# The Q-Space Deformed Wave Equation

Authors: Mahjoub A. Elamin; Narjess Turki Khalifa; Sami H. Altoum
Journal: Complexity
 (2022)

---

## Abstract

Letq∈0,1. We know that, when q tends to 1, we recover “classical” quantum mechanics. However, when q is not equal to one, we have a theory of quantum mechanics in a spacetime, i.e., a theory where the vacuum has a nonzero energy density. As a q-analogue on the space variable of the wave equation, we introduce the q-space wave equation. Solutions of these equations are given for q∈0,1 and for q⟶0. Since some states have more spin degrees of freedom than the physical particle has, the q-space wave equations remove the additional degrees of freedom.


---

## Body

## 1. Introduction

The wave equation is a second-order linear hyperbolic partial differential equation that describes the propagation of a variety of waves, such as sound or water waves. It arises in different fields such as acoustics, electromagnetics, or fluid dynamics. In its simplest, the wave equation takes the form:(1)∂2u∂t2=c2∂2u∂x2,where x is the coordinate, t is time, u=ux,t is the displacement, and c is the wave velocity. Due to the ambiguity in the direction of the wave velocity, c2=+c2=−c2, the equation does not contain information about the wave direction and therefore has solutions propagating in both the forward +x and backward −x directions. In 1746, D’ Alembert discovered the one-dimensional wave equation, and he gave the equation of motion of the string as the one-dimensional wave equation in 1747, after ten years Euler discovered the three-dimensional wave equation. This equation is a good description for a wide range of phenomena because it is typically used to model small oscillations about an equilibrium, for which systems can often be well approximated by Hooke law. Solutions to the wave equation are of course important in fluid dynamics, but they also play an important role in electromagnetism, optics, gravitational physics, and heat transfer. The problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond D’Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In the area of combinatorics and quantum calculus, the q-derivative, or Jackson derivative, is a q-analogue of the ordinary derivative, introduced by Frank Hilton Jackson [1, 2]. In the last decades, the q-calculus has developed into an interdisciplinary subjects, the q-calculus has in the last twenty years served as a bridge between mathematics and physics. The majority of scientists in the world who use the q-calculus today are physicists. The field has expanded explosively, due to the fact that applications of basic hypergeometric series to the diverse subjects of combinatorics, quantum theory, number theory, and statistical mechanics. The formulation of fractional calculus began shortly after the classical calculus was established. Since its definition is based on the concept of a noninteger order either integral or derivative, the fractional calculus had been considered as a subject in pure mathematics with no real applications for a long time. However, the role of fractional calculus has been changed in recent decades. Its applications take place in many fields of mathematical sciences, extended from the fractional calculus. The fractional q-calculus is the q-extension of the ordinary fractional calculus. Many results of the study on the theory of the q-calculus operators in recent decades have been applied in various areas such as problems in the ordinary fractional calculus, optimal control, solutions of the q-difference equations, q-differential equations, q-integral equations, and q-transform analysis, and also in the geometric function theory of complex analysis. q-calculus has some applications in physics. In theories of quantum gravity, q can be thought of as a parameter related to the exponential of the cosmological constant. So, when q=1, there is no gravity, and we recover ‘classical’ quantum mechanics. However, when q is not equal to one, we have a theory of quantum mechanics in a spacetime with constant curvature, i.e., a theory where the vacuum has a nonzero energy density, see [3–5]. In this setting, we study a q-analogue of the wave equation. More precisely, we are interested by the following problem:(2)∂2u∂t2=c2Dq,x2u,where Dq,x is defined using the q-derivative (see the Preliminaries).


