The terms containing b and lP are the direct consequences of spacetime fuzziness and fractal nature of spacetime in Planck scale. It is evident that in ordinary picture, these extra terms will be disappeared since in usual quantum mechanics we should set b ! 0 and lP ! 0. Now Eq. (21) generalizes to c 2 !12 ð6l2P k 0 6bh2 k 0 þ 60bh2 l2P k 30 Þ t2 2 ðDxÞt ¼ ðDxÞ0 1 þ : ð34Þ a2 Note that when we set b ! 0 and lP ! 0 in the last two equations, we find the case corresponding to x = kc which contains no broadening for wave profile(corresponding to Eq. (14)). This is a restricted situation. To find more reliable picture which contains usual quantum mechanical broadening in the absence of quantum gravitational effects (i.e. corresponding to Eqs. (20) and (21)), we proceed as follows: Suppose that 1 k ¼ kgðkÞ;
ð35Þ
where is a generalization of (25) to separate quantum mechanical and quantum gravitational contributions from each other. In this situation we can write gðkÞ ¼ gq ðkÞ þ gqg ðkÞ; therefore we have c x ¼ ¼ ckgðkÞ ¼ ckgq ðkÞ þ ckgqg ðkÞ: k One can decompose this equation to the following two pieces: xq ¼ ckgq ðkÞ and
xqg ¼ ckgqg ðkÞ
ð36Þ
ð37Þ
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for quantum mechanical and quantum gravitational contributions respectively. In the same manner one can define dxq 1 d2 xq ¼ vg and ¼ g: 2 2 dk k
dk k
Now suppose that gqg ðkÞ ¼ a1 l2p k 2 þ a2 l4p k 4 þ Oðl6p k 6 Þ:
ð38Þ
With this form of gqg(k), we find
dgqg ðkÞ dxqg v0g ¼ ¼ c gqg ðkÞ þ k ; dk dk k
k
ð39Þ
which leads to v0g ¼ c½3a1 l2p k 2 þ 5a2 l4p k 4 :
ð40Þ
On the other hand, for g 0 we have 1 d2 xqg c 0 ¼ ½6a1 l2p k þ 20a2 l4p k 3 ¼ c½3a1 l2p k þ 10a2 l4p k 3 : g ¼ 2 dk 2 k 2
ð41Þ
Now Eqs. (20) and (21) will generalize to ! !12 a½x ðvg þ v0g Þt2 p2 2 exp jf ðx; tÞj ¼ a2 þ ðg þ g0 Þ2 t2 2ða2 þ ðg þ g0 Þ2 t2 Þ
ð42Þ
and ðDxÞt ¼ ðDxÞ
ðg þ g0 Þ2 t2 1þ a2
!12 ð43Þ
respectively. Since ðg þ g0 Þ2 ¼ g2 þ 2gg0 þ g02 ¼ g2 þ 6ga1 ck l2p þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p ; we find for profile of the wave packet jf ðx; tÞj2 ¼
p2 2 2 2 a þ ½g þ 6ga1 ck lp þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p t2
!12 exp
aðx ½vg þ c½3a1 l2p k 2 þ 5a2 l4p k 4 tÞ2
!
2ða2 þ ½g2 þ 6ga1 ck l2p þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p t2 Þ
:
ð44Þ
Finally, wave packet broadening is given by ðDxÞt ¼ ðDxÞ
½g2 þ 6ga1 ck l2p þ ð20ga2 ck 3 þ 9a21 c2 k 2 Þl4p t2 1þ a2
!12 :
ð45Þ
It is obvious that in this situation in the limit of b ! 0 and lP ! 0, we find the usual quantum mechanical results consisting ordinary dispersion of wave profile(or wave broadening). Fig. 1 shows the phenomenon of wave packet broadening in three candidate models. 3.3. Coherent states in fuzzy spacetime Within ordinary quantum mechanics, the coherent state was introduced by Schro¨dinger as the quantum state of the harmonic oscillator which minimizes the uncertainty equally distributed in both position x and the momentum p. By definition, coherent state is the normalized state jki 2 H, which is the eigenstate of annihilation operator and satisfies the following equation: ajki ¼ kjki where hkjki ¼ 1 and
ð46Þ
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 34 (2007) 224–234
231
3 a 2.5
2
y
1.5
b
1
0.5
–20
–10
c
10
20 x
Fig. 1. Wave packet broadening. The profile of wave packet (a) in a non-dispersive medium, (b) in usual quantum mechanical case and (c) in quantum gravity regime. Vertical axis measures y = jf(x, t)j2 in a given time. Obviously, wave packet has more broadening within fuzzy spacetime.
