LAGRANGIAN FORMULATION OF MAXWELL S FIELD IN FRACTIONAL D DIMENSIONAL SPACE-TIME

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1 THEORETICAL PHYSICS LAGRANGIAN FORMULATION OF MAXWELL S FIELD IN FRACTIONAL D DIMENSIONAL SPACE-TIME SAMI I. MUSLIH 1,, MADHAT SADDALLAH 2, DUMITRU BALEANU 3,, EQAB RABEI 4 1 Department of Mechanical Engineering, Southern Illinois University, Carbondale, Illinois , USA 2 Al-Azhar University-Gaza 3 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530, Ankara, Turkey dumitru@cankaya.edu.tr 4 Physics Department, Al-al-Bayt University, Mafraq, Jordan Received August 5, 2009 The Lagrangian formulation for field systems is obtained in fractional space-time fractional dimensions D = D space + D time. The equations of motion for Maxwell s field are obtained. It is shown that the form of Maxwell s equations in fractional dimensional space are not invariant and they can be solved in the same manner as in the integer space-time dimensions. Key words: Lagrangian approach, fractional D dimensional space-time. 1. INTRODUCTION Differential operators of fractional order have outstanding significance for the modelling of many memory-dependent phenomena in physics, engineering, finance, as well as other fields [1 8]. During the last years a huge effort was devoted to connect the fractional dimension introduced at the beginning of the last century and the fractal geometry [6, 9 19]. Non-integer dimension can also be described by the analytic continuation of the dimension in Gaussian integrals [20 23]. Dynamical equations involving the fractional derivatives describe the evolution of physical systems with loss, the fractional exponent of the derivative being a measure of the fraction of the states of the dynamical system that are preserved during the evolution time. Dynamical systems with fractional order are non-conservatives On leave of absence from Al-Azhar University-Gaza, smuslih@ictp.it On leave of absence from Institute of Space Sciences, P.O.BOX, MG-23, R 76900, Magurele- Bucharest, Romania, baleanu@venus.nipne.ro Rom. Journ. Phys., Vol. 55, Nos. 7 8, P , Bucharest, 2010

2 660 Sami I. Muslih et al. 2 and are broadly used for describing intermediary physical processes and critical phenomena in non-equilibrium complex non-linear systems. There are much interest to study the equations of motion in non-integer dimensional space [24 34]. In [24, 25] it has been proposed a method to replace the real confining structure with an effective space, where the measure of its anisotropy is given by non-integer dimensions. The Euler-Lagrange equations of motion in non-integer dimensions and solutions of Schrödinger equation were obtained in three variables system in [26]. Another important progress in this field was reported in [35], where the fractional Euler-Lagrange equations of motion for classical fields are obtained as a generalization to the simplest variational problem as proposed in [36] is presented. For the above mentioned reasons obtaining the Euler-Lagrange equations of motion for Maxwell s field in fractional dimensions is an interesting issue to be investigated. The plan of the paper is as follows: In Section two the Euler-Lagrange equations in non-integer dimensions are briefly reviewed. In Section three the Maxwell s equation in D = D s + D t dimensional fractional space-time is investigated. Section four is devoted to our conclusions. 2. EULER-LAGRANGE EQUATIONS IN D s + D t DIMENSIONAL FRACTIONAL SPACE-TIME In this section we shall give a brief review on the Euler-Lagrange equations of motion for fields in non-integer dimensions [26]. A covariant form of the action would involve a Lagrangian density L via S = Ω Ld D+1 x = Ld D xdt where Ω is the boundary for all coordinates. The Lagrangian density L is defined as, L = L(φ, µ φ) and with L = Ld D x. The corresponding covariant Euler-Lagrange equations are L φ L µ = 0, (1) where φ is the field variable and µ is space and time derivative. For non-integer space-time coordinates, the action function for N degrees of freedom is S = = d Dt t d Ds xl(φ, µ φ)), Ω N d αt t d α i x i L(φ, µ φ), (2) i=1

