- Email: [email protected]
Spacetime non-commutativity, generalized uncertainty principle and the fine structure constant
Chaos, Solitons and Fractals 31 (2007) 777–781 www.elsevier.com/locate/chaos
Spacetime non-commutativity, generalized uncertainty principle and the fine structure constant Kourosh Nozari *, Behnaz Fazlpour Department of Physics, Faculty of Basic Science, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran Accepted 21 April 2006
Communicated by Prof. L. Mavek-Cvnjac
Abstract Quantum gravitational effects and spacetime non-commutativity should affect the value of the fine structure constant. In this paper, using generalized uncertainty principle, we calculate the modified fine structure constant in noncommutative spacetime. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Spacetime has a non-commutative structure [1–4]. This non-commutativity has novel implications for the rest of the physics. One of these implications is the possible modification of the fine structure constant. Since fine structure constant contains important information regarding the relative strength of fundamental interactions, its numerical value and relative modifications have their own importance. Recently, from a string theoretical point of view, it has been revealed that in extreme quantum gravity regime, standard uncertainty relation of Heisenberg should be modified to incorporate quantum gravitational effects [5–8]. These extra terms in uncertainty principle are referred as gravitational uncertainties. Here we are going to consider the effects of spacetime non-commutativity on the value of fine structure constant. We use generalized uncertainty principle as our primary input and calculate modified numerical value of fine structure constant. We will show that the value of this modification is quantum gravity model dependent and tends to zero when non-commutative and quantum gravity parameters tend to zero. Finally, we discuss the relation between our approach and the other existing approaches to the issue. 2. Fine structure constant in non-commutative space A non-commutative space can be realized by the coordinate operators satisfying ½^xi ; ^xj ¼ ihij ; *
i; j ¼ 1; 2; 3;
Corresponding author. E-mail address: [email protected] (K. Nozari).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.050
ð1Þ
778
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
where ^x’s are the coordinate operators and hij is the non-commutativity parameter with dimension (length)2. In noncommutative spaces, one has the following commutation relations ½^xi ; ^xj ¼ ihij ;
½^xi ; ^pj ¼ ihdij ;
½^pi ; ^pj ¼ 0:
ð2Þ
From a string theoretical point of view, the usual uncertainty principle of Heisenberg, DxDp P h2, should be modified when gravitational uncertainties are taken into account. A possible form of these generalized uncertainty principle (GUP) is given by [9] h DxDp P ½1 þ bðDpÞ2 þ c; 2
ð3Þ
which leads to ½^xi ; ^pj ¼ ihdij ð1 þ bp^2 Þ;
ð4Þ 2
where we have supposed c ¼ bh^pi . The extra term in the right hand side of (4) is important in high energy regime where we can write approximately Dp p. Note that one can recover these information from a non-commutative picture of spacetime coordinates as well as from loop quantum gravity approach [10]. Now within GUP framework, non-commutativity of spacetime reads ½^xi ; ^pj ¼ ihdij ð1 þ bp^2 Þ;
½^xi ; ^xj ¼ ihij ;
½^pi ; ^pj ¼ 0:
ð5Þ
Now, one can show that there is a new coordinate system in which xi ¼ ^xi þ
1 hij ^pj ; 2hð1 þ b^p2 Þ
pi ¼ ^pi ;
ð6Þ
where the new variables satisfy the usual canonical commutation relations ½xi ; xj ¼ 0;
½xi ; pj ¼ ihdij ð1 þ bp2 Þ;
½pi ; pj ¼ 0:
ð7Þ
The radius of the nth orbit of Bohr’s hydrogen atom in commutative space is given by rn ¼
4p0 n2 h2 : me2
ð8Þ
In non-commutative space, we propose the following relation for the radius of the nth orbit of Bohr’s hydrogen atom: ^rn ¼
