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On 1 d fractional supersymmetric theory

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  • 1. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol.17, 201312ON 1D FRACTIONAL SUPERSYMMETRIC THEORYH. Chaqsare1*, A. EL Boukili1, B. Ettaki2,M.B. Sedra1and J. Zerouaoui11Ibn Tofail University, Faculty of SciencesPhysics Department, (LHESIR), Kénitra, Morocco2Ecole des Sciences de l’InformationBP 6204, Agdal, Rabat, Morocco* E-mail of the corresponding author: h. chaqsare@gmail.comAbstractFollowing our previous work on fractional supersymmetry (FSUSY) [1,2], we focus here our contribute to the studyof the superspace formulation in D1 that is invariant under FSUSY where 3=F and defined by HQ =3, weextend our formulation in the end of our paper to arbitrary F with 3>F .Key-words Fractional superspace - Fractional Supersymmetry of order F - Fractional Supercharge - CovariantDerivative1. IntroductionMotivating by the results founded in [1,2], the aim of this paper is to develop a superspace formulation in D1 QFTthat is invariant under fractional supersymmetry (FSS). In such construction, the Hamiltonian H is expressed as thethF power of a conserved fractional supercharge: HQF= , with 0=],[ QH and 3≥F . Here, we shall reformulatethese results in fractional superspace, using generalized Grassmann variable of order F satisfying 0=Fθ .Additionally, we construct the Noether fractional supercharges in the case where 3=F .The presentation of this paper is as follow: In section 2, we present the Fractional Superspace and FractionalSupersymmetry 3=F . In section 3, we will give the Fractional supercharges and Euler-Lagrange equations for3=F . In section 4, we will generalise the FSUSY in arbitrary order and finally, we give a conclusion.2. Fractional Superspace and Fractional Supersymmetry 3=FThe FSUSY of order 3 are generated by the Hamiltonian H generator of the time translation, and Q the generatorof FSUSY transformations . they satisfies:HQHQ =;0=],[ 3(1)In fields quantum theory, this symmetry can be realized on generalized superspace ),( θt , where t is the time, andθ is a real generalized Grassmann variable. The latter variable and his derivative ∂∂∂=θsatisfies:0=;0= 33∂θ
  • 2. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol.17, 201313Ι∂−∂ =θθ θθ q (2)2θθ ∂≡∫dThe introduction of the ε (parameter of the transformation associated to Q ) et f (parameter of the transformationassociated to H ) in the case where 3=F give the following transformations [3]:)(= 22εθθε +−−′ qftt (3)εθθ +′ =where ε verify:0=3εεθθε q= (4)where 32=πieq . The q-commutation relation between the two variables ε and θ ensures that:• if 0== 33θε then 0=)( 3εθ + ;• the reality of the time is not affected by the FSUSY transformation;• the FSUSY transformations q-commute with covariant derivative;• the FSUSY transformations satisfied the Leibnitz rules.We now can introduce the scalar superfield Φ of order 3:2221210=),( ϕθθϕϕθ qqt ++Φ (5)where 10 ,ϕϕ et 2ϕ are the extension of the bosonic and the fermionic field. These fields verifies:θϕθϕ )(=)( 00 ttθϕθϕ )(=)( 121 tqt (6)θϕθϕ )(=)( 22 tqtWe now can see that *= ΦΦ . Using relations (3), we get easily the FSUSY transformations upon the fields:11/20 = εϕδϕ q21/21 = εϕδϕ −−q (7)012 = ϕεδϕ −∂−Then, let us consider the two basic objects Q and D , which represent respectively the FSUSY generator and thecovariant derivative [1]2212)()(=θθθθθ∂∂+∂∂−∂− − qqQ22212)()()(=θθθθθ∂∂+∂∂−∂− − qD (8)Using the equations (2) and (4), we can prove that:
