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Fractional Dirac operators and deformed field theory on Clifford algebra
Chaos, Solitons and Fractals 42 (2009) 2614–2622
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Fractional Dirac operators and deformed field theory on Clifford algebra Rami Ahmad El-Nabulsi Department of Nuclear and Energy Engineering, Cheju National University, Ara-dong 1, Jeju 690-756, South Korea
a r t i c l e
i n f o
Article history: Accepted 1 April 2009
a b s t r a c t Fractional Dirac equations are constructed and fractional Dirac operators on Clifford algebra in four dimensional are introduced within the framework of the fractional calculus of variations recently introduced by the author. Many interesting consequences are revealed and discussed in some details. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction We believe today that fractional derivatives and integrals describe more accurately a complex physical system. Fractional Differential and Integral Operators (FDIO) are incorporated correctly and successfully in many classical and quantum dynamical theories where they were proved practical for exploring new fractal concepts [1–5]. They opened new branches of considerations and fills in the gaps of traditional standard calculus in ways that as of yet, no one completely assimilates or understands. Fractional calculus has been applied to many problems in physics, which ranges from classical to quantum physics. It is considered nowadays as a triumphant device for describing complex systems which are far-from equilibrium and displaying scale-invariant properties, dissipation and long-range correlations that cannot be illustrated using traditional analytic functions and ordinary differential operators. From historical point of view, despite the fact that fractional operators has been studied for over 300 years now, it has been regarded mainly as a mathematical curiosity until about 1992, where dynamical equations involving fractional derivatives and integrals were pretty much restricted to the realm of mathematics. Physicists and mathematicians have begun to explore the realm of applications of fractional calculus with ever new developments rapidly taking place in different fields of science, in particular, physics beyond the standard model, e.g. a growing body of empirical evidence supports the importance of fractional integral in quantum dynamics where important data series might be fractionally integrated. Recently, in an attempt to investigate about the characteristic properties of fractional field theories, it has been argued that the onset of large and determined quantum vacuum fluctuations, beside strong-gravity effects emerging from the short distance behavior of quantum field theory necessitate the use of fractional differentiable operators. Moreover, the macroscopic description of phenomena in terms of conventional differential and integral operators breaks down due of dynamical instabilities developed on long time scales, i.e. unstable vacuum fluctuations leading to selforganized criticality and therefore, this is one of the main arguments for using fractional differential and integral operators within the context of field theory. Owing it to their apparent scale-invariant features fractional operators provide a useful tool for dealing more accurately with complex dynamics having multiple scales, generated in the deep ultraviolet (UV) regime of quantum field theory. It is noteworthy that the main argument for dealing with fractional operators concerns the fact that it may represent an analytic framework suitable for the description of physical phenomena that are likely to arise in the TeV realm of particle physics [6]. In reality, fractional field theory is a subject of strong current interest and is a relatively new one. In 2000, Raspini has l proposed a fractional Dirac equation of order 2/3 and establish the relation between the corresponding ca -matrix algebra and the generalized Clifford algebras [7]. Raspini approach was generalized by Zàvada in 2002 and found that relativistic covariant equations generated by taking the nth root of the d’Alembert operator are fractional wave equations with an
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.002
R.A. El-Nabulsi / Chaos, Solitons and Fractals 42 (2009) 2614–2622
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inherent SU(n) symmetry [8].1 In 2004, free fractional fields were revealed by Lim and Muniandy [9,10]. The principle of local gauge invariance was applied to fractional fields in 2005. One attractive element of this approach concerns the derivation of an analytic mass formula of a non-relativistic fractal charged particle moving in a constant magnetic fractional field [11–14]. Hermann apply the concept of fractional derivative to derive a fractal Schrödinger type wave equation by the quantization of the classical non-relativistic Hamiltonian. This equation was considered by the author as an alternative tool for a suitable explanation of the charmonium spectrum usually described by a phenomenological potential. Recently, fractional dynamics of a toy model based on complex scalar fields was explored by Goldfain [15–17] and many appealing consequences emerged beyond the energy range of the standard model for particle physics by offering a beautiful and simple solution to the gauge hierarchy problem. Within the same skeleton, it was argued that a bridge between the hierarchy problem and Cantorian fractal spacetime on energy scales comparable to the Planck mass is possible. This is made successful in the context of El Naschie E-Infinity Cantorian spacetime which appears to be obviously a new framework for understanding Nature globally [18–30]. More recently, multi-dimensional fractional action-like problem of the Calculus of Variations (CoV) [31] was addressed within the context of fractional field theories. It represents the first attempt to explain two main features of supersymmetry without introducing supersymmetry. It was observed the manifestation of the ultralight Hubble mass in the Affleck–Dine mechanism and the time-variation of the gauge coupling constant. Exceedingly small correction to the standard theory in terms of ‘‘fractional weak boson mass” was obtained [32]. In fact, the study of fractional problems of the CoV [31–37] is a subject of current strong investigations. Different forms of Euler–Lagrange equations were obtained in literature depending on the form of the action and the type of fractional derivative used. The major problem with most of the fractional approaches treated in the literature is the presence of non-local fractional differential operators and the adjoint of a fractional differential operator used to describe the dynamics is not the negative of itself. Further, the derived Euler Further, the derived Euler–Lagrange equations depend on left and right fractal derivatives, even when the dynamics depend only on one of them. Other complicated problems arise during the mathematical manipulations as the appearance of a very complicated Leibniz rule (the derivative of product of functions) and the nonpresence of any fractional analogue of the chain rule. The formulation of the fractional problems of the CoV still needs more elaboration as the problem is deeply related to the fractional quantization procedure and to the presence of non-local fractional differential operators. We introduce the main notations, conventions and assumptions that underlie the remainder of the present work: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
In the notation t ? f(t), t is a dummy variable. _ q; sÞ; q; _ q; s are here dummy variables. _ q; sÞ ! f ðq; Exactly, the same function can be written, for example ðq; _ q; sÞ, the partial derivative of f with respect to the first argument is denoted by @L=@ q. _ _ q; sÞ ! f ðq; For ðq; For a given scalar field /, @ a / @/=@a. Einstein summation convention is applied throughout. The analysis is carried out entirely at the classical level. Spacetime variables and scalar fields are properly normalized as dimensionless observables. The Poincare indices are denoted by l,m = 0, 1, 2, 3. The Minkowski metric glm has signature (+, , , ) so that g00 = + 1. We work in units ⁄ = c = 1.
The definition of the fractional order derivative and integral are not unique where several definitions exist, e.g. Grunwald–Letnikov, Caputo, Weyl, Feller, Erdelyi–Kober, Riesz fractional derivatives, fractional Liouville operators which have been popularized when fractional integration is performed in the dynamical systems under study. Following our previous work, we use in this paper the left fractional Riemann–Liouville integral which is the most widely used definition of an integral of fractional order is via an integral transform defined as a
a It
f ðtÞ ¼
1 CðaÞ
Z
t
ðsÞðt sÞa1 ds;
0 < a < 1:
t0
In the present paper, we will discuss the fractional Lagrangian scalar and spinor field theory for Lagrangians with derivatives up to first order. We caution that our contribution is meant to serve as an informal introduction and not as a rigorous and comprehensive treatment of the topic discussed through this paper. The paper is organized as follows: In Section 2, we will review briefly the basic concepts of fractional calculus of variations in one, two and multi-dimensional cases. In Section 3, we will explore the four-dimensional field theory where both the scalar and the spinor fields are discussed. The fractional Dirac equations and the fractional Dirac operators were derived in the same section. Section 3 is devoted for conclusions and perspectives. 2. Fractional action-like variational approach (FALVA) In 2007, the author introduced that four-dimensional FALVA problem as follows [31]: 1
It should be pointed here that the work of Zàvada appears first as a preprint (hep-th/0003126v1) before publication of Raspini.
