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Supersymmetric Behaviour of Tachyons

Supersymmetric Behaviour of Tachyons

Indian Journal of Pure & A pplied Ph ys ics Vol. 3 ~. June 2000. pp . 440-445

Supersymmetric behaviour of

M e Pant, P S Bisht, 0 P S Negi & B S Rajpul* Department of Ph ys ics. Kum

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 s associated w ith tac hyons. it h;IS been shown tha t th e spinorial 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 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 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 --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. ) 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 -() to th e necessity of constru cting a self- consistent qu an­ 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 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 and ) 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 Q" chan ges bos ons into 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 , 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 . Eq. (4) is no more in variant under either - d" -:) {( =dl ... (6) K type of superiuminal transformati ons • ' -'. Hence the where square bra cket shows that \\ 'e ha ve onl y two Casimir operator P" = PI, PP cann ot be ta ken as the four-dimensional slices or R('-space. W hen an y refer­ invariant operat or for tachyons in ou r usual space-time ence f rame desc ribes brad yons 8 , we ha ve to as sume M 442 INDIAN J PURE & APPL PHYS, VOL 38, JUNE 2000

(1,3) = [f, x,y,Z) (R4_ space) so that the coordinates tv and space and momentum are scalars. The Hamiltonian is tl of B are not observable or coupled together giving f = space-dependent. (t, 7-+ t,. ,- + fz -7 ) 1/'- . 0 n the other han d , w hen a f rame 4 Wess-Zumino Model for Tachyons describes a bradyon at rest in 5 it will describe a tachyon The authors choose the following Lagrangian density T(with velocity v~ 00)] in 5' with M' (1,3) space for the field theoretical realization of the super Poincare M' (1,3) x', y', z') (x, f .. , t,., ) ... (7a) =(tx', = tz algebra associated with tachyons in T-space: lx, f" ( ) We define M' (1,3) =(x, 1 as T-space or M(3, I) 2 space where y and z are not observable (or coupled L = 1/2 (opA)(o~ A) +1/2 m A" . . ( '7 ') 112) I . . bl 2 toget Iler gl v1l1g r = x- +y-+z- -. t IS not POSSl e to + 1/2 (opB (o~B) + 1/2 m B2+1 /2 \jJ «(J - 111.) \jf ... (II) accelerate a particle from subluminal to superluminal where A and B are two spin-O fields (one is scalar and world. As such the spaces R4_ and T - are two observa­ the other pseudoscalar) and \jf is spin-l/2 Majorana tion slices of R6_ space. Bradyons and tachyons cannot field. be described individually in R6_ space as this space is 6 The equation of motion are obtained by applying the not fully consistent with special theory ofrelativit/ .37. Euler- Lagrange equation in T-space: Subluminal and superluminal transformation in R6_ aL IOi- op(oL = 0 ... ( 12) space lose their meaning in the sense that these transfor­ lo(o~,;) mations do not represent either bradyons or tachyons in where iEIA, B, \jf, \jf } 4 this space. So, a bradyonic R = M( 1,3) space now maps Case (i) = A, The equation of motion for unde r SLT'S to a tachyonic T= M(3, I) space or vice­ is obtained as the K.G. equation for tachyons: versa I.e. [0_111.2] A =0 ... (13) (1,3) SLT (3, I) . ..(7b) Case (ii) = B, gives rise to the K.G. equation for the ~ pseudoscalar field B of tachyons: The space has been considered as true localization T - [O-m"] B = 0 .. . ( 14) space for tachyons in view of their localizability, Her­ Case (iii) = \jf, the equation of motion for Majorana miticity, Lorentz invariance and unitarity of the repre­ field associated with tachyons as, sentations of field operators. [d - 112] \jf (yV m) \jf 0 ... ( 15) Klein-Gordon equation = ov- = Case (iv) = \j!, we get the equation of motion for [ovc)" - 111." 1(x) = 0 ... (8) conjugate tachyon spinor \jf as: and o,.\j!y"- m\jf = 0 ... ( 16) lyVo.. - I7l] (x)=O ... (9) The authors consider the following variation of the or fields which transform tachyonic fermions and (-m) (x) = () tachyonic into each other: M ="£ \jf(x) for tachyons retain their forms only if we switch over 8B = -i "£ y'i\jf(x) from R4_ space to T- space i.e. 8\jf = -(~ + m) (A- iy'i B)E lop} = I olox~'} = lolor, -V; } 8\jJ="£(A-iYB)(~-m) ... (17) whe re where E is 't' independent Grassmann variable. ~ I V', } = I aIOf" aldfy, alOfz } Transformations (17) are called Wess-Zumino Sll­ persymmetric transformations"; and the e ne rgy relation for tachyons is described The supercurrent four vector for tachyons may by equation: JP now be defined as: p 7. = lEI" + n? ... ( 10) 2 As such the Casimir operator P7. = p~,P;' = _m is taken i' = i/2"£(A - i/ B) (~-IIl) Y''l/ .. .( 18) as an invariant operator for t,k:hyons in T- space and The conserved Majorana spinor current Ki' for one can according ly formulate the other generators of tachyons is defined as: super- Poincare algebra for tachyons. Thus, it is possible i' = l/li ~ giving rise to: to deal with of tachyons in T'l_ space only when energy and time are vector quantities while f('-' = il2/..(A - i/ B) (~-111) y'\jf ... ( 19) PANT el 01.: SUPERSYMMETRI C BEHAVIOUR OF TACHY O S 443

