Indian Journal of Pure & A ppli ed Phys ics Vol. 3 ~. June 2000. pp. 440-445

Supersymmetric behaviour of tachyons

M e Pa nt, P S Bisht , 0 P S Neg i & B S Raj pul *

Department of Ph ys ics. Kum<lun Uni versi ty. A lmOn! Campus. A lmora 263 (lO I

Recei ved 27 M arch 200n

The Supersymmelr ic fi elds of tachyons arc constructed in terms of Wess-ZulTl ino model of SupcrsYllllTletry . Starti ng from

Lagrangian densit y and Supersymmetric transformati ons or fields associated w ith tachyons. it h;IS been shown that the spinori al

charge operator for tachyons in T-space satisfy the same cO llllllutati on re lations of super Poinca re algehra.

I Introduction During the past few years, there has been continuin g

interest I·.! in hi gher dimensional kinematica l models for proper and unified theory of sublumi nal (bradyon ic) and superluminal (tachyonic) obj ects·,·7 The problem of rep­resentati on and locali zati on of ex tended particles and tachyo ni c objects may b solved onl y by the use of hi gher dimensi onal space. Several attempts of ex tending spec ial theory of relati vity to superlulllinal realm in

I I· I.X·HI I d . 11·1, d usua 'our space e to man y controverSies an no sati sfactory theory for tachyons could be made ac­cept abl e so far.

The lack of ex perimental search l .!.17 for tachyons led to the necess ity of constructing a self-consistent qu an­tu m fi eld theory o r tachyons which might yie ld their quantal properties relevant for their production and de­tecti on. The continuing in terest on tachyon.-showed that these partic les are not ill contradi cti on to the special theory of relati vity . A basic di sagree ment in the vari ous models of tac hyons was the spin-stati sti cs relationship .

I') " 0 " I d Tanaka Dhar-Sudarsan - , Aron-Sudarslwn- assume the spin for tachyons as that of bradyons. On the other hand . Feinberg). Halllamoto~ ~ showed that Spin-statis­tics for tachyons is reversed to that for bradyons. Rajput

I k 71. " 6 I . b d L . . an e co-wor ' ers- - c escrJ e a orentz IIlVarJant quan-tum fi e ld theory of tachyons of vari ous spins and showed that tachyons arc not locali zed in space. Recentl y, the scalar and spinoI' fi e ld theo ry of free tachyons ha ve been developed alld it is shown th at tachyo ns are locali zed in time. T he locali zati on space for the desc ripti on of tac hyo ns is T-space where the ro les of space and time (;lIld those of momentum and energy) are interchanged on pass in g from brad yons to tachyons. The initial values

':' Vice Chance llor. Kum;llIn Universi ty . Nai nil ;i1

for tachyons li e on the hyperplane 1 \" = () j whil e those for tachyom: lie on 11 = OJ . The space R.!- (i.e. the .. -localizati on space for bradyons) and T-(the time repre­sentati on space for tachyons) demonstra te the structural sy mmetry between these two and direc tl y lead to space­time du ality between superiuminal and sllbiliminal ob­

jects . S . I F ' B 17-17 lIpersy mmetry, I. e. t le "erml- ose sy mmetry- -,

is one of the most fasc inating di scoveries in the hi story of physics . The local extension of supersy mmetric theo­ri es (i.e. supergravity) provides a natural framework for the unificati on of fundamental interacti ons of e lemen­tary pa rt ic les. It was bel ieved ea rlie r" tha t spin-() tachyons are quanti zed onl y with anti -commutati on re­lati ons but their locali zati on in space creates problem with Lorentz in variance. Now it is made clear that tachyons are local ized inti me and the i I' loca l izati on space T -behaves as that of bradyons and it is not poss i­bl e to acce lerate directl y a parti c le fro m R.!- space to T-space (or subluminal to superluminal) .]n the present paper, attecnpts ha ve been made to demonstrate the .. supersy mmetri c behavi our of tachyons. It is emphas ised that the Casimir operator pl = PIt Pp. is no more in vari ant in usual space-time (R4_ space) whil e it is an in variant operator in T -space. Starti ng from the Lagran gian de ll-sity for the fi e ld theoreti ca l rea l izati on of super-Poi ncare al gebra of tachyons, the supersy mmetri c transforma­ti ons" (Wess-Zumino mode l) ha ve been obtained CO I1 -

s iste nt w ith tac hyo ni c fi e ld equati ons fo r scalar , pseudoscal ar and Majorana spinoI' fi e lcl s. The spillori al .... charges for tachyons are obtained from the Fourier expan s iom' of scalar, pseud oscalar and Maj o rana tachyon Dirac spinors and it has bee n shown th ;lt these

PANT ('( al.: SUPERSYMMETRIC BEHAVIOUR OF TACHYONS 44 t

charge operators are the real ization operators for the

generators of super Poinca re group.