pffiffiffiffiffiffiffiffi 4p0 n2 h2 ^rn^rn ¼ ; me2
ð9Þ
where ^rn satisfies in (6). Changing the variables ^xi , ^pi to xi, pi in (9), we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi hij pj hik pk 4p0 n2 h2 xi ¼ : xi 2 2 2 2hð1 þ bp Þ 2hð1 þ bp Þ me
ð10Þ
Assuming that the space–space non-commutative parameter hij to be very small, Eq. (10) reads rn
xi hij pj hij his pj ps 4p0 n2 h2 þ 2 ¼ ; 2 2 2hð1 þ bp Þrn 8h ð1 þ bp2 Þ rn me2
ð11Þ
where hij ¼ 12 ijk hk . Using the identity ijr iks ¼ djk drs djs drk ;
ð12Þ
we have r2n
~ L:~ h 4p0 n2 h2 1 rn ðh2 p2 ð~ h:~ pÞ2 Þ þ Oðh3 Þ þ ¼ 0; þ 2 me 4hð1 þ bp2 Þ 32h2 ð1 þ bp2 Þ2
ð13Þ
p:~ p and h2 ¼ ~ h:~ h. This equation has two solutions which are where Lk = ijkxipj, p2 ¼ ~ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ~ L:~ h 2p0 n2 h2 u 2p0 n2 h2 1 t rn ¼ þ þ ðh2 p2 ð~ h:~ pÞ2 Þ 2 2 4hð1 þ bp2 Þ 32 me me h2 ð1 þ bp2 Þ2
ð14Þ
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 ~ L:~ h 2p0 n2 h2 u 1 t 2p0 n h rn ¼ þ ðh2 p2 ð~ h:~ pÞ2 Þ: 2 2 4hð1 þ bp2 Þ 32 me me h2 ð1 þ bp2 Þ2
779
ð15Þ
In the limit h ! 0, i.e. within commutative space, one should recover the usual relation for rn (for n = 0 we set r0 a0: the Bohr’s radius), a0 ¼
4p0 h2 ¼ 5:29 1011 m: me2
ð16Þ
Apparently, the solution (15) does not approach to this value when h ! 0. Therefore, the Bohr radius in non-commutative space can be obtained by Eq. (14). If we put h3 = h and assuming that the remaining components of h all vanish (which can be done by a rotation or a re-definition of the coordinates), then ~ L ~ h ¼ Lz h and ~ h ~ p ¼ hpz . Therefore, Eq. (14) can be rewritten as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 2p0 n2 h2 u Lz h 1 t 2p0 n h ð17Þ þ þ ðp2 p2z Þh2 : rn ¼ 2 me2 me2 4hð1 þ bp2 Þ 32 h ð1 þ bp2 Þ2 The electron in the first Bohr orbit of hydrogen atom has zero angular momentum, i.e. Lz = 0. Therefore, substituting n = 1, Lz = 0 and p2 p2z ¼ p2x þ p2y , we obtain the following relation for Bohr radius, aˆ0, in non-commutative space: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2p h2 2 2p0 h 1 0 t ^a0 ¼ þ ðp2x þ p2y Þh2 : ð18Þ me2 me2 32h2 ð1 þ bp2 Þ2 In non-commutative space, the orbital speed of the electron ^v in the first Bohr orbit of hydrogen atom can be obtained 2 2 and the electrostatic force F ¼ 4pe ^r2 . If we put ^r1 to be equal to Bohr radius ^ a0 by equality of the centripetal force F ¼ mv ^r1 0 1 in non-commutative space, then we find m^v2 e2 ¼ : ^a0 4p0 ^a20
ð19Þ
This equation gives ^v2 ¼
e2 : 4p0 m^a0
ð20Þ
The ratio of the speed ^v to the speed of light, that is, ^v=c, is the fine structure constant ^ a in non-commutative space 1=2 e2 ^a ¼ : ð21Þ 4p0 mc2 ^a0 Substituting Bohr radius from (18) into (21) we find the fine structure constant in non-commutative spacetime ffi!1=2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 e2 2 1 me2 2 2 2 ^a ¼ 1þ 1 ðpx þ py Þh : 4p0 hc 2ð1 þ bp2 Þ2 8p0 h3
ð22Þ
2
Using Eq. (16) and a ¼ 4pe0 hc ¼ 7:30 103 (where a is commutative space fine structure constant), we can rewrite Eq. (22) as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11=2 u pffiffiffi 0 2 u ðv2x þ v2y Þh2 e 2 @ A ^a ¼ 1 þ t1 : ð23Þ 4p0 hc 8a2 c2 a40 ð1 þ bp2 Þ2 Since p = mv, this relation can be written as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11=2 u pffiffiffi 0 u ðv2x þ v2y Þh2 e2 2 @ A ^a ¼ 1 þ t1 : 4p0 hc 8a2 c2 a40 ð1 þ bðmcÞ2 ðcv Þ2 Þ2
ð24Þ
780
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
From the principle of special relativity, since the maximum value of ðv2x þ v2y Þ=c2 is unity, we can set it equal to one. Now, we can expand (24) to obtain ! e2 h2 4 ^a ’ 1þ þ Oðh Þ þ : ð25Þ 4p0 hc 64a2 a40 ð1 þ bðmcÞ2 Þ2 Using Eq. (25), the fine structure constant in non-commutative spacetime is greater than the fine structure constant in commutative space. Since m = 9.11 · 1031 kg, e = 1.6 · 1019 C, c = 3.00 · 108 m s1, 0 = 8.85 · 1012 C2 N1 m2, and G = 6.67 · 1011 m3 s2 kg1, we can calculate the numerical value of ^ a as follows: ^a ’ 7:3 103 ½1 þ 3:74 1043 h2 5:59h2 b;
