New covariantly quantizable action for superparticle

Nuclear Physics B (Proc. Suppl.) 18B (1990) 180-190 North-Holland

180

NEW COVARIANTLY QUANTIZABLE ACTION FOR SUPERPARTICLE

R. Kallosh 1

Theory Division, CERN CH-1211 Geneva 23, Switzerland A new classical gauge action for superparticle is suggested. This action has a double set of infinite reducible gauge symmetries. The Brink-Schwarz action can be obtained from the new one by partial gauge-fixing. The new action can be quantized in the covariant gauge and in the unitary gauge, where it coincides respectively with the covariant gauge and the light-cone gauge of the Brink-Schwarz superparticle. The corresponding regular BRST operator breaks explicitely the infinite number of twisted N=2 supersymmetries of the covariant gauge-fixed action down to one N=I SUSY and gives the correct cohomology.

1

Introduction

It is a great pleasure to participate in the meeting honoring Raymond Stora on his 60th birthday and to give a talk with the purpose to describe the BRST-operator for the superparticle. For a rather general class of gauge theories this famous operator was discovered by Carlo Becchi, Alain Rouet and Raymond Stora in France and by Igor Tyutin in USSR. The BRST-ccnstruction is the most powerful and beautiful way to understand quantum theory of &,m~,e symmetric systems. However, until recently it was impossible to find a consistent Lorentz-covariant BRST-quantization of the manifestly space-time supersymmetric objects, like superparticle, superstrings and supermembranes, because these theories do not fit into the general class of gauge theories, considered before. We are happy that now, with the appropriate classical action for the superparticle being discovered, the BRST-quantization has been peformed for the new type of gauge systems. It is thus confirmed that the BccchiRouet-Stora-Tyutin construction is not only powerful and beautifid but also flexible enough to allow the modifications which solve the problem of quantization of most intricate gauge systems. The covariant quantization of the superstring with the Green-Schwarz action [1] and of the superparticle with the Brink-Schwarz action [2] is a longstanding problem. It was IOn leave of absence from: Lebedev Physical Institute, Moscow, I17924, USSR Address after 1 September 1990 : Phys. Dept., Stanford University, Varian Bldg. , Stanford CA 94305

0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-llolland)

R. Kallosh~New covadantly quantizable action for saperparticle

181

pointed out by Bengtsson and Cederwal [3] that the main problem in covariant quantizatioff of manifestly supersymmetric superstring and superparticle theories is related to the existence of the second-class constraints, mixed with the first-class ones. Indeed, this was the cause of problems in searching the appropriate BRST operator. However after understanding the issue of the infinite reducibility [4], the covariant gauge-fixed actions have been found, both for superparticle [51- [61 and for superstring [71. The counting of propagating degrees of freedom in these theories in covariant gauges coincided with that in the light-eone gauge• For the covariant superstring action it was shown that the conformal anomaly is vanishing [7]. This was the consequence of the equivalence between the spinorial representation of the OSp(10/4) supergroup and the one of the SO(6) group [8]. It has been also noted [9] that the covariant action for the superstring can be reproduced from the requirement of the equivalence of the corresponding vacuum functional to the one in the light-cone one, simultaneously with the requirement of vanishing conformal anomaly. However the solution of the quantization problem proposed in [5] - [7] was not complete, since despite the discovery of a nice gauge fixed action for each of these theories, it was impossible to construct the corresponding BRST operators with the correct cohomology by any known method of quantization. Recently such a BRST operator has been found for the superparticle theory [10]. It has been constructed from all fields entering the gauge-fixed action in the free covariant gauge. The existence of such an operator was indicated by the fact that the counting of the propagating degrees of freedom in the gauge-fixed action was correct. However, the procedure of construction of this operator in [10], starting from the classical action for the superparticle given by Brink and Schwarz [2], was a kind of art rather than a regular procedure. The purpose of this paper is to suggest a new action which gives a correct description of the superparticle. This action has only first-class constraints which are infinite reducible. After a partial gauge-fixing one gets the Brink-Schwarz action with the mixed first- and secondclass constraints. The new action can be quantized in a regular way. By the regular way we mean the possibility to use the full power of the Batalin-Fradkin-Viikovisky Lagrangian and/or Hamiltonian quantization Of gauge theories with finite reducibility [11] - [13] supplemented with our recent experience of dealing with infinite reducibility in the case that the corresponding set of fields forms a representation of a supergroup [141, [8].

