BRST charge for the Brink-Casalbuoni-Schwarz superparticle

Volume 226, number 1,2

PHYSICS LETTERS B

3 August 1989

BRST CHARGE FOR THE B R I N K - C A S A L B U O N I - S C H W A R Z SUPERPARTICLE

A.R. M I K O V I C 1 Department of Physics and Astronomy, University of Maryland at College Park, College Park, MD 20742, USA Received 1 April 1989; revised manuscript received 14 April 1989

We calculate the BRST charge for the Brink-Casalbuoni-Schwarz superpanicle in the hamiltonian formalism. In spite of the non-covariant definition of the Dirac brackets the expression for the BRST charge is manifestly covariant. Implications for "the covariant canonical quantizafion are discussed.

1. Introduction

2. Hamiltonian formulation

One can attempt to covariantly quantize the BrinkCasalbuoni-Schwarz (BCS) superparticle [1] either in the hamiltonian or the lagrangian formalism. The most general hamiltonian quantization procedure was formulated by Batalin and Fradkin [2], while the most general lagrangian procedure was formulated by Batalin and Vilkovisky [3]. If these two procedures are related, then the hamiltonian formulation is more fundamental, since by integrating over the m o m e n t a one gets the lagrangian formulation. Also, the path integral measure and unitarity follow from the hamiltonian formalism. The BRST charge is very important for the covariant quantization and for the second quantization (field theory actions) [4]. In this paper we calculate the BRST charge for the BCS superparticle in the hamiltonian formulation. Although the lagrangian formulation is manifestly covariant, it is not clear how to define the BRST charge. On the other hand, there exists a general expression for the BRST charge in the hamiltonian formalism; however, in the case of the BCS superparticle the manifest Lorentz covariance seems to be lost. Surprisingly, the BRST charge turns out to be covariant, which makes the idea of covariant canonical quantization of the original superparticle (i.e. without any modification ) plausible.

The action for the BCS superparticle [ 1 ] can be written in the hamiltonian form [ 5 ] as

SBcs= f d r ( P J C + n ~ O ~ - ½ 2 p 2 - 2 " d ~ , ) ,

(2.1)

where

d~ = ~ + ip~/~O/~

(2.2)

is the supersymmetric covariant derivative and P~t~=-7"~,P,,. The superparticle coordinates are (x a, 0 ~) where (p~, ~z~) are the corresponding canonically conjugate momenta, a is a D dimensional vector index, while c~, fl are the irreducible D dimensional spinor indices. ~'~,~/~ and ),,y~ are the corresponding gamma matrices, which are D dimensional generalizations [4] of the D = 4 sigma matrices. Unless we say otherwise, we will assume that D = 10 since then the superparticle describes the zero-slope limit of the GS superstring. The D = 10 analysis can be straightforwardly generalized to other D's. One can easily show that the constraint d~ = 0 is the mixture of 8 first class and 8 second class constraints [5]. The firstclass part can be covariantly separated as

.~"=p"/~d/~ .

(2.3)

The second-class part cannot be covariantly separated [ 5 ] and it can be written as Work supported by NSF grant PHY 87-46846. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

=~ +~/~d/~,

(2.4) 89

Volume 226, number 1,2

PHYSICS LETTERSB

where y+ = ( 1/x/2 ) (~o + ~9). By evaluating the Poisson brackets {(¢~, ~ } = 2 i x f 2 p + ~+'*~,

(2.5)

one can define the Dirac bracket as (~} C,~p{ (~P, B},

{A,B}D={A,B}-{A,

(2.6)

where 1 (1

~)

(2.7)

2-77+ 0 The blocks in the matrix (2.7) correspond to 8~ and 8~ subspaces of the little group SO (8). The usual canonical commutation relations are not preserved by the Dirac bracket (2.6). For example

Also the covariant momenta p, and d= satisfy {p, d}D = 0 ,

{d., d#}D=ip.~--

i

2 x/2 P~;,7+ ~'~P6~. P+

(2.8a) (2.88)

However, the algebra of the first-class constraints ~ = ½p2and .~ =~bd is preserved: {d, ~ } D = 0 ,

{~,

"~}D =4ip ~ ¢ "

(2.9)

This is not an accident, but a general feature of the constrained systems. It is a consequence of the fact that the Dirac bracket of any quantity with a firstclass constraint is the same as the corresponding Poisson bracket, if both evaluated on the constraint surface. In other words, this is just the statement that the Dirac bracket has to preserve the gauge symmetries of the system. (A nice discussion of the constraint systems with the second-class constraints can be found in ref. [ 6 ]. ) Another important feature of our system is reducibility, since the first-class constraints are linearly dependent. They satisfy 2~4d, _ .~ Bp~ = 0 .