2 Representation of Supeq)oincare Aigebl"a The super Poincare algebra is the ex tension of Poin­

care al gebra. It has 14 generators i.e., 4 generators of translation P", 6 generators of L orentz tran sformati ons

Mp", and 4 spinoI' charges Q" (Majorana spinors) and Q". The spin - 1/2 generator Q" changes bosons into fermi ons and is known as the grading representation of Poincare

algebra . Becau se of the spinorial character Q", the su­

persy mmetri c al gebra invo lves both commutation and anti- commutation relati ons in the foll owing mannel.28

;

IPp , Pvl = 0

I Mpv, ppJ = -i (l1 !'"P"" +ll vp P,,) IMpv, Pp,, 1 =- i(Tlpp Mv" +Tlv" Mpp -llp" Mvp -ll vp Mp(i) IPp, Q"J = () ... ( 1)

IM!II" Qui = -( 0 p\, )"I>QI> I Qu, QI> ) = 2 (Y ')u(.Pp

w ith Gp\' = il41 yp,y,· 1

where I I represent s commutator and I ) denotes anti ­

commutator, yp are Dirac matri ces in Majorana repre­

sentation, 1l !1I' = (+ 1,- 1,- 1,- 1) and natural unit s are used th roughout the notati ons.

Supersymmetri c al gebra Eq. ( I ) corresponds to the particles moving slower th an li ght veloc it y which are

named as br;ldyons in usual four dimensional space-t ime . R~ () . I' b d I '.r, ( I.e. -space) . n pass ing rom ra yons to tac l yons .

superlumin al Lorentz tran sformati ons are needed be­tween these two. Cas imir operator p2 = ppp P is defin ed

as: I'" = PpPP = /1( 2 ( for brad yons) (II < I )

= () ( for Lu xons) ( II = I ) = - /I /- (for tachyons) ( II > I )

Thu s for tach yons we have : p2 = £" + /1/ 2

while that fo r bradyons £2 = p" +/1/

... (2)

... (3 )

.. . (4 )

whe re I II is denoted as the rest ma ss of the partic le in its own frame or rcl'erence .

It is well knownq

that arbit rary ini tial values 1'01'

tachyons can not be presented Oil t he hyperp lane I I = ()} anclthat it is imposs ible to construct wavepackets local ­ized in space. Eq. (4) is no more in variant under either type of superiumin al transformati onsK

• ' -'. Hence the

Casimir operator P" = PI, PP cannot be taken as the

invariant operat or fo r tachyons in ou r usual space-time

which is named as R4-space consistin g of one time and

three space coordinates.

3 Localization Space for Tachyons Special theory of relati vity has been ex tended strai ght

forwardl y from bradyons to tachyons via two types of

superiuminal Lorentz tran sformati ons (SLT'S ,) wh ich are call ed complexH and real ones') These two types of

SLT'S lead to se veral diffi c ulties' 2. 1~ assoc iated w ith

tachyon localization in fourdimensional space- time (R.J_

space). Secondl y, it is not poss ible2.J to derive reduction,

second quanti zation and interacti on of fi elds associated with tachyons in R.J-space unless the constraints like space- time reciprocity') and reinterpretation principle20

are imposed. The generators of either type of SL T'S are not integrabl e and it is shown that tachyons are not loca li zed in space and the Cauchy data for tachyons cannot be described on the hyperpl ane 11= Ol. On the other hand, it is speculated",·:q that tachyons are loca l­

ized in time and their evolution operator is space de­

pendent. Thus it is emphasised ear lier " " that spec ial theory should be extended to superluminal observer (tachyons) onl y by increasing the number of space- time dimensions from four to six dimensional space- time.