2

The

action

Consider the following classical superparticle action

L,t = P~,X#-IPag+ ~.~{AP'P(~,p - AP')'~)(p- (Ap'v +Op,v ¢ + )~P+")'+' - Op+z,p+, ¢)%}

• O)

p=O

The independent fields in this action are the usual fields of the bosonic particle action (Pu, X u, g) and also the anti-commuting ten-dimensional spinors of positive chirality, (0p,p)" and those of negative chirality, (Av'v)~, and p starts from zero and goes to infinity. We will use the following notations 2: 2The action, given in the Appendix of [15] can be obtained from our action at Kp = 0,p = 1,2,3, ...oo

182

R. Kallosh / New covariantly quantizable action for superparticle Kv =

,Xv,pg

Fp

0p+Q

=

v+'

,

(2)

where QV and (~v Qp =

,Xp'p- ~'0p,p

Op =

~"'~+~'0p,p

(3)

are the charges of a twisted N=2 supersymmetry:

{Q.,Q~} = -2 ¢~,~ {0",0 ~} = +2 ~,~.,~

(4)

1 Hd = x P~g + KP¢p + F P %

(5)

The classical Hamiltonian z,

is given by the set of first-class constraints T~=

{p2, Kp, F p}

(6)

times the Lagrange multipliers

~={g,

Cp,

..}

(7)

{T~, T~} = fhTo

(s)

The only nontrivial part of the superalgebra

is the anticommutator {K q, F p} = p2(6q,,+'_ 6q,v)

(9)

Thus our action has a very simple form of the type L d = Pi;1i - T a ( p , q ) ¢ " = Viii i - Hot

(10)

In (10) we denoted the set of coordinates {X", 0p,v} by qi and the set of canonical conjugate momenta {P", Av'p} by Pi. The action has a global super-Poincare symmetry, which in this case means the space-time Lorentz symmetry as well as N=I supersymmetry, realized as follows. ~ susyO0,o

--"

f.

~susy,~ 0'0

=

--e

,5~,~X ~" =

e'fl'O

gs~8~g =

2¢0e

(11)

Note that the global symmetry of the quadratic part of the action is much higher then that of the full action. The quadratic part has infinite number of twisted N=2 supersymmetries, which are broken by the constraints terms in the action down to one N=I SUSY. These global symmetries will, in fact, survive in the covariant gauge-fixed action and the physical states

183

R. Kallosh / N e w covariantly quantizable action for superpartide

will form the representations of one simple N = I SUSY due to the special properties of the BRST-operator. The classical action (1) is much more complicated in the second order form, when the equation of motion for P , is soh, ed. It is however instructive to present it here with the purpose to have some insight into the possibilities to make a generalization to superstring theory. oo

1 1 "-'~ Lo, = ~o(X --

~ {x~"-r"¢,, - (o,,.,~ - o~+,.,.-,)-t",7,,})' p=O

+

~1~,.,0...-

(~,.~ + ~,+'.,+')~.}

(12)

p=O

3

Gauge symmetries

of the action

The gauge action of the type of (10) has a set of gauge symmetries, defined by the firstclass constraints T=, their algebra and their zero modes. The set of gauge transformations, generated by the first-class constraints is given by the following eqs.

@i----" {Pi, T=}~ = ~qi = {qi, Ta}~a ~d2a =

(13)

( ~ d t + f~cd2c)~ b

These symmetries are infinite reducible in the following sense. The first type class constraints have zero modes T.Z~I = O, (14) the n-stage zero modes have their own zero modes for each n, Za: -' Za:+, Jr f22~:brb ---- 0

,

(15)

and the procedure of finding the zero modes never stops. The existence of the first stage zero modes given by (14) means that the action has additional gauge symmetries generated by

z:,: ~Pi = 0

~qi = 0

,

,

~¢= = Z~1~"'

(16)

In our particular case the gauge symmetries of the action generated by the first dass type constraints are gP"

=

0

6X"

= =

)~v'vT'%"v + Ov,vT"(~ p - ~v-,) + ~ P " (~. - ~ . _ , ) ~' ,

~¢p

=

kv

6~v

=

iv

~,~

,

,

(17)

184

R. Kallosh / New covariantly quantizable action for superparticle The additional symmetries generated by the first stage zero modes are

&7p = ~Pp , 6g = 2pp(Op,p -Op+l,p+l) + 2~v)ff 'p

(18)

These transformations have their own zero modes, etc.