Tio(Zl)i°il ~.~0 ,

io=1,..., mo,

(23"--I)13--2/s 1(Zs')i;--If; ]T=O ~ 0 ,

s = 2 ..... L , i s = l .... , m s ,

(2.11)

where the T are the first-class constraints and the Z are the zero-modes. The symbol ~ means the equality up to a linear superposition of second-class constraints with regular coefficients. One can easily show that the BCS superparticle is of the infinite stage of reducibility, since (z1)'%

= (2d~.

-p~,

(22s)i2s-li2s=pOe2s

),

1~2~,

( Z 2 s + 1 ) % ..... = p . . . . . 2. . . .

s = l , 2 .... o r .

i

{0'~' 0~}D = 2 , ~ p ~ 7+~"

{p,p}D = 0 ,

3 August 1989

(2.10)

We are going to define the infinite case by using the expressions for the finite L and then taking the limit L-~ oo. This definition is consistent with the counting of the physical degrees freedom of the superparticle. In general, the number of the physical coordinates of a reducible system of stage L is [ 2 ] (n*)+ = n + - m + - y o ( L ) +

,

(2.13)

where +_ indices denote the number of bosonic and fermionic objects respectively and 7o(L) = m o - [ml - (m2 - . . . ) ] L

= ~ (-1)'ms.

(2.14)

s~O

2n is the number of the primary canonical variables, 2m is the number of the second-class constraints and yo(L) is the number of the independent first-class constraints. In our case n+ = 10, n_ = 16, (70) = 1 and (Yo)-= ~ (-1)'.16=½.16=8,

(2.15)

s=O

where we have used the power series 1 / ( l + x ) = 1 -x+x 2 - x 3 + ... to regularize the 5~Go ( - 1 )'. Since m + = 0 and m_ = 4 we have (n*)+ = 1 0 - 0 - 1 = 9 ,

In general, a constrained system is said to be of the "Lth stage of reducibility" [2] if

(2.12)

(n*)_ = 1 6 - 4 - 8 - 4

, (2.16)

which is the right number of the physical coordinates for the BCS superparticle. 90

Volume 226, number 1,2

PHYSICS LETTERS B

3 August 1989

3. G h o s t s and the B R S T c h a r g e {-Qmin~ ,Qrnin }D ~ 0

The ghosts associated with a reducibile system are determined from eqs. (2.1 1 ). At each level s = 0, 1, ..., L one introduces ghosts and antighosts (and their m o m e n t a ) respectively: (C~ ;', ~ s ~ , ) ,

(C~. ", .+~~,.,) ,

(3.1)

and "extraghosts" for s>_- 1 (C~V",~';~),

s ' = l .... , s .

(3.2)

together with the boundary conditions ~,1,.,1~ t , = . = 0

0.0~f2..,,

where B.

IL .. L/

= ]1 ~z. [~2;, l~

(3.4)

s'=l,...,s.

For the statistics and the ghost-number assigments see ref. [2]. In the case of the superparticle the ghosts which follow from (2.12) and the coordinates can be organized in the following way: (x", Co eo)

= (Z,)"

',,.

(3.9)

,,=,,

(n+ 1 )~2.+1.4,....4,~.1 OC,~ 8~o = B , [.4,...-4.1,

(2~ ", zr';,),

(2, s';~, z~+.s';+),

,

=T,,~

The nilpotency condition implies {g2, g2}v= 0 and -

and for s~> 1 additional ghosts

+

oC,'-,a~'~;

One also has to introduce the Lagrange multipliers and their m o m e n t a (3.3)

0~2...... I

f l . . 1/,

O

(3.10)

+ t2,,_,, ~,'+ .... '"~Jl)

X ( - 1 )':;'; ..... + " - ; ' . n

[

+ ~. ( p - t - 1 ) ~2 . +.

,,.. b., 8~2._;, ' ........ '"

OC ~

;,=ll X ( - 1 )>:;'; . . . . . . + . - v

(3.11)

(3.5) see refs. [ 2 , 8 ] . ~ A = ~ . + S . w h e r e ~+ is 0 o r 1 d e p e n d ing on whether i,. is a bosonic or fermionic index, re-

and ( OG, Coo~(Oa, ..., C2sa (2so~ C2s Io' ...C2~ 2sa ,

spectively. Derivatives in (3.11) are all left and is graded antisymmetrization, whose symmetry properties are the same as the symmetries of the product .G, ~ , [ E( ~4 ) = +A+ 1 ]. ( 3.1 0) is a very useful equation since it determines Q.+~ as a function of ~2o, ..., ~2,,. Starting from ~2o and ~21 [which can be determined from the boundary conditions (3.9) ] one can calculate all subsequent ~2~'s. In the case of the superparticle we start from X[.4,..+4,,~

c2++~ " ~++~ " c2++t ~...c2+++~ 2~+~", ...) •

(3.6)

In this form it is easy to recognize (3.5) as a vector of O S p ( D - 1 , 112) and (3.6) as a spinor of O S p ( D - 1 , 112) decomposed with respect to a S O ( D - 1, 1 ) ® S p ( 2 ) subgroup ( D = 10). Therefore Siegel's BRST formalism which is based on the orthosimplectic groups [ 7 ] gives the same set of ghosts as BFV formalism. The BRST charge for a reducible system [2] can be written as if2 ~---ff2min

£20 =Co~¢+ Co~, ~ " .