ð26Þ
where can be written as ^a ’ a½1 þ 3:74 1043 h2 5:59h2 b:
ð27Þ
The second term in the right hand side has pure non-commutative geometrical origin, while the third term includes both non-commutativity and the quantum gravitational effect. Since h l2p [11,12] where lp ’ 1.6 · 1035 and b 1 in extreme quantum gravity regime, we find ^a ’ a½1 þ 2:45 1096 3:66 10139 :
ð28Þ
The possible variation of the fine structure ‘‘constant’’ has been investigated from several experimental and observational evidences. For example constraint imposed by cosmic back ground radiation, Bloch oscillations of ultra-cold atoms, most precise single redshift experiments, and several other approaches has been investigated to show the variation of fine structure constant and its implications.
3. Discussion There are several attempt to incorporate spacetime non-commutativity and quantum gravitational effects in the calculation of the value of fine structure constant. Some of these attempts consider only the effect of non-commutativity [12], but it is natural to incorporate the quantum gravitational effects also. Here, we have proceeded in this direction. There is an elegant approach to investigate possible modification of the fine structure constant within E1 space of El Naschie [13]. In this approach one can write 1 ð29Þ ¼ 137 þ /5 ð1 /5 Þ ¼ 137:082039325; ^a or equivalently, ^a ’ 7:2949016875 103 :
ð30Þ
Comparison between this result which is based on fractal geometry of spacetime and our calculations will reveal some important link between different approaches to quantum gravity problems. Note that models based on time variation of speed of light also show possible variation of fine structure ‘‘constant’’ [14,15]. Our proposed model here is consistent with these mentioned scenarios. The possible relations between our finding and the results of El Naschie will be investigated in our forthcoming paper.
References [1] Seiberg N, Witten E. String theory and noncommutative geometry. JHEP 1999;9909:032. [2] Connes A. A short survey of noncommutative geometry. J Math Phys 2000;41:3832–66. [3] Castro C. Noncommutative geometry, negative probabilities and Cantorian-Fractal spacetime. Chaos, Solitons & Fractals 2001;12:101–4. [4] Aschieri P et al. Noncommutative geometry and gravity. Class Quant Grav 2006;23:1883–912. [5] Gross DJ, Mende PF. String theory beyond the Planck scale. Nucl Phys B 1988;303:407. [6] Konishi K, Paffuti G, Provero P. Minimum physical length and the generalized uncertainty principle in string theory. Phys Lett B 1990;234:276. [7] Amati D, Ciafaloni M, Veneziano G. Can spacetime be probed below the string size? Phys Lett B 1989;216:41. [8] Garay LJ. Quantum gravity and minimum length. Int J Mod Phys A 1995;10:145.
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
781
[9] Kempf A et al. Hilbert space representation of the minimal length uncertainty relation. Phys Rev D 1995;52:1108–18. [10] Amelino-Camelia G et al. The search for quantum gravity signals. In: AIP conference, Proceedings of the 2nd Mexican meeting on mathematical and experimental physics. Available from: arXiv:gr-qc/0501053. [11] Nicolini P et al. Noncommutative geometry inspired Schwarzschild black hole. Phys Lett B 2006;632:547–51. [12] Nasseri F. Fine structure constant in noncommutative spaces. Available from: arXiv:hep-th/0512139. [13] El Naschie MS. New elementary particles as a possible product of a disintegrating symplistic vacuum. Chaos, Solitons & Fractals 2004;20:905–13. [14] Mota DF. Variations of the fine structure constant in space and time. PhD Thesis. DAMTP, University of Cambridge. Available from: arXiv:astro-ph/0401631. [15] Mota DF, Barrow JD. Local and global variations of the fine structure constant. Mon Not Roy Astron Soc 2004;349:291.