4

Quantization

The classical Lagrangian is given by eq. (1) and is rewritten in (10) in the form adjusted to the use of the Batalin-Fradkin effective canonical action [11]. This action in a very general class of gauges is given by the following expression:

Sg.j. -- / d l ( Z { PminOrnin -}" PnonminOnonmin } - H,~)

(19)

The first term in (19) presents a set of minimal fields, including the original classical pairs (p, q) and the set of ghosts (75, C), which serve to realize the complete algebra of the firstclass constraints and their zero modes. The second term in (19) presents a set of nonminimal pairs, which consists of Lagrange multipliers (~r, ¢) and pairs of additional ghosts (anti-ghosts) (C, P). The unitarizing Hamiltonian is given by H,=H+{ql,

fi}

,

(20)

where ~ is the gauge fermion defining the gauge condition and 12 is the nilpotent BRST operator, (21)

f'l = f~min(P, q; 7), C) "4- ~non,nin

The complete information about the first-class constraints T~, all their zero modes Z=~"_~ and all structure functions of the algebra of the type jfa._~ a.+l is codified in the minimal part of the BRST operator. The nonminimal part of the BRST operator is universal and for any theory is given by the following equation: = , (22) The heart of the theory is f/m;n. For our particular theory given in (1) and in (10) we are able to present the exact result in a closed form, despite" the theory is infinite reducible. First of all, we are going to specify the notations. It turns out that the minimal set of fields for the theory under consideration coincides exactly with the full system of classical fields and ghosts introduced in the covariant quantization of the Brink-Schwarz superparticle [5] [6]. We will use the notations from [8]. The set of ten-dimensional spinors of alternating Grassmann parity and chirality forms two conjugate spinorial representations of the orthosymplectic supergroup OSp(10/4), O(j,m)

,

A(j, - m )

,

(23)

R. Kallosh / New covariantly quantizable action for superparticle

185

where O <_j < c¢

,

(24)

I m l< j

This is the SO(10) × SU(2) content of the metaplectic representations of OSp(10]4) [8]. The states on the horizontal lines of tile triangles, given in the Figs. to ref. [8] fit into SU(2) multiplets IJ, m >, < j, -ml. When 2j is even the field 0 is a fermion of a positive chirality, while when 2j is odd it is a boson of opposite chirality. For A-fidds fermions with even 2j have negative chirality, bosons with odd 2j have positive chirality. The precise relation of new notations with our previous ones in refs. [6], [10] with 0 < q _< p is the following: On,q

O(j, m)

~(j,-m)

~n+,,q+, = O(j = (p + q + 1)/2, rn = (q -- p - 1)/2)

=

=

= O(j = ( p + q ) / 2 , ra = ( p - q ) / 2 ) ,

{

~'q

= A(j = ( p + q ) / 2 , - - m = ( p -- q ) [ 2) ,

•,~p-t-l,q+l

"-

)t(j

---

( p -1L q + 1 ) / 2 , - - m

(ZS)

= (q -- p -- 1 ) / 2 )

Using the (j, m) labelling of the fields, which corresponds to an assignment of the fields to a representation of the supergroup OSp(10/4), we are able to present the BRST operator of the theory in a relatively simple form. We will use the following notations: em= +1/2 f o r m ) _ O em = --1/2 f o r m < O Zm = 2Cm(--1) 'n(2"=-1}

(26)

The function Z,~, defined ahove, has the following property: Z,~ = Z-,~-112

(27)

The dependence of the BRST-operator on spinorial fields will enter through the special combinations: s(j,m) so(j,m)

= ~ 0 ( j , m ) - 20(j + e ~ , m + 1/2)b = s(j + em,m + 1/2) - 6(m + 1/2)s(j + 1/2,0) + ~(rn + 1)20(j,O)b

(28)

Now everything is prepared to write down the BRST operator ft,~in in terms of all spinors, furnishing 2 conjugate representations of OSp(IO/4). The procedure behind this is pretty regular, one just uses the Batalin-Fradkin prescription [11]. The result was, in fact, deduced in [10], just because it was expected to exist due to the group theory equivalence between the 0 S p ( 1 0 / 4 ) and S0(6) representations. Still its origin from the point of view of quantization was mysterious. But now, taking into account all first type class constraints and their zero modes, one gets 1 2 ~ m=j fl,~i, = ~cP + ~_, ~ A ( j , - m ) s " ( j , m ) , (29) j---om=-j

186

R. Kallosh /New covadantly quantizable action for superparticle

where

(30)

s"(j,m) = so(j,m) + A(j + 1/2,m + 1/2)Z,.