3 3 Lagrangian formulation of Maxwell s field 661 where, φ and µ φ are functions of (t,x 1,...,x N ) and µ = ( t, x i ), with i is running from 1 to N and the fractional volume element d D x and the fractional line element are given respectively as [1, 8] d D x = µ d αµ, (3) and D s = d αµ = παµ/2 x αµ 1 d, (4) Γ(α µ /2) N α i, D t = α t. In this paper we will consider the limits of α µ as 0 < i=1 α µ 1, such that 0 < D N + 1. The case for α µ > 1 will be considered in our future work. Putting δs = 0, the Euler-Lagrange equations of motion in non-integer dimensions is given by [26] L(φ, µ φ) φ µ L(φ, µ φ) (α µν δ µν )(x ( 1) ) ν L(φ, µφ) = 0, (5) with δ µν is a diagonal unit matrix, (x ( 1) ) µ = column(t 1,(x 1 ),...,(x N )), and α µν are the diagonal elements of a matrix which include both time and spatial dimensions (α = dimension(α t,α 1,...,α N )), the spatial dimension of the system is specified by D s = T r(α) α t. 3. MAXWELL S EQUATION IN D s + D t DIMENSIONAL FRACTIONAL SPACE-TIME The Lagrangian density for Maxwell s field is given by L = 1 4 F µνf µν + J µ A µ, (6) where F µν are antisymmetric field stress tensor and defined as F µν = µ A ν ν A µ, and J µ are the four component current density. The case of α t = 1 is considered in our calculations which means that their is no time-decaying friction term. In this case the Euler-Lagrange equation of motion (5) for this system reads as J µ + (α µν δ µν )(x ( 1) ) ν F µν = 0. (7) Calculations show that the third term in this equation vanish for all α. Hence, the Maxwell s equations in fractional dimensional space are given by = J µ. (8)

4 662 Sami I. Muslih et al. 4 This means that these equations have the same form as those obtained in the integer space-time dimensions and should take into consideration that Maxwell s equations (8), should be solved simultaneously in fractional dimensional space-time. We can generalize this formulation to the case of non-abelian field equation and Yang-Mills field. The Lagrangian density for massive Yang-Mills field is given by where F µν α are defined as L = 1 4 F α µνf µν α M 2 A µ αa α µ + J µ A α µ, (9) F µν α = µ A ν α ν A µ α + gf αβγ A µ β Aν γ. (10) Here, f αβγ are the structure constant of the Lie-Algebra and g represents the coupling constant. The Euler-Lagrange equations for massive Yang-Mills field is calculated as given below α M 2 A µ α + (α µν δ µν )(x ( 1) ) ν F µν α J µ = 0. (11) Calculations, show that equation (11) leads to α = J µ + M 2 A µ α (α µ 1) gf αβγ A µ β Aµ γ. (12) For all α µ = 1, we obtain the same equations for Yang-Mills field in the integer space-time dimensions, but for any of α µ differs from 1 we have additional term (α µ 1) gf αβγ A µ β Aµ γ, which means that the term (αµ 1) gf αβγ A µ β Aµ γ is the displacement current density J Disp. Hence, the new form of Yang-Mills field equation is written as α = J µ (α µ 1) gf αβγ A µ β Aµ γ + M 2 A µ α. (13) 4. CONCLUSIONS In this paper we have obtained the Euler-Lagrange equations of motion for field systems in fractional D dimensional space-time. As an example we have obtained the equations of motion for Maxwell s field and for massive Yang-Mills field in the fractional space. It is observed that these equations are invariant under Mandelbrot scaling, while, the equations of motion for Yang-Mills field are not invariant. For α µ = 1 we have recovered the classical case. Acknowledgements: One of the authors S.M. would like to thank Fulbright fellowship Program for financial support and also he would like to thank Southern Illinois University-Carbondale, IL, USA for hospitality.

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