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Problem 2.1. Find the stationary points of the four-dimensional integral functional
Sa ½qðn; n 2 XÞ ¼ Q4
Z Z Z Z
1
i¼1
Cðai Þ
_ LðqðxÞ; qðxÞ; xÞ
XðnÞ
4 Y ðni xi Þai 1 dx;
ð1Þ
i¼1
where the admissible paths are smooth functions q : X R4 ? M satisfying giving Dirichlet boundary conditions on @ X. Here x = (x1, x2, x3, x4) is the intrinsic time vector, n = (n1, n2, n3, n4) 2 X is the observer time vector, x 2 X(n) # X with xi – ni (i = 1, 2, 3, 4), dx = dx1dx2dx3dx4 and a = (a1, a2, a3, a4), 0 < ai < 1 and ðqx1 ; qx2 ; qx3 ; qx4 ; q; x1 ; x2 ; x3 ; x4 Þ ! Lðqx1 ; qx2 ; qx3 ; qx4 ; q; x1 ; x2 ; x3 ; x4 Þis a sufficiently smooth Lagrangian function. In four-dimensions, we use the following notation: the 3D intrinsic space notations (x1, x2, x3) = (x, y, z) and the intrinsic time notation (x4) = (s) where we associated, respectively, for each xi ? ai, ni. Theorem 2.1. If q() are solutions to the previous problem, i.e. q() are critical points of the function (1), then q() satisfy the following Euler–Lagrange equations:
" ! !# 4 X d @L 1 ai @L @L þ ¼ 0: d @q @q @q s n x i i x x i i i¼1
ð2Þ
The previous arguments can be repeated mutadis-mutandis to fractional field theory for both the scalar and spinor fields [11]. 2.1. Scalar field We shall consider a fractional classical field theory in four-dimensional spacetime in which the dynamical system is described by N scalar field component /A(x), A = 1, 2, . . . , N – the dependent variables – which are functions of the independent variables ~ x ¼ ðs; x; y; zÞ. We assume that a Lagrangian density functions can be defined as a function of (x0 T, x1 x, 2 3 x y, x z), /A(x) and its derivatives up to first order. Definition 2.1.1. The fractional scalar action integral for the Lagrangian L(x, y, z, s, /A, @ x/A, @ y/A, @ z/A, @ s/A) is defined as:
S½/1 ; . . . ; /N ¼
1 CðaÞ
Z
t1
dsðt sÞa1
t0
1 CðaÞCðbÞCðcÞ
Z R3
ðn1 xÞa1 ðn2 yÞb1
ðn3 zÞc1 Lðx; y; z; s; /A ; @ x /A ; @ y /A ; @ z /A ; @ s /A Þdxdydz:
ð3Þ
From now on, we will assume for mathematical simplicity that a = b = c = a. Theorem 2.1.1. The fractional Euler–Lagrange field equation associated to the fractional scalar action (3) is
@L @L @L @ @L @ @L @ @L @ @L @l @/A @ð@ l /A Þ @/A @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ 1 @L 1 @L 1 @L 1 @L ¼ ða 1Þ þ þ þ : n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ
ð4Þ
All basic differential equations of physics have a variational structure. One important equation is the wave equation. One expects then a deformed wave equation if the previous fractional formalism is applied to the wave problem. In fact, by applying the fractional Euler–Lagrange equation (4) to the massless Lagrangian density L(/, /s, /x, /y, /z) with xÞ where ðs; ~ xÞ 2 M4 (the four-dimensional Minkowski space), the following equation certain generalized coordinate /ðs; ~ arises consequently:
/ þ ða 1Þ /s /x y /z ¼ 0; / T X Y Z
ð5Þ
denotes the d’Alembertian operator with the signature metric (, +, +, +). where Here T = s t, X = x n1, Y = y n2, Z = z n3. Corollary 2.1.1. The Noether’s symmetry theorem which asserts that ‘‘to each continuous symmetry there corresponds a quantity which is conserved” is violated. Proof. Suppose d/ is the symmetry transformation. Then there is some 4-vector Nl such that
dL ¼
@ Ns @ Nx @ Ny @ Nz þ þ þ : @s @x @y @z
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Using the fact:
dL ¼ Therefore
@L @ @L @ @L @ @L @ @L d/A þ @ s d/A þ @ x d/A þ @ y d/A þ @ z d/A : @/A @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ
@ @L @ @L @ @L @ @L @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ 1 @L 1 @L 1 @L 1 @L þða 1Þ þ þ þ d/A n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ @ @L @ @L @ @L @ @L þ @ s d/A þ @ x d/A þ @ y d/A þ @ z d/A @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ @ @L @ @L @ @L @ @L ¼ d/ þ d/ þ d/ þ d/ @ s @ð@ s /A Þ A @x @ð@ x /A Þ A @y @ð@ y /A Þ A @z @ð@ z /A Þ A
dL ¼
And hence:
@ @L @ @L @ @L @ @L d/A Ns þ d/A Nx þ d/A Ny þ d/A Nz @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ 1 @L 1 @L 1 @L 1 @L d/A : þ ða 1Þ þ þ þ n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ
If we identify the quantity: 3 X i¼0
Ji
@ @L @ @L @ @L @ @L d/A Ns þ d/A Nx þ d/A Ny þ d/A Nz ; @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ
by the Noether current associated to the symmetry d/A, then 3 X
J i ¼ ð1 aÞ
1 @L 1 @L 1 @L 1 @L þ þ þ n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ
i¼0
and consequently, the current is not conserved and Noether’s theorem is violated.
d/A ;
h
2.2. Spinor field ! Rþ be the Dirac spinor electrodynamics complex field with its Consider a smooth manifold M and let the wðxÞ : RNþ1 þ where b is the infin ! w0 ¼ eib w which are invariant under the transformation w ? w0 = eibw and w complex conjugate w, itesimal arbitrary function of x [38]. Definition 2.2.1. Given the Lagrangian L = L(x, y, z, s, wi, @ xwi, @ ywi, @ zwi, @ swi), i = 1, . . . , N where wðxÞ : RNþ1 ! Rþ is the Dirac þ spinor electrodynamics complex field. For the Lagrangian density L, the fractional spinor action integral is defined by:
S½w1 ; . . . ; wN ¼
1 CðaÞ
4 Z
t1
t0
dsðt sÞa1
Z R3
½ðn1 xÞðn2 yÞðn3 zÞa1 Lðx; y; z; s; wi ; @ x wi ; @ y wi ; @ z wi ; @ s wi Þdxdydz:
ð6Þ
Theorem 2.2.1. The fractional Euler–Lagrange field equations associated to the fractional spinor action (6) are:
@L @ @L @ @L @ @L @ @L @w @ s @ð@ s wÞ @x @ð@ x wÞ @y @ð@ y wÞ @z @ð@ z wÞ 1 @L 1 @L 1 @L 1 @L ¼ ða 1Þ þ þ þ ; n1 x @ð@ x wÞ n2 y @ð@ y wÞ n3 z @ð@ z wÞ t s @ð@ s wÞ
@L @ @L @ @L @ @L @ @L @ s @ð@ s wÞ @x @ð@ x wÞ @y @ð@ y wÞ @z @ð@ z wÞ @w 1 @L 1 @L 1 @L 1 @L ¼ ða 1Þ þ þ þ : n1 x @ð@ x wÞ n2 y @ð@ y wÞ n3 z @ð@ z wÞ t s @ð@ s wÞ
ð7Þ
ð8Þ
In the next section, we discuss the implications of these results in spinor field theory.