The minimal BRST-transformations of the 0-fields are very simple in these notations. Taking into account that [O(j,m),A(j',m') } = ~(j,j')6(m, -rn'), (31) we get for 6BRsTO(j, m.) = [O(j, m ), Qmin }

(32)

~BRsTO(j, m) = s'(j, m)

(33)

s'(j,m) = so(j,m) + X(j,m)

(34)

the following expression: where and

X(j,m) = [A(j + l/2,m + 1/2 ) + A(j

-

(35)

ll2,rn + l/2)]Z,n

The operator fl~i,~ (29) can be also rewritten in the following form 1

2

oo ,n=j

1-

ft,,,i,, = ~cP + ~., ~_, A(j,-m)(so(j,m) + ~A(j, rn))

(36)

j=Om=-j

It is nilpotent and the solution to its cohomology is the part of the N=I supersymmetry charge q0 = Ao _ s(0,0), (37) which cannot be represented through ~Tq°. This is explained in detail in [10].

5 5.1

Different gauges Brlnk-Schwarz

action

as a partially

gauge-fixed

new

action

One can take the classical action given in (1) and make a partial gauge fixing

Lv.g.! = Ld + rcp(t~P - Q'+')

(3s)

/~P = O p _ Qp+,

(39)

The operator has the following commutation relations with the constraint F": {F p,/~q} = - 4 ~?

(40)

This means that adding the gauge-fixing function as in (38) breakes half of gauge symmetries, generated by F p, since only half of the terms in the commutator of F n with Fq, produce the invertible right hand side. Namely, {7 + F~, 7 +/~q} = -27 + P+

(41)

R. Kallosh ~New covariaufly quantizable action for superparticle

187

In this way also the mixing of the first- and second-class constraints is introduced and the structure of the zero modes of the classical summetries is changed drastically. To proceed one integrats over 1/p and ~rp and gets 1

2

Lp.g.l. = P~X~ - ~ g P + Y~ {~P'P0p,~ - (p ~?M'~ }

(42)

p=O

under the conditions O p + Q"+~ = 0

, (~P - QP+x = o,

(43)

which are equivalent to the following ones

0

,p>o

(44)

Inserting these conditions into the remaining part of the partially gauge-fixed action we get L = pu(.~. _ 0o,o.yu0o,o) _ l ( g _ 20o,a(o)p2

(45)

Performing the shift in metric/Sg = -200,o(0, we get the Brink-Schwarz action [2] for the superparticle. 1

L s s = P~(.~" - Oo,oTUOo,o) - -~gP

2

(46)

Thus we end up without {-symmetries and only with the h=-symmetry, acting on the fields, remaining in the action.

5.2

Semi-light-cone gauge

One can start again with the classical action given in (1) and perform this time a complete gauge fixing, a Ls.l.c. = L c t + ~rp(Qp - Qp+l) + b~ + 7r(9 - 1 ) + pT+Oo,o

(47)

After integrating over r/p, ~rp and p one gets the semi-light-cone gauge action [16], Ls.t.~. = P j ( ~ - 1 P 2 - Oo,o'y-P+Oo,o + bb

(48)

Z

5.3

Covariant gauge

To get the covariant gauge starting from the classical action (1) one can use eelS. (19) and equation (20). In our case, as it was already explained, the set of minimal spinorial fields is given by two conjugate OSp(lO/4) multiplets A ( j , m ) and O(j,-rn). The set of non-minimal

aSince after the partial gauge fixing, presented in the previous subsection, we get tile BS action containing the second-class constraints, one should include into the path integral the me,x~ureof integration (dctP+) -4.

R. Kallosh / New covariantly quantizable action for superparticle

188

fields consists of two sets of fields of opposite statistics, namely the Lagrange multipliers (~r, ¢) and anti-ghosts (C, P) for all stages of reducibility. The unitarizing Hamiltonian in our case is given simply by the equation (20) with H = 0. Our main purpose now is to find a gauge-fixing function in such a way that all non-minimal fields become non-propagating. We take the gauge-fixing functior, in the form = ~ { 6 ' ( X ( ¢ ) + 6) + 75¢}

(49)

The sum in (49) means the sum over the full set of anti-ghosts and Lagrange multipliers. For the unitarizing Hamiltonian one gets, using equations (20),(21) and (49), -

6X

S,~ = ~_.pnonminO, r,onrnin + T ( p , q ; ~ , C ) ¢ + ~rx(~b) - T'(C-~-~ + ~),

(50)

where the full set of all reducible constraints is denoted by T, T =

Thus He indeed contains the Pt0-term for all non-minimal fields, which is necessary to make them non-propagating after substituting He given in equation (50), into equation (19). As a result one gets the gauge-fixed action for the given class of gauges, where only minimal fields are propagating.

Sg.l" - - f

-

dt E{PminQmin

-

~ix

T¢ - rX(¢) + 7~(C~, + ~)

.

