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Lorentz-covariant quantization of the superparticle
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
LORENTZ-COVARIANT QUANTIZATION OF THE SUPERPARTICLE U. L I N D S T R O M Institute of Theoretical Physics, Universityof Stockholm, Vanadisv~igen9, S- 113 46 Stockholm, Sweden M. R O C E K 1, W. SIEGEL 1, p. VAN N I E U W E N H U I Z E N 1 Institute for TheoreticalPhysics, State Universityof New York, Stony Brook, NY 11794-3840, USA and A.E. VAN DE VEN 2 Department of Physics, New York University, New York, NY 10003, USA Received 21 March 1989
We use the Batalin-Vilkovisky formalism to quantize the superparticle in a gauge which manifestly preserves supersymmetry and Lorentz covariance, and makes the gauge-fixedaction completely quadratic.
The classical action for the Casalbuoni-BrinkSchwarz superparticle [ 1 ] in first-order form is So=
dr [ p ( J c - ~ O ) - ½ g p 2 ] ,
(1)
and g~ves rise to the following field equations: /~=0,
~0=0,
pZ=0,
gp-~+OyO=O.
(2)
The action ( 1 ) is invariant under the following general coordinate transformations and local x transformations [ 2 ]: ~0=0, ~x=@,
,~O=t~x,
[6(~), 6(x)] =0,
[&(tq ), 6(x2) ] =6(~12 = 4te2/~Xl ).
(4)
We quantize this theory following the general formalism of Batalin and Vilkovisky [3 ], by adding terms linear in "antifields" to the action. The scaling dimensions of the classical fields, the coordinate ghost Z and the x ghost cl, and their antifields can be chosen as follows:
[g]=l, [g*]=0,
[p,x,O]=O, Ix]=0, [p*, x*, 0*] = l,
[cl]=0,
[Z*]=I,
[cT] = 1. (5)
~x=O~,~O,
6p=O, 6p=O, ~g=~,
[6(~,), 6(~2)] =0,
~g= 40t¢.
The corresponding ghost numbers are (3)
The general coordinate transformation in (3) is not the naive 8 x = ~ , 6p=p~, 60=0~, and 6g=~,~+g~; the relation is explained below. The classical gauge algebra is [modulo the field equations (2) ] Work supported in part by NSF grant No. PHY 85-07627. 2 Worksupported in part by NSF grant No. PHY 87-15995. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
gh(g,p,x,O)=O,
gh(z, cl) = 1,
gh(g*,p*,x*,O*)=-l,
gh(z*, cI') = - 2.
(6)
The terms linear in the antifields are of two types: terms linear in the antifields o f the classical fields g, p, x, 0, which follow from the classical transformations (3), and terms linear in the ghost antifields, which follow from the algebra (4): 285
Volume 224, number 3
PHYSICS LETTERSB
L1 =x*(PX+ Oy~c, ) + O*(~cl ) +g*(40cl +;() +Z*(2¢l/~Cl ).
(7)
We note that Ll is invariant under transformations of the 1¢ ghost ~c~ =/~x2 modulo the classical field equations. This implies that the system has ghostsfor-ghosts; indeed, there is an infinite tower of such ghosts [4 ], and at this level they can all be incorporated by adding L l (higher levels) = ~ c*gk6+l.
29 June 1989
We now gauge fix by introducing a gauge fixing function ~ a n d replacing the antifields ¢] by 0 ~/O¢ A. We choose ~=;~(g--1)+~
~ g~+~+l~{.
(12)
i=0j=0
The first few terms in ( 11 ) and ( 12 ) are L .... i.
=2*n~+cl*nl+(c22*~t22+c~'*~')+....
~P=2(g-1)+el0+
( C-2' C' l 1 -{- C2C 0 )
-1 ' 2 -2 " 1 (c3c2 +c3c2 +e~c ° ) + ....
(8)
(13)
i=1
We obtain the BRST-invariant action (or equivalently, the BRST charge) S=So+SI +Sz+... from the master equation; S: follows from
(S~,Sl)+ 2(So, S2)=O .
(9)
Evaluating (Sb $2) and collecting all the terms proportional to the classical field equations, we find L2 =
2g*(x* ( gl 7el + 07c2) + 0"c2 + 4Z*gl c2
+,=, ~ c~c,+~).