where A is a real parameter which has to be determined u( IV , S ) U ( IV , r ) = b" such that the spinoI' charge JK :: d' ksa ti sfies the same v ( IV , S ) Jl ( IV , r ) = - b" u( IV ,s) u(IV,r)=O anti -commutation relati on as Q" . As such in the Wess­ ... (24) Zumino model, the spinoI' current density ( 19) sati sfies V( IV ,s)v( IV,r )=() th e continui ty equ at ion. We get the following equal space anti-commutation relati ons: () ~t KP = 0 The spinoI' charges Q,t are now defined as: {d(w,r),cl(w,s) } = b" b(w- w') ... (25) () ., For scalar field A and pseudoscalar field B used for Q,t = JK II d --r ... (20) Wess-Zumino model of tachyons, following reduced ., ex pansions (pl ane wave expansions) are taken into ac­ where d . -r = d I , d Ir d I: count, i.e. and k:> 112 A [ i A - i y' B ) (~ - 111 ) ! l \jJ ]" ... (2 1) 3/2 (2 )- 112 1 ( ) iEI .;. ( ) ili' (? 6) A()x = I I(?_n ) J{, , 3w paw e +0 w e l .. . _ with 0 = I ,2,3,4. Thus if the Wess-Zumino model is considered as supersymmetri c fi eld theory, the spinoI' ihl B(x) = 1/(2n )3/2 cL'w(2p)" 112 [/)(w )e- +a; (w) /"1.. .(27) charges Eq. (20) with charge density Eq. (2 1) mu st J sa ti sfy the commutation and anti commutation relati ons where 0 (w), a)(w) and /)(w) b ;(w) sati sfy the following of Super Poincare algebra Eq. ( I). equal space commutation relations: Thus th e authors use th ei I' second quantized reduced la (w), o\ W' )] =b(W- w') .1 expansions of spin-O and spin - 112 [R ef. 25-261 [a (w), o(w')J=[(/ (w), (/ ;(w')l=O tac hyo ni c fi eld operators in energy time representati on .~ (T'-space) .lt is emphas ised that Majorana spinoI' as well [17 (w), /) f(w')J=b(w- w') I as scalar and pseudoscalar fi eld s associated with [17 (w), /)(w')]=[/)'; (w), 17;( w') 1=0 J .. . (28) tac hyo ns get plane wave expansions onl y in T- space Substituting Eqs (22), (26) and (27) in to Eq. (2 1) on whil e in usual space (R-l- space), there always ex ists a usin g Eq. (20), we get: momentum violatin g inequality I pi :::0: rn lead in g to trun­ Q" = i A(117/2)I I2 L J(t' wi c (w) d' (W,.I) U (w,.I) ., cated Dirac-b functions. - D (w) d (w ,s) LtC W,s) III ... (29) The Maj orana spin oI' fo r tachyons in r-space may now be ex pressed as the foll owin g pl ane wave expan­ where sion: C (w) = a (w) 1 - iy'b(w) D (w ) = a j (w) 1- iy' b-i(w) ... (3() \I, M(x)= 1/ (2n)V} I Jd .1 \1l( II1 I £)111 Takin g Hermitian conju gate of Eq. (29) we have, Q ;o = iA (/1/12)1 /2 L J(P w[v (w,s) d (W,.I) c ; (w) ! d ( II' .I ) " ( II' ,.1 ) e - i IJ 1 + d -' " ( II' , S ) e - il" I ! ... (22) .' - II ; (w ,s) d' (w ,s) D;(w)l, ... (] I) K with \I,M (x) = \I/ ' (x) and the Dirac adjoint is: p and E 1= E t - fJ x = Pp x = -\ Q II (Q ; (YO)II rI \II = dl\·... (!I v,. d\llc = -iA (/1/12)"2 L, Jd \v Iv ; (w ,s) d (w,s) C; (w) Yll I'p = (I',-E), _\)t = (x, I" I\,. I/. ) Bec;lu se energy and time are considered as vec tor - II ; (w,s) d (w,s) D; ( IV) Y,I II ... (32) quant it ics for tachyo ns so t he frequencies associated Now usin g Eq. n O) we have: with are al so tak en as vec tors whil e momentum C' (w)Yo=Yo D(w) and space are taken as scabrs in T'-space. As slI ch th e Maiorana spi nor for tachyo ns in ~o l ves and ( D i (w) Yo) = Yo C (w) ... (33) creati on and annihilation operators d (w,s) and d(w,.I). Thus in sertin g Eq. (33) into Eq. (32) we get: The annihilation operat or is ex pressed as: Q" = - i Vnd2) 112 L d\,' {v (w ,s) d (W ,.I) D ( w ) , J d( w ,.I)= I /(2 n )\12 (lII/fi)I I2 J (dll / "1II ; (w,.I) \If(X) ... (23) -u (w,s) d ' (w ,s) C (w) 1 .. .(34) Us in g th e fo ll owi ng relat ions defined for spinors " The authors now demonstrate the following relations:

