Fractional spin - a property of particles described with a fractional Schroedinger equation.pdf

a r X i v : 0 8 0 5 .3 4 3 4 v 1 [ p h y s i c s .g e n - p h ] 2 2 M a y 2 0 0 8 Fractional spin - a property of particles described with a fractional Schro?dinger equation Richard Herrmann GigaHedron, Farnweg 71, D-63225 Langen, Germany E-mail: herrmann@ Abstract. It is shown, that the requirement of invariance under spatial rotations reveales an intrinsic fractional extended translation-rotation-like property for particles described with the fractional Schro?dinger equation, which we call fractional spin. PACS numbers: 03.65.-w,05.30.pr 1. Introduction The fractional calculus [1],[4] provides a set of axioms and methods to extend the coordinate and corresponding derivative definitions in a reasonable way from integer order n to arbitrary order α: {xn, ?n ?xn } → {xα, ?α ?xα } (1) The definition of the fractional order derivative is not unique, several definitions e.g. the Riemann, Caputo, Weyl, Riesz, Gru?nwald fractional derivative definition coexist [6]-[13]. To keep this paper as general as possible, we do not apply a specific representation of the fractional derivative operator. We will only assume, that an appropriate mapping on real numbers of coordinates x and fractional coordinates xα and functions f and fractional derivatives g exists and that a Leibniz product rule is defined properly. Therefore we use xα as a short hand notation for e.g. sign(x)|x|α as demonstrated in [5] and ?αx = ? α/?xα as a short hand notation for e.g. the fractional left and right Liouville derivative (Dα+, D α ? ): (Dα+f)(x) = 1 Γ(1? α) ? ?x ∫ x ?∞ dξ (x? ξ)?αf(ξ) (2) (Dα ? f)(x) = 1 Γ(1? α) ? ?x ∫ ∞ x dξ (ξ ? x)?αf(ξ) (3) which may be combined via ?αx f(x) = Dα+ ?D α ? 2 sin(απ/2) f(x) (4) = Γ(1 + α) cos(απ/2) π ∫ ∞ 0 f(x+ ξ)? f(x? ξ) ξα+1 dξ (5) 0 ≤ α 1 Fractional Spin 2 For this derivative definition, the invariance of the scalar product follows:∫ ∞ ?∞ ( ?α ?xα ? f?(x) ) g(x)dx = ? ∫ ∞ ?∞ f(x)? ( ?α ?xα g(x) ) dx (6) where ? denotes the complex conjugate. The Leibniz