null spinning strings

Nuclear Physics B338 (1990) 143-187North-Holland

NULL SPINNING STRINGS

J . GAMBOA*

Instituto de Fisica, Uniuersitad de Säo Paulo, CP 20516, Sâo Paulo, CEP 01498, Brasil

CUPATITZIO RAMiREZ**

Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16,D-6900 Heidelberg, FRG

M. RUIZ-ALTABA ***

CERN, Theory Division, CH-1211 Genève 23, Switzerland

Received 19 July 1989(Revised 18 December 1989)

We quantize in detail the closed string with vanishing tension, the null closed string . ForWeyl ordering of quantum operators, the spectrum is continuous and there is no criticaldimension . For normal ordering, however, the usual critical dimension holds and the spectrum isfinite : the field content is the same as in the field theory (infinite tension) limit of usual (closed)strings . The content of bosonic null strings is massless gravity. We carry through the localsupersymmetrization of this system and find similar conclusions, normal ordered null spinningstrings contain in their spectrum only the massless supergravity multiplet . However, normalordered bosonic null strings do not have a consistent quantum interpretation, whereas nullspinning strings do . It points to the likely origin of some spontaneous symmetry breakdownwhich generates a non-zero tension, and an infinite tower of massive states. We also consider thehamiltonian formalism for quantum null (super)strings.

1 . Introduction

An approach to a better understanding of the beauty and subtleties of stringtheory is to simplify it by studying its zero-tension limit [1-8] (see also [9,10]) .Taking the string tension T=(27ra')-' to zero corresponds in fact to the veryhigh-energy limit of usual strings, where the mass gap between any two string

* 47551 ::JGamboa (hepnet); on leave from Centro de Estudios Cientificos de Santiago, Casilla 16443,Santiago 9, Chile .

** Q79 at dhdurzl .bitnet.*** RuizAltb at cernvm.cern .ch .

0550-3213/90/$03 .50 © Elsevier Science Publishers B.V .(North-Holland)

144

J. Gamboa et al. / Null spinning strings

modes tends to zero . Grasping the very high-energy regime of string theory mightcontribute to clarifying the fundamental structure of quantum strings and conse-quently the short-distance structure of space-time and the origin of the universe asdiscussed in refs . [11-19] where computations in the perturbative approach havebeen performed .

String dynamics in this regime seem highly non-trivial, with two main approachesin the recent literature . The first one considers high-energy scattering of strings[11-15], where all orders in perturbation theory must be taken into account . Thefixed momentum-transfer scattering considered in refs . [11,12] reproduces classicalgeneral relativity effects at high energies . An analysis of the infinite-energyscattering at fixed-angle [13-15] resulted in an infinite number of linear relationsbetween the scattering amplitudes of different string states, pointing to a very highdegree of symmetry at this level .The second approach is to study the quantum statistics of string ensembles at

high temperatures [16-18] . A phase transition occurs at the Hagedorn temperatureand the high-energy limit of strings is then obtained by adding more energy to thesystem . However, it is unclear how strings behave beyond the phase transition .Indeed, according to ref. [16] a tachyon instability arises for T > TH , independentlyof gravitational instabilities, whereas in the weak coupling approach [17] theHagedorn temperature is truly a limiting temperature (as originally proposed byHagedorn [20]) so that any excess energy flows mainly into a long string, consistentwith the bootstrap condition [21] . In refs . [18,22,23] it has been shown that, in thelarge N limit of the 1/N expansion of QCD, the effective string tension vanishesat the Hagedorn temperature, i.e . deconfinement takes place . The same phe-nomenon arises in a simple compactified string theory, for which the string tensionvanishes at a certain critical radius R,, [23] . These issues are closely connectedwith the duality symmetry of string theory, which relates the temperature T of astring gas to its inverse 1/T [16], or equivalently the compactification radiusR to 1 /R [24] .The bosonic null string is a one-dimensional extended object with zero tension,

introduced and studied by Schild [1] several years ago. Classically, it describes asystem of free massless point particles subject to kinematical constraints, that istheir potential energy vanishes and consequently, the system is unstable . In theproper-time gauge, the kinematical constraints mean that the velocities of thepoint particles are perpendicular to the string and that they lie on a D - 2dimensional sphere . Moreover, there is a rather stringy constraint which requiresthe momentum flow along the string to vanish . This last constraint also remainsunsolved in the light cone gauge (it is a zero-mode) and corresponds in fact to theequality of the masses between the left and right sectors of the usual closed strings ;it will play an important role in the quantization of null strings . Null strings havethe peculiarity that, although their action can be written in a reparametrizationand Weyl invariant way, their physical solutions do not display the full two-dimen-

J. Garnboa et al. / Null spinning strings

145

sional conformal invariance . In fact, this is a reflection of the singularity of thegeometry involved.The quantization of null bosonic strings was carried out by Lizzi et al. [2], who

found that there are no critical dimensions for the null string, that the massspectrum is continuous and that the corresponding wave functions are plane waves.Thus their results were consistent with the view of the null strings as a system offree massless point particles . However, it seems that they did not consider the factthat the momentum along the string does not vanish for plane waves, and henceLorentz covariance is broken . The existence of critical dimensions actually de-pends on the ordering prescription, with very different results according to thechoice of vacuum [7] . More recently, Marquard and Scholl [25] have obtained thecritical dimension considering the null tension limit perturbatively . Their approach(as ours) can be easily generalized for higher-dimensional extended objects,allowing in this way for the computation of their critical dimensions .A related issue concerns the instability [26,271 of membrane theories [28] . It

turns out that, at the classical level, the potential for a tensionful membrane hasflat directions and hence, a spherical membrane will naturally stretch itself into astring where no energy is paid for an increase in length, i.e . it will tend to a nullstring. De Wit et al . have shown that quantum corrections to the classical potentialdestroy the flat directions, and thus a quantum bosonic membrane has a discretespectrum and will not degenerate into a null string . However, quantum supersym-metric membranes retain the flat directions and their spectrum is continuous .Hence, quantum supermembranes are unstable : they degenerate into null super-symmetric strings . In recent work [29] it has been shown that supermembranes in anontrivial background do not show this inconsistency, enjoying a discrete spectrumand a stable vacuum . Another application of null strings concerns the dust stringsconsidered as classical sources of gravity [30] .

In this article we consider the issue of the consistent quantization of null stringtheory. Bosonic null strings can be consistently quantized with either Weyl ornormal orderings . With Weyl ordering, the mass spectrum is continuous and thereis no critical dimension, fully consistent with Lizzi et al . [2] although our wavefunctions have a more complicated structure, essentially given by Bessel functions .Quantization with normal order is consistent only at D = 26 and the physical statesare eigenstates of the momentum along the string, corresponding to a constantnon-vanishing eigenvalue . This anomaly severely restricts the physical spectrum,which contains only massless particles, with the spin content of 26-dimensionalgravity . Unfortunately, the Hilbert space is not positive definite .We also tackle the local supersymmetrization of null strings in the lagrangian

formalism which has never been discussed in the literature . It turns out that thesupersymmetrization works only after some nontrivial reparametrizations and thatthe resulting supersymmetric null string is equivalent, up to kinematical con-straints, to a system of N = 2 free massless spinning particles [31] .

146


The quantization of the supersymmetric case is similar to that of the simplerbosonic null string : with Weyl ordering, the mass spectrum remains continuous,and there are no critical dimensions . With normal ordering, the quantum theory isconsistent only at the critical dimension D = 10 and the full spectrum, after GSOprojection, contains only the massless spectrum of type II superstrings, i .e . N = 2supergravity . In this case the quantum Hilbert space is well defined, i .e . it has onlystates with positive norm, thus reflecting the well-known fact that only tachyon-freesupersymmetric strings can be consistently quantized .We shall review the lagrangian approach for the well-known bosonic null strings

in sect . 2 . We shall then fix the light-cone gauge, canonically quantize, and find thephysical spectrum for the Weyl- and normal-ordered quantum null strings (sects . 3and 4), commenting on the origin of the differences with the authors of ref . [2] . Insect . 5 we establish the lagrangian formulation of the null spinning string. In sect . 6we discuss its light cone quantization and consider the issue of space-timesupersymmetry . We briefly consider interactions à la Mandelstam in sect . 7, andthen study in sect . 8 the null string constraints from an algebraic point of view .General considerations allow us to relate null strings with chiral bosons . Wesupersymmetrize the null string in the hamiltonian approach, and discover the nullspinning string constraint algebra [5] . Sect . 9 details the BRST quantization of thenull spinning string; we have minimized the overlap of this section with our work inref. [7] . Finally, the conclusions yield some insight into the likely equivalencebetween our null spinning string and the recently formulated null superstring [8],as well as some further comments on other approaches in the literature where thetensionless limit of strings seems to play a role . Some technical results concerningthe null string wave functions in sect . 3 are collected in appendix A.

2 . The bosonic null string

The formal problem of writing down an action for the tensionless string can beovercome exactly as in the massless limit of the free particle action . Suppose wehave a generic action

S=TfL(x)dx,

where T is a parameter of non-vanishing dimension . This action is equivalent to

zS' = 2f

z+zT2 ) dx,

where z is an auxiliary field, whose equation of motion inserted back into S' yields


147

S . The limit T - 0 of S is thus given by

where g = det g,nn, gmn = ô,nX" an X", with m, n = 0, 1, A, = 0, 1, . . . , D - 1 . In or-der to maintain conformal invariance, z must transform as an invariant densityunder world-sheet reparametrizations and its Weyl weight must be zero . Theequation of motion-_ of z gives g = 0, i .e . the world-sheet is a null surface fromwhich actually came the name of null strings [1] . Computing from So the momen-tum conjugate to X", we get

which satisfies the constraints

so that

The equations of motion are then

1 L2so=21Z dx .

The action for the null string follows from using for L the Nambu action[1, 3,4,32,331 :

1 g 2So = --2f-dor

p. = 1 [X,2X- -X. X,]z (2.2)

X,2P-X'=0, P2=-=0 .

(2.3)g

The dynamics of the system described by this action can be studied after the gaugefixing

1

1So = 2IdT d~ zX2X'2

and

X X' = 0 .

X2X,2 = 0,

ô~(iX,2Xwl+ôQ(iX2Xu'I

=0 .z J z

148

J. Gamboa et al. / Nu!! spinning strings

These equations have non-trivial solutions only if we take

X(0,) 2 =0 .

Observe that the singular geometry of the world-sheet does not allow the fullconformal symmetry of the string, but only a subgroup of it .

The gauge just derived is the closest analogue to the conformal gauge for usualstrings, and we shall call it Schild's (or null) gauge . The metric takes, then, thecanonical null form ( ô

°) instead of the canonical conformal ( -â

° ) . Theremaining gauge degrees of freedom are subject to

~° = 0, where ;='

arethe infinitesimal reparametrizations of the world-sheet . This can be used, togetherwith the constraints, to fix the light-cone gauge . We shall do that in sect. 3 and weshall subsequently quantize .

Physically, X"(T, o-) describes a set of free massless point particles (one forevery point Q along the string) whose velocity is perpendicular to the string [1-3j .The crucial differences with respect to the usual string can be traced to thedifference in the constraints (2.6), which for the tensionful string read

X2 +T2X' 2 =X .X , =0 .