(w,.I) and Jl (w ,s): I C,lil (w),D,,(w')=( b 1"1 b " - Yj; 'I Y;, )b(w-w') ... (35) 444 INDIAN 1 PURE & APPL PHYS, VOL 38, JUNE 2000

and oth er of C(w) and D (w) are taken as Acknowledgment vanishing. One of the authors, (MCP) is thankful to th e UGC, Thus the foll owin g an li-commutation relati on be­ New De lhi , for financial ass istance. tween Q" and Q (/ are obtained after taking into account Refcl'cnccs th e tedious calculations: 1 Recailli E. Ril' N uol'{) Cilll. 9 ( 191-16 ) I : Found PIn', 17 ( 19 R7) (Q II , Q" 1= 1/2 fc2 1f (d\v (y'pp) II" (a ; (w) (/ (w) 23 9. 2 Cole E A B. /VI/()I'O CilllC'n/o A. 60 ( 19RO) I; PhI'S Lell A. 95 + !J(w) !J (w) + L d ' ( w ,r) d ( w ,r)} ... (36) ( 19R3) 2l-! 2. The energy momentum operator for Wess-Zumino 3 Chandola H C & RajplIl B S. ./ Mmh Phr.l. 26 ( 19X5) 2()X and references th erein. model is de fin ed as: 4 Bora Sliehi & RajplIt 13 S. IJ/ WI/IIlIO . 44 ( 19<))) 50 I . ... (37) 5 Feinherg G.IJ/trs/(I'I'. 15S1 ( I %7) l OX'). where Tpr is the usual canonical energy momentulll 6 Bilnaiuk 0 M P. Deshp,lIlde V K & Sudarshan E C G. AIIIN ./ PhI's. ( 1962) 7 1R. obtai ned from the Lagrangian densit y rEg. ( I I) I. 7 Negi 0 P S & Rajpul 13 S. LI'II N It (1I'I1 Cillli'll/O. 32 (19X I ) I 17: The energy - momentum operator can now be expressed PhI'S LI'II n. I 13 ( 19R 2) I X3. as foll ows: X Rec,lIlli E & Mignani R. Nil' NIIIJI'1i Cillli'lI lo. 4 ( I Sl74) 2() I . p ll W p ll (w) (((IV) /J "(w) (w) = f (d ~ 1(/ + h 9 A nlipp" A F. PhI'S Rei' IJ. II ( 1975) 724. + L d ' (w,r) d (w,rj / ... (38) 10 A nlippa A F. Everell A E. PIli'S Ni'l' n. 4 (1971) 2 190. I I Mariwala K H. A lii(,/,./ PIn's. 37 ( I (9) 12X I. As such for fc = 2, Eg. (36) now changes to: - II 12 Corhen H C. Im./ Thl'o r IJliys. 15 ( 1976) 7( 13. ( Q II , Q I> ) = 2 P (YII)III> 1:1 T eli M T , Lell N lIol'o Cillll'II/II, 22 ( 1976) 4X9. This resembles with th e anti-commlllation relations 14 A lvager T & Kres ller M N. I>hrs HI'I·. 17 1 ( 19613) 1357. of super Poincare algebra. So, th e Wess-Zumino model I :; Da vies M M. Krestler M & A lvagerT. PhI'S N" I'. I X:I ( I %9) in r l-space can now be visualized as th e Supersy mmet­ 11 32. ric field theory of tachyons as it satisfies th e commuta­ 16 Barllel D F & Lahana M D. Pln's NI,\, n. () ( 1(72) 1X 17. tion and anti-commutation relati ons of Super Poincare 17 End ers A & N imlz G . ./ I>lil's/{IIII'. 12 ( 1992) I ()<)3; 3 1199:1) alge bra. Supersymmetri c behaviou r of tachyons trans­ IOXSI. IX Hei llllannW & i ll1t zG. PliysLI'IIA.I96( 1994 ) 15 4. fo rm~ the (spin 112) Spi no I' ta c hyons to (S pin-O) tachyo ns or vice-versa. 19 Tanaka S. Prog Th l'or IJlirs. 24 ( 10(0) 177. Thus th e supersy mmetri c be ha viour of tachyons is 20 Dhar.J & Slidarshan E C G. IJli.l's Nl'I'. 1741 1961-!) I X()X. visuali zed in TI-s pace o nl y where th ey behave as 21 Arons M E & Sud ars h;lIl E C G. Plirs RI'I'. In ( I %X) I ()22. 22 HamalllolO S. Pmg Tlll'or I'hl·s. 4X (1972) 103 7. bradyo ns. As such, the gauge hi erarchy problems asso­ 23 R