3. Fractional Dirac equations and fractional Dirac operators In quantum field theory, the electron is represented by the 4-component complex Dirac spinor wð~ r; sÞ. The free standard Lagrangian density is defined in quantum field theory by
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L¼w
@ @ @ @ m w; ic0 þ c1 þ c2 þ c3 @s @x @y @z
ð9Þ
where c0,1,2,3 are the 4 4 Dirac matrices and m is a parameter with the dimension of the mass. The fractional Euler–Lagrange equations (7) and (8) yield, respectively:
@w @w @w @w c1 c2 c3 c ¼ 0; i c0 þ c1 þ c2 þ c3 þða 1Þ þ þ þ 0 w þ mw @s @x @y @z n1 x n2 y n3 z t s
ð10Þ
@ @ @ @ m w ¼ 0: þ c1 þ c2 ic0 þ c3 @s @x @y @z
ð11Þ
Eq. (10) may be written like:
@ a1 @ a1 @ a1 @ a1 ¼ 0: þ c2 þ c3 w þ mw þ c1 i c0 þ þ þ þ @s t s @x n1 x @y n2 y @z n3 z
ð12Þ
It is obvious that the fractional differential operator
c0
@ a1 @ a1 @ a1 @ a1 þ þ þ þ þ c2 þ c3 þ c1 @s t s @x n1 x @y n2 y @z n3 z @ @ @ @ c0 c1 c2 c3 þ c1 þ c2 þ ; c0 þ c3 þ ða 1Þ þ þ @s @x @y @z t s n1 x n2 y n3 z |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
standard
ð13Þ ð14Þ
fractional
does not appear in Eq. (12). In other words, fractional differential operator operates only on the complex conjugate scalar field. The Dirac operator is therefore modified. Definition 3.1. The fractional Dirac operator D0
D0
@ a1 @ a1 @ a1 @ a1 þ þ þ þ þ c2 þ c3 : þ c1 @s t s @x n1 x @y n2 y @z n3 z
ð15Þ
Moreover, the combination of Eqs. (10) and (11) do not yield the correct dispersion relation psps + pxpx + pypy + pzpz = m2 for a given scalar field /. ~ p being the momentum. In order to remedy the problem we propose the following: Proposition 3.1. The modified fractional spinor Lagrangian is defined by
c @ s þ c @ x þ c @ y þ c @ z Þw mww þ ða 1Þ La ðwÞ ¼ wði 0 1 2 3
c0 ts
þ
c1 n1 x
þ
c2 n2 y
þ
c3 n3 z
iw:
ð16Þ
The second term on the right of Eq. (16) is defined as the fractional perturbed spinor Lagrangian part. Corollary 3.1. The fractional Dirac equations corresponding to the Lagrangian La(w) are:
@ a1 @ a1 @ a1 @ a1 ¼ 0; i c0 þ þ þ þ þ c2 þ c3 w þ mw þ c1 @s t s @x n1 x @y n2 y @z n3 z
ð17Þ
@ a1 @ a1 @ a1 @ a1 þ i c0 þ þ þ þ c2 þ c3 w mw ¼ 0: þ c1 @s t s @x n1 x @y n2 y @z n3 z
ð18Þ
Proof. In fact, it is easy to prove from the definition of La(w) that:
@L ¼ mw; @w @L @ @ @ @ c1 c2 c3 c0 c þ c c c a 1Þ ¼ i þ þ w mw þ ið þ þ þ ; 0 1 2 3 @s @x @y @z n1 x n2 y n3 z t s @w @ @L @ @L @ @L @ @L @w @w @w @w þ ic1 þ ic2 þ ic3 ; þ þ þ ¼ ic0 @ s @ð@ s wÞ @x @ð@ x wÞ @y @ð@ y wÞ @z @ð@ z wÞ @s @x @y @z @ @L @ @L @ @L @ @L þ @x @ð@ x wÞ þ @y @ð@ y wÞ þ @z @ð@ z wÞ ¼ 0; @ s @ð@ s wÞ 1 @L 1 @L 1 @L 1 @L c1 þ c2 þ c3 þ c0 ; þ þ þ ¼ iw n1 x @ð@ x wÞ n2 y @ð@ y wÞ n3 z @ð@ z wÞ t s @ð@ s wÞ n1 x n2 y n3 z t s 1 @L 1 @L 1 @L 1 @L þ n2 y @ð@ y wÞ þ n3 z @ð@ z wÞ þ t s @ð@ s wÞ ¼ 0; n1 x @ð@ x wÞ and thus Eqs. (17) and (18) are easily derived. h
R.A. El-Nabulsi / Chaos, Solitons and Fractals 42 (2009) 2614–2622
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Corollary 3.2. The current 3 X
c0 w þ w c1 w þ w c2 w þ w c3 w; Ji Jl , w
ð19Þ
i¼0
is not conserved. and adding give: Proof. Multiplying Eq. (17) from the right by w and Eq. (18) from the left by w
c w c w c w @ @ @ @ wc w w w w ðwc0 wÞ þ ðw c1 wÞ þ ðw c2 wÞ þ ðw c3 wÞ ¼ 2ð1 aÞ 0 þ 1 þ 2 þ 3 ; @s @x @y @z t s n1 x n2 y n3 z and thus conservation occurs for a = 1 which corresponds to the standard case. Now that the fractional Dirac equations are derived, we can return to the fractional Dirac operator on C 1 ðRn ; CN Þ. It is worth-mentioning that D20
2 2 2 2 2 @2 2 a1 2a1 @ 2 @ 2 a1 2 a1 @ 2 @ 2 a1 þ c þ 2 c þ c þ c þ 2 c c þ c þ 0 0 1 1 1 2 2 t s t s @s n1 x n1 x @x n2 y @ s2 @x2 @y2 2 2 a1 @ @ a1 a1 @ @ @ ða 1Þ ða 1Þ þ 2c22 þ 2c23 þ c23 2 þ c23 þ 2c0 c1 þ 2c0 c1 n2 y @y n3 z n3 z @z @ s @x t s n1 x @z
D20
þ 2c0 c2
@ @ ða 1Þ ða 1Þ @ @ ða 1Þ ða 1Þ þ 2c0 c2 þ 2c0 c3 þ 2c0 c3 @ s @y t s n2 y @ s @z t s n3 z
@ @ ða 1Þ ða 1Þ @ @ ða 1Þ ða 1Þ @ @ þ 2c1 c2 þ 2c1 c3 þ 2c1 c3 þ 2c2 c3 @x @y n1 x n2 y @x @z n1 x n2 z @y @z ! ða 1Þ ða 1Þ @2 @2 @2 @2 1 @ 1 @ 1 @ 1 @ þ 2c2 c3 2 2 2 þ 2ða 1Þ ;¼ 2 n2 y n3 z t s @ s n1 x @x n2 y @y n3 z @z @ s @x @y @z " # 2 2 2 2 1 1 1 1 þ ða 1Þ2 x n1 y n2 z n3 st @ @ @ @ @ @ @ @ @ @ þ 2 c1 c2 þ c1 c3 þ c2 c3 þ 2c0 c1 þ c2 þ c3 @x @y @z @ s @x @y @x @z @y @z c1 c2 c1 c3 c2 c3 ða 1Þ2 c1 c c þ 2ða 1Þ2 þ þ þ 2c0 þ 2 þ 3 ; ð20Þ ðn1 xÞðn2 yÞ ðn1 xÞðn3 zÞ ðn2 yÞðn3 zÞ ðt sÞ n1 x n2 y n3 z þ 2c1 c2
where we have used the fact that c20 ¼ 1 and c21 ¼ c22 ¼ c23 ¼ 1 [39]. If in addition {ci, cj} = 2dij which is called the Clifford algebra, then Eq. (20) may be simplified to:
1 @ 1 @ 1 @ 1 @ t s @ s n1 x @x n2 y @y n3 z @z " 2 2 2 2 # 1 1 1 1 þ ða 1Þ2 x n1 y n2 z n3 st c1 c2 c1 c3 c2 c3 þ 2ða 1Þ2 þ þ ðn1 xÞðn2 yÞ ðn1 xÞðn3 zÞ ðn2 yÞðn3 zÞ ða 1Þ2 c1 c2 c3 þ 2c0 þ þ ; ðt sÞ n1 x n2 y n3 z
D20
which is the sum of the Laplacian D and the fractional parts.