The resulting space for brad yons and tachyons has been identified as the R6-or M (3,3) space where space and time playa symmetrical role. Both space and tillle (also

momentum and energy) are considered as vector quan­titi cs in J?6_ space. Superluminal transformati on in R('-

space hetween two frames Sand S' mov ing wi th \' > are defined as foll ows :

/= ± rex-lit)

\,'=± I ~ ,::' = ±I, w here r = ( I '~ - I r ln

I', = ± r(t-lIx)

I' ) . = ±VI 1' / . = ± :: ... (:'l )

T hese tran sformat ions lead to the mi x ing of space and

time coordinates for transcendental tachyons (V -:) 00 ),

Eq. (5) takes the following fonn:

+dl -:) dl .. = d:: +

j ,

+dl -:) dl ' =dy+ \' , l + d1-> dl .

= d.r + , , - d.r -:) d.r' =dl -,

- d~ -:) d~~ =dl -\'

- d" -:) {( =dl - ... (6)

where square bracket shows that \\ 'e ha ve onl y two four-dimensional sl ices or R('-space. W hen an y refer­

ence f rame describes brad yons 8 , we have to assume 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

tl of B are not observable or coupled together giving f = 7 , 7 1/' 0 h d h f (t, -+ t,. - + fz - ) - . n the other an , w en a rame

describes a bradyon at rest in 5 it will describe a tachyon

T(with velocity v~ 00)] in 5' with M' (1,3) space

M' (1,3) = (tx', x', y', z') = (x, f .. , t,., tz) ... (7a)

We define M' (1,3) =(x, lx, f" (1) as T-space or M(3, I) space where y and z are not observable (or coupled

I . . ( '7 ') 112) I . . bl toget ler gl v1l1g r = x- +y-+z- -. t IS not POSSl e to

accelerate a particle from subluminal to superluminal

world. As such the spaces R4_ and T - are two observa­

tion s lices of R6_ space. Bradyons and tachyons cannot

be described individually in R6_ space as this space is

not fully consistent with special theory ofrelativit/6.37.

Subluminal and superluminal transformation in R6_ space lose their meaning in the sense that these transfor­

mations do not represent either bradyons or tachyons in

this space. So, a bradyonic R4= M( 1,3) space now maps

unde r SL T'S to a tachyonic T= M(3, I) space or vice­

versa I.e.

(1,3) SL T (3, I) . .. (7b) ~

The space T -has been considered as true localization

space for tachyons in view of their localizability, Her­

miticity, Lorentz invariance and unitarity of the repre­

sentations of field operators.

Klein-Gordon equation

[ovc)" - 111." 1 <!>(x) = 0 ... (8)

and Dirac equation

lyVo .. - I7l ] <!>(x)=O

or

(<!>-m) <!>(x) = ()

... (9)

for tachyons retain their forms only if we switch over

from R4_ space to T- space i.e.

lop} = I olox~'} = lolor, -V; } whe re

~ I V', } = I aIOf" aldfy, alOfz }

and the mass e ne rgy relation for tachyons is described by equation:

p7. = lEI" + n? ... ( 10) As such the Casimir operator P7. = p~,P;' = _m 2 is taken

as an invariant operator for t,k:hyons in T- space and

one can according ly formulate the other generators of

super- Poincare algebra for tachyons. Thus, it is possible to deal with supersymmetry of tachyons in T'l_ space

only when energy and time are vector quantities while

space and momentum are scalars . The Hamiltonian is

space-dependent.

4 Wess-Zumino Model for Tachyons The authors choose the following Lagrangian density

for the field theoretical realization of the super Poincare

algebra associated with tachyons in T-space:

L = 1/2 (opA)(o~ A) +1/2 m2A"

+ 1/2 (opB (o~B) + 1/2 m2B2+1 /2 \jJ «(J - 111.) \jf ... (II)

where A and B are two spin-O fields (one is scalar and

the other pseudoscalar) and \jf is spin-l/2 Majorana

field.

The equation of motion are obtained by applying the

Euler- Lagrange equation in T-space:

aL IO<!>i- op(oL lo(o~,<!>;) = 0 ... ( 12)

where <!>iEIA, B, \jf, \jf }

Case (i) <!> = A, The equation of motion for scalar field

is obtained as the K.G. equation for tachyons : [0_111.2] A =0 ... (13)

Case (ii) <!> = B, gives rise to the K.G. equation for the

pseudoscalar field B of tachyons: [O-m"] B = 0 .. . ( 14)

Case (iii) <!> = \jf, the equation of motion for Majorana

field associated with tachyons as,

[d - 112] \jf = (yVov-m) \jf = 0 ... ( 15)

Case (iv) <!> = \j!, we get the equation of motion for

conjugate tachyon spinor \jf as:

o,.\j!y"- m\jf = 0 ... ( 16) The authors consider the following variation of the

fields which transform tachyonic fermions and

tachyonic bosons into each other:

M ="£ \jf(x)

8B = -i "£ y'i\jf(x)

8\jf = -(~ + m) (A- iy'i B)E

8\jJ="£(A-iYB)(~-m) ... (17)

where E is ' t ' independent Grassmann variable .