Spacetime non-commutativity, generalized uncertainty principle and the fine structure constant Kourosh Nozari *, Behnaz Fazlpour Department of Physics, Faculty of Basic Science, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran Accepted 21 April 2006
Communicated by Prof. L. Mavek-Cvnjac
Abstract Quantum gravitational effects and spacetime non-commutativity should affect the value of the fine structure constant. In this paper, using generalized uncertainty principle, we calculate the modified fine structure constant in noncommutative spacetime. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Spacetime has a non-commutative structure [1–4]. This non-commutativity has novel implications for the rest of the physics. One of these implications is the possible modification of the fine structure constant. Since fine structure constant contains important information regarding the relative strength of fundamental interactions, its numerical value and relative modifications have their own importance. Recently, from a string theoretical point of view, it has been revealed that in extreme quantum gravity regime, standard uncertainty relation of Heisenberg should be modified to incorporate quantum gravitational effects [5–8]. These extra terms in uncertainty principle are referred as gravitational uncertainties. Here we are going to consider the effects of spacetime non-commutativity on the value of fine structure constant. We use generalized uncertainty principle as our primary input and calculate modified numerical value of fine structure constant. We will show that the value of this modification is quantum gravity model dependent and tends to zero when non-commutative and quantum gravity parameters tend to zero. Finally, we discuss the relation between our approach and the other existing approaches to the issue. 2. Fine structure constant in non-commutative space A non-commutative space can be realized by the coordinate operators satisfying ½^xi ; ^xj ¼ ihij ; *
i; j ¼ 1; 2; 3;
Corresponding author. E-mail address: [email protected] (K. Nozari).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.050
ð1Þ
778
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
where ^x’s are the coordinate operators and hij is the non-commutativity parameter with dimension (length)2. In noncommutative spaces, one has the following commutation relations ½^xi ; ^xj ¼ ihij ;
½^xi ; ^pj ¼ ihdij ;
½^pi ; ^pj ¼ 0:
ð2Þ
From a string theoretical point of view, the usual uncertainty principle of Heisenberg, DxDp P h2, should be modified when gravitational uncertainties are taken into account. A possible form of these generalized uncertainty principle (GUP) is given by [9] h DxDp P ½1 þ bðDpÞ2 þ c; 2
ð3Þ
which leads to ½^xi ; ^pj ¼ ihdij ð1 þ bp^2 Þ;
ð4Þ 2
where we have supposed c ¼ bh^pi . The extra term in the right hand side of (4) is important in high energy regime where we can write approximately Dp p. Note that one can recover these information from a non-commutative picture of spacetime coordinates as well as from loop quantum gravity approach [10]. Now within GUP framework, non-commutativity of spacetime reads ½^xi ; ^pj ¼ ihdij ð1 þ bp^2 Þ;
½^xi ; ^xj ¼ ihij ;
½^pi ; ^pj ¼ 0:
ð5Þ
Now, one can show that there is a new coordinate system in which xi ¼ ^xi þ
1 hij ^pj ; 2hð1 þ b^p2 Þ
pi ¼ ^pi ;
ð6Þ
where the new variables satisfy the usual canonical commutation relations ½xi ; xj ¼ 0;
½xi ; pj ¼ ihdij ð1 þ bp2 Þ;
½pi ; pj ¼ 0:
ð7Þ
The radius of the nth orbit of Bohr’s hydrogen atom in commutative space is given by rn ¼
4p0 n2 h2 : me2
ð8Þ
In non-commutative space, we propose the following relation for the radius of the nth orbit of Bohr’s hydrogen atom: ^rn ¼
pffiffiffiffiffiffiffiffi 4p0 n2 h2 ^rn^rn ¼ ; me2
ð9Þ
