Fractional Superspace Formulation of Generalized Mechanics

Fractional supersymmetry denotes a generalisation of supersymmetry which may be constructed using a single real generalised Grassmann variable, θ =θ, θ n = 0, for arbitrary integer n = 2, 3, .... An explicit formula is given in the case of general n for the transformations that leave the theory invariant, and it is shown that these transformations possess interesting group properties. It is shown also that the two generalised derivatives that enter the theory have a geometric interpretation as generators of left and right transformations of the fractional supersymmetry group. Careful attention is paid to some technically important issues, including differentiation, that arise as a result of the peculiar nature of quantities such as θ.

show abstract

“…grade zero real superfields f , whose expansion in powers of θ involves three real terms (see e.g. [11,15,16,18])…”

Section: Derivatives With Respect To θmentioning

confidence: 99%

“…To conclude, we note the further well-known results (cf. [11,15,18]) 9) which are most easily seen as identities by applying them to arbitrary f .…”

Section: Covariant Derivative Objectsmentioning

confidence: 99%

Group theoretical foundations of fractional supersymmetry

Azcárraga

Macfarlane

1996

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…(25) and (26), where now b Ϯ are ordinary boson operators. They satisfy the commutation relation [X Ϫ , X ϩ ] ϭ 1.…”

Section: Special Case Of W Kmentioning

confidence: 99%

Two approaches to fractional supersymmetrical quantum mechanics

Daoud

Kibler

2003

Int J of Quantum Chemistry

View full text Add to dashboard Cite

Two complementary approaches of ᏺ ϭ 2 fractional supersymmetrical quantum mechanics of order k are studied in this article. The first, based on a generalized Weyl-Heisenberg algebra W k (which comprises the affine quantum algebra U q (sl 2 ) with q k ϭ 1 as a special case), apparently contains solely one bosonic degree of freedom. The second uses generalized bosonic and k-fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetrical oscillator of order k.

show abstract

“…However, mathematicians have obtained in spirit of usual Clifford algebras, new algebras defined from n-linear relation and leading to an underlying Z~-graded structures, the so-called generalized CJifford algebras which emerge naturally from various contexts [8,9,10]. The latter endow a differential structure on non-commutative variables which allows to build a theory beyond supersymmetry [11,12]. The content of this paper is organized as follows: In sect 1, we sketch briefly the basic and useful properties of GCA, then we construct the quantum tori Lie algebra in sect 2.…”

Section: Introductionmentioning

confidence: 99%

“…If we substitute the equation in the second line i.e F~ = 1 to F~ --0 the obtaincd algebra becomes generalized Grassmann algebras G(r,n). The latter is the fundamental tool in the concepts of fractional statistics, fractional supersymmetry [11] and even in 2D fractional conformal theory [12].…”

Section: Introductionmentioning

confidence: 99%