Transformations (17) are called Wess-Zumino Sll­

persymmetric transformations";

The supercurrent four vector JP for tachyons may now be defined as :

i' = i/2"£(A - i/ B) (~-IIl) Y''l/ .. . ( 18) The conserved Majorana spinor current Ki' for

tachyons is defined as:

i' = l/li ~ giving rise to:

f('-' = il2/..(A - i/ B) (~-111) y'\jf ... ( 19)

PANT el 01.: SUPERSYMM ETRIC BEHAVIOUR OF TACHYO S 443

where A is a real parameter which has to be determined

such that the spinoI' charge J K:: d 'k satisfies the same

anti -commutation re lati on as Q" . As such in the Wess­Zumino model, the spinoI' current density ( 19) sati sfies the continui ty equ at ion.

() ~t KP = 0 The spinoI' charges Q,t are now defined as:

J () ., Q,t = K II d --r ... (20)

., where d . -r = d I , d Ir d I :

and k:> 112 A [ i A - i y' B ) (~ - 111 ) ! l \jJ ]" ... (2 1)

with 0 = I ,2,3,4. Thus if the Wess-Zumino model is considered as supersym metri c fi e ld theory, the spinoI' charges Eq. (20) with charge density Eq. (2 1) must sa ti sfy the commutation and anti commutation relations of Super Poincare algebra Eq. ( I ).

Thus the authors use thei I' second quanti zed reduced expa nsions of sp in-O and sp in - 112 [Ref. 25-261 tachyonic fi e ld operators in energy time representation (T' -space) .lt is emphas ised that Majorana spinoI' as we ll as sca lar and pseudosca la r fi e ld s associated with tachyons get plane wave expans ions onl y in T- space while in usua l space (R-l-space), there always ex ists a

momentum violatin g inequalit y I pi :::0: rn lead in g to trun­

cated Dirac-b functions. The Majorana spinoI' fo r tachyons in r-space may

now be ex pressed as the foll ow ing pl ane wave expan­sion:

\I, M(x)= 1/ (2n)V} I J d .1 \1l( II1 I £) 111

! d ( II' .I ) " ( II' ,.1 ) e - i IJ 1 + d -' " ( II' , S ) e - il" I ! ... (22)

with \I,M (x) = \I/ K' (x)

and E 1= E t - fJ x = Pp xp

-\ rI \II = dl\ · ... (!I v,. d\llc

I'p = (I',-E), _\)t = (x, I" I \,. I/. )

Bec;lu se energy and time are considered as vec tor quant it ics for tachyons so t he frequencies associated with energies are al so taken as vec tors whil e momentum and space are taken as scabrs in T'-space.

As slIch the Maiorana spi nor for tachyons in ~o l ves creati on and annihilation operators d (w,s) and d(w,.I). The annihilati on operat or is ex pressed as:

d( w ,.I)= I /(2 n )\12 (lII/fi)I I2J (dl l / "1 II ; (w,.I) \If(X) ... (23)

Using the fo ll owi ng re lat ions defined for spinors " (w,.I) and Jl (w ,s):

u ( IV , S ) U ( IV , r ) = b" v ( IV , S ) Jl ( IV , r ) = - b" u( IV ,s) u(IV,r)=O

V( IV ,s)v( IV,r )=() ... (24)

We get the following equal space anti-commutation relati ons:

{d(w,r),cl(w,s) } = b" b(w- w') ... (25) For scalar field A and pseudoscalar field B used for

Wess-Zumino model of tachyons, following reduced ex pansions (plane wave expansions) are taken into ac­count, i.e.