(2 .7)

Having set T= 0, we have lost the ("nearest neighbour") interaction - that is whythe null string is made out of free particles instead of coupled harmonic oscillators .These particles, nevertheless, do not form plane waves: they still satisfy theinherently stringy X- X' = 0 constraint . Note that there are no boundary condi-tions for the open null string in Schild's gauge [6] : no momentum flows along rr atany point, so in particular no momentum flows out at the end-points either . Thisfact has as a consequence that for open strings, functions of Q cannot be expandedin Fourier series so that their boundary values are well defined ; this is a problemparticularly when we consider constraints because setting them to zero will not be

We may further fix the gauge z = X'2, so that the momentum is P"the action is simply

and

so = fX2 . (2 .4)

The fields X'4(-r, Q) then satisfy the equation of motion

X"(a) =0 (2.5)

subject to the first-class constraints

X2 =X . X' =0 . (2.6)


149

equivalent to setting their Fourier expansion to zero . The closed null stringsatisfies, of course, the customary periodicity. Therefore, unless explicitly statedotherwise, we shall deal throughout this article with closed null strings.

3. Light-cone quantization of the bosonic null string

The safest way to study the physical content of a gauge theory is to avoid thesubtleties of covariant quantization by considering the theory with all the gaugedegrees of freedom fixed. In general, this procedure is not manifestly covariantunder the original global symmetries, which are now non-linearly realized . Therequirement that there be a realization of these symmetries in the quantizedtheory provides us with a consistency test for the theory . For usual strings withnon-zero tension, this amounts to requiring Lorentz covariance in the light-conegauge, which proves to hold only at the critical dimension . It also restricts the massspectrum of the theory . From sect . 2, we write

The light-cone gauge is defined by the condition X+= (27r)-1/zp+-r, with p+ aconstant . The constraints can be solved in this gauge as

by

2p+P Z(Q),

Y-'=

p+r P . Y"

(3 .3)

where the vector products on the right-hand side include only the transversecomponents P' and Y', I = 1, . . . , d = D - 2.

It is convenient to Fourier-expand the independent (transverse) degrees offreedom. Periodicity of the bosonic co-ordinates in o E [0, 2-rr] implies, rathertrivially, that

PW( o-) =

1

Epm

eim'

Y"(O-) =

ym e imu .

(3 .4)l

27r M Œ ZL

2Tr me7L

The zero-mode of the second constraint in (3.2) is given in the light-cone gauge

1 (P -- J P-Y'do,=i F, mp_�,-ym=0,

(3 .5)2Tr mE7L

X"(,r,or) = P"(o,)-r+Y"(o,), (3 .1)

where Pl'(Q) and Y"(o-) are subject to the constraints

PZ (o.) = P( 07 ) . Y'(O.) =0 . (3 .2)

150


that is, it remains unsolved and can be interpreted classically as the requirementthat the momentum flowing along the string, in the o- direction, vanish . It reflectsthe symmetry of the null string under rigid Q-rotations . The condition PU = 0 is the(closed!) null-string analogue of the usual string constraint LO - L o = 0 .The mass M of the string is given by the expression [2]

l zM Z = -~fdo-P(Q) P',Pn1 ~,

n~0

where we have used the constraint p2(0,)=0 and also the fact that, in thelight-cone gauge, the only mode of P+ which does not vanish is the zero-mode(P + ) .As usual, the quantization prescription is

[Yo 'P+1

[Y1(O-),P'(O-')j =i6"S(O--a-') .

(3 .7)

In modes, the second commutator reads as

[Yt

~

=i 6'J sn , Pm~ n+m,0'

After quantization, the condition Pa = 0 will be replaced by the requirementthat physical states be eigenstates of PQ with some eigenvalue, fixed by thedefinition of operator ordering. Thus we have to solve the eigenvalue problem ofthe mass operator and Pa. Expand P, and MZ as follows :

Pa=l 1: n(p-n ' yn -pn 'y_n) = Y, nsn ,n>0

n>0

It turns out that the commutators [MZ , Pa ], [M2 , S n ], [M Z , Tn J, [Pa, Sn,o [Pa, Tn 1,

[ Sn, Tyn], [Sn,S,n], and [Tn,Tj vanish . Moreover,

[P"YmI -m[Sm,Ym ] =mym

[P" Pm ~ = m[Sm>PmJ = mpm,

[MZ ,Ym] - [Tm,Ym ]- -2LPm,

[ MZ ,Pm~ - [ Tm,PmJ =0

(3 .6)

(3 .8)

MZ-2 Fa P-n*Pn =2 Y, Tn .n>0 n>0

(3 .10)

where the commutators of M Z hold for m 0 0 . Therefore, if we take the massoperator plus the zero-modes, i.e . 2pß , it will give us the hamilton operator [2] .Thus, the problem of finding mass eigenstates which are also eigenstates of PQ canbe reduced to that of simultaneously diagonalizing Sn and Tn :

The wave function of a generic state will be written as a product of wavefunctions, a factor for each n >_ 0 :

is such that the operators

S=VI d� , -v;

are hermitian .


151

SnWn = sn*,n

and

TnWn = tn1Pn .

(3 .11)

y t =

t

y t

= (VI)*,n vn

n n

r

a

aP-n

-tav~ n '

Pn

-ta(Vn)*

_ rI Ifn .

(3 .12)n=0

The zero-modes pô and y(I) are not contained in Pv and MZ. To them correspondsa plane wave describing the motion of the string's center of mass,

e lk o v ,, .n=0

In what follows, we shall discuss the wave functions 4n(n > 0) . We will firstquantize with the Weyl order which essentially amounts to realizing the commuta-tion relations in the usual way, i.e . representing the momentum by the derivativewith respect to the coordinate :

where vn E (Cd and v,* is its complex conjugate . For conciseness, we drop the levellabel n and restore it at the end, when we assemble the results .The wave functions depend on v ' and v; . The scalar product of two wave

functions 0 and

(e, p) = fddvddv*rp*(v,v*) 1ll(v,v *)

(3.l5)

(n>0)

(3 .13)

(n > 0)

(3.14)

T= -avtaq

(3.l6)

152


In order to construct the eigenstates of S, we start from wave functions Yrgdepending only on the scalar x = v - v* and satisfying

The general solution to the eigenstate problem of S will be given by tensorial wavefunctions

satisfying tracelessness, i .e .

SIYg = 0 .

(3.l7)

i, . . .iq=N(vi, . . .vigvJ* . . . v*q

) tl'Yg(x)

(3.18)Jq

s~a gfEI. . . ..q =0

(VaE [1,q],d E [1,9]) .

(3 .l9)

The notation ( . . . )t , means precisely that only the traceless part of the tensorproduct of v's and v *'s is retained . Of course the traces are just states with asmaller number of v's and v*'s.These wave functions are, by construction, eigenfunctions of S :

S~P il . .i q =s,yi, . . .iq . s=q-q.

(3.20)JJ . . .Jii Ji . . .J4

Furthermore, the Laplace operator T in eq . (3.16) acts as

T'Yj,'.'. . .q -(v' . . . v qvj* . . .vj* ) tl(T

g -hWg)

so that tlrg(x) must satisfy the differential equation

= tJf11 . . .eq h=q+q

(3 .21)JJ . . . Jq ,

x 1Pg'(x) + (h+d)1Yg(x) +tPg (x) =0

(3 .22)

whose solution can be expressed in terms of Bessel functions (see appendix A) asfollows (for convenience of notation, we set t = m2 and call r = f the lengthof v) :

lrg(r)

= N�,r l-h-dJh+d-1( 2 mr) .

(3 .23)

This function is bounded as r --o oo only for m2 >_ 0, and thus the mass spectrum ispositive . The normalization constant N�, depends on m in such a way that form = 0 the wave function does not vanish (see appendix A for details) .The wave functions (3.18), with ~Pg given by (3.23), will depend on the space-time

dimension (d + 2) and on the three quantum numbers m, s = q - q and h =q +q,the first of which is real, the last two integral (note that both q, q > 0 so that h is a


153

non-negative integer) . For fixed m, the degeneracy is infinite and contains allpossible integer spins, and even for fixed s the degeneracy is infinite and does notdepend on the value of s, the degenerate states being labelled by h. Theconsistency of the Hilbert space generated by these wave functions is ensured bythe fact that the following orthogonality relationships hold (appendix A):

where

S

S-,rS(i, . . .t~) S(i l . . .)4)1

S(m_m),

(3 .24)l

h . . .t~,m ,

1, . . .

q ,;m

qq~5 4q'[

(k, . . .

),)

(t, . . .l4)

tl

S0 , . . .i ) =6(i, 5 ;2

8i 9 )(k,_ ZC9)

(k,

k2 . . .

k q)

is a symmetrized product of Kronecker deltas . Thus we can build normalizablewave packets as superpositions of the wave functions constructed above . Ingeneral, the wave functions of the bosonic null string are infinite products oftensorial wave functions satisfying (3 .11), with values of P, and M Z given by

PQ- n(qn-qn), M2= mn,n>0

n>0

where qn , 4n are non-negative integers and m2, are non-negative real numbers. Themass spectrum is, in principle, continuous, ranging from zero to infinity . Clearly, itsdegeneracy is huge and the spin content is independent of the possible value ofPP, which has to be fixed by the requirement of Lorentz covariance .

It is perhaps worth pausing to consider the result that the mass spectrum for thenull string is continuous . Since our theory had no parameter to start with, the onlyother foreseeable alternative was that the spectrum would have consisted only ofmassless states . From the point of view of the limit a' --~- cc of usual string theories,it is clear that all the states in the leading Regge trajectory collapse to zero mass .In fact, any state of finite mass collapses to zero mass . But since there is an infinityof daughter trajectories, for any real value of a', however large, there are stillinfinitely many states with arbitrarily high mass . We believe this reasoning explainsheuristically our rigorous result : the mass spectrum is continuous, not massless . Infact, this is a consequence of the ordering chosen so far, p's to the right of y's,which essentially agrees with the Weyl ordering used in ref. [2]. We shall now seethat, with normal ordering, only a finite number of massless states survive in theHilbert space.

4. Lorentz covariance of the light-cone quantization

(3 .25)

Up to now we have neglected the problems arising from ordering ambiguities .Now we must take them into account . There are two inequivalent orderings ofinterest . The first is the one already used in sect. 3 and amounts to putting the

154


momenta on the right of the coordinate modes . This order is obviously equivalentto the Weyl order, as can be seen from (3.9) . Weyl ordering was used by Lizzi et al .[2] and, as already shown by them, it does not give rise to a critical dimension .Zheltukhin agrees with them as well [4] . It is rather straightforward to check thatcomputations involving this ordering are actually equivalent to classical computa-tions . With this order we have shown that, up to Lorentz covariance, the Hilbertspace of the physical states (and the observables acting on them) is consistent .However, there is no compelling reason, like zero-point energy, which would tell usa priori that this ordering is the good one . An alternative ordering prescription wasused by us in ref. [7], where we chose to put the positive-indexed modes of P andY to the right of the negative-indexed ones (this order is often referred to as"normal") . As shown in ref. [7], this order implies the existence of a criticaldimension for null strings, precisely the same critical dimension as for theirtensionful siblings . We now want to discuss these issues in some detail, in theframework of the required Lorentz covariance .The generators for Lorentz transformations are

where we defined

__ _127r I-Ir

Y- :(PnY'=n - PnY-n) :

(4.1)n

As usual, the only problematic commutator is [Mt- , M'- ], where

Z

1_1 ~-+M_t - POYo + p+ ~

~. - PnY_ � : - +L :PnY I n :

no n

P

n

Pn =P+Pn = 2

Pn-m pmm

1tnp +yn

if n =# 0

m

1 v

As for usual strings, they constitute a realization of the constraint algebra, i .e . :

[Pm,Pn ) = o,[Ym, PJ = (n - m)Pm+n ,

[Ym , Yj = (n - m)Ym+n + «m)8m+n ,

(4.2)

(4.4)

where we have allowed for the most general central term 4(m) = a,m3 + almcompatible with the Jacobi identities .