ð21Þ
h
Definition 3.2 [40]. Let c : Cln ! EndðCN Þ be a representation of the Clifford algebra of Cln on CN . Let {xi} be the standard coordinates on Rn and {ei} the standard basis. The fractional Dirac operator D0
D0
n X i¼1
X n @f 1 a @f 1 a ¼ ; cðei Þ þ ei þ @xi @xi Yi Yi i¼1
ð22Þ
for some function f. HereY1 = n1 x, Y2 = n2 y, Y3 = n3 z, Y4 = t s, etc., and ‘‘” is another way of representing the action of c(ei) – the Clifford multiplication.
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Notes that the fractional Dirac operator over flat space defined by Eq. (22) is deviated from the standard definition by the presence of the fractional term and thus, one may rewrite Eq. (22) like: n X
D0
ei
i¼1
n n X X @f þ ð1 aÞ ei Y 1 Df þ ð1 aÞ ei Y 1 i i : @xi i¼1 i¼1
ð23Þ
Proposition 3.2. The fractional Dirac operator D0
D0
ei
i¼1
@f @xi
ð24Þ
; a
where
@f @xi
a
@f 1 a þ ; @xi Yi
ð25Þ
is the fractional differential operator. On Euclidean space En, Eq. (23) holds. Further, Eq. (23) may be compared to the standard Dirac operator derived on curved spaces expressed according to the standard (a = 1) Bochner–Weitzenböck formula [41]:
D20
X k
R @ 2k þ ; 4
ð26Þ
P where k @ 2k is the flat covariant Laplacian on spinor fields and R the scalar curvature [41]. The Ricci scalar curvature may be identified to the fractional part and thus R ? 0 for a = 1 or for late time s 1 and larger distances x, y, z 1. This is to say that the fractional classical part may be identified to the scalar curvature in Riemann geometry. From Eq. (26), one expects that the Bochner–Weitzenböck formula will be modified. For this, assume for the moment a ~ the induced connection on E. The standard conRiemann manifold M and let r be the Levi-Civita connection on TM and r ~ r ~ is defined by nection Laplacian r
~ r ~s ¼ r
X
~ 2 s; r ei ;ei
ð27Þ
i
~ is the formal adjoint of r ~ with where s 2 C(E), C(E) are locally functions from U ? M, U is a coordinate neighborhood M. r 1 N n respect to the metric (,). In fact, C(E) is the correct analogue of C ðR ; C Þ. Then, we have:
D20
X
~ e Þ ðr ~e Þ ¼ e i e j ðr i a j a
i;j
X
~ e ;e Þ ; ei ej ðr i j a
ð28Þ
i;j
~e Þ r ~ e ;e þ ð1 aÞY 1 is the fractal Levi-Civita connection. where ðr i;j i;j a i j Proposition 3.3. Let D0
~ r ~ þ R þ ð1 aÞ D20
X
ei ej Y 1 i;j :
ð29Þ
i;j
Proof. In fact:
D20
X
h i X X ~ e e þ ð1 aÞY 1 ¼ ~ e þ ð1 aÞ ei ej r ei ej r ei ej Y 1 i;j i;j ; i; j i;j
i;j
¼
X i
~ e ;e þ r i i
X
i;j
~ e ;e þ ð1 aÞ ei ej R i j
i
X
i;j
ei ej Y 1 i;j ;
i;j
~ ; property: where we have used the induced Riemann curvature R
~2 r ~Y r ~X r ~2 ¼ r ~ Xr ~ Yr ~ ½X;Y R ~ X;Y ; r X;Y Y;X with [X,Y] = rXY rYX. Making use of the fact
X i
~ e ;e ¼ ei ej R i j
1 X 1X R Rliij ei ej ei þ Rljij ei ej ej el ¼ Ricjj ; 8 i;j;l 4 j 4
gives the modified or fractional Bochner-Weitzenböck formula in curved space:
~ r ~ þ R þ ð1 aÞ D20
X i;j
2 ei ej Y 1 i;j ¼ D þ ð1 aÞ
X i;j
ei ej Y 1 i;j ;
R.A. El-Nabulsi / Chaos, Solitons and Fractals 42 (2009) 2614–2622
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Definition 3.3. Let D0
~ þ R; ~ r D20
ð30Þ
and
Ra ¼ 4ða 1Þ
X
ei ej Y 1 i;j :
ð31Þ
i;j
~ r ~. If for instance, R Ra , R ! 0 and D20
Oldham KB, Spanier J. The fractional calculus. New York, London: Acad. Press; 1974. Samko S, Kilbas A, Marichev O. Fractional integrals and derivatives: theory and applications. New York: Gordon and Breach; 1993. Miller KS, Ross B. An Introduction to the fractional calculus and fractional differential equations. New York: John Wiley & Sons Inc.; 1993. Podlubny I. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. New York, London: Academic Press; 1999. Hilfer R, editor. Applications of fractional calculus in physics. New Jersey, London, elmh6b Hong Kong: Word Scientific Publishing Co.; 2000. Goldfain E. Fractional dynamics and the TeV regime of field theory particle. Commun Nonlin Sci Numer Simul 2008;13(3):666. Raspini A. Simple solutions of the fractional Dirac equation of order 2/3. Phys Scr 2000;64:20. Zàvada PJ. Relativistic wave equations with fractional derivatives and pseudo-differential operators. J Appl Math 2002;2(4):163. Lim C, Muniandy SV. On some possible generalizations of fractional Brownian motion. Phys Lett A 2004;266(2):140–5. Lim C, Muniandy SV. Stochastic quantization of nonlocal fields. Phys Lett A 2004;324(5–6):396. Hermann R. Gauge invariance in fractional field theories. arXiv:0708.2262. Hermann V. Continuous differential operators and a new interpretation of the charmonium spectrum. arXiv:nucl-th/0508033. Herrmann R. Fractional spin – a property of particles described with a fractional Schrödinger equation. arXiv:0805.3434 [physics.gen-ph]. Hermann R. Continuous differential operators and a new interpretation of the charmonium spectrum. arXiv:nucl-th/0508033. Goldfain E. Fractional dynamics and the standard model for particle physics. Commun Nonlin Sci Numer Simul 2008;13:1397. Goldfain E. Complexity in quantum field theory and physics beyond the standard model. Chaos, Solitons & Fractals 2006;28:913. Goldfain E. Fractional dynamics, Cantorian spacetime and the gauge hierarchy problem. Chaos, Solitons & Fractals 2004;22:513. El Naschie MS. A guide to the mathematics of E-Infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955. El Naschie MS. Hilbert space, the number of Higgs particles and the quantum two-slit experiment. Chaos, Solitons & Fractals 2006;27(1):9. El Naschie MS. Hilbert, Fock and Cantorian spaces in the quantum two-slit gedanken experiment. Chaos, Solitons & Fractals 2006;27(1):39. El Naschie MS. On the nature of complex time, diffusion and the two slit experiment. Chaos, Solitons & Fractals 1995;5(6):1031. El Naschie MS. On the uncertainty of Cantorian geometry and the two slit experiment. Chaos, Solitons & Fractals 1998;9(3):517. El-Naschie MS. Fractal gravity and symmetry breaking in a hierarchical Cantorian space. Chaos, Solitons & Fractals 1997;8(11):1865. El Naschie MS. On’t Hooft dimensional regularization in E-Infinity space. Chaos, Solitons & Fractals 2001;12(5):851. El Naschie MS. Quantum mechanics and the possibility of Cantorian space-time. Chaos, Solitons & Fractals 1992;1(5):485. El Naschie MS. On the cohomology and instantons number of E-Infinity Cantorian space-time. Chaos, Solitons & Fractals 2005;26(1):13. El Naschie MS. A review of E Infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19(1):209. El Naschie MS. On the unification of the fundamental forces and complex time in the e(1) space. Chaos, Solitons & Fractals 2000;11:1149. El Naschie MS. On an eleven dimensional E-Infinity fractal spacetime theory. Int J Nonlin Sci Numer Simul 2006;7(4):407. El Naschie MS. The ‘‘Discrete” charm of certain eleven dimensional spacetime theories. Int J Nonlin Sci Numer Simul 2006;7(4):477. El-Nabulsi RA, Torres DFM. Fractional action-like variational problems. J Math Phys 2008;49:053521. and references therein. El-Nabulsi RA. Fractional dynamics, fractional weak bosons masses and physics beyond the standard model. Chaos, Solitons & Fractals, in press. El-Nabulsi RA. Fractional field theories from multi-dimensional fractional variational problems. Int J Geom Methods Mod Phys 2008;5(6):863. and references therein.