## 2. Preliminaries

We recall some basic notations of the language ofq-calculus (see [6–9]). The natural number n has the following q-deformation for q∈0,1:(3)nq≔1+q+q2+⋯+qn−1,with0q=0.Occasionally, we shall write∞q for the limit of these numbers: 1/1−q. One can get easily that(4)nq=1−qn1−q,q∈0,1,for n∈N. We can give this definition for any real number λ. In this case, we call λq a q-real. The q factorials and q binomial coefficients are defined naturally as(5)nq!:=1q⋅2q⋯nqwith0q:=1,for q∈0,1 and analytic f:C⟶C define operators Z and Dq as follows (see [8, 9]):(6)Zfz:=zfz,Dqfz=fz−fqzz1−q,z≠0f′0,Dq,x1 and Dq,x2 are given by(7)Dq,x1gx1,x2:=Dqg.,x2x1,Dq,x2gx1,x2:=Dqgx1,.x2.The operatorDq has the following properties:(i)
limq⟶1Dqfz=f′z,(ii)
Dqzn=nqzn−1,(iii)
Dqfzgz=Dqfzgz+fqzDqgz,(iv)
Dqfz/gz=Dqfzgz−fzDqgz/gzgqz.It is well known [8] that the operators Dq and Z satisfy(8)DqZ−qZDq=1.One of theq-analogues of classical exponential function ex is(9)eqx=∑k=0∞xkkq!.We see that(10)eqx=11−1−qxq∞,where(11)1−yq∞=∏k=0∞1−qky.Theq-exponential functions satisfy the following property:(12)Dqeqx=eqx.Note that forq∈0,1, the series expansion of eqx has radius of convergence 1/1−q.


## 3.q-Space Wave Equation

In the reminder of this paper, we takeq∈0,1. In this section, we introduce the following equation:(13)∂2u∂t2=c2Dq,x2u.Equation (13) will be called q−space wave equation.Theorem 1.
For0<q<1, we have(1)
ugiven by(14)ut,x=Ae−tc2L+Betc2LeqxLis solution of Eq. (13), for any constantsAandB, whereL>0.(2)
ugiven by(15)ut,x=Acost−c2L+Bsint−c2Leqjx−Lis solution of Eq. (13), for any constantsAandB, whereL<0andj2=−1.Proof.
For0<q<1, we get(16)Dq,xut,x=ut,x−ut,qxx1−q:=ht,x,(17)Dq,x2ut,x=ht,x−ht,qxx1−q.
Putting(18)ut,x=RtSx.
We obtain(19)SxR′′t=c2RtDq,x2Sx.
Then,(20)Dq,x2SxSx=R′′tc2Rt=L.
For a constantL, this implies(21)R′′t=Lc2Rt.(22)Dq,x2Sx=LSx.
Using Eq. (17) in Eq. (22), we obtain(23)qx−1+qSqx+Sq2xqx21−q2=LSx,qSx−1+qSqx+Sq2x=qx21−q2LSx.
This gives(24)q1−x21−q2LSx=1+qSqx−Sq2x.
From which, we obtain(25)q1−q2x21−q2LSqx=1−qSq2x−Sq3x⋮q1−q2nx21−q2LSqnx=1+qSqn+1x−Sqn+2xq1−q2n+2x21−q2LSqn+1x=1+qSqn+2x−Sqn+3x.
Then, we deduce that(26)q1−αxSx+qSqx−q3αxSqx⋮+qSqnx−q2n+1αxSqnx+qSqn+1x−q2n+3αxSqn+1x=Sqx+qSqx−Sq2x+Sq2x+qSq2x−Sq3x⋮+Sqn+1x+qSqn+1x−Sqn+2x+Sqn+2x+qSqn+2x−Sqn+3x,(27)q1−αxSx−αxq3Sqx+⋯+q2n+3Sqn+1x=Sqx+qSqn+2x−Sqn+3x,where(28)αx=x21−q2L.
ForS given by(29)Sx=∑i=0∞aixi.
We get(30)αxq3∑i=0∞aiqixi+⋯+q2n+3∑aiqn+1ixi=q31−q2L∑i=0∞aiqi1+qi+⋯+q2+inxi+2=q31−q2L∑i=0∞aiqi1−q2+in+11−q2+ixi+2:=β.
Asn⟶∞, the last term gives(31)β⟶q31−q2L∑i=0∞ai−2qi1−qixi.
Then, Eq. (27) and Eq. (31), we get(32)q1−x21−q2L∑i=0∞aixi−β=∑i=0∞aiqixi+q−1a0.
Therefore, we get(33)∑i=0∞qaixi−q1−qL∑i=0∞aixi+2−β=∑i=0∞aiqixi+q−1a0,
which gives(34)qa0+qa1x+∑i=0∞qaixi−q1−q2L∑i=0∞ai−2xi−β=a0+a1qx+∑i=0∞aiqixi+q−1a0.
This implies that(35)qai−q1−q2Lai−2−q1−q2Lqi1−qiai−2=aiqi.
Hence, we obtain(36)q1−qi−1ai=q1−q2L1+qi1−qiai−2.
Then,(37)qi−1qai=q1−qL1−qi+qi1−qiai−2.
This implies that(38)i−1qai=L1−q1−qiai−2,
which gives(39)i−1qai=L1iqai−2.
Therefore, we get(40)ai=Liqi−1qai−2i≥2.
IfL>0, then(41)ai=Liiq!.
Therefore, we obtain(42)ai+2=Li+2i+2q!=Liiq!⋅L2i+2qi+1q=aiLi+2qi+1q.
This verifies Eq. (40). Then, we get(43)Sx=∑i=0∞aixi=∑i=0∞Lxiiq!=eqxL.
IfL<0, we take(44)ai=j−Liiq!,j=−1.
Hence, we obtain(45)ai+2=ji+2−Li+2=j−Liiq!j2−Li+2qi+1q=ai−−Li+2qi+1q=Li+2qi+1qai.
This verifies Eq. (40). Then, we obtain(46)Sx=∑i=0∞aixi=eqjx−L.
Hence, we obtain forL>0(47)ut,x=Ae−tc2L+Betc2LeqxL.
IfL<0, we obtain(48)ut,x=Acost−c2L+Bsint−c2Leqjx−L,where A and B are constants.