  • 3. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol.17, 201314zQD ∂== 33DQqQD 2= (9)ΦΦ εε δδ DD =whereΦΦ Qεδε = (10)The invariant action under the FSUSY transformations in equations (7) is:dtLDdtdqS =2= 12∫∫ ΦΦ∂−θ][2= 122212022ϕϕϕϕϕθθ &&& qqdtdq−+−∫ (11)][21= 1222120 ϕϕϕϕϕ &&& qqdt +−∫3. Fractional supercharges and Euler-Lagrange equationsFollowing [3] and [6], we can introduce the generalized momenta conjugate to iϕ000 = ϕϕπ &&∂∂≡L211 =2 ϕϕπ qL−∂∂≡&(12)1222 =2 ϕϕπ qL&∂∂≡If we consider Φ& and ΦD as independent variables, we can prove that the generalized momenta conjugate are thecomponents of the fractional superspace momentum conjugate to ),( θtΦΦΦ∂∂≡Π DLqt =2),(&θ (13)which is decomposed as)(221220=2121202==),( iiiiqqqqt −−∑−+−Π πθθϕϕϕθθ & (14)Note that ΠΠ =*. If wish to add0=],[ ΦΠ∫ θd0=],[ ΦΠ∫ &θd (15)0=],[ ΦΦ& (16)we must require
  • 4. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol.17, 201315ijijji −≠ 3if= ϕϕϕϕ && (17)1221 = ϕϕϕϕ q (18)Focus on the internal-space part of the Lagrangian 1222122= ϕϕϕϕ &&qqL +− . The Lagrangian variation]..[2]..[2= 121222121 δϕϕϕϕδδϕϕϕϕδδ &&&& +++−qqL]..[2]..[2= 2121221221 ϕδϕϕϕδϕδϕϕϕδ &&&& qqqq+++− (19)and knowing that12221 2=;2= ϕϕϕϕ&&qLqL−∂∂∂∂12221 2=;2= ϕϕϕϕqLqL&& ∂∂−∂∂(20)Then, the Lagrangian variation will be:)()(=22221111ϕϕδϕδϕϕϕδϕδϕδ&&&&∂∂+∂∂+∂∂+∂∂ LLLLL (21)From this equation, it is easy to show that the generalized Euler-Lagrange equations which follow from a least-actionprinciple are:0=)(11 ϕϕ &∂∂−∂∂ LdtdL0=)(22 ϕϕ &∂∂−∂∂ LdtdL(22)Therefore, the quantity0=;=21=dtdCXLCiii−∂∂∑ ϕδϕ&(23)is a constant of motion when the lagrangian varies under a transformation iδϕ by the total derivative dtdXL /=δwhere 102121= ϕϕ&qX . The particular case of the Hamiltonian when ii ϕδϕ &= is:2020=21== ϕϕϕ &&& LLHiii−∂∂∑ (24)For iδϕ given by (7) and corresponding X , we find the following fractional supercharge associated with thesymmetry transformations:)21(2= 220121ϕϕϕ +&qQ (25)
  • 5. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol.17, 201316Note, like in [6], that QQ εε =)( *, i.e., qQQ =*, using (17), we can prove that Qq 21−is a real charge.4. FSUSY of arbitrary orderIn this section, we will give a generalisation of the FSUSY of order F where 3≥F . For this, we introduce theexpression of the superfield ),( θtΦ of order F .iiiFiqt ϕθθ 2210==),( ∑−Φ (26)where θ is a real generalized Grassmann variable satisfied 0=Fθ and q is the F-th root of unity ( Fieqπ2= ). Thesuperfield components verifies the following commutation relations:θϕθϕ iii q−=iiFiiFi q ϕϕϕϕ −− = (27)the first relation in (27) implies that ΦΦ =*. while the second relationship is used to introduce the followingcommutation relation:iFiiFi q ϕϕϕϕ 1= −− && (28)for 2=F , the equations (27) and (28) reduces to the usual results of supersymmetry θϕθϕ 11 = − , 0=21ϕ and1111 = ϕϕϕϕ && − while 0000 = ϕϕϕϕ && .Les FSUSY transformations of order F are generated by the generator of the FSUSY Q whose expression is:])()([= 122130=1 −−−−−−−∂∂+∂∂−∂− ∑ FFiFFiFitFqqAQθθθθθθ (29)where FFFFFqA12)1}!({)(=−−−−−− . This implies that:tFQ ∂= (30)Acting on ),( θtΦ , we have),(=),( θεθδ tQt ΦΦ (31)which gives on components:122221)(11}!{)(= +−+−−− iiiiFi FqqAq εϕδϕ1222)(221)(212 1}!{)(= −−−−−−− − FFFFFF FqqA εϕδϕ (32)011221)(1 )(= ϕεδϕ &−−−−− − FFFF qAqq