(10)
We find (Sl, $2) = ($2, $2) =0, and hence there are no further contributions in the "minimal" sector. We now add the "nonminimal" sector in the usual way:
Ci hi,
(11)
i=lj=l
where c o =c, and c o = 0, and the remaining cj and all the he, as well as 2 and n~ are nonminimal. The scale dimension of all the c{ can be chosen to vanish, and their ghost numbers are gh(c{)= i-2j. The scale dimension of all the corresponding antifields c{* is then 1, and the ghost numbers are g h ( c F ) = - i + 2 j - 1 . The scale dimension of)~ is + 1, and of 2* is 0, while the ghost number of 2 is - 1, and of 2* is 0. Plotting the fields on graphs of ghost generation versus ghost number, they organize into the following pyramids:
After elimination of the antifields, the nonminimal terms become L ....
in.
= ( g - 1 ) h e - ( ~ +b-)n{
l 7~ - - ( C~0
~l 2 q-el ~! n21 q - c 2 n l ) q- . . . . q-C3n2
(14)
The gauge choices g - 1 = 0 and 0= 0 fix the classical symmetries. Since ~0=~x, one can only gauge half of 0 away on-shell [3(~0)~p2K=0]. The gauge choice 0 = 0 indeed fixes half of these symmetries on-shell (/k0=0). (On the other hand, 0 = 0 would be too strong a gauge choice, and would not preserve manifest supersymmetry.) The gauge choices cl'°=c,'l = 0 fix the gauge invariances of the usual Faddeev-Popov ghost action cHl~kcl, o and further terms in (13) and (14) fix further ghost-for-ghost symmetries. The complete action after gauge fixing becomes
L~, =p.i;- @ 0 - ½gp2+ ~(j~ + 40c o ) -
~
c~i+1,/~ i ÷ , ~ c0t + l + ( g - - 1 ) ~ - - ( ~ + O - ) ~ l
i=0
i=2j=l
i=0
(15) After defining ~¢ = nij + hi_2, j - 1 and shifting
he-*he+ lp2, ffl ~Ttl +g0--4c°2, ~t]~t] +2c°2-~c° . fori>_-2,
(16)
the action takes on a free-field form i
L=p.;c--½p2+f~+n¢(g--1)--Onl -- ~ • ci_,~i-JTr¢. ~t~
~
~;
~2
el
n~ c 2
i=2j=l
co
c~'
c °"
(17) The essential points for obtaining these results were
286
Volume 224, number 3
PHYSICS LETTERS B
our choice o f gauge-fixing in (12) a n d the basis for the gauge transformations. There are further local invariances, for example,
OO=~b, 6 x = - ( 6 6 ) 7 0 ,
Op=6g=0,
(18)
and we could have also taken these into account by adding ghosts for them. However, for nonvanishing m o m e n t u m this s y m m e t r y can be expressed as the sum o f a x - s y m m e t r y and a field equation, 0= ~
1
[/k(~0) + ¢ ( ~ O ) l
(19)
for arbitrary n. It is therefore redundant, and we believe that it makes no difference whether one includes these extra symmetries or not. The transform a t i o n ( 1 8 ) can also be used to obtain the usual d i f f e o m o r p h i s m s from those given in (3). The effect on the complete L can be obtained by unitary transformations using antibrackets:
S' = e ~ S ,
29 June 1989 5gAB-- ( A , B ) .
(20)
The above results for the superparticle can be ext e n d e d to the superstring. Work along these lines is in progress.
References [ 1] R. Casalbuoni, Phys. Lett. B 62 (1976) 49; L. Brink and J,H. Schwarz, Phys. Lett. B 100 ( 1981 ) 310. [2] W. Siegel, Phys. Lett. B 128 (1983) 397. [ 3 ] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 ( 1981 ) 27; B 120 (1983) 166; Phys. Rev. D 28 (1983) 2567; D 30 (1984) 508; Nucl. Phys. B 234 (1984) 106; J. Math. Phys. 26 (1985) 172. [4] J. Thierry-Mieg,Cont. Rend. Sc. Paris, t. 299, Serie II (1984) p. 1309; E. Nissimov and S. Pacheva, Phys. Lett. B 189 (1987) 57; L. Brink, M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (1987) 505; R. Kallosh, JETP Left. 45 (1987) 365; Phys. Lett. B 195 (1987) 369.