## 4. The Limit Caseq⟶0 of the q−Space Wave Equation

In this section, we study the following equation:(49)∂2u∂t2=c2Dq,x2u,where(50)D0,xux,y=ux,y−u0,yx.Theorem 2.
(1)
We have(51)ut,x=a01−Lx2Acost−c2L+Bsint−c2Lis solution of Equation (49), with any constantsa0, AandB, wherek<0.(2)
We have(52)ut,x=a01−Lx2Aetc2L+Be−tc2L,ifx≠1LandL>0,SxAetc2L+Be−tc2L,ifx=1LandL>0,.is solution of Equation (49), with any constantsa0, AandB, wheref0=0andk>0.Proof.
Forut,x given by(53)ut,x=RtSx.
Therefore, Equation (49) gives(54)SxR′′t=c2RtD0,x2Sx.
Then, we obtain(55)D0,x2SxSx=R′′tc2Rt=L,Lis constant.
Therefore, Equation (55) gives(56)R′′t=c2LRt,(57)D0,x2Sx=LSx.
The solution of Equation (56) is given by(58)Rt=Aetc2L+Be−tc2LifL>0,and if L<0, we obtain(59)Rt=Acost−c2L+Bsint−c2L.for any constants A and B. Since we have(60)D0,xSx=Sx−S0x:=F0x,we obtain(61)D0,x2Sx=F0x−F00x,=Sx−S0/x−0x,=Sx−S0x2.
Using Eq. (57), we get(62)Sx−S0x2=LSx,which implies that(63)Sx−S0=Lx2Sx,which gives(64)1−Lx2Sx=S0.
Then, we get(65)Sx=S01−Lx2forL<0.
IfL>0, we obtain(66)Sx=S01−Lx2ifx≠1L,and for x=1/L, we should take S0=0. Then, we get(67)ut,x=a01−Lx2Acost−c2L+Bsint−c2L.
For any constantsa0, A and B, where L<0. On the other hand, we get(68)ut,x=a01−Lx2Aetc2L+Be−tc2L,ifx≠1LandL>0,SxAetc2L+Be−tc2L,ifx≠1LandL>0,
For any constantsa0, A and B, where S0=0 and L>0. This completes the proof.


## 5. Conclusion

In this paper, theq-space wave equation as well as the limit case q⟶0 of the q-space wave equation are studied. We expect to study the quantum white noise [10–13] case which is now attractive in mathematical physics area. Also, the numerical study, the graphical presentations of the obtained solutions and the representations of these results will be given in future works. The two- and the three-dimensional types of these equations are in our focus [14, 15].



---

*Source: 8753231-2022-07-22.xml*