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[34] El-Nabulsi RA. Fractional Lagrangian formulation of general relativity and emergence of complex, spinoriel and noncommutative gravity. Int J Geom Methods Mod Phys 2009;6(1):25. [35] El-Nabulsi RA. Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2008.10.022. [36] El-Nabulsi RA. On the fractional minimal length Heisenberg–Weyl uncertainty relation from fractional Riccati generalized momentum operator. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2008.10.031. [37] El-Nabulsi RA, Torres DFM. Necessary optimality condition for fractional action-like variational approach with Riemann–Liouville derivatives of order (alfa,betta). Math Methods Appl Sci 2007;30(15):1931. [38] Ryder LH. Quantum field theory. Cambridge University Press; 1985. [39] Cheng TP, Li LF. Gauge theory of elementary particle physics. Oxford: Clarendon Press; 1984. [40] Jiang W. Dirac operators. Part III of the Cambridge mathematical tripos. [41] Esposito G. Dirac operators and spectral geometry. Cambridge Lecture Notes in Physics. Cambridge University Press; 1998.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Fractional Dirac operators and deformed field theory on Clifford algebra Rami Ahmad El-Nabulsi Department of Nuclear and Energy Engineering, Cheju National University, Ara-dong 1, Jeju 690-756, South Korea
a r t i c l e
i n f o
Article history: Accepted 1 April 2009
a b s t r a c t Fractional Dirac equations are constructed and fractional Dirac operators on Clifford algebra in four dimensional are introduced within the framework of the fractional calculus of variations recently introduced by the author. Many interesting consequences are revealed and discussed in some details. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction We believe today that fractional derivatives and integrals describe more accurately a complex physical system. Fractional Differential and Integral Operators (FDIO) are incorporated correctly and successfully in many classical and quantum dynamical theories where they were proved practical for exploring new fractal concepts [1–5]. They opened new branches of considerations and fills in the gaps of traditional standard calculus in ways that as of yet, no one completely assimilates or understands. Fractional calculus has been applied to many problems in physics, which ranges from classical to quantum physics. It is considered nowadays as a triumphant device for describing complex systems which are far-from equilibrium and displaying scale-invariant properties, dissipation and long-range correlations that cannot be illustrated using traditional analytic functions and ordinary differential operators. From historical point of view, despite the fact that fractional operators has been studied for over 300 years now, it has been regarded mainly as a mathematical curiosity until about 1992, where dynamical equations involving fractional derivatives and integrals were pretty much restricted to the realm of mathematics. Physicists and mathematicians have begun to explore the realm of applications of fractional calculus with ever new developments rapidly taking place in different fields of science, in particular, physics beyond the standard model, e.g. a growing body of empirical evidence supports the importance of fractional integral in quantum dynamics where important data series might be fractionally integrated. Recently, in an attempt to investigate about the characteristic properties of fractional field theories, it has been argued that the onset of large and determined quantum vacuum fluctuations, beside strong-gravity effects emerging from the short distance behavior of quantum field theory necessitate the use of fractional differentiable operators. Moreover, the macroscopic description of phenomena in terms of conventional differential and integral operators breaks down due of dynamical instabilities developed on long time scales, i.e. unstable vacuum fluctuations leading to selforganized criticality and therefore, this is one of the main arguments for using fractional differential and integral operators within the context of field theory. Owing it to their apparent scale-invariant features fractional operators provide a useful tool for dealing more accurately with complex dynamics having multiple scales, generated in the deep ultraviolet (UV) regime of quantum field theory. It is noteworthy that the main argument for dealing with fractional operators concerns the fact that it may represent an analytic framework suitable for the description of physical phenomena that are likely to arise in the TeV realm of particle physics [6]. In reality, fractional field theory is a subject of strong current interest and is a relatively new one. In 2000, Raspini has l proposed a fractional Dirac equation of order 2/3 and establish the relation between the corresponding ca -matrix algebra and the generalized Clifford algebras [7]. Raspini approach was generalized by Zàvada in 2002 and found that relativistic covariant equations generated by taking the nth root of the d’Alembert operator are fractional wave equations with an
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.002
R.A. El-Nabulsi / Chaos, Solitons and Fractals 42 (2009) 2614–2622
2615
inherent SU(n) symmetry [8].1 In 2004, free fractional fields were revealed by Lim and Muniandy [9,10]. The principle of local gauge invariance was applied to fractional fields in 2005. One attractive element of this approach concerns the derivation of an analytic mass formula of a non-relativistic fractal charged particle moving in a constant magnetic fractional field [11–14]. Hermann apply the concept of fractional derivative to derive a fractal Schrödinger type wave equation by the quantization of the classical non-relativistic Hamiltonian. This equation was considered by the author as an alternative tool for a suitable explanation of the charmonium spectrum usually described by a phenomenological potential. Recently, fractional dynamics of a toy model based on complex scalar fields was explored by Goldfain [15–17] and many appealing consequences emerged beyond the energy range of the standard model for particle physics by offering a beautiful and simple solution to the gauge hierarchy problem. Within the same skeleton, it was argued that a bridge between the hierarchy problem and Cantorian fractal spacetime on energy scales comparable to the Planck mass is possible. This is made successful in the context of El Naschie E-Infinity Cantorian spacetime which appears to be obviously a new framework for understanding Nature globally [18–30]. More recently, multi-dimensional fractional action-like problem of the Calculus of Variations (CoV) [31] was addressed within the context of fractional field theories. It represents the first attempt to explain two main features of supersymmetry without introducing supersymmetry. It was observed the manifestation of the ultralight Hubble mass in the Affleck–Dine mechanism and the time-variation of the gauge coupling constant. Exceedingly small correction to the standard theory in terms of ‘‘fractional weak boson mass” was obtained [32]. In fact, the study of fractional problems of the CoV [31–37] is a subject of current strong investigations. Different forms of Euler–Lagrange equations were obtained in literature depending on the form of the action and the type of fractional derivative used. The major problem with most of the fractional approaches treated in the literature is the presence of non-local fractional differential operators and the adjoint of a fractional differential operator used to describe the dynamics is not the negative of itself. Further, the derived Euler Further, the derived Euler–Lagrange equations depend on left and right fractal derivatives, even when the dynamics depend only on one of them. Other complicated problems arise during the mathematical manipulations as the appearance of a very complicated Leibniz rule (the derivative of product of functions) and the nonpresence of any fractional analogue of the chain rule. The formulation of the fractional problems of the CoV still needs more elaboration as the problem is deeply related to the fractional quantization procedure and to the presence of non-local fractional differential operators. We introduce the main notations, conventions and assumptions that underlie the remainder of the present work: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
In the notation t ? f(t), t is a dummy variable. _ q; sÞ; q; _ q; s are here dummy variables. _ q; sÞ ! f ðq; Exactly, the same function can be written, for example ðq; _ q; sÞ, the partial derivative of f with respect to the first argument is denoted by @L=@ q. _ _ q; sÞ ! f ðq; For ðq; For a given scalar field /, @ a / @/=@a. Einstein summation convention is applied throughout. The analysis is carried out entirely at the classical level. Spacetime variables and scalar fields are properly normalized as dimensionless observables. The Poincare indices are denoted by l,m = 0, 1, 2, 3. The Minkowski metric glm has signature (+, , , ) so that g00 = + 1. We work in units ⁄ = c = 1.