By inserting this into (3.10) for n = 0 and taking the boundary conditions (3.9) into account we get

~2]=

L

"~'+ £ s=O

~

"' C s';~

(3.12)

- 2cl "'d,,, - 2icll, cll/sp "/J ,

(3.7)

s=l s':l

Omen is defined as Omin =

~ ~'~nAl"~4n~n...~At , n=O

(3.8)

where ~ , = ( ~ ° i o , ~ i ...... ~l/~ ) and it satisfies the nilpotency condition

-Q~.... =c2~+ i . . . . . . p . . . . . . . . . . .

s = 1, 2 .....

(3.13)

as only non-zero £2~'s. Then by inserting (3.13) into (3.10) for n = 1 we find that the only non-zero t22's are 9]

Volume 226, number 1,2

=~2,+ ~

~(2 ~

~

PHYSICS LETTERS B .(2.....

=-c2,+2,>.

(3.141

By inserting (3.12)-(3.14) into (3.10) one can easily show that all £23's vanish. Similarly, all ~4's and ~2s's vanish, which implies that all higher ~,,'s vanish, so that the expansion (3.8) terminates after n = 2 . Therefore we get ~m~n = CO~¢ + COd.~ ~ - 2ic0~ C0~p ~

0

+ ~ C,*l~@s+ ~ (--1)'-]G+2~°~@ ". s=

I

s=

(3.15)

1

4. Discussion and conclusions Note that expression (3.15) is covariant, in spite of the fact that the Dirac bracket (2.6) is non-covariant. The reason for this is the relation (2.8a) and the fact that the Dirac bracket of any quantity with a firstclass constraint is the same as the Poisson bracket. Therefore one avoids the non-covariant expressions in the calculation of ~mm. Thus one may hope that the expression for the generating functional will be covariant, since Z~, =

~ F e x p [iS~,],

References

~ ~2--H--{gt, g?}D) . (4.2)

~u is the gauge fixing fermion and H is the hamiltonian [2]. It can be easily shown that H = 0 for the superparticle and since C2is covariant one can choose gt such that &, is covariant. However, the path integral measure ~ F contains a factor [ 6,2 ]

3(~") (sdetl { ~ ", ~¢~}1 )~/2

(4.3)

which is non-covariant. At this point it is not clear any more whether it is possible to covariantly quan-

92

tize the BCS superparticle in the canonical formalism. Note that one would come up with the affirmative answer in the framework of the lagrangian formalism, since the path integral measure in that formalism does not contain the factor (4.3). If one believes that the hamiltonian formalism is related to the lagrangian formalism (by integrating over the momenta one should get the lagrangian formulation ) then there should exists the change of coordinates and momenta such that the new coordinates and momenta have canonical Dirac brackets and the corresponding path integral measure should be covariant. A possible way to covariantly quantize the BCS superparticle (and superstring), without adding new coordinates and in the canonical formalism, is to describe it in terms of the first-class constraints only. That is possible if one uses the following set of constraints: { ~p ~ 2, p~/~d~, d, de} [ 9 ]. The authors of ref. [ 10] have discussed the construction of the BRST charge of the superparticle defined by the .~' and .8 constraints only. They got the same expression as (3.15 ); however, the expression is incorrect, since they missed an additional zero mode ( Z z s + 1)'2'i . . . . = d~2~. This is a non-trivial zero mode, since d~ is not a constraint in their case.

(4.1)

where

Sv,= f at ( ~ P ~ / + ~ C ' +

3 August 1989

[1] R. Casalbuoni, Phys. Lett. B62 (1976) 49; L. Brink and J.H. Schwarz, Phys. Lett. B 100 ( 1981 ) 310. [ 2 ] I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 ( 1983 ) 157. [3] I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D 28 (1983) 2567. [4] W. Siegel, Introduction to string field theory (World Scientific, Singapore, 1988). [5] L. Brink, M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (19871 505. [6] P. Senjanovi6, Ann. Phys. 100 (1976) 227. [7]W. Siegel, Universal supersymmetry by adding 4 + 4 dimensions to the light cone, University of Maryland preprint UMDEPP 88-231 (May 19881. [8] M. Henneaux, Phys. Rep, 126 ( 19851. [ 9 ] A.R. Mikovid and W. Siegel, Phys. Len. B 2(19 (1988) 47. [ 10 ] A.H. Diaz and J. Zanelli, Phys. Lett. B 20? ( 1088 ) 347.