where ^rn satisfies in (6). Changing the variables ^xi , ^pi to xi, pi in (9), we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi hij pj hik pk 4p0 n2 h2 xi ¼ : xi 2 2 2 2hð1 þ bp Þ 2hð1 þ bp Þ me
ð10Þ
Assuming that the space–space non-commutative parameter hij to be very small, Eq. (10) reads rn
xi hij pj hij his pj ps 4p0 n2 h2 þ 2 ¼ ; 2 2 2hð1 þ bp Þrn 8h ð1 þ bp2 Þ rn me2
ð11Þ
where hij ¼ 12 ijk hk . Using the identity ijr iks ¼ djk drs djs drk ;
ð12Þ
we have r2n
~ L:~ h 4p0 n2 h2 1 rn ðh2 p2 ð~ h:~ pÞ2 Þ þ Oðh3 Þ þ ¼ 0; þ 2 me 4hð1 þ bp2 Þ 32h2 ð1 þ bp2 Þ2
ð13Þ
p:~ p and h2 ¼ ~ h:~ h. This equation has two solutions which are where Lk = ijkxipj, p2 ¼ ~ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ~ L:~ h 2p0 n2 h2 u 2p0 n2 h2 1 t rn ¼ þ þ ðh2 p2 ð~ h:~ pÞ2 Þ 2 2 4hð1 þ bp2 Þ 32 me me h2 ð1 þ bp2 Þ2
ð14Þ
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
and
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 ~ L:~ h 2p0 n2 h2 u 1 t 2p0 n h rn ¼ þ ðh2 p2 ð~ h:~ pÞ2 Þ: 2 2 4hð1 þ bp2 Þ 32 me me h2 ð1 þ bp2 Þ2
779
ð15Þ
In the limit h ! 0, i.e. within commutative space, one should recover the usual relation for rn (for n = 0 we set r0 a0: the Bohr’s radius), a0 ¼
4p0 h2 ¼ 5:29 1011 m: me2
ð16Þ
Apparently, the solution (15) does not approach to this value when h ! 0. Therefore, the Bohr radius in non-commutative space can be obtained by Eq. (14). If we put h3 = h and assuming that the remaining components of h all vanish (which can be done by a rotation or a re-definition of the coordinates), then ~ L ~ h ¼ Lz h and ~ h ~ p ¼ hpz . Therefore, Eq. (14) can be rewritten as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 2 2 2p0 n2 h2 u Lz h 1 t 2p0 n h ð17Þ þ þ ðp2 p2z Þh2 : rn ¼ 2 me2 me2 4hð1 þ bp2 Þ 32 h ð1 þ bp2 Þ2 The electron in the first Bohr orbit of hydrogen atom has zero angular momentum, i.e. Lz = 0. Therefore, substituting n = 1, Lz = 0 and p2 p2z ¼ p2x þ p2y , we obtain the following relation for Bohr radius, aˆ0, in non-commutative space: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2p h2 2 2p0 h 1 0 t ^a0 ¼ þ ðp2x þ p2y Þh2 : ð18Þ me2 me2 32h2 ð1 þ bp2 Þ2 In non-commutative space, the orbital speed of the electron ^v in the first Bohr orbit of hydrogen atom can be obtained 2 2 and the electrostatic force F ¼ 4pe ^r2 . If we put ^r1 to be equal to Bohr radius ^ a0 by equality of the centripetal force F ¼ mv ^r1 0 1 in non-commutative space, then we find m^v2 e2 ¼ : ^a0 4p0 ^a20
ð19Þ
This equation gives ^v2 ¼
e2 : 4p0 m^a0
ð20Þ
The ratio of the speed ^v to the speed of light, that is, ^v=c, is the fine structure constant ^ a in non-commutative space 1=2 e2 ^a ¼ : ð21Þ 4p0 mc2 ^a0 Substituting Bohr radius from (18) into (21) we find the fine structure constant in non-commutative spacetime ffi!1=2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 e2 2 1 me2 2 2 2 ^a ¼ 1þ 1 ðpx þ py Þh : 4p0 hc 2ð1 þ bp2 Þ2 8p0 h3
ð22Þ
2
Using Eq. (16) and a ¼ 4pe0 hc ¼ 7:30 103 (where a is commutative space fine structure constant), we can rewrite Eq. (22) as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11=2 u pffiffiffi 0 2 u ðv2x þ v2y Þh2 e 2 @ A ^a ¼ 1 þ t1 : ð23Þ 4p0 hc 8a2 c2 a40 ð1 þ bp2 Þ2 Since p = mv, this relation can be written as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11=2 u pffiffiffi 0 u ðv2x þ v2y Þh2 e2 2 @ A ^a ¼ 1 þ t1 : 4p0 hc 8a2 c2 a40 ð1 þ bðmcÞ2 ðcv Þ2 Þ2
ð24Þ
780
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
From the principle of special relativity, since the maximum value of ðv2x þ v2y Þ=c2 is unity, we can set it equal to one. Now, we can expand (24) to obtain ! e2 h2 4 ^a ’ 1þ þ Oðh Þ þ : ð25Þ 4p0 hc 64a2 a40 ð1 þ bðmcÞ2 Þ2 Using Eq. (25), the fine structure constant in non-commutative spacetime is greater than the fine structure constant in commutative space. Since m = 9.11 · 1031 kg, e = 1.6 · 1019 C, c = 3.00 · 108 m s1, 0 = 8.85 · 1012 C2 N1 m2, and G = 6.67 · 1011 m3 s2 kg1, we can calculate the numerical value of ^ a as follows: ^a ’ 7:3 103 ½1 þ 3:74 1043 h2 5:59h2 b;