The definition of the fractional order derivative and integral are not unique where several definitions exist, e.g. Grunwald–Letnikov, Caputo, Weyl, Feller, Erdelyi–Kober, Riesz fractional derivatives, fractional Liouville operators which have been popularized when fractional integration is performed in the dynamical systems under study. Following our previous work, we use in this paper the left fractional Riemann–Liouville integral which is the most widely used definition of an integral of fractional order is via an integral transform defined as a
a It
f ðtÞ ¼
1 CðaÞ
Z
t
ðsÞðt sÞa1 ds;
0 < a < 1:
t0
In the present paper, we will discuss the fractional Lagrangian scalar and spinor field theory for Lagrangians with derivatives up to first order. We caution that our contribution is meant to serve as an informal introduction and not as a rigorous and comprehensive treatment of the topic discussed through this paper. The paper is organized as follows: In Section 2, we will review briefly the basic concepts of fractional calculus of variations in one, two and multi-dimensional cases. In Section 3, we will explore the four-dimensional field theory where both the scalar and the spinor fields are discussed. The fractional Dirac equations and the fractional Dirac operators were derived in the same section. Section 3 is devoted for conclusions and perspectives. 2. Fractional action-like variational approach (FALVA) In 2007, the author introduced that four-dimensional FALVA problem as follows [31]: 1
It should be pointed here that the work of Zàvada appears first as a preprint (hep-th/0003126v1) before publication of Raspini.
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Problem 2.1. Find the stationary points of the four-dimensional integral functional
Sa ½qðn; n 2 XÞ ¼ Q4
Z Z Z Z
1
i¼1
Cðai Þ
_ LðqðxÞ; qðxÞ; xÞ
XðnÞ
4 Y ðni xi Þai 1 dx;
ð1Þ
i¼1
where the admissible paths are smooth functions q : X R4 ? M satisfying giving Dirichlet boundary conditions on @ X. Here x = (x1, x2, x3, x4) is the intrinsic time vector, n = (n1, n2, n3, n4) 2 X is the observer time vector, x 2 X(n) # X with xi – ni (i = 1, 2, 3, 4), dx = dx1dx2dx3dx4 and a = (a1, a2, a3, a4), 0 < ai < 1 and ðqx1 ; qx2 ; qx3 ; qx4 ; q; x1 ; x2 ; x3 ; x4 Þ ! Lðqx1 ; qx2 ; qx3 ; qx4 ; q; x1 ; x2 ; x3 ; x4 Þis a sufficiently smooth Lagrangian function. In four-dimensions, we use the following notation: the 3D intrinsic space notations (x1, x2, x3) = (x, y, z) and the intrinsic time notation (x4) = (s) where we associated, respectively, for each xi ? ai, ni. Theorem 2.1. If q() are solutions to the previous problem, i.e. q() are critical points of the function (1), then q() satisfy the following Euler–Lagrange equations:
" ! !# 4 X d @L 1 ai @L @L þ ¼ 0: d @q @q @q s n x i i x x i i i¼1
ð2Þ
The previous arguments can be repeated mutadis-mutandis to fractional field theory for both the scalar and spinor fields [11]. 2.1. Scalar field We shall consider a fractional classical field theory in four-dimensional spacetime in which the dynamical system is described by N scalar field component /A(x), A = 1, 2, . . . , N – the dependent variables – which are functions of the independent variables ~ x ¼ ðs; x; y; zÞ. We assume that a Lagrangian density functions can be defined as a function of (x0 T, x1 x, 2 3 x y, x z), /A(x) and its derivatives up to first order. Definition 2.1.1. The fractional scalar action integral for the Lagrangian L(x, y, z, s, /A, @ x/A, @ y/A, @ z/A, @ s/A) is defined as:
S½/1 ; . . . ; /N ¼
1 CðaÞ
Z
t1
dsðt sÞa1
t0
1 CðaÞCðbÞCðcÞ
Z R3
ðn1 xÞa1 ðn2 yÞb1
ðn3 zÞc1 Lðx; y; z; s; /A ; @ x /A ; @ y /A ; @ z /A ; @ s /A Þdxdydz:
ð3Þ
From now on, we will assume for mathematical simplicity that a = b = c = a. Theorem 2.1.1. The fractional Euler–Lagrange field equation associated to the fractional scalar action (3) is
@L @L @L @ @L @ @L @ @L @ @L @l @/A @ð@ l /A Þ @/A @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ 1 @L 1 @L 1 @L 1 @L ¼ ða 1Þ þ þ þ : n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ
ð4Þ
All basic differential equations of physics have a variational structure. One important equation is the wave equation. One expects then a deformed wave equation if the previous fractional formalism is applied to the wave problem. In fact, by applying the fractional Euler–Lagrange equation (4) to the massless Lagrangian density L(/, /s, /x, /y, /z) with xÞ where ðs; ~ xÞ 2 M4 (the four-dimensional Minkowski space), the following equation certain generalized coordinate /ðs; ~ arises consequently:
/ þ ða 1Þ /s /x y /z ¼ 0; / T X Y Z
ð5Þ
denotes the d’Alembertian operator with the signature metric (, +, +, +). where Here T = s t, X = x n1, Y = y n2, Z = z n3. Corollary 2.1.1. The Noether’s symmetry theorem which asserts that ‘‘to each continuous symmetry there corresponds a quantity which is conserved” is violated. Proof. Suppose d/ is the symmetry transformation. Then there is some 4-vector Nl such that
dL ¼
@ Ns @ Nx @ Ny @ Nz þ þ þ : @s @x @y @z
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Using the fact:
dL ¼ Therefore
@L @ @L @ @L @ @L @ @L d/A þ @ s d/A þ @ x d/A þ @ y d/A þ @ z d/A : @/A @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ
@ @L @ @L @ @L @ @L @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ 1 @L 1 @L 1 @L 1 @L þða 1Þ þ þ þ d/A n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ @ @L @ @L @ @L @ @L þ @ s d/A þ @ x d/A þ @ y d/A þ @ z d/A @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ @ @L @ @L @ @L @ @L ¼ d/ þ d/ þ d/ þ d/ @ s @ð@ s /A Þ A @x @ð@ x /A Þ A @y @ð@ y /A Þ A @z @ð@ z /A Þ A
dL ¼
And hence:
@ @L @ @L @ @L @ @L d/A Ns þ d/A Nx þ d/A Ny þ d/A Nz @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ 1 @L 1 @L 1 @L 1 @L d/A : þ ða 1Þ þ þ þ n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ
If we identify the quantity: 3 X i¼0
Ji
@ @L @ @L @ @L @ @L d/A Ns þ d/A Nx þ d/A Ny þ d/A Nz ; @ s @ð@ s /A Þ @x @ð@ x /A Þ @y @ð@ y /A Þ @z @ð@ z /A Þ
by the Noether current associated to the symmetry d/A, then 3 X
J i ¼ ð1 aÞ
1 @L 1 @L 1 @L 1 @L þ þ þ n1 x @ð@ x /A Þ n2 y @ð@ y /A Þ n3 z @ð@ z /A Þ t s @ð@ s /A Þ
i¼0
and consequently, the current is not conserved and Noether’s theorem is violated.