  • 6. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol.17, 201317while 3}{0,1,..., −∈ Fi ,qqFF−−−−11=1}{1and ε is real infinitesimal parameter verify:iii qq εϕεϕεθθε =;= (33)To build invariant action under FSUSY transformations need the introduction of the fractional covariant derivativecommuting with Qε .])()([= 1221130=1 −−−−−−−−−∂∂+∂∂−∂− ∑ FFiFFiiFFitFqBDθθθθθθ (34)where ])1}!({)...(1)[(=112321FFFFFFFFqqqB−−−−−−−− . Q et D satisfies the following relations:tFDqQDDQ ∂=;= (35)DqDqQQ =;= εεεεAfter defining the two generators of FSUSY, we can now give the expression of the action S invariant under theFSUSY transformations (32)ΦΦ−− −∫ &DdtdFBS Fθ11}!{21=iFiiFiiFiFiFiFFqqqqFdtdFS −−−−−−−∑∫ −+−ϕϕϕθθ && )(22)(2221=20111}!{{1}!2{1=}1}!{ 11221)(121ϕϕ &−−−−− FFFqqqF (36)iFiiFiiFiFiFiqqqqFdtS −−−−−∑∫ −+ ϕϕϕ && )(22)(2221=20 1}!{{21=}1}!{ 11221)(121ϕϕ &−−−−− FFFqqqF (37)ConclusionIn this paper, we have extended the results founded in [1] and [2] of fractional symmetry (FSUSY) from D2 to D1 ,and following [3] and [6], we are giving the fractional supercharge in 3=F . In the last section, we gave thegeneralized formulation of the generator of the FSUSY Q and fractional covariant derivative D .References:1. H. Chaqsare, B. Ettaki, J. Zerouaoui (2013), On D = 2(1/3, 1/3) Fractional Supersymmetric Theory III, Adv.Studies Theor. Phys., Vol. 7, no. 5, 229 - 236.2. H. Chaqsare, M. B. Sedra, J. Zerouaoui (2012), kW - Algebra and Fractional Supersymmetry, Adv. Studies Theor.Phys., Vol. 6, no. 16, 755 - 764.
  • 7. Advances in Physics Theories and Applications www.iiste.orgISSN 2224-719X (Paper) ISSN 2225-0638 (Online)Vol.17, 2013183. Rausch de Traubenberg (1987), Algèbres de Clifford Supersymétrie et symétries nZ Applications en Théorie desChamps, Habilitation à diriger des Recherches, Laboratoire de Physique Théorique, University Louis Pasteur. .4. E.H. Saidi, M.B. Sedra and J. Zerouaoui (1995), Class. Quant. Grav. 12 , 1567-15805. A. Perez, M. Rausch de Traubenberg (1996), P. Simon, Nucl. Phys. B 482 : 3256. S. Durand (1993), Mod. Phys. Lett. A8, 2323-2334 .7. E.H. Saidi, M.B. Sedra and J. Zerouaoui (1995), Class. Quantum Grav. 12 2705-27148. V.A. Fateev and A.B. Zamalodchicov, Sov. Phys. JETP.62 (1985); 215 and 63, 913 (1986)9. A. Leclair and C. Vafa (1992), CLNS 92/1152, HUTP-92/A04510. Rausch de Traubenberg (1998), Nucl. Phys. B517:48511. M. B. Sedra, J. Zerouaoui (2008), Adv. Studies Theor. Phys., Vol.4, 2010, N 5; Vol.3, 2009, N 9-12; Vol.3,2009, N 5-8; Vol.2, N2012. A. Elfallah, E.H. Saidi, J. Zerouaoui (1999), Phys.Lett. B 468 : 8613. M. Daoud and M.R. Kibler (2004), Physics Letters A, Volume 321, Issue 3, Pages 147-15114. E.H. Saidi, H. Chaqsare and J. Zerouaoui (2012), Adv. Studies Theor. Phys., Vol. 6, no. 21, 1025 - 1038
  • 8. This academic article was published by The International Institute for Science,Technology and Education (IISTE). The IISTE is a pioneer in the Open AccessPublishing service based in the U.S. and Europe. The aim of the institute isAccelerating Global Knowledge Sharing.More information about the publisher can be found in the IISTE’s homepage:http://www.iiste.orgCALL FOR PAPERSThe IISTE is currently hosting more than 30 peer-reviewed academic journals andcollaborating with academic institutions around the world. There’s no deadline forsubmission. Prospective authors of IISTE journals can find the submissioninstruction on the following page: http://www.iiste.org/Journals/The IISTE editorial team promises to the review and publish all the qualifiedsubmissions in a fast manner. All the journals articles are available online to thereaders all over the world without financial, legal, or technical barriers other thanthose inseparable from gaining access to the internet itself. Printed version of thejournals is also available upon request of readers and authors.IISTE Knowledge Sharing PartnersEBSCO, Index Copernicus, Ulrichs Periodicals Directory, JournalTOCS, PKP OpenArchives Harvester, Bielefeld Academic Search Engine, ElektronischeZeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe DigtialLibrary , NewJour, Google Scholar