A classical computation yields


155

2 1[mi- , MJ ] = - + y -Pli, Pm1PQ .

P

m>O m

Hence, even classically, solutions with non-vanishing Pa break the Lorentz covari-ance. As already mentioned, calculations with the Weyl order give the same resultsas classical calculations and, therefore, in this case the central term in (4.4)vanishes and closure of the Lorentz algebra implies that Pa must vanish whenacting on physical states :

P,Iphys) = Y_ n(qn - gn)Iphys) =0 .

(4.6)n>0

Therefore, in this case Lorentz covariance holds in all dimensions and there are nocritical dimensions . Further, from ref . [7], we know that the normal order, with allthe positive-indexed modes on the right of the negative-indexed ones, leads to atheory with critical dimension D = 26 . From (3.10) we see that the normal orderrequires that Pv take only negative values :

PQ = i Y_ n(p-n - yn

y-n - pn) 'n>0

The Fock space is created out of a vacuum 10) annihilated by yn and pn, for npositive . Thus

PP10) =yn la) =PnlO) = 0,

n > 0

(4.8)

as well as M Z I0) = 0 . In this case, the central term is given by [7]

computation we obtain for the Lorentz algebra

- E

-+ -

+ 2aP +

m>o~(~

2 1-- -P['mPm](P,-a) .p +m>0 m

(4 .5)

(4 .7)

e(n)= 6(ns -n) .

(4 .9)

Letting a denote the eigenvalue of P, on physical states, after a rather tedious

(4.10)

Clearly, the r.h .s . vanishes if and only if d = 24 and a = - 2 . Physical states willbe those eigenstates of PU with eigenvalue P, = - 2 which are, at the same time,

156


mass eigenstates . It is easy to see that the only states which fulfill these conditionsare the massless states PI 21k>, pI IPJilk>, and (p, iy'_ i

-PJlyf i)lk), i .e . withspin content 0, 1, and 2 (dilaton, Kalb-Ramond, graviton), k is the momentum ofthe center-of-mass of the spin . Indeed, all the eigenstates of PQ = - 2 are pt-2 1k),PI iPJ ilk), pI 1 YJ I lk), Y-21k), and yI , y'k) . Their mass matrix is given by

M2 =

which has zero eigenvalues with the above-mentioned eigenstates. The physicalinterpretation of this result is unclear because the first two of these eigenstateshave zero norm whereas the antisymmetric one has negative norm . In order toclarify the order issue, we "diagonalize" M2 and P, by the redefinitions:

am

F2 (Pm + Zmym ),

such that yn =An - Bn and pn =An + Bn + Cn, with

and

An = 2Ean-m . am , Bn = 2Lbn-m . bm , Cn=2Lran-m .bm . (4.13)m

m

m

The following commutators hold :

bm =-(Pm - Zmym

[J __

t

J'' __

IJam,an 1 -[bm,bnl -mS Srn+n,0

(4.12)

(4 .14)

[An, Am ] = (m -n)Am+n +,.Ql(n)Sm +n,0,

[Bn, Bm] = (n - m)Bm+n + .~6(n)8m +n,0,

[Cn,Cm] = (n -m)(Am+n -Bm+n) - [,V(n) + ,~W(n)jam+n,O *

(4 .15)

The corresponding transformations are

A ,' r

i

tAn , a'MJ

mare+n ,[Bn ,bm] _ -mbm+n,

[Cn , am] = mbm+n ,[Cn, bn,] = -mat� +n'

(4.16)

, 0 0 0 0 0)0 0 0 0 0'0 1 0 0 0 (4 .11)1 0 0 0 00 0 2 0 0

Furthermore, P, =A 0 - Bo and M Z =A 0 + Bo + Co are given by the zero-modes .With normal ordering (positive modes of a and b to the right), we have

d~(n) - ~(n) 12 (n' - n)

and the corresponding central term in (4.4) will be

have zero norm because


157

de(n) = ,V(n) + 66(n) = 6 (n' - n) .

(4.17)

However, from the commutators (4.14) we see that this quantization is actually notunitary. We can circumvent this problem by redefining bn - d� = ib�. Some signschange so that now, with the same ordering, the mass eigenstates

(a,, ± dm )IOi

(4.18)

(a,,+d,'� )t=a' Td,'�

(4 .19)and so on. In fact, at this point we are back to the situation leading to the massmatrix (4.11) . The remaining possibility is to order all operators with the positivemodes of a 'M to the right of the negative modes of b� (or vice versa) ; in this case

and there is no contribution to the central term of (4.4) . It can be easily verifiedthat this order is in fact equivalent to the Weyl order, which yields triviallyvanishing central terms, i .e . V(n) = ~d(n) = 0 . Note that also for usual stringsWeyl order yields no anomalous central term and thus no critical dimensions .

Therefore, the issue of the orderings and critical dimensions of null stringsresembles to some extent the situation of the usual strings, i .e . there are essentiallytwo orderings : the normal order which gives a theory with critical dimensionD = 26 and the Weyl order which gives a theory without critical dimensions .However, contrary to the case of usual strings where the quantization with Weylorder is not consistent, in the case of null strings it is consistent .

Quantization with normal order leads to very nice results but, unfortunately,there are physical states with zero-norm . In fact, this may not be as terrible as itappears at first sight. The normal-ordered null string is the rigorously obtainedquantum theory in the limit T - 0 of normal strings, and we have found that purestringy consistency implies that there are zero-norm "physical" states . From thiswe could conjecture that there are states of the usual tensionful bosonic stringwhose norm vanishes in the limit* T -> 0 . This agrees to some extent with the

* We are particularly grateful to J . Labastida and A. Morozov for most enjoyable discussions on thispoint .

158


conjectured behavior of usual strings at very high energies [14], and the most likelyphysical interpretation of this fact is that it is not meaningful to constructasymptotic in- and out-states with null strings, i .e . string theory at infinitely highenergies is not a usual quantum field theory [11,14,16,34,35] .

Null strings do not describe a system of harmonic oscillators as usual strings do,but rather a system of massless point particles related by purely kinematicalconstraints . In the light-cone gauge, there is a reminder of these constraints in thefact that the net momentum flow along the string must take a definite value, zeroclassically and with hermitian ordering, - 2 with normal ordering . It is veryinteresting that the normal-ordered null string develops this sort of chirality (thenet momentum along the string is non-zero, so left is distinguished from right) .This intriguing spontaneous generation of chirality is probably the clue to theinstability of null strings . Indeed, even if we have not written it down, the curious- 2 for the P, eigenvalue must have energy units . Our theory had no dimensionfulparameters to start with, and although a continuous mass spectrum does notrequire any mass parameter, a discrete momentum eigenvalue does . In the end,since physical states are massless, there is still no mass scale whatsoever in thetheory . So, one could conjecture that the normal-ordered null string (i) is unstable,and (ü) it generates a "vacuum expectation value" for "left-right" asymmetry. Itremains beyond the scope of this paper to study the dynamics of the flow to thestable chiral phase, presumably realized by usual (heterotic?) strings .

5 . Lagrangian formulation of the null spinning string

Let us now consider the supersymmetric null string, starting from the action fortwo-dimensional supergravity and taking the string tension to zero following thegeneral procedure outlined in sect . 2 . The superspace approach clarifies thegauge-fixing process, and is best suited for the analysis of the spectrum . Ournotation closely follows Howe's for the usual spinning string [36] .

In complete analogy to the bosonic action (2.1), consider the action

1dZQ dzB ZE z ( 2aXW

_~eaXaf

,(5 .1)

which is invariant under local world-sheet supersymmetric, superfeedl, and globalspace-time Poincar6 transformations . The superfield Z transforms as an invariantdensity, whose components are

Z =Z + B apa - 2102b

(5 .2)


159

From the spinning string action in the WZ gauge

S=Tfd 2o-d 20E-9"X" -9"X~,

=Tfe[ 1P2 -290E - 2i0ymdnX

+B 2 (gmn amXan X-i+lly rn arn4f+B 2- I gl 2X.Y ny mXn

Xnymyn tpdMX)]

we obtain the null spinning string action, with the structure of a o--model :

e 2S", _ - fd 2Q d2B

z ~- 4I# 2Xmy ny mXn -ZXmynym~anX

Observe that this lagrangian contains as auxiliary fields (i .e . non-propagatingLagrange multipliers) the zweibein em, the last component B" of X"`, and all thecomponents of Z: z, p" and b . Moreover, there is a kinetic term for the gravitino

Xm which is actually pure gauge. Indeed, we may fix the gauge such that

where the spinor 0 is the y-trace of Xrn :

and therefore

+ ( -,",2gmn,",l ,LL v + lllp y my n,",v) amXj.an X, + lgj2tpgym arn`Yg

+( -,Jl 2 ngv++G1'0y)Bw B y +2t q'ym0 °B,~ amX,

+ z # 2PYmq,W amXW + z02P0WBw- 4z (b- zP2

/ q,2q,2

1

Xm =Ym(b,

(5 .4)

20" \ Y miim) "

irm 'InXm = 0'

(5 .

We shall fully fix the conformal gauge, consisting of the above gauge for Xrn andthe choice of a diagonal zweibein,

em" = fm S°rn '

(5 .3)

(5 .5)

(5 .7)

160


It is, however, best to keep the off-diagonal components of the zweibein, denotedas go and g l, and set them to zero only after deriving the equations of motion . Weneed the equations of motion

aL oaAL. ~ conformal

gauge

=0 (5.s)

for the auxiliary fields Ai = (Xm, em, Z, B) = (Xm, fo, fl, go, gl, z, P, b, B), in orderto get the constraints .

Since we wish to exhibit all the constraints as primary constraints, it is mostconvenient to effect a redefinition of the fields before solving for the equations ofmotion . This field redefinition amounts to a rescaling of the Grassmann variables0« -, fo 1/20« followed by a further rescaling of the field z . Explicitly, the redefini-tions are

Xg _, XW ,

0' --,

fo o N ,

Bu ---)foB" ,

z-fOZ,

P«-

fop«,

b -fo b .