d/A ;
h
2.2. Spinor field ! Rþ be the Dirac spinor electrodynamics complex field with its Consider a smooth manifold M and let the wðxÞ : RNþ1 þ where b is the infin ! w0 ¼ eib w which are invariant under the transformation w ? w0 = eibw and w complex conjugate w, itesimal arbitrary function of x [38]. Definition 2.2.1. Given the Lagrangian L = L(x, y, z, s, wi, @ xwi, @ ywi, @ zwi, @ swi), i = 1, . . . , N where wðxÞ : RNþ1 ! Rþ is the Dirac þ spinor electrodynamics complex field. For the Lagrangian density L, the fractional spinor action integral is defined by:
S½w1 ; . . . ; wN ¼
1 CðaÞ
4 Z
t1
t0
dsðt sÞa1
Z R3
½ðn1 xÞðn2 yÞðn3 zÞa1 Lðx; y; z; s; wi ; @ x wi ; @ y wi ; @ z wi ; @ s wi Þdxdydz:
ð6Þ
Theorem 2.2.1. The fractional Euler–Lagrange field equations associated to the fractional spinor action (6) are:
@L @ @L @ @L @ @L @ @L @w @ s @ð@ s wÞ @x @ð@ x wÞ @y @ð@ y wÞ @z @ð@ z wÞ 1 @L 1 @L 1 @L 1 @L ¼ ða 1Þ þ þ þ ; n1 x @ð@ x wÞ n2 y @ð@ y wÞ n3 z @ð@ z wÞ t s @ð@ s wÞ
@L @ @L @ @L @ @L @ @L @ s @ð@ s wÞ @x @ð@ x wÞ @y @ð@ y wÞ @z @ð@ z wÞ @w 1 @L 1 @L 1 @L 1 @L ¼ ða 1Þ þ þ þ : n1 x @ð@ x wÞ n2 y @ð@ y wÞ n3 z @ð@ z wÞ t s @ð@ s wÞ
ð7Þ
ð8Þ
In the next section, we discuss the implications of these results in spinor field theory.
3. Fractional Dirac equations and fractional Dirac operators In quantum field theory, the electron is represented by the 4-component complex Dirac spinor wð~ r; sÞ. The free standard Lagrangian density is defined in quantum field theory by
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R.A. El-Nabulsi / Chaos, Solitons and Fractals 42 (2009) 2614–2622
L¼w
@ @ @ @ m w; ic0 þ c1 þ c2 þ c3 @s @x @y @z
ð9Þ
where c0,1,2,3 are the 4 4 Dirac matrices and m is a parameter with the dimension of the mass. The fractional Euler–Lagrange equations (7) and (8) yield, respectively:
@w @w @w @w c1 c2 c3 c ¼ 0; i c0 þ c1 þ c2 þ c3 þða 1Þ þ þ þ 0 w þ mw @s @x @y @z n1 x n2 y n3 z t s
ð10Þ
@ @ @ @ m w ¼ 0: þ c1 þ c2 ic0 þ c3 @s @x @y @z
ð11Þ
Eq. (10) may be written like:
@ a1 @ a1 @ a1 @ a1 ¼ 0: þ c2 þ c3 w þ mw þ c1 i c0 þ þ þ þ @s t s @x n1 x @y n2 y @z n3 z
ð12Þ
It is obvious that the fractional differential operator
c0
@ a1 @ a1 @ a1 @ a1 þ þ þ þ þ c2 þ c3 þ c1 @s t s @x n1 x @y n2 y @z n3 z @ @ @ @ c0 c1 c2 c3 þ c1 þ c2 þ ; c0 þ c3 þ ða 1Þ þ þ @s @x @y @z t s n1 x n2 y n3 z |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
standard
ð13Þ ð14Þ
fractional
does not appear in Eq. (12). In other words, fractional differential operator operates only on the complex conjugate scalar field. The Dirac operator is therefore modified. Definition 3.1. The fractional Dirac operator D0
D0
@ a1 @ a1 @ a1 @ a1 þ þ þ þ þ c2 þ c3 : þ c1 @s t s @x n1 x @y n2 y @z n3 z
ð15Þ
Moreover, the combination of Eqs. (10) and (11) do not yield the correct dispersion relation psps + pxpx + pypy + pzpz = m2 for a given scalar field /. ~ p being the momentum. In order to remedy the problem we propose the following: Proposition 3.1. The modified fractional spinor Lagrangian is defined by
c @ s þ c @ x þ c @ y þ c @ z Þw mww þ ða 1Þ La ðwÞ ¼ wði 0 1 2 3
c0 ts
þ
c1 n1 x
þ
c2 n2 y
þ
c3 n3 z
iw:
ð16Þ
The second term on the right of Eq. (16) is defined as the fractional perturbed spinor Lagrangian part. Corollary 3.1. The fractional Dirac equations corresponding to the Lagrangian La(w) are:
@ a1 @ a1 @ a1 @ a1 ¼ 0; i c0 þ þ þ þ þ c2 þ c3 w þ mw þ c1 @s t s @x n1 x @y n2 y @z n3 z
ð17Þ
@ a1 @ a1 @ a1 @ a1 þ i c0 þ þ þ þ c2 þ c3 w mw ¼ 0: þ c1 @s t s @x n1 x @y n2 y @z n3 z
ð18Þ
Proof. In fact, it is easy to prove from the definition of La(w) that:
@L ¼ mw; @w @L @ @ @ @ c1 c2 c3 c0 c þ c c c a 1Þ ¼ i þ þ w mw þ ið þ þ þ ; 0 1 2 3 @s @x @y @z n1 x n2 y n3 z t s @w @ @L @ @L @ @L @ @L @w @w @w @w þ ic1 þ ic2 þ ic3 ; þ þ þ ¼ ic0 @ s @ð@ s wÞ @x @ð@ x wÞ @y @ð@ y wÞ @z @ð@ z wÞ @s @x @y @z @ @L @ @L @ @L @ @L þ @x @ð@ x wÞ þ @y @ð@ y wÞ þ @z @ð@ z wÞ ¼ 0; @ s @ð@ s wÞ 1 @L 1 @L 1 @L 1 @L c1 þ c2 þ c3 þ c0 ; þ þ þ ¼ iw n1 x @ð@ x wÞ n2 y @ð@ y wÞ n3 z @ð@ z wÞ t s @ð@ s wÞ n1 x n2 y n3 z t s 1 @L 1 @L 1 @L 1 @L þ n2 y @ð@ y wÞ þ n3 z @ð@ z wÞ þ t s @ð@ s wÞ ¼ 0; n1 x @ð@ x wÞ and thus Eqs. (17) and (18) are easily derived. h
R.A. El-Nabulsi / Chaos, Solitons and Fractals 42 (2009) 2614–2622
2619
Corollary 3.2. The current 3 X
c0 w þ w c1 w þ w c2 w þ w c3 w; Ji Jl , w
ð19Þ
i¼0
is not conserved. and adding give: Proof. Multiplying Eq. (17) from the right by w and Eq. (18) from the left by w
c w c w c w @ @ @ @ wc w w w w ðwc0 wÞ þ ðw c1 wÞ þ ðw c2 wÞ þ ðw c3 wÞ ¼ 2ð1 aÞ 0 þ 1 þ 2 þ 3 ; @s @x @y @z t s n1 x n2 y n3 z and thus conservation occurs for a = 1 which corresponds to the standard case. Now that the fractional Dirac equations are derived, we can return to the fractional Dirac operator on C 1 ðRn ; CN Þ. It is worth-mentioning that D20
