Indian Journal of Pure & A pplied Ph ys ics Vol. 3 ~. June 2000. pp . 440-445 Supersymmetric behaviour of tachyons M e Pant, P S Bisht, 0 P S Negi & B S Rajpul* Department of Ph ys ics. Kum<l un Uni ve rsity. A lmOn! Ca mpus. A lmora 263 (lO I Rece i ved 27 M arch 200n The Supersymmelric fi elds of tachyons arc constructed in terms of Wess-ZulTl ino model of SupcrsYllllTletry . St arting from Lagran gian density and Supersy mmetric transformati ons or field s associated w ith tac hyons. it h;IS been shown tha t th e spinorial charge opera tor for tachyons in T-space satisfy th e sa me cO llllllutati on re lati ons of super Poinca re algehra. I Introduction for tachy ons li e on the hyperplane 1 \" = () j whil e th ose During the past few years, th ere ha s been continuing for tachyom: li e on 11 = OJ . The space R.!- (i.e. the ..- interest I·.! in hi gher dimensional kin ematical models for localizati on space for brady ons) and T-(the time repre­ proper and unified theory of sublumi nal (bradyon ic) and sentation space for tachyons) demonstra te the structural superluminal (tachyonic) obj ects·,·7 The problem of rep­ sy mmetry between th ese two and directl y lead to space­ resentati on and locali zati on of ex tended particles and time du ality between superiuminal and sllbiliminal ob­ ta chyo ni c obj ects may b solved onl y by th e use of jects . hi gher dimensi onal space. Several attempts of ex tending SlIp ersy mmetry, .I. e. t Ile F"erml-' B ose symmetry--17-17 , spec ial th eory of relati vity to superlulllinal realm in is one of th e most fa scinating di scoveries in the hi story usua I I·'o ur space I. X·HI Ie d to man y controverS.ie s 11·1, an d of physics. The local extension of supersy mmetri c theo­ no sati sfactory th eory for tachyons could be made ac­ ri es (i. e. supergravity) provides a natural framework for cept abl e so far. the unificati on of fundamental interacti ons of elemen­ The lack of ex perimental search l .!.17 for tachyons led tary pa rt icles. It was bel ieved earlier" tha t spin-() to th e necessity of constru cting a self- consistent qu an­ tachyon s are quantized onl y with anti -commutation re­ tu m fi eld theory or tachyons whi ch might yield th eir quantal properti es relevant for th eir produ cti on and de­ lati ons but their locali zati on in space creates probl em tecti on. The continuing in terest on tachyon.-showed that with Lorentz in variance. Now it is made clear th at th ese particles are not ill contradicti on to the special tachyons are local ized inti me an d thei I' local ization th eory of relati vity . A basic di sagreement in the vari ous space T -behaves as that of bradyons and it is not poss i­ models of tac hyons was th e spin-statisti cs relationship. bl e to accelerate directl y a particle fro m R.!- space to Tanaka I') Dhar-Sudarsan "-0 , Aron-Sudarslwn"- I assume d T-space (or subluminal to superluminal) .]n the present the spin for tachyons as that of bradyons. On the other paper, attecnpt s ha ve been made to demonstrate th e .. hand . Feinberg). Halll a m o t o ~ ~ sh owed that Spin-statis­ supersy mmetri c behavi our of tachyons. It is emphas ised ti cs for tachyons is reversed to that for bradyons. Rajput that the Casimir operator pl = PIt Pp. is no more in vari ant an eI co-wor k' ers71- . "-6 cI esc rJ. be d a L orentz IIlVarJant. quan- in usual space-time (R4_ space) whil e it is an in variant tum fi eld theory of tachyons of vari ous spins and sh owed operator in T -space. Starti ng from the Lagran gian dell- that tachyons arc not locali zed in space. Recentl y, the sity for the fi eld theoreti cal real izati on of super-Poi ncare sc alar and spinoI' fi eld theory of free tachyons ha ve been al gebra of tachyons, th e supersy mmetri c transforma­ developed alld it is shown th at tachyo ns are locali zed in ti ons" (Wess- Zumino model) ha ve been obtained CO I1 - time. The locali zati on space for th e descripti on of s iste nt with tac hyoni c fi eld equations for scalar, tac hyo ns is T-space where th e ro les of space and time pseud oscal ar and Majorana spino I' fi e lcl s. The spill ori al .... (;lIl d th ose of momentum and energy) are int erchanged charges for tachyons are obtained from the Fouri er on pass in g from bradyons to tachyons. The initial values expansiom' of scalar, pseud oscalar and Maj orana ':' Vice Cha nce llor. Kum;llIn University . Nai nil;i1 tachyon Dirac spinors and it ha s been shown th ;lt these PANT ('( al.: SUPERSYMMETRIC BEHAVIOUR OF TACHYONS 44 t charge operators are th e real ization operators for the which is named as R4-s pace consisting of one time and generators of super Poinca re group. three space coordinates. 