A() I I(? )3/2 J ,3 (2 )- 112 1 ( ) iEI .;. ( ) ili'l

(?6) x = _n {, w paw e +0 w e .. . _

B(x) = 1/(2n )3/2 J cL'w(2p)" 112 [/)(w )e-ihl+a ; (w) /"1 .. . (27)

where 0 (w), a)(w) and /)(w) b ;(w) sati sfy the following equal space commutation relations:

la (w), o\ W')] =b(W- w') .1 [a (w), o(w') J=[(/ (w), (/ ;(w')l=O .~ [17 (w), /) f(w')J=b(w- w') I [17 (w), /)(w')]=[/)'; (w), 17; (w') 1=0 J .. . (28)

Substituting Eqs (22), (26) and (27) in to Eq. (2 1) on usin g Eq. (20), we get:

Q" = i A(117/2)I I2L J (t' wi c (w) d' (W,.I) U (w,.I) .,

- D (w) d (w ,s) LtCW,s) III ... (29)

where

C (w) = a (w) 1 - iy'b(w) D (w ) = a

j (w) 1- iy' b-i(w) ... (3()

Taking Hermitian conjugate of Eq. (29) we have,

Q ;o = iA (/1/12)1 /2 L J (P w[ v (w,s) d (W,.I) c ; (w) .'

- II ; (w ,s) d' (w ,s) D;(w)l, ... (] I )

and the Dirac adjoint is:

Q II = (Q ; (YO)II

= -iA (/1/12)"2 L, J d\v I v; (w ,s) d (w,s) C; (w) Yll

- II ; (w,s) d (w,s) D; ( IV) Y,I II Now using Eq. n O) we have:

C ' (w)Yo=Yo D(w)

and ( D i (w) Yo) = Yo C (w) Thus in sertin g Eq. (33) into Eq. (32) we get:

... (32)

... (33)

Q" = - i Vnd2) 112 L J d\,' {v (w ,s) d (W,.I ) D (w ) ,

-u (w,s) d ' (w ,s) C (w) 1 .. .(34)

The authors now demonstrate the following re lations:

I C,lil (w),D,,(w')=( b 1"1 b " - Y j; 'I Y;, )b(w-w') ... (35)

444 INDIAN 1 PURE & APPL PHYS, VOL 38, JUNE 2000

and other commutators of C(w) and D (w) are taken as

vanishing. Thus the foll ow ing anli-commutation re lati on be­

tween Q" and Q (/ are obta ined after taking into account

the tediou s calculations:

(Q II , Q" 1= 1/2 fc21f (d\v (y'pp) II" (a ; (w) (/ (w)

+ !J(w) !J (w) + L d ' (w ,r) d (w ,r)} ... (36 )

The energy momentum operator for Wess-Zumino

model is de fined as:

... (37)

where Tpr is the usual canonical energy momentulll

tensor obtai ned from the Lagrangian density rEg. ( I I) I. The energy - momentum operator can now be expressed

as fo ll ows:

p ll = f (d~ W p ll 1(/ (w) (((IV) + /J"(w) h (w)

+ L d ' (w, r) d (w, rj / ... (38)

As such for fc = 2, Eg. (36) now changes to : - II ( Q II , Q I> ) = 2 P (YII)III>

This resembles with the anti-commlllation re lations

of super Poincare algebra. So, the Wess-Zumino mode l

in r l-space can now be visualized as the Supersy mmet­

ric field theo ry of tachyons as it satisfies the commuta­

tion and anti-commutation re lati ons of Super Poincare

a lgebra. Supersymmetric behaviou r of tachyons trans­

form~ the (sp in 112) Spi no I' tac hyo ns to (S pin-O)

tachyons or vice-versa. Thus the supersy mmetri c be ha viour of tachyons is

visuali zed in TI-s pace o nl y where th ey behave as

bradyons. As such, the gauge hierarchy problems asso­

c iated with supersymmetric th eory of tachyons are re­

moved within the veloc it y range" = c (li ght ve loc ity) to

I ' = 00 . Infinit e speed tach yon behaves as the zero speed

hrad yo n. Superluminal space-time transformations do not ha ve

;II1 Y ro le in supersymmetri c gauge transformations

whi ch transform bosons 10 ferilli ons and vice versa. Thus it is not agreed with the authors)22 that spin-O

braclyo ns be visua li zed as spin-I 12 tac hyons. SupersYIll­

metric fi e ld of tachyons arc discussed onl y in terms o f

thei r locali zati on space. Here, the authors ha ve dealt onl y with the case of free

tachyons, the tachyon-tacll yon and tachyon-bradyon in ­

tcr;lct ioll wi t hin th e framework of Supersy mmet ric fi eld

theory. wi ll also be discussed in the forthcomin g paper.

Acknowledgment One of the authors , (MCP) is thankful to th e UGC,

New De lhi , for financial ass istance.

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