2 2 2 2 2 @2 2 a1 2a1 @ 2 @ 2 a1 2 a1 @ 2 @ 2 a1 þ c þ 2 c þ c þ c þ 2 c c þ c þ 0 0 1 1 1 2 2 t s t s @s n1 x n1 x @x n2 y @ s2 @x2 @y2 2 2 a1 @ @ a1 a1 @ @ @ ða 1Þ ða 1Þ þ 2c22 þ 2c23 þ c23 2 þ c23 þ 2c0 c1 þ 2c0 c1 n2 y @y n3 z n3 z @z @ s @x t s n1 x @z
D20
þ 2c0 c2
@ @ ða 1Þ ða 1Þ @ @ ða 1Þ ða 1Þ þ 2c0 c2 þ 2c0 c3 þ 2c0 c3 @ s @y t s n2 y @ s @z t s n3 z
@ @ ða 1Þ ða 1Þ @ @ ða 1Þ ða 1Þ @ @ þ 2c1 c2 þ 2c1 c3 þ 2c1 c3 þ 2c2 c3 @x @y n1 x n2 y @x @z n1 x n2 z @y @z ! ða 1Þ ða 1Þ @2 @2 @2 @2 1 @ 1 @ 1 @ 1 @ þ 2c2 c3 2 2 2 þ 2ða 1Þ ;¼ 2 n2 y n3 z t s @ s n1 x @x n2 y @y n3 z @z @ s @x @y @z " # 2 2 2 2 1 1 1 1 þ ða 1Þ2 x n1 y n2 z n3 st @ @ @ @ @ @ @ @ @ @ þ 2 c1 c2 þ c1 c3 þ c2 c3 þ 2c0 c1 þ c2 þ c3 @x @y @z @ s @x @y @x @z @y @z c1 c2 c1 c3 c2 c3 ða 1Þ2 c1 c c þ 2ða 1Þ2 þ þ þ 2c0 þ 2 þ 3 ; ð20Þ ðn1 xÞðn2 yÞ ðn1 xÞðn3 zÞ ðn2 yÞðn3 zÞ ðt sÞ n1 x n2 y n3 z þ 2c1 c2
where we have used the fact that c20 ¼ 1 and c21 ¼ c22 ¼ c23 ¼ 1 [39]. If in addition {ci, cj} = 2dij which is called the Clifford algebra, then Eq. (20) may be simplified to:
1 @ 1 @ 1 @ 1 @ t s @ s n1 x @x n2 y @y n3 z @z " 2 2 2 2 # 1 1 1 1 þ ða 1Þ2 x n1 y n2 z n3 st c1 c2 c1 c3 c2 c3 þ 2ða 1Þ2 þ þ ðn1 xÞðn2 yÞ ðn1 xÞðn3 zÞ ðn2 yÞðn3 zÞ ða 1Þ2 c1 c2 c3 þ 2c0 þ þ ; ðt sÞ n1 x n2 y n3 z
D20
which is the sum of the Laplacian D and the fractional parts.
ð21Þ
h
Definition 3.2 [40]. Let c : Cln ! EndðCN Þ be a representation of the Clifford algebra of Cln on CN . Let {xi} be the standard coordinates on Rn and {ei} the standard basis. The fractional Dirac operator D0
D0
n X i¼1
X n @f 1 a @f 1 a ¼ ; cðei Þ þ ei þ @xi @xi Yi Yi i¼1
ð22Þ
for some function f. HereY1 = n1 x, Y2 = n2 y, Y3 = n3 z, Y4 = t s, etc., and ‘‘” is another way of representing the action of c(ei) – the Clifford multiplication.
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Notes that the fractional Dirac operator over flat space defined by Eq. (22) is deviated from the standard definition by the presence of the fractional term and thus, one may rewrite Eq. (22) like: n X
D0
ei
i¼1
n n X X @f þ ð1 aÞ ei Y 1 Df þ ð1 aÞ ei Y 1 i i : @xi i¼1 i¼1
ð23Þ
Proposition 3.2. The fractional Dirac operator D0
D0
ei
i¼1
@f @xi
ð24Þ
; a
where
@f @xi
a
@f 1 a þ ; @xi Yi
ð25Þ
is the fractional differential operator. On Euclidean space En, Eq. (23) holds. Further, Eq. (23) may be compared to the standard Dirac operator derived on curved spaces expressed according to the standard (a = 1) Bochner–Weitzenböck formula [41]:
D20
X k
R @ 2k þ ; 4
ð26Þ
P where k @ 2k is the flat covariant Laplacian on spinor fields and R the scalar curvature [41]. The Ricci scalar curvature may be identified to the fractional part and thus R ? 0 for a = 1 or for late time s 1 and larger distances x, y, z 1. This is to say that the fractional classical part may be identified to the scalar curvature in Riemann geometry. From Eq. (26), one expects that the Bochner–Weitzenböck formula will be modified. For this, assume for the moment a ~ the induced connection on E. The standard conRiemann manifold M and let r be the Levi-Civita connection on TM and r ~ r ~ is defined by nection Laplacian r
~ r ~s ¼ r
X
~ 2 s; r ei ;ei
ð27Þ
i
~ is the formal adjoint of r ~ with where s 2 C(E), C(E) are locally functions from U ? M, U is a coordinate neighborhood M. r 1 N n respect to the metric (,). In fact, C(E) is the correct analogue of C ðR ; C Þ. Then, we have:
D20
X
~ e Þ ðr ~e Þ ¼ e i e j ðr i a j a
i;j
X
~ e ;e Þ ; ei ej ðr i j a
ð28Þ
i;j
~e Þ r ~ e ;e þ ð1 aÞY 1 is the fractal Levi-Civita connection. where ðr i;j i;j a i j Proposition 3.3. Let D0
~ r ~ þ R þ ð1 aÞ D20
X
ei ej Y 1 i;j :
ð29Þ
i;j
Proof. In fact:
D20
X
h i X X ~ e e þ ð1 aÞY 1 ¼ ~ e þ ð1 aÞ ei ej r ei ej r ei ej Y 1 i;j i;j ; i; j i;j
i;j
¼
X i
~ e ;e þ r i i
X
i;j
~ e ;e þ ð1 aÞ ei ej R i j
i
X
i;j
ei ej Y 1 i;j ;
i;j
~ ; property: where we have used the induced Riemann curvature R
~2 r ~Y r ~X r ~2 ¼ r ~ Xr ~ Yr ~ ½X;Y R ~ X;Y ; r X;Y Y;X with [X,Y] = rXY rYX. Making use of the fact
X i
~ e ;e ¼ ei ej R i j
1 X 1X R Rliij ei ej ei þ Rljij ei ej ej el ¼ Ricjj ; 8 i;j;l 4 j 4
gives the modified or fractional Bochner-Weitzenböck formula in curved space:
~ r ~ þ R þ ð1 aÞ D20
X i;j
2 ei ej Y 1 i;j ¼ D þ ð1 aÞ
X i;j
ei ej Y 1 i;j ;
R.A. El-Nabulsi / Chaos, Solitons and Fractals 42 (2009) 2614–2622
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Definition 3.3. Let D0
~ þ R; ~ r D20
ð30Þ
and
Ra ¼ 4ða 1Þ
X
ei ej Y 1 i;j :
ð31Þ
i;j
~ r ~. If for instance, R Ra , R ! 0 and D20
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