(5 .9)

In terms of the new fields, the lagrangian becomes

e 2

i ,~', ~~~

Z I_

,t,2gmn7,lAv+,1N.,ym7nq,v)ÔmXAÔnX,+fO 0 24y mam4'

+ 2(- Z,O,w+

uq,")BAB,,+t(ZBOymgG+ 1~ZPY

(9n,X+1V

2POBf0

fo z

fo

4z ( fo

z1 P2) 1#202 -( 4 ,imyny mxn +

2

Xmynym4'ÔnX )W 2

r

I .f0

The equations for b and Xm in the conformal gauge are

2z

2f0f2

1p2q12= 0

2y

,

(5 .10)

I

f

( )-"Pa +f

(y 5 )«Xs.12f

dL o


161

so that fo = 0 and X~ra = 0. Note that setting f, = 0 or qJ2 = 0 is of no practicalinterest . In the conformal gauge, fo = 0 means that the metric is singular and thusthe original fictitious conformal invariance is broken . The canonical form for themetric is to °) instead of ( - 1 1) . The equations of motion for the remainingauxiliary fields are then

°

if+GZgGy,~

_ 0,_-dg z

aL

aL°

z L2iB p'OX' + 4 g 1 q' + a ( 2 (PIPB +

âb~2)

=0

(515)- -y1iy

-J

, .

dLo 2f1[_

~1 2Xz-i~Z y°+ ~J2B2-(B+~)Z1=0,

df,

z

aLo

dpa

(5 .14)

(5 .16)

dLo _ 2fZ, [,,2B" + (BiP)~p "] = 0 .

(5 .18)dB" z

Multiplying the last equation by qi« and using the fact that

a q'13 = Z6~pgZ , weobtain B" = 0.

Therefore, the lagrangian we find is

plus the constraints

Xa =Xgfai

+ 21~l a i~lâ = 0 .

(5 .19)

(5 .20)

The null spinning string describes an N = 2 massless spinning superparticle [31] atevery point of the string . The null spinning string is a line of such N= 2 masslessspinning particles, subject to the constraints (5.20) .

162


With a bit more work, the lagrangian (5 .19) can be cast in an even simpler form .Indeed, the gauge freedom remaining after fixing the gauge (5.7) is given by

whereas ~o and the Lorentz and superWeyl transformations are unrestricted (~a isthe supersymmetry parameter) . The transformation law for f2 4, ß ,2/z is not hard tofind :

' + i~~y50=0,

.Î4 - z.h ~, _ 0 ,

2~- rrtamlflV'2~+ I - ~~ fI WZ

whereby the gauge freedom in ~° may be used to set

~' =

= 0. In this gauge, the lagrangian reads (compare with (2.4))

and the equations of motion are

° =~' = ~'x=0, e',=~o .

(5.21)

(5 .22)

z = 2f,

(5.23)

Moreover, superWeyl invariance can be used to set 0 = 0 and f = 1, thus

1 2 iL0= 2X

+ 2 ~ra a

(5 .24)

XA = ~g = 0,

(5.25)

whereas the (primary and first-class) constraints are given by (5.20) . The remaininggauge freedom is restricted by

(5 .26)

We have succeeded in constructing an action for the null spinning string whoseinterpretation is rather simple . Just as the bosonic null string looked like a bunchof free massless bosons, the spinning null string contains, in addition, free masslessfermions . Of the four constraints (5 .20), two are fermionic and two are bosonic.One of these latter ones is exactly the transverse constraint of the bosonic case,whereas the longitudinal constraint has acquired a purely fermionic piece . Weshall take the opportunity to study in some depth these constraints and theiralgebra in sect . 7 .


163

6. Light-cone quantization of the null spinning string

Canonical quantization of the lagrangian (5.24) is straightforward after using theleft-over gauge freedom (5 .26) to fix a light-cone gauge [37] in order to retain thephysical degrees of freedom . The most general solution to (5.25) does not involveany oscillators,

Xw(T, ~) = Pw(~)T + Y~`(~) ,

'('r, 0-) = +fa(O-) -

(6 .1)

Since the fermionic variables have no r-dependence, they do not contribute to themomentum of the string . Hence the fermionic creation operators will increase andenrich the degeneracy of the spectrum, without contributing to the mass. On theother hand, fermions do contribute to the anomaly and hence both to the criticaldimension and to the net momentum along the string, P,In the light-cone gauge (X+ = (21x7) -t/2P + r, qa = 0), the solution to the con-

straints (5.20) is

2arP_= 2p+ PZ,

27r

iY

_=

P+

(P-Y,+2 (6.2)

where u - v = uY, I = l, . . . , D - 2 . From the third equation above, we find thatthe net momentum along the spinning null string is

1

j7, (

1

. g$ , ~ = puB + puF .2Tr

We may now proceed to the canonical quantization and postulate the following(graded) brackets* for the independent degrees of freedom

~Pj(~),YJ(~~)~ _ - t8tJ8(Q-~~),

[qja(O-), ~j"j(0")] =5a"SIJS( or - cr'),

(6 .4)

As for the bosonic case, we consider only closed null strings . The modeexpansion for the bosonic variables is given by (3 .4), whereas, in complete analogyto the usual string, the fermionic coordinates are double-valued . We have twosectors, depending on whether the fermionic coordinates satisfy periodic bound-

* Here we are actually using the Dirac bracket, since the fact that the momentum of a fermionicvariable is the variable itself yields a second-class constraint which we have already trivially resolved[38] .

164


ary conditions (Ramond sector [39])

or antiperiodic ones (Neveu-Schwarz sector [40])