jki ¼ ejkj
2
=2
1 X 2 kn y pffiffiffiffi jni ¼ ejkj =2 eka j0i: n! n¼0
ð47Þ
Actually k can be complex because a (annihilation operator) is not Hermitian. Now we are going to consider the situation in a fuzzy spacetime. There are two approaches for definition of generalized coherent states: non-commutative geometry considerations and fuzzy sphere approach. The non-commutative plane is defined by the relations [25,26] ½^xj ; ^xk ¼ ihj;k ;
h > 0; k; j ¼ 1; 2:
ð48Þ
The coherent states are defined as eigenvalues of a destruction operator and saturate the Heisenberg uncertainty Dx1Dx2 = h/2. The situation of the fuzzy sphere is more complicated. It is a matrix model defined by the following relations [52] iR ½^xk ; ^xl ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi klm^xm jðj þ 1Þ
and
dlk ^xl^xk ¼ R2
ð49Þ
with j integer or half-integer and k, l, m = 1, 2, 3. Here we use re-scaled variables such that R = 1. There is no state saturating simultaneously all the Heisenberg uncertainties Dx1 Dx2 ¼
1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jh^x3 ij; 2 jðj þ 1Þ
Dx2 Dx3 ¼
1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jh^x1 ij; 2 jðj þ 1Þ
Dx3 Dx1 ¼
1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jh^x2 ij: 2 jðj þ 1Þ
ð50Þ
As a consequence, one has to resort to other criteria to define coherent states in this context. One way to tackle this problem is based on the construction of a deformation of the creation-destruction operators. In this approach, one uses the stereographic projection to define the operators z ¼ ð^x1 i^x2 Þð1 ^x3 Þ1 ;
zþ ¼ ð1 ^x3 Þ1 ð^x1 þ i^x2 Þ;
ð51Þ
which obey the commutation relation ½z; zþ ¼ F ðzzþ Þ; where
ð52Þ
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1 a F ðzz Þ ¼ av 1 þ jzj2 v 1 þ jzj2 ; 2 2 þ
n ¼ 1 þ ajzj2 ;
and
2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 24 a 1 a 5 ; 1þ þ v¼ a 2n n 2n
1 a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : jðj þ 1Þ
ð53Þ
Now, there exists a function f which relates the operators z and z+ to the usual creation and destruction operators aˆ+ and aˆ respectively z ¼ f ð^aþ ^a þ 1Þ^a:
ð54Þ
Now the generalized coherent states jfi are taken to be the eigenstates of the destruction operator z and read as follows: jfi ¼ N ðjfj2 Þ1=2 exp½ff 1 ð^aþ ^aÞ^aþ f 1 ð^aþ ^aÞj0i;
ð55Þ
2 1/2
the function N(jfj ) enforcing the normalization of the wave function. The vacuum j0i is annihilated by aˆ. This approach to definition of coherent states opens the way for more investigation of the issue specially comparison between alternative scenarios of quantum gravity.
4. Summary and conclusion Today, loop quantum gravity, string theory and non-commutative geometry are three main approaches to quantum gravity. A common feature of these scenarios is the existence of a minimal length scale on the order of Planck length which restricts the accuracy of position measurements. In another words, one cannot probe distances smaller than Planck length. In this framework, in the Planck-scale regime spacetime points have a finite resolution. This finite resolution of spacetime points can be described within foamy or fractal picture of underlying geometry and usually is referred to spacetime fuzziness. There are signatures of this spacetime fuzziness from non-standard features of underlying Planck scale quantum optics which have been supported in recent observations of gamma-ray bursts spectra. These effects could led us to observable cosmological predictions of the discrete nature of quantum spacetime. In this paper, we have studied the effects of fuzzy structure of spacetime in the spirit of quantum theory. Due to the fuzzy nature of spacetime structure, the notion of locality breaks down and therefore the usual picture of position space representation is no longer suitable for description of quantum states. In this situation, a quasi-position representation of quantum states seems to be helpful in order to retain the role of position space representation via a maximally localized picture. We have used this maximally localized states to investigate the effects of spacetime fuzziness on the wave packets propagation. As a result of the fuzzy structure of spacetime, wave packets suffer more broadening relative to standard situation during their propagation. This extra broadening can be detected in principle and this feature provides a possible experimental test of underlying quantum gravity proposal. Interferometric noise experiments are suitable candidates of such test experiments. Finally, coherent states of quantum mechanical systems have been re-examined within non-commutative geometry and stereographic projection algorithm. Obviously, fuzzy nature of spacetime requires a redefinition of coherent states and generalization of these states to more realistic situation in quantum gravity era. Finally, it should be noted that spacetime fuzziness has several other important consequences such as break down of local Lorentz invariance and modification of dispersion relations which recently have attracted considerable attentions.
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