ð26Þ
where can be written as ^a ’ a½1 þ 3:74 1043 h2 5:59h2 b:
ð27Þ
The second term in the right hand side has pure non-commutative geometrical origin, while the third term includes both non-commutativity and the quantum gravitational effect. Since h l2p [11,12] where lp ’ 1.6 · 1035 and b 1 in extreme quantum gravity regime, we find ^a ’ a½1 þ 2:45 1096 3:66 10139 :
ð28Þ
The possible variation of the fine structure ‘‘constant’’ has been investigated from several experimental and observational evidences. For example constraint imposed by cosmic back ground radiation, Bloch oscillations of ultra-cold atoms, most precise single redshift experiments, and several other approaches has been investigated to show the variation of fine structure constant and its implications.
3. Discussion There are several attempt to incorporate spacetime non-commutativity and quantum gravitational effects in the calculation of the value of fine structure constant. Some of these attempts consider only the effect of non-commutativity [12], but it is natural to incorporate the quantum gravitational effects also. Here, we have proceeded in this direction. There is an elegant approach to investigate possible modification of the fine structure constant within E1 space of El Naschie [13]. In this approach one can write 1 ð29Þ ¼ 137 þ /5 ð1 /5 Þ ¼ 137:082039325; ^a or equivalently, ^a ’ 7:2949016875 103 :
ð30Þ
Comparison between this result which is based on fractal geometry of spacetime and our calculations will reveal some important link between different approaches to quantum gravity problems. Note that models based on time variation of speed of light also show possible variation of fine structure ‘‘constant’’ [14,15]. Our proposed model here is consistent with these mentioned scenarios. The possible relations between our finding and the results of El Naschie will be investigated in our forthcoming paper.
References [1] Seiberg N, Witten E. String theory and noncommutative geometry. JHEP 1999;9909:032. [2] Connes A. A short survey of noncommutative geometry. J Math Phys 2000;41:3832–66. [3] Castro C. Noncommutative geometry, negative probabilities and Cantorian-Fractal spacetime. Chaos, Solitons & Fractals 2001;12:101–4. [4] Aschieri P et al. Noncommutative geometry and gravity. Class Quant Grav 2006;23:1883–912. [5] Gross DJ, Mende PF. String theory beyond the Planck scale. Nucl Phys B 1988;303:407. [6] Konishi K, Paffuti G, Provero P. Minimum physical length and the generalized uncertainty principle in string theory. Phys Lett B 1990;234:276. [7] Amati D, Ciafaloni M, Veneziano G. Can spacetime be probed below the string size? Phys Lett B 1989;216:41. [8] Garay LJ. Quantum gravity and minimum length. Int J Mod Phys A 1995;10:145.
K. Nozari, B. Fazlpour / Chaos, Solitons and Fractals 31 (2007) 777–781
781
[9] Kempf A et al. Hilbert space representation of the minimal length uncertainty relation. Phys Rev D 1995;52:1108–18. [10] Amelino-Camelia G et al. The search for quantum gravity signals. In: AIP conference, Proceedings of the 2nd Mexican meeting on mathematical and experimental physics. Available from: arXiv:gr-qc/0501053. [11] Nicolini P et al. Noncommutative geometry inspired Schwarzschild black hole. Phys Lett B 2006;632:547–51. [12] Nasseri F. Fine structure constant in noncommutative spaces. Available from: arXiv:hep-th/0512139. [13] El Naschie MS. New elementary particles as a possible product of a disintegrating symplistic vacuum. Chaos, Solitons & Fractals 2004;20:905–13. [14] Mota DF. Variations of the fine structure constant in space and time. PhD Thesis. DAMTP, University of Cambridge. Available from: arXiv:astro-ph/0401631. [15] Mota DF, Barrow JD. Local and global variations of the fine structure constant. Mon Not Roy Astron Soc 2004;349:291.