(52)

For all spinorial Lagrange multipliers we choose the gauge-fixing fimction to be Xspin (¢spin) -- Cspin

,

(53)

and the gauge-fixing fimction for the multiplier to the reparametrization constraint to be X,~p(¢~, = g) = g - 1

(54)

Integrating over the Lagrange multipliers to gauges r and over T~one gets the expected free covariant gauge-fixed action, and all non-minimal fields drop from it.

Sg.I, =

1

2

oo rn---j

dt{P,.f( ~' + b6- -~P + ~

~

.~(j,-m)0(j,m)}

,

(55)

j=O m---j

In eq. (55) for our particular theory we are left besides clasical fields with ghosts C = {c,O(j,m)} and their canonical momenta ~5 = {b, A(j,-m)} This action was proposed before in [5] and [6], starting from the Brink-Schwarz superparticle action. The BRST-operator in this gauge is exactly the Q,ni~, given in eq. (29).

6

Conclusion

Thus we have found the covariantly quantizable classical action for the superparticle. Its quantization reproduces in a regular way the covariant gauge-fixed action [5], [6] and the BRST operator [10] coinciding with those of the Brink-Schwarz superparticle. However, as different

R. Kallosh/ New covariantlyquantizableaction for saperparticle

from the previous direct attempts to obtain the BRST operator with the correct cohomology, this new action leads quite straightforwardly to the fu'_lyquantized theory. In arbitrary gauge the BRST operator defines the gauge-fixed action, in a simplest covariant gauge all fields are free and spinors package into two conjugate representations of tile 0Sp(10/4) supergroup. The new action has two types of gauge symmetries: ~-symmetry, whidi is a reflection of N=I space-time supersymmetry in the world-sheet and also ~-symmetry, which is a local in the world-sheet partner of a set of space-time twisted N=2 supersymmetries. Because of that the new action, described in the paper, has only first-class infinite-reducible constraints, and therefore its covariant quantization is simple and confirms the correctness of the covariant gauge-fixed action [5]- [6] and of tile BRST operator obtained ill [10]. The main question now arises: Is it possible to find a generalization of the superpartide classical action given in eq." (1) for the manifestly space-time supersymmetric string theory?

References [1] M.B. Green and J.H. Schwarz, Phys. Lett. B136 (1984) 367. [2] L. Brink and J.H. Schwarz, Phys. Lett. B100 (1981) 310. [3] I.Bengtsson and M.Cederwall, "Covariant Superstrings do not admit Covariant Gauge Fixing", Goteborg preprint 84-21 (1984). [4] R. Kallosh, Phys. Lett. B195 (1987) 369. [5] W. Siegel, in Strings 89 (World Scientific, Singapore, 1990) p. 211; M.B. Green and C.M. ttull, in Strings 89 (World Scientific, Singapore, 1990) p. 478; U. LindstrSm, P. van Nieuwenhuizen, M. RoZ:ek, W. Siegel and A. van de Ven, Phys. Lett. B224 (1989) 285. [6] E.A. Bergshoeff and R.E. Kallosh, Phys. Left. B240 (1990) 105. [7] R.Kallosh, Phys. Lett. B224 (1989) 273; Phys. Lett. B225 (1989) 49; M. Green and C. ltnli, Phys. Lett. B225 (1989) 57; J. Gates, M. Grisaru, U. LindstrSm, P. van Nieuwenhuizen, M. Ro~:ek, W.Siegel and A. van de Ven, Phys. Lett. B225 (1989} 44; E.A. Bergshoeff and R.E. Kallosh, Nucl. Phys. B333(1990) 605. [8] I. Bars and R.E. Kallosh, Phys. Lett. B233 (1989) 117. [9] , L. Brink, "Covariant superstrings from the light-cone gauge', Goteborg preprint 89-45 (1989). [10] E.A. Bergshoeff, R.E. Kallosh and A. Van Proeyen, CERN preprint TIl.5788190. [11] I.A.Batalin and E.S. Fradkin, Phys. Lett. B122 (1983) i57. [12] I. Batalin and G. Vilkovisky, Phys. Rev. D28 (1983) 2567. [13] M. Henneaux, Phys. Rep. 126 (1985) 1.

189

190

R. Kallosh / New covariantlyquantizableaction for superparticle

[14] M.B. Green and C.M. Hull, Phys. Lett. B229 (1989) 215. [15] A.Mikovi~, M Ro~ek, W. Siegel, P.van Nieuwenhuizen, J.Yamron, A.E. van de Ven, Phys.Lett. B235 (1990) 106. [16] S.Carlip, Nucl.Phys. B284(1987) 365; R. Kallosh and A. Morozov, Int. J. Mod. Phys. A3 (1988) 133.