287
PHYSICS LETTERS B
29 June 1989
LORENTZ-COVARIANT QUANTIZATION OF THE SUPERPARTICLE U. L I N D S T R O M Institute of Theoretical Physics, Universityof Stockholm, Vanadisv~igen9, S- 113 46 Stockholm, Sweden M. R O C E K 1, W. SIEGEL 1, p. VAN N I E U W E N H U I Z E N 1 Institute for TheoreticalPhysics, State Universityof New York, Stony Brook, NY 11794-3840, USA and A.E. VAN DE VEN 2 Department of Physics, New York University, New York, NY 10003, USA Received 21 March 1989
We use the Batalin-Vilkovisky formalism to quantize the superparticle in a gauge which manifestly preserves supersymmetry and Lorentz covariance, and makes the gauge-fixedaction completely quadratic.
The classical action for the Casalbuoni-BrinkSchwarz superparticle [ 1 ] in first-order form is So=
dr [ p ( J c - ~ O ) - ½ g p 2 ] ,
(1)
and g~ves rise to the following field equations: /~=0,
~0=0,
pZ=0,
gp-~+OyO=O.
(2)
The action ( 1 ) is invariant under the following general coordinate transformations and local x transformations [ 2 ]: ~0=0, ~x=@,
,~O=t~x,
[6(~), 6(x)] =0,
[&(tq ), 6(x2) ] =6(~12 = 4te2/~Xl ).
(4)
We quantize this theory following the general formalism of Batalin and Vilkovisky [3 ], by adding terms linear in "antifields" to the action. The scaling dimensions of the classical fields, the coordinate ghost Z and the x ghost cl, and their antifields can be chosen as follows:
[g]=l, [g*]=0,
[p,x,O]=O, Ix]=0, [p*, x*, 0*] = l,
[cl]=0,
[Z*]=I,
[cT] = 1. (5)
~x=O~,~O,
6p=O, 6p=O, ~g=~,
[6(~,), 6(~2)] =0,
~g= 40t¢.
The corresponding ghost numbers are (3)
The general coordinate transformation in (3) is not the naive 8 x = ~ , 6p=p~, 60=0~, and 6g=~,~+g~; the relation is explained below. The classical gauge algebra is [modulo the field equations (2) ] Work supported in part by NSF grant No. PHY 85-07627. 2 Worksupported in part by NSF grant No. PHY 87-15995. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
gh(g,p,x,O)=O,
gh(z, cl) = 1,
gh(g*,p*,x*,O*)=-l,
gh(z*, cI') = - 2.
(6)
The terms linear in the antifields are of two types: terms linear in the antifields o f the classical fields g, p, x, 0, which follow from the classical transformations (3), and terms linear in the ghost antifields, which follow from the algebra (4): 285
Volume 224, number 3
PHYSICS LETTERSB
L1 =x*(PX+ Oy~c, ) + O*(~cl ) +g*(40cl +;() +Z*(2¢l/~Cl ).
(7)
We note that Ll is invariant under transformations of the 1¢ ghost ~c~ =/~x2 modulo the classical field equations. This implies that the system has ghostsfor-ghosts; indeed, there is an infinite tower of such ghosts [4 ], and at this level they can all be incorporated by adding L l (higher levels) = ~ c*gk6+l.
29 June 1989
We now gauge fix by introducing a gauge fixing function ~ a n d replacing the antifields ¢] by 0 ~/O¢ A. We choose ~=;~(g--1)+~
~ g~+~+l~{.
(12)
i=0j=0
The first few terms in ( 11 ) and ( 12 ) are L .... i.
=2*n~+cl*nl+(c22*~t22+c~'*~')+....
~P=2(g-1)+el0+
( C-2' C' l 1 -{- C2C 0 )
-1 ' 2 -2 " 1 (c3c2 +c3c2 +e~c ° ) + ....
(8)
(13)
i=1
We obtain the BRST-invariant action (or equivalently, the BRST charge) S=So+SI +Sz+... from the master equation; S: follows from
(S~,Sl)+ 2(So, S2)=O .
(9)
Evaluating (Sb $2) and collecting all the terms proportional to the classical field equations, we find L2 =
2g*(x* ( gl 7el + 07c2) + 0"c2 + 4Z*gl c2
+,=, ~ c~c,+~).
(10)
We find (Sl, $2) = ($2, $2) =0, and hence there are no further contributions in the "minimal" sector. We now add the "nonminimal" sector in the usual way:
Ci hi,
(11)
i=lj=l
where c o =c, and c o = 0, and the remaining cj and all the he, as well as 2 and n~ are nonminimal. The scale dimension of all the c{ can be chosen to vanish, and their ghost numbers