I' = 00 . Infinit e speed tach yon behaves as th e zero speed IJ/"·I. 1<) ( 19X I ) l OX hrad yo n. 2) Panl M C. l3i shl P S. cgi 0 P S & Raj pIII 13 S. IlIiIillll ./IJII /'{ , Superluminal space-time transformations do not ha ve & AIIIII PIn's. 1997 (Pre prinl ). ;II1 Y ro le in supers ymmetri c gauge transformations 2() I)alll M C. Bi ~l ll P S. Negi 0 P S & RajpIIl Jj .. elill ./ IJ/II·.I • ( I SI<)7 ) whi ch transform bosons 10 ferilli ons and vice versa. 27 Fa y fel P & Ferr" r" S. IJlirs NCJlllrI C. :12 (I 1)ln) 241). Thus it is not agreed with the authors)22 th at spin-O 2X Miller-Kirsler, II .I W &. Wiedemann 1\ . .\""/1('1'1'.1'1111111'1/'.1': All braclyo ns be visua li zed as spin-I 12 tac hyo ns. SupersYIll­ ill /rollllclioll lI ,i,h ('olln'/lllIal 11 11 11 ('("cIIIII/ioIlS 111' /(11'11-. (\Vorld metric fi eld of tachyons arc discussed onl y in terms of Sci eillific. I-long Kong). ISlX7 . thei r locali zati on space. 29 Ferrar" Se rgio. SIIII{'/'SI'IIII1II'I/T. (World SciL! lilific. Iiollg Kong). I <)X7 lColicclion ol reprinlsl. Vol I & I I. Here, th e authors ha ve dealt onl y with the case of free 30 W ess & Bagel'. SIIIil'I SI'IIIIIII'I/T IIlId .1·lIlwrgm l·in·. (Princeloll tachyo ns, th e tachyon-tacll yon and tachyon-bradyon in ­ UnivCl'sily Press . Prin ce loll. USA). 1 9X~. tcr;lct ioll wi t hin th e framework of Supersymmetri c fi eld 31 N illes 1-1 P. SUf!cr.IT/III1/I'I/T. slll/I'rgrl/l'i l l' ,\': /){1I1il'i1' /llirsiCl. theory. wi ll also be discussed in th e forthcoming paper. IJliys Hl'llorl. 1 1 10 ( 1 9X~) I . PANT 1'1 al.: SUPERSYMMETRIC BEHAVIOUR OF TACHYONS 445

32 Sohn iu ~ M F. IlI lrodl.lcl;OIl SIlII(,/"SI"II/II/l'lry, Ph )"s i?I'I'Orl. 12R 35 Pa llt M C. Bisht P S & Neg i 0 P S . .I Ph},s A. ( 1997). (1985) 3') . Communicated .

33 Wess J & Zumino B. Nucl Pln·s II . 70 ( 1974) 39. 36 Chandola H C & Rajput B S. NU()lJO CiIlll'IlIO 13 72 (1982) 21 .

34 So ucekJ . .IPhys &M(llhCI'II. 14( 198 1) 1629. 37 Strnad J, L(' II N U(MI Ci1ll1'1I10, 26 ( 1979) 535.