3 Localization Space for Tachyons 2 Representation of Supeq)oincare Aigebl"a Special theo ry of relati vity has been ex tended st rai ght The super Poincare algebra is the ex tension of Poin­ forwardly from bradyons to tachyons via two types of care al gebra. It has 14 generators i. e., 4 generators of superiuminal L orentz tran sformations (S LT'S ,) which translation P", 6 generators of L orentz tran sformations are called complexH and real ones') These two types of Mp", and 4 spinoI' charges Q" (Majorana spinors) and Q". SLT'S lead to se veral assoc iated w ith The spin- 1/2 generator Q" chan ges bos ons into fermions diffi c ulti es' 2. 1~ and is kn own as th e grading rep resentation of Poincare ta chyon localization in fourdimensional space- time (R.J _ 2 algebra . Becau se of th e spinorial character Q", the su­ space). Secondl y, it is not poss ible .J to derive reduction, persy mmetric al gebra involves both commutation and second quantiza tion and intera cti on of fi elds as sociated 8 with tach yons in R.J -space unless the constraints like anti- co mmutation relations in the following mannel.2 ; space- time reciprocity') and reinterpretation principle20 IPp , Pvl = 0 are imposed. The generators of either type of SLT'S are IM pv, ppJ = -i (l1 !'" P"" +llvp P,,) not integrabl e and it is shown that tachyons are not IMpv, Pp,, 1 =- i(Tlpp Mv" +Tlv" Mpp -llp" Mvp -llvp Mp(i) loca lized in space and the Cauchy data for tach yons IPp, Q"J = () ...( 1) ca nn ot be described on the hyperplane 11= Ol. On th e IM !II" Qui = -( 0 p\, )"I> QI> other hand, it is spec ulated",·:q that tachyons are loca l­ I Qu, Q I> ) = 2 (Y') u(.Pp ized in time and th eir evolution operator is space de­ w ith Gp\' = il41y p,y,· 1 pend ent. Thus it is emphasised earlier " " that spec ial th eo ry should be extend ed to superluminal observe r where I I represe nt s co mmutator and I ) denotes anti­ (tachyons) onl y by increasing th e number of space- time commutator, yp are Dirac matrices in Majorana rep re­ dimensions from four to six dimensional space- time. sentation, 1l!1I' = (+ 1,- 1,- 1,- 1) and natural units are used The resulting space for brad yons and tachyons ha s been th ro ughout the notations. identified as th e R6-or M (3,3) space where space and Supersy mmetric al gebra Eq. ( I ) corresponds to th e time playa sy mmetrical role. Both space and tillle (also particles moving slower th an light velocity which are momentum and energy) are considered as vector quan­ named as br;ldyons in usual four dimensional space-time titics in J? 6_ space. Superluminal transformation in R('- . R ~ () . I' b d I '.r, (I. e. -space) . n pass ing rom ra yons to tac l yons . space hetwee n two frames Sand S' moving with \' > superluminal Lorentz tran sformations are needed be­ are defined as follows: tween th ese two. Cas imir operat or p2 = pppP is defined /= ± rex-lit) as: \,'=± I ~ ,::' = ±I, w here r = ( I '~ - I r ln I'" = PpPP = /1( 2 (for brad yons) (II < I ) I', = ± r(t-lIx) = () (for Luxons) (II = I ) I') . = ±VI 1' / . = ± :: ... (:'l ) = - /I /- (for tachyons) (II > I ) ... (2) T hese tran sformat ions lead to th e mi x in g of space and Thus for ta ch yons we have : time coordinates for transcendental ta chyons (V -:) 00 ), p2 = £" + /1/2 ... (3 ) Eq. (5) tak es the following fonn: while th at fo r bradyons +dl -:) dl .. = d :: + £2 = p" +/1 / .. .(4 ) , +dl -:) dl ' =dy+ \' , whe re I II is denoted as the rest ma ss of the particle in its own frame or rcl'erence . , , = d.r + j q + d1-> dl . It is we ll kn own that arbit rary initial values 1'01' l- d.r -:) d.r' =dl - , ta chyons can not be present ed Oil t he hyperplane I I = ()} =dl - anclthat it is imposs ible to construct wavepackets local­ - d ~ -:) d~~ \' - ized in space .