1

~~~=

27r

E `Vam eima "

(6 .5)mE 7L

I _

,1,I iro .a

0

2Tf

War erEZ +-

The (mass)2 operator is exactly the same as in the purely bosonic case, eq . (3.9) .The bosonic part Pô of P, is also given by eq . (3.9) . The expansion for Pâ is

PF= 1

a

ao 2

Y- a* ' -aa=1,2 7LaE

17 + z

where the index a takes integer or half-odd values depending on the sector(Ramond, R, or Neveu-Schwarz, N) in which W,(o-) is realized . From the commu-tators (6.4) we obtain the following non-vanishing commutators for the string"modes" :

[.Y0 I P+ 1

-t~

Pm7Yn ] - -I5IJ5m+n+

[4'aaI4IPb] =5"5al3sa+b>

(6.8)

[M2

~~, t

j =0

[P

,~~I

1

= a,~, I`l'aa

>

v `Naa

4'aa

(6.6)

from which it is easy to establish that the fermonic operators qIm are ladderoperators for PU , but do not change the mass :

(6 .9)

That is, the operators qaa will increase (a > 0) or decrease (a < 0) the value of P,.The wave functions for the spinning string are thus obtained from the bosonicwave functions (3.18) by applying creation and annihilation fermionic operators tothem .

Concretely, the commutation relations (6.8) can be realized in two ways, depend-ing on whether Oa > 0 or qa , 0 are taken as coordinate variables . This ambiguityamounts to defining the normal order by putting positive-indexed modes to theright of negative-indexed ones, or vice versa . Explicitly, we realize the commuta-


165

tion relations with the help of Grassmann variables 17

aOaa = 71, ,

~a-a = -'977a

and the two orderings mentioned above correspond to imposing (6.10) on eitherpositive or negative a for every value of a, separately .

Clearly, the large variety of possible ordering schemes corresponds to a largevariety of quantum theories with the same classical limit . These theories differamong themselves, quite pointedly, in the value for the central extension of (theVirasoro subalgebra of) the null string superalgebra . This situation is actuallyentirely analogous to that in usual string theory, where D = 26 (or 10 in thesupersymmetric case) arises only as a consequence of nonnal ordering . It isdifficult to provide a criterion for choosing one ordering over another, and all thearbitrariness is only increased when we come to null strings, for which there is nopotential and hence no zero-point fluctuations in the ordinary sense .

Let us work out the consequences of Lorentz covariance for null spinningstrings . The generators of the Lorentz group are

1mg,

2 7r f~d~[Pw(~)Yv(~) -p~(~)YW(~) +t a(~)

a(~)]

(6.11)

or, in modes, MI" =L" + KI" with Lam- given by (4.1) and

K~`° - i E

Y- qlmOa-ma=1,2 m

In order to avoid confusion we shall work in a fixed sector (R-R, R-N, N-R orN-N), i .e . to each value of a corresponds a fixed sector R or N. All modeexpressions are understood to be appropriately ordered, of course .The realization of the constraints is now

a

,yn -ynB

Z Ta,n+a ~ `#a,-a ,a=1,2 a

pn =pnB ,Oaa=p+Oa,a = Epm ' *a,m-a ,m

(6 .10)

(6 .12)

(6.13)

166


where PB and YB are given by (4.3) . We have

l0a,a, `Vli,b) = 2Pa+b 5oli ,

where A(m) =AB(m) +AF(m) is the ordering prescription-dependent centralterm . The value of A F also depends on the sector, i .e . on whether the fermionmodes are integer or half-odd .

Consider the contributions to AF coming from one of the two fermionicvariables, i .e . for a = 1 or 2 . Each of the two fermionic terms of the first equationin (6.13) constitutes, on their own, a Virasoro algebra :

where

[Yn,Yn1] = (m - n)Ym+n +A(m)â~ +,, ,

[Yn,`V.,a] =(m 2I`Va,a+n~

[ ua,n>ua,m] = (m - n)ua,m+n +A,(n) ,

(6.l5)

As in ref. [7], we obtain the following central terms :

dAa(n) _ ±24(m3+2m),

AR_R(m) _d4M3,

AN_N(m) = d

4(m3 _ M),

A R_

aua, n -2 *a, n +a ' *a,-a

a

N( m) =AN-R ( m) =4(ms-

2m l.

dAN(n) _ ± 24 (m3

-m) ,

(6 .14)

(6 .16)

(6.17)

where the plus (minus) sign corresponds to the order with the positive (negative)modes to the right . Therefore, as well as in the bosonic case, the total fermioniccentral term AF(m) depends on the relative ordering of the two fermionicvariables and, essentially, we have two inequivalent orderings : the normal orderwhere all the positive modes are on the right of the negative ones and the Weylorder which in fact amounts to taking for a = 1 the normal order, and for a = 2the antinormal order (i .e . the negative modes on the right) . For the Weyl order,there is no central term . With normal order, for the total central term A(m) _AB(m) +A F(m) we have the following results :

(6.18)

We did not consider mixed orderings, e.g . normal for the bosonic sector and Weylfor the fermionic sector, because they lead to unphysical critical dimensions . Usingthis for the Lorentz algebra, we find that

1

A(m) 2[MI-, M J 1 =

E (p t p, - pJ p t )[

2 - 2m - - Pv I - (6 .19)p+2n> o

-m

m

-m

m m m

Thus, if we insert the central terms (6 .18) we will get the critical dimension D = 10and the PQ intercept values : aR_R = 0, aN_N = -1 and aR-NaN-R

=- z .

We may now establish what the physical content of the theory of spinning nullstrings will be : with Weyl ordering the mass spectrum is continuous from zero toinfinity, and the wave functions are the bosonic wave functions of sect . 3, on whichfermionic creation operators act . Thus at any given mass value one finds infinitiesof bosonic and fermionic degrees of freedom, such that both sectors contain allspin values . From the naive point of view of counting degrees of freedom, there istrivial space-time supersymmetry .With normal order the situation is much more simple and interesting : just as for

the bosonic string, the spectrum contains only massless states (the fermionicvariables do not contribute to the mass) :

N-N

R-R :

R-N

N-R:


167

p t- 1 10>,

1,-1/2 ß,-1 /210>

IAB>

~J i; 1 /21A>

1z,- 1 /21A> (6.20)

where A and B indices are 8-dimensional spinor indices for the Ramond zero-modes, realized as I' matrices . In this case only the state p1 ,10) of the N-Nsector has zero norm, all the other states generated by fermionic creation opera-tors have positive norm . However, this spectrum is not supersymmetric and theonly way we know to achieve it is the GSO projection [41] . In fact, for null stringsthe GSO projection also has the consequence of ensuring the quantum consistencyof the theory by eliminating the zero-norm states . Thus, after the GSO projectionthe content of the null spinning strings will be given by the massless sector of thetype II superstrings, i .e . :

Therefore the null spinning string spectrum is t4at of N = 2 (type II) 10-dimen-

N-N ~1 i.-i/2 (~z,-1/210>

R-R : JAB)

R-N : qfi; i/21A>

N-R : 02-1/2IA> . (6 .21)

168


sional supergravity . After this beautiful result, the reader is probably anxious toturn the page and find phenomenologically sound null string theories . Alas, wehave not yet constructed a heterotic null string, and it ought to be quite tricky :there are no left- and right-movers in the null string .The extension to extended supersymmetric constructions (N> 1) remains to be

fully explored, although a comment is in order [7] . We have refrained fromcomplexifying our real fermions (technically, we deal with N = 1 and not N = 2),in contrast to the massless spinning particle studied by Brink et al . [31] : once theoperators qt are complexified, their (anticommuting) zero-mode can be realizedeither fermionically or bosonically, thus enlarging the spectrum . This allowed theauthors of ref. [31] to include the graviton in the spinning particle spectrum . Forus, however, enlarged supersymmetry does not appear so crucial, since we alreadyhave massless bosonic states with two symmetric space-time indices .

7 . Interactions for null strings

One would guess that the simplest way to introduce interactions among nullstrings would be to find reasonable vertex operators for a variety of states, and tocompute the averages of some of these vertex operators weighted by the action .Unfortunately, however, the action for the null string is essentially the determinantof the induced world-sheet metric, which is zero on-shell . We have not been ableto produce an appealing vertex-operator formalism for null (super)strings, thoughthat is hardly evidence that it does not exist. And yet, it might well be that there isno operator formalism for null (super)string scattering, if the theory is inherentlyand trivially free, so amplitudes would not be computable as for usual strings . Weconcentrate, for simplicity, on the bosonic case, and follow Mandelstam in thelight-cone gauge [42] .

In this marvellous gauge, all states are physical and the world-sheet can berepresented by a rectangle, whose horizontal axis is r and whose ordinate is 0.The height of this rectangle is p + , whereas the length is, when we visualizeS-matrix elements, tending to infinity . The strip may have horizontal cuts in it,representing splitting and joining of strings . Actually, since we are dealing withclosed strings, the total number of edges at any given T must be even and theclosedness is enforced by identifying them pairwise .The identification of in and out states at the left and right ends of the strip

presents no particular problems due to the zero-norm of these states (in thesupersymmetric case, in and out states have positive norm and thus this subtlety isavoided) . A scattering diagram can be computed by evaluating [42,43] the determi-nant of the propagator on the strip, subject to the appropriate boundary condi-tions . For usual strings, the relevant operator is av - a7 . In the null case, theoperator of interest is simply -d,, whose determinant can be evaluated on thestrip and is almost unaffected by the number and position of horizontal cuts (cuts


169

in the T direction) . The only dependence on the cuts comes from (i) the couplingconstant g associated to every point where a cut ends or begins, and (ü) the totalnumber of cuts at different heights (different values of a). Point (i) is trivial, butpoint (ü) deserves some explanation . The determinant of the operator -aT issimply the product of the eigenvalues of the eigenfunctions of this operator on thestrip . These functions are of the form

and

fnl o-, T) = etn7/T h( a )

where we have assumed T E [ - T/2, T/2] . Now, h(Q) must be expanded in a basis .If there are no cuts whatsoever on the strip,

h(Q) = e2mmalp'

(7.2)

det(-a2) = fl fl n2 .

(7 .3)mez nez

The product of all integers squared can be evaluated using ~-function regulariza-tion, and it is some number . The problem comes from the second infinite product .By way of comparison, consider a light-cone process where two null strings mergeinto one, i .e . a strip with one cut from the left end to somewhere in the middle .The basis for the h(o-) functions is now given by only those functions (7.2) whichhave a node at Q = p i /p + , i .e . only those functions which are periodic in all threeranges Q E [0,p + ], [0, pi ], and [pi , p + ] . We have not yet studied in detail how toregularize these products nor have we found explicitly the dependence of theanswer on the number of cuts, i .e . on the genus and number of external states . Anobvious way to regularize the whole thing is to introduce a small tension to modifythe null string propagator and make it into the usual one . We then expect Grossand Mende's results to apply [13] . Of course it would be more appealing to showdirectly that the ratio of any diagram with cuts to that of the uncut strip is zero,and thus

This is completely consistent with all our intuition about null strings but we still donot have a proof of it . The interesting question of what is the physically relevantlimit (does g - - when T ---> 0? how fast? may we introduce a finite coupling fornull strings?, etc .), and which are the consistent first-quantized interactions forWeyl and normal ordered null strings, will be addressed elsewhere .

It is clear that the result S = 1 should follow rather trivially, if true, from thenull string formalism . A relevant and useful check on this statement should comefrom the study of null string field theory . Below we give the explicit form of theBRST charge for null strings . We have not investigated whether the symmetries of

170


the theory allow for the introduction of cubic (and/or quartic?, higher order?)interactions in the second quantization of null (spinning) strings .

8. Hamiltonian constraints for the null string

In this section we turn to the constrained hamiltonian formalism and deal withthe null string as a dynamical system defined by the first-class constraints . For thenull bosonic string, the constraints (2.6) are, in the hamiltonian formulation[44-46],

,r, = P-aa x,

(8 .1)

where X"(7-, o, ) is the position of the string in space-time, and P"(7,0") is itsconjugate momentum .We shall supersymmetrize the above constraints and recover, from a different

perspective, the spinning string discussed in previous sections . Before doing so, it isof great interest to squeeze everything we can from the constraints (8.1) . Inparticular, let us be very general and assume that we know two things about adynamical system with X and P as phase-space coordinates . Firstly, the (first-class)constraints (8 .1) are satisfied . Secondly, the coordinates X and P depend on twointernal smooth parameters but their dynamics does not . In other words, thesecond input is "world-sheet covariance" or reparametrization invariance . Thehamiltonian is then a pure constraint,

,r=A A~l + p, r, ,

(8.2)

where the Lagrange multipliers A and l.c are arbitrary . Now impose a thirdassumption, concerning the formulation of this (very general) theory in the la-grangian formulation : assume that the momentum is linear in the parameterderivatives of the space-time coordinate :

P=aX+,ex, .

(8 .3)

Substituting this ansatz into the expression for the lagrangian density _Z, and thenidentifying P = a simple system of equations for a, ß, /e, and A arises, withtwo different solutions, either a = 0, or else a = 1/2A.

In the first case, P =,6X' is independent of X. With the gauge choice /3 = ßc(3 +A = 1, we find chiral bosons [47] with hamiltonian A P - X' . In the second case,/3 = -A,/2A and we find null strings in a general gauge in which the induced worldsheet metric is

wz

i ), of zero determinant indeed . Schild's gauge corresponds to

the choice ju = 0 .

This relationship between chiral bosons and null strings is interesting but,unfortunately, we have not found any way of exploiting it fruitfully : it is notpossible to go smoothly from one system to the other.Now we shall supersymmetrize, as advertised, the null bosonic string in the

constrained hamiltonian formalism . The equal-time Poisson bracket is quite canon-ical, since the relationship between P and X does not involve any second-classconstraint :

so that


171

[P'(tr),X°(O-')I =77"''8((T -(T') .

(8 .4)

It is easy to check that the constraints (8.1) satisfy the following Poisson bracketalgebra* :

(8.s)

This is the (bosonic) null string constraint algebra.For the usual string, ,YL ± 2TJ,9, = (P 11 ± Ta, X")2 are perfect squares, so one

can choose light-cone coordinates and carry through the supersymmetrization inthe left and right directions of the light-cone [39,401. The supersymmetrization ofthe null string constraints (8.1) is not as straightforward, because no linearcombination of the constraints is a perfect square and hence, the square-rootmethod is not immediately applicable . Following the super-numeral variablesproposed by Collins and Tucker [48], we implement the supersymmetrization ofthe null string in the normal and tangential directions to its dynamical evolution .Accordingly, we define the supercurrents or fermionic constraints by

~(0-) =P~(0-)'I'â (0-)

(8 .6)

where the subindex a = 1 ,1 denotes the normal and tangential directions . Theanticommuting variables IF" and Ti transform as vectors in space-time and asspinors on the world-sheet. They satisfy a Clifford algebra

[

a(07 ),

v(O"')]= isab71"8(0--Q'),

1,f 11r

(8.7)

[-""a~(0~ ),"ab(0')] ='A~1 5abs(0'-0~

')

(8 .8)

* The bracket [, ] is graded according to the variables it acts upon . We use the bracket throughout,the transition to quantum brackets will take place below without a change in bracket notation .

172


and, obviously,

In order to close the superalgebra, 7~1, must be modified . The correct extension isgiven by [5,71

=P a'x(O') - 2 (

l(0-)

â

l(0-) + WAU) - TPI(O'))

(8 .10)

so that

Xj(cr')] =z(~(~)+2--I,(Q'))a~s(o--o,'),

[mil((r), l=(ßl(0-)+-r1(0-'))av5(or-0-'),

[YI(O-)IX'(0')]=(XI(O-)+ M(O' - 0') (8 .11)

The supersymmetric null string is described by the superalgebra in eqs . (8.8),(8.9) and (8.11) . Closure of the superalgebra guarantees that -,"a and '-"-~a_

can beimposed consistently as first-class constraints . The fermionic constraints

gen-erate local N= 1 supersymmetry transformations 8A = [A, E'-,' ] in the temporal(a = 1) and spatial (a = 1) directions . Explicitly,

a 11ra ( O~) =Ea(O~)p~(O~),

5XA(tr) = EEa(O')Pa(O'),

ap~'(O') =0 .a

(8 .12)

We shall restrict ourselves to the N= 1 case, although by adding several Cliffordalgebras this construction can be extended to N> 1 supersymmetry [7] as in thesuperparticle case [49] .The total hamiltonian for this general coordinate invariant system is

H= fdo ~,(A°

+k-"'a),

(8.13)a

where .la are bosonic Lagrange multipliers and .la are fermionic .The boundary conditions for the null spinning string are derived by requiring the

bosonic variables Aa , jXa to satisfy the same boundary conditions as if no fermionswere present . For closed strings, the bosonic variables are periodic in Q E [0, 2-rr],and the T's are the only variables whose boundary conditions must be determined,by requiring that the functionals fAa Ya and ha_la be well defined, i .e . they

contain no surface terms . The variation of the functional

with terms of theform f1Yaa~Y, produces the boundary term

which implies T"(°,)= ±1Yâ

leading to two different sectors for eachfermionic variable . The corresponding boundary conditions on the fermionicLagrange multipliers âa are also periodic or anti-periodic .The Fourier expansion is accordingly

(0)

2Trn

xn__

e

F- Z

ino

p"( 7 ) -

2?T nLr pne

Vfa ( 0-)

Y'n_

W eino2Tr

with a = L ,1 . The non-vanishing brackets are


173

8îYl +'Y,

zv~8~Y1

~l0,

r

v l _= "v

LPMI xnJ- -t~ sm+n,0~

~~~ a",

,/'bv

= 5ab

".v 8[4'm ~Wn ,+

.Im

Pn-k ' Pk ,kc7L

`Yn1 =t E kPn-k'xk-zke7L

__~~~ a

Pn-k 4'k >

kE S ~ +=

k ~~~ a

'~,a

4'n-k * `Vk

(8.14)

(8 .15)

(8 .16)

Note that pl-n =p,"t, and similarly for xn and

nW.The algebra of constraints for the null spinning string can be neatly expressed in

terms of the above operator eigenmodes . Using convention G((T) = (2 rr)-1Y-Gn e`nafor all the constraints G, the constraint modes are

(8 .17)

where in the last expression the sum runs over integers or half-odd numbersdepending on the sector . Care must be exercised to (normal) order all operators .Note that the constraint modes are hermitian .

174


The algebra satisfied by these constraint modes (in any of the sectors) is given by

'] _(n-m) ,_1nn+n+(b3m3+btm)Sm+n,O,

[ -rm, -Y,'] = (n - m)~m+n +(a,m3

+ a1m) (Sm+n,O ,

[ ~l , «al = 0,

rL

oot ,

a] _ (n

m

2 m+n

r

a

b

abl

1

_ (

2

b1L`~m , -n

_ (S

~m+n

263m + 2j Sm +n,0 (8 .18)

where the most general central terms allowed by the Jacobi identities are shown.The null super-Virasoro algebra (8.18) is interesting on its own, and the central

extensions give a clue as to the quantum consistency of the theory . Using the modeexpansions (8.17), it is easy to check that the parameters b 3 and b, of the aboveextensions are zero for the null spinning string, whereas a 3 and a, reveal theconformal anomaly of the system and will be dealt with at length below. To find aphysical realization of the algebra (8.18) with a non-trivial central extension bremains an open problem . This is equally true, in fact, for the purely bosonic case,which looks like the usual (anomalous) Virasoro algebra plus the "tail" providedby the AP', modes, which almost behave as faithful representations of the X,Virasoro algebra except that, tragically, they commute among themselves .

9. BRST quantization of the null spinning string

We quantize the null spinning string using the BRST invariance [501 in its pathintegral BFV formalism [51,52], both because of its elegance [53] and because theformalism is most suited to the constrained hamiltonian approach . The hamilto-nian (8.13) is

H=

Y-

f77'du(Aa

+Àa

a) .

(9 .1)a=L,1 0

The primary first-class constraints -;K, and

involve the variables X", PI

and ~Pl' and span the classical algebra found above . The phase space of the theoryincludes the Lagrange multipliers .la and .la also as canonical variables . Their


175

conjugate momenta are 7ra(Q) and ïrQ (or), with non-zero graded brackets

[Ira(0.),Ab(o-')1 = [ ira(O_),Àb(U')] = -aa3(or-0.') .

The first-class constraints of the theory are the original "Y's and ms's and the7r's . As usual, enlarge the phase space by introducing ghosts ,lla and ~a for theoriginal constraints r, and -1a , and ghosts Na and Na for the Lagrangemomenta -rra and "ira . Introduce also the corresponding conjugate ghost momenta`~PVi Il aa~

a,

a and 1Îa , with non-zero brackets

[,~%

( 07 ), 97 b(Q,)] = [`~;a(Q), ~ b(O ' )]

= [Il,(u),Nb(or')] = [Ùa(o-),Nb(e')] = -saS(0_- O~ ')' (9 .3)

Note the untilded ghosts and their momenta are odd (anticommuting) .The classical BRST charge is a sum of two terms [54,551,

where S2 ° is the sum of products (ghost) X (constraint)

S2 = S2° + S2',

(9.4)

S2 ° = fdQ(r~a +47 a~+Narra +Na7ra)

(9 .5)a=1,1

and S2' takes care of the non-trivial constraint algebra,

f2' = - 2(-)Fn97b7lcCcb`~,ia

(9 .6)

The structure functions Ce can be read off from the algebra [G,, Gb ] = CabG,with Ga any of the first-class constraints (the X's, .,'s and "rr's), and Eb = 0 (1) ifthe constraint Gb is bosonic (fermionic) . In the case at hand, the non-vanishingstructure functions following from the classical superalgebra are as follows :

Cl l = C;1 = p( Q - (T �)+ S(Q' -Q �)] aQ S(o--Q') ,

Cb=iâ(tr - 0,)S(0-,-0-")Sab,

Ca1=[15(Q-07 ") +S(Q'-Q �) ]a~ô(o--0.,)ab . (9 .7)

Explicitly, then,

S2 1 = - fdado-W(Q-o-')+77,(O-)n'(O-')

X0- )+91(o-')]+

711(Q') ~a(Q)[2 Pa( O- )+Pa(0~

')]~a=1,1

fdo- ~a(Q)~a(Q)~~ (a) .a=1,1

(9 .a)

(9.8)

176


Using the null spinning string constraint algebra and the commutation relationsamong all the fields, it is straightforward to check that the classical BRST charge isnilpotent . It is somewhat easier to analyze the effect of the central extensions(resulting from an ordering prescription for quantum operators) in terms ofFourier modes, using the algebra (8 .18). In obvious notation, one finds that

0, f2l

~ ( b3n3 +bin ) 77n 771,+(a3n3 +a,n)rlnr11nJ

nEZL

2

b)~a~,

a=1,1

lnE

l%pm

k 'xk +a

which proves nilpotency of the BRST charge ,fl iff the anomalous central termsvanish .

For the quantum BRST charge, one has [7]

~=a~a

+aa +(rrt-n ) ~

l 77i~l

+ i(m-n) i7' 7l'~'7%k

-k

?7k

k

m

n

m-n

2

m

n

-m-n

nl a 1 a

_ 1 a a

1 1

(9 .10)+ m - - 1~7m?7n`

m-n

2 71m71n`~m-n+GY?7~,

where all repeated indices are summed over, all operator products are assumed tobe ordered according to some prescription, and a is an unknown constant, whichtakes into account that the zero mode 7/ 0' presents ordering problems .The single Virasoro subalgebra ([Lm , Ln ] _ (m - n)Lm+n +Am8m+n) for the

full system is generated by

2~ k

Omm -k ' Wk + (k +m) 71k~m-k

=1,7

+ (k +

r"lkM ) -ce8n, . (9.11)

Quantum nilpotency of the BRST charge ,(2 is equivalent to the vanishing of theanomaly Am. In the case of Weyl ordering, Am = a = 0 and there is no criticaldimension [2] . For the case with normal ordering, the anomaly can be computedusing standard methods [7] : it vanishes only if the minkowskian space-time dimen-sion is D = 10, as in the tensionful case . The value of the constant a depends onthe fermion boundary conditions in the 1 and 1 directions; if these are(RR, NN, RN NR), then a = (0, - 1, - 1/2,- 1/2) respectively.


177

Before concluding this section, we should like to derive directly from the BRSTcharge the action (or rather, an action) for null (supersymmetric) strings. Let uswork with the Q-dependence, rather than the modes. The generating functionalreads as

Z = f_'2X.

a 2a° ecra _1~'Na 2"na) X (tilded) expl if12 -,--' dtl

,r,

(9 .12)

where we choose the following BRST-invariant boundary conditions [52] at*t i = tl,t2 :

7a( ti) =7]a( ti) =Ha( ti) = 'fra(ti) =71a(ti) = na(ti) = 0 .

(9.13)

In terms of an arbitrary odd and imaginary gauge-fixing function T, the effectivelagrangian is

°= f do- [W,~l] +X*P+

E

4ra . 11ra+( A'a7Ta+71a~pa+1VaHa + [tilded])] .a=1, L

(9 .14)

Following Henneaux [52], we choose as a gauge-fixing function something of theform

1P= E fd o (~,Aa + naXa + [tilded]),

(9 .15)a

where Xa and Xa do not involve any ghosts . Even with this restriction, there is stillan enormous variety of possible gauge fixings . The simplest one would appear tobe merely Xa =Xa = 0, but in fact the most convenient gauge is (to our taste)Teitelboim's proper-time gauge [56-58], generalized to the current situation :

_X1=0,

1X 1 = - A ,Xa=

1-Aa

(9 .l6)E

E

where E is an arbitrary parameter . A possible variation of this generalization is totake X L = 0 instead, and X a and X i as shown.

* Because of these boundary conditions, the ghosts II are often referred to as v7, and accordingly themomenta N are labelled as ~. We shall not indulge in such notation .

178


It is simple to find the variation of W under a BRST transformation :

1

1_ 1= Aarfdo

a +rt°~-~aNa-~aNa +-A'Tr, + -k17a + -N 1 17,E E E

1

f+ -N°ÎIa +~1a~7a.

1 1 + fdudQ'8'(o--Q')~i1'(Q) ~ a( u')( ya( O- )E

+ 2ya(Q')) -~a(Q)~'l~l)(2-~ea ( e) + "~Oa(u'))

- (77'(u)AI(u')

17'(0")A'(u))~(0-) + (À'(0-)771(u') +À1(u)771(U'»

X G'-~~O l(u) + -~-P1(0)1

We now perform a resealing [59]

ar, --> E7r, ,

N' - EN' ,

4ra --'> ETra ,

IVa -__, EjVa

(9.18)in the generating functional (9.12), after which the parameter E is taken to zero .The effective lagrangian is then, in terms of the phase-space variables,

'= fdo,

P+4̀a 11̀ a+rtlrrl+~ a` a+7la~a+N1II1- .l1Pz

i-A'(P*X'+2tYa ' 1YQ)-äaP-'Ya -N 1 .~1 - .t'-rr 1 -Aa-ira + HI N'

+liaNa+ ~ai7a,~11 + fdudo-'â'(Q-a')[a'( u)~a(u ')(~a(u)+2~a( O"))

+(rt'(Q)171(0"') +À1(Q)77'(Q')) G~e 1(U) +~~( 0-'»+ (°7'(0"')À'(O-)

- 77 I (Or)A t W))~i(07) -~a(0-)771(0")(z~a(~) + .~a(~'))] .

The functional integral in the Lagrange momenta Tra , ,Fra can now be performed,yielding a bunch of 5-functions of the Lagrange multipliers (note that fermionic8-functions are well defined)

8( .11)S(A1)8(~l')S(J~') ,

8(1V 1)S(N1)S(N')S(N') .

(9 .17)

(9 .19)

(9.20)

which allow us to eliminate the 7--dependence of .l1 and to eliminate ~1 , A' and.l' altogether . The ghost momenta 17a , lia associated to the Lagrange multiplierscan be integrated out as well, yielding similarly

(9.21)


179

The gaussian integration over the coordinate momenta PI can also be performed,taking care that the invariant world-sheet measure is in this gauge dT du A 1 andthat the bosonic zero modes for A 1 may take only positive values [60] . Theresulting vacuum functional is

Z = f .~X" ~1' Wa _2A' IN 1 _9T1 a .2_~'a 271 a ~~a S( .l 1 ) 8 (1V 1 )

Xexp{ift2dTfdu~

1

1 X2 + 1Ya 1Ya +71a.~a +~7a

4A

The ghost integrals involve an operator matrix whose determinant contains A 1and information about the topology (trivial in the free case) of the world-sheet . Itsevaluation (with ~-function regularization, for instance) requires a careful distinc-tion between the Ramond and Neveu-Schwarz sectors and will be presentedelsewhere . It can be written formally as

Z= f_2X" _6YT, _2A 1 5(~ 1)F[A 1 ]

Xexp~if r2dT fdu~

1

1 X2 +4Ya . T

aJ~,

(9 .23)4A

where we expect the functional F[A 1] to eliminate the negative norm-states fromthe original degrees of freedom. Evidently, the action for the matter fields isreduced to that for free bosonic and fermionic coordinates, with the multiplier A1playing the role of a conformal factor .

10 . Conclusions

- (A+A (9 .22)

In this work, we have studied a system with more structure than the pointparticle, and yet simpler than the usual strings with non-zero tension because thetransverse constraint ,~'1 is merely P 2 . Classically, from the viewpoint of a givenpoint of the string, this means that the nearest-neighbour interaction, the harmonicoscillator potential, is absent. Hence every point on the string is pretty much free,except of course that the longitudinal constraint *i enforces its velocity to beperpendicular to the string itself and implies invariance under 0--translations . In atheory of higher-dimensional objects with a Nambu action, the limit of vanishingtension has even more dramatic consequences: the non-linear interaction termpresent in the transverse constraint 1 disappears, and thus quantization ispossible [25, 61, 62] .

180


It could appear as if the T ---> 0 limit of string theories is, modulo technicalities, asimple limiting case . The opposite limit, T- ' - 0, has been used in the past to findthe field theory limit of string theories, because it represents the Regge slope goingto zero, hence the boosting to infinite mass of all the massive states . Naively, thelimit T - 0 corresponds to the slope of the Regge trajectories becoming infinite,i .e . projecting all the massive states onto the M Z = 0 axis . However, due to theinfinity of equally spaced massive states, one could also argue that this limit givesrise to a positive and continuous spectrum. Surprisingly, our analysis above isconsistent with both possibilities, depending on the chosen vacuum : Weyl orderingimplies the second scenario, whereas with normal ordering only the same finite(and small) number of massless states as were present in the field theory limitsurvive . This most intriguing feature of the (normal-ordered) null (super)stringresembles closely the duality symmetry observed for closed strings relating thecharacteristic compactification length R to its inverse 1/R [24] or equivalently thetemperature in string thermodynamics T- 1/T [16] .Thus, the quantization of the Schild action and its supersymmetric extension

yield two very different theories depending on the operator ordering prescriptionadopted . Note that the only two possibilities for the spectrum of a theory withoutany mass scale are realized in these two scenarios : massless and continuousspectra . The massless case carries the bonus of a finite spectrum.Weyl ordering yields (contrary to the usual string case) a consistent theory

which, nevertheless, does not appear to be the limit of the usual string : there is noanomaly, hence no critical dimension . Its continuous mass spectrum points to thepossible interpretation of the supersymmetric case as a degenerate supermem-brane . Further, the infinite degeneracy of its continuous spectrum probably allowsfor space-time supersymmetry regardless of the GSO projection . An implementa-tion of consistent interactions would perhaps help to clear this issue .With normal ordering, the Virasoro anomaly shows up to yield a definite critical

dimension and a definite (non-zero) intercept for the generator of rigid rotationsalong the string, PP . The closed null (super)string constraint algebra contains onlyone Virasoro subalgebra, and the appearance of a central term reflects itself on ashift in the momentum along the string, not the mass squared operator . Theindependence of the string parametrization on the chosen origin for the space-likestring coordinate a is a natural condition for the quantum consistency of theHilbert space, merely the zero-mode of the longitudinal constraint : it is notsomething extra put in by hand . The net quantum non-zero twist developed bysome of the sectors as a consequence of the non-zero "intercept", plays essentiallythe same role as the intercept of the tensionful strings, namely it determines thespin content of the massless states .In this context, supersymmetry appears essential for the quantum consistency of

strings ; whereas null bosonic strings suffer from zero-norm states, null superstringshave a well-defined Hilbert space, and the GSO projection plays a crucial role in


181

eliminating the zero-norm states . In any case, the study of null string interactionswill clarify their applicability to the very high-energy limit of string theories .Classically, we would expect only frontal interactions so that the structure of thequantum theory would be extremely simple as stressed, for instance, by Gross [14] .Another area worth investigating is the null string second-quantized formalism

[63], the starting point for which is the nilpotent BRST charge of sect . 9 . Note thatin usual string field theory, one is led to contemplate interactions among stringswith different values of a'= T- ' [64,65] . Whether the formalism already devel-oped allows us to consider strings with a' = - remains to be explored . Note thatthe BRST charge of the Heisenberg algebra [66] which yields massless particles inarbitrary representations of the Lorentz group [67] is not the same as ours, whichhas much more structure.

Allowing for some free thought, one might conjecture that null (super)strings areto usual strings as massless fields are to massive ones : their crucial purpose is torealize the gauge degrees of freedom . In strict analogy to the particle case, asymmetry of the null (supersymmetric) string, so far unidentified, could yield afinite non-zero tension after its breaking [14] .Another open question concerns the thermodynamics of null (supersymmetric)

strings themselves, not the thermodynamics of usual strings at high temperatures(we have found indeed the conjectured finite spectrum [16]). In principle, thelimiting or Hagedorn temperature is zero, so one would expect that the interactingtheory, if it exists, must be established previous to any relevant explorations in thisdirection . Nevertheless, the simple fact that the degeneracy of states is indepen-dent of the mass level (for the free Weyl-ordered null string) already allows us toexplore the beast (see also ref . [68]) . The partition function for the null string lookslike

Z=f~dmg(m)e -ßm=gfxdme-Rm=gl,6,

(10 .1)o

a

where g is a big overall infinity. The free energy is continuous, the specific heat isconstant, and ( 4E 2 )/(E )Z = 1 . These and related issues are currently underinvestigation . We expect null superstrings to help significantly in the quest forunderstanding quantum gravity and many other stringy miracles .

We would like to thank L . Alvarez-Gaum6, A.P . Balachandran, J . Barcelos-Neto,J . van der Bij, C . Gômez, R. Grimm, J . Labastida, A. Morozov, S . Ouvry, A.V .Ramallo, V.O . Rivelles, M. Scholl, J . Zanelli and A.A . Zheltukhin for interestingdiscussions and comments . M.R.-A . thanks J . Wess at the Institut für TheoretischePhysik, Karlsruhe, and M. Schmidt at the Institut für Theoretische Physik, Heidel-berg, for hospitality at one stage of this work . C.R . thanks the CERN TheoryDivision for hospitality at another stage of this work . J.G.'s work was partially

182


financed through a CAPES fellowship . C.R . i s supported by the DeutscheForschungsgemeinschaft .

Appendix A

In sect . 3 we have shown that the wave functions of the null string (in the Weylordering) are constructed on the solution to the differential equation

1'V(r) + - [2(h+d) - 1]V(r) +4m 2 W(r) =0

(A.1)r

obtained from (3.22) by the rescaling x - r = 6_ . It is well known that equations ofthis type have solutions which can be expressed in terms of Bessel functions [69] .In general, there are solutions for any complex value of m 2 , but we are interestedin normalizable wave functions ; i .e . m 2 has to be real because it is the eigenvalueof a hermitian operator (3.16) . Thus the wave functions corresponding to differentvalues of m 2 must be orthogonal . Since m 2 is a real and not discrete variable,normalizability must be understood in the larger sense that we can build normaliz-able (in the strict sense) wave-packets . In other words, the orthonormality condi-tions for the wave functions 'Y,~ ;,n = ~Vm read

( 1Y., lYm,) = 8(m - m') .

(A.2)

In fact, if we take Bessel functions of the first kind, the corresponding solutionsimmediately fulfill this condition for m, m' :Fl- 0 . The special m = 0 will be dealtwith separately below, although it will turn out that it can be obtained smoothlyfrom W, as m - 0.For simplicity, consider first the case h = 0 (i .e . q = q = 0) . The solution to (A.1)

is

m(r) = Nnr l -dJd-1(2mr)

-Nmmd-~ ( -1)k(mr)Zk

k-o k!(k + d - 1) !(A .3)

We see that the tachyonic solutions (with m2 < 0) are positive for all values ofr > 0 and hence cannot possibly be normalizable . Thus the physical solutions havem>0.

be computed with the help of the standard result

fxJ�(ax)J�(bx) dxx

Making free use of distributions,

Thus, for m, m'> 0, we get


183

The scalar product of two wave functions is given by (3.15) . In sphericalcoordinates, it reads

(Ilrme

lIfm,) =NmNm,122df ~drr 2d-1 ''Ym (r)Y m̀,(r)n

=NmNm�O2df-drrJd_,(2mr)Jd_,(2m'r) ,

(A.4)0

where ~22d = 27rd/(d - 1)! is the area of the unit sphere . The integral in (A.4) can

a2 -b 2 [aJ� +l( ax)Jn( bx)- bJ,(ax)J,+i(bx)] . (A.5)

f~xJ,(ax)J,(bx) dx= lim

2A

2 [aJn+,(an)J,(bn) - bJ�(an)J� +i(bn)]0 A---a- b

1 1

[sin(a-b)rl

� cos(a+b)Ill

"tr ab nl-

a-b

-(-1)

a+b

where we used J�(0) = 0 for n E N and the asymptotic form

2 rJ� (x) =

V 7rx cos[x-(2n+1)

7r

4]

asx~oc .

(A.7)

,rr d

N 2

( m,Vm)

(d - 1)! rn8(m-m ,

(A.8)

184


where the normalization constant is

Thus for m o 0 we have

Nm

which would mean that ql;n - o(r) = 0.In fact, the massless m = 0 case of eq . (A.1) deserves special attention. The

general solution is

where A and B are constants. We have

111o =Ar2(l _d) +B,

(A .9)

(A .10)

(wo, Vfm) =Nmo2df~dr (Ar2-d +Brd)Jd_,(2mr)

(A.11)0

which can be computed as (A.4) above using

rr dm (d_3)/2

( 'Ifo , 11m) = -AN,

fdxxn+ijn(x) = zn+IJn+i(x) ,

fdxx-n+ljn(x) = _X-n+1Jn-1(x)

+BU2

02d

lim

N~Ad- '/2 cos}~mA - (2d + 1) 7

.

(A.12)7F ~A--- m3/4

L

4

For m 0 0, the second term is already zero so that the r.h .s . vanishes only if A = 0 .Therefore, for m = 0, the solution is a constant :

To =B .

(A.13)

Hence, if we take for m :)4- 0 the wave function (A.3), with Nn given by (A.9),and for m = 0 we take (A.13), the wave functions fulfill (6.5) . Actually, the wavefunctions Pm do not have to be normalized to "1" as in (6.5); for arbitrary Nn theprobability density of the mass eigenvalue m in the normalized wave-packet

W(r) - f'p(m) Wn(r) dr

(A.14)


185

would not be p(m) but rather p(m) = Nrn p(m), such that p EL2(IR+ ). Hence, if wetake Nn = Nlmd-t with N = (d - 1)!B, the limit of (A.3) for m -0 will be just(A.13) .The general wave functions with q, q =t- 0 take the form (3.18) :

where ~P., Jr) is the solution of (A.1)

Using (A.4) and (A.8), we get (3.24) .

Wjii. .,'.'J'

m=N(V" . . .V'qVj* . . .Vj*9 )IITm h(r),

(A.15)1

'Ifm,h( r ) = Nmrl-d-hjd+h-,(2mr) .

The scalar product of two such functions is

i~ . . .i 4 ki . . .k q ,

'r, . . .rA,;m-)=Nn,Nm,fo-drrI -h -hJd+h-I(2mr)Jd+h-t(2mrr)

X fd,f2(U l* . . . Vi* UJi . . . UJA)tl(Uk'. . . Vkq,Vr* . . . Ur*I

)tl (A .17)

The integral is over a 2d-dimensional euclidean space with vector length squared

r 2 = vV5 , Vi*

so the solid angle integral in (A.17) yields

Id,f2(Vr* . . .

qVi* Vli, . .Vlq)tl

1

:7 ,(V kl . . .VkgVÎ . . .V) tl

2h

(k, k 2= cr

(Sqq ' 8qq, [ s(i

si2

. .

References

Skq)S(J1 5i2

Srq)1iq)

(11

12 - . .

jq) It,

(A.16)

(A.18)

(A.19)

[11 A. Schild, Phys . Rev. D16 (1977) 1722[2] F. Lizzi, B. Rai, G. Sparano and A. Srivastava, Phys. Lett . B182 (1986) 326[3] A. Karlhede and U. Lindström, Class. Quantum Grav . 3 (1986) L73[4] A.A. Zheltukhin, Pis'ma Lett . 46 (1987) 208; Yad. Fiz. 48 (1988) 487[5] J. Gamboa and M. Ruiz-Altaba, La Cuerda de Schild Supersim6trica, in Proceedings, Sexto

Simposio Chileno de Fisica, ed . M. Massman et al . (Socialismo o Barbarie, Santiago de Chile,1988) p. 258

[61 R. Amorim and J. Barcelos-Neto, Z. Phys . C38 (1988) 643[7] J. Gamboa, C. Ramirez and M. Ruiz-Altaba, Phys. Lett . B225 (1989) 335

186


[8] J. Barcelos-Neto and M. Ruiz-Altaba, Phys . Lett. B228 (1989) 193[9] A.P . Balachandran, F. Lizzi and G. Sparano, Nucl . Phys . B265 (1986) 608

[10] A.A . Zheltukhin, Teor . Mat. Fiz. 77 (1988) 377[ll] G. Veneziano, Europhysics Lett . 2 (1986) 133;

D. Amati, M. Ciafaloni andG. Veneziano, Phys . Lett . B216 (1989) 41 ; Int. J. Mod. Phys . 3A (1988)1615

[12] D. Amati, G. Ciafaloni and G. Veneziano, Phys . Lett. B197 (1987) 81[13] D.J . Gross and P. Mende, Phys . Lett . B197 (1987) 129; Nucl . Phys . B303 (1988) 407[14] D.J . Gross, Phys. Rev. Lett . 60 (1988) 1229[15] D.J . Gross and J.L . Manes, Nucl . Phys . B326 (1988) 73[16] J.J . Attick and E. Witten, Nucl. Phys . B310 (1988) 291[17] M.J . Bowick and S.B . Giddings, Nucl . Phys. B325 (1989) 631;

S.B . Giddings, Phys . Lett. B226 (1989) 55[18] P. Salomonson and B. Skagerstam, Nucl . Phys . B268 (1986) 349[19] E. Alvarez, Phys . Rev. D31 (1985) 418[20] R. Hagedorn, Nuovo Cim. 3 (1965) 147[21] S. Frautschi, Phys . Rev. D3 (1971) 2821[22] P. Olesen, Phys . Lett . B160 (1985) 144[23] R.D . Pisarski and O. Alvarez, Phys . Rev. D26 (1982) 3735 ;

P. Olesen, Phys . Lett . B160 (1985) 408; Nucl . Phys . B267 (1986) 539[24] See, e.g., K. Kikkawa and M. Yamasaki, Phys . Lett . B149 (1987) 357;

J. Lauer, J. Mas and H.P. Nilles, Phys . Lett . B226 (1989) 251[25] U. Marquard and M. Scholl, Phys . Lett . B209 (1988) 434; Phys . Lett. B227 (1989) 227, 234[26] B. de Wit, J. Hoppe and H. Nicolai, Nucl . Phys . B305 (1988) 545[27] B. de Wit, M. Liischer and H. Nicolai, Nucl . Phys . B320 (1989) 135[28] E. Bergshoeff, E. Sezgin and P.K . Townsend, Phys . Lett . B189 (1987) 75 ; Ann. Phys . (N.Y .) 185

(1988) 330[29] M.J . Duff, C.N . Pope and E. Sezgin, Phys . Lett . B225 (1989) 219[30] S. Deser and R. Jackiw, Ann. Phys . (N.Y .) 192 (1989) 352[31] L. Brink, P. di Vecchia and P. Howe, Nucl . Phys. B118 (1977) 76[32] J. Gamboa and V. Tapia, A generic action for strings, Trieste preprint ICTP-IC/87/265 (1987)

unpublished[33] J. Gamboa and M. Ruiz-Altaba, Phys. Lett . B205 (1988) 405[34] I . Klebanov and L. Susskind, Nucl . Phys . B309 (1988) 175[35] G. 't Hooft, Proc . Moscow Seminar on Quantum Gravity (USSR Academy of Sciences, May 1987)[36] P. Howe, J. Phys . A12 (1979) 393[37] P. Goddard, J. Goldstone, C. Rebbi and C.B . Thorn, Nucl . Phys . B56 (1973) 109[38] R. Casalbuoni, Nuovo Cim. A33 (1976) 115[39] P. Ramond, Phys. Rev. D3 (1971) 2415[40] A. Neveu and J.H. Schwarz, Nucl . Phys . B31 (1971) 86; Phys . Rev. D4 (1971) 1109[41] F. Gliozzi, J. Scherk and D. Olive, Phys . Lett . B65 (1976) 282; Nucl . Phys . B122 (1977) 253[42] S. Mandelstam, in Strings and unification, ed . M.B . Green and D.J . Gross (World Scientific,

Singapore, 1987)[43] For an in-depth review, see M.B . Green, J.H . Schwarz and E. Witten, Superstring Theory

(Cambridge University Press, New York, 1987)[44] P.A.M . Dirac, Can. J. Math . 2 (1950) 129; Lectures on quantum mechanics (Academic Press, New

York, 1965)[45] A. Hanson, T. Regge and C. Teitelboim, Hamiltonian constrained systems, Accademia dei Lincei

(1976) Roma[46] K. Sundermeier, Constrained hamiltonian systems, Lecture Notes in Physics (Springer, Heidelberg,

1985)[47] R. Floreanini and R. Jackiw, Phys . Rev. Lett. 59 (1987) 1873[48] P.A. Collins and R.W . Tucker, Phys . Lett . B64 (1976) 207; Nucl . Phys . B121 (1977) 307[49] P. Howe, S. Penati, M. Pernici and P.K . Townsend, Phys . Lett . B215 (1988) 555


187

[50] C . Becchi, A . Rouet and R . Stora, Phys . Lett . B52 (1974) 344; Ann . Phys . (N.Y.) 98 (1976) 287 ;LV . Tyupin, Lebedev report FIAN-39 (1979) unpublished

[51] E.S. Fradkin and G.A . Vilkovisky, Phys . Lett . B55 (1975) 224 ;I .A . Batalin and G.A . Vilkovisky, Phys . Lett . B69 (1977) 309 ;E.S . Fradkin and T.E . Fradkina, Phys . Lett . B72 (1978) 343

[52] M . Henneaux, Phys . Reports 126 (1985) 1 . It includes very complete references[53] C. Teitelboim, in Proc. La Plata Latin American School of Physics (ELAF 87), ed. L . Epele and

C . Garcia-Canal (World Scientific, Singapore, 1988)[54] M . Kato and O.K. Ogawa, Nucl . Phys . B212 (1983) 443[55] S . Hwang, Phys . Rev . D28 (1983) 2614[56] C. Teitelboim, in Superspace and supergravity, ed . S.W . Hawking and M . Roéek (Cambridge

University Press, New York, 1981) ; Phys . Lett . B96 (1980) 77 ; Phys . Rev. D28 (1983) 297[57] B.S. Skagerstam and A . Stern, Nucl . Phys . B294 (1987) 636 ;

T . Lee, Ann . Phys. (N.Y .) 183 (1988) 191[58] J . Gamboa and V . Rivelles . Spinning self-dual particles, S5to Paulo preprint (1988); BRST

formalism and the quantization of hamiltonian systems with first-class constraints, in Proc . Conf.Honoring J . Douglas, ed . T.M . Rassias (World Scientific, Singapore, 1989)

[59] E.S . Fradkin and G.A . Vilkovisky, Quantization of relativistic systems with constraints : equivalenceof canonical and covariant formalisms in quantum theory of gravitational field, CERN preprintTH-2332 (1977) unpublished

[60] C. Teitelboim and M. Henneaux, Ann . Phys . (N.Y .) 143 (1982) 127[61] I .A . Bandos and A.A . Zheltukhin, Multidimensional null membranes : absence of critical dimen-

sions, Kharkov preprint (1989) Yad . Fiz ., to be published[62] J . Gamboa, C . Ramirez and M . Ruiz-Altaba, Phys . Lett . B231 (1989) 57[63] P. Ramond and M . Ruiz-Altaba, in Proc . II Mexican School of Particles and fields, ed . J .L. Lucio

and A . Zepeda (World Scientific, Singapore, 1987) and references therein[64] H . Hata, K. Itoh, T . Kugo, H . Kunitomo and K. Ogawa, Progr . Theor . Phys . 78 (1987) 453[65] A . Neveu and P . West, Nucl . Phys . B293 (1987) 266[66] S . Ouvry and J . Stern, Phys . Lett . B177 (1986) 335[67] J . Labastida, Phys . Rev . Lett. 58 (1987) 531 ; Nucl . Phys . B322 (1989) 185[68] F. Lizzi and G . Sparano, Phys . Lett . B232 (1989) 311[69] S . Gradshteyn and I.M . Ryzhik, Table of integrals, series and products (Academic Press, New

York, 1980) p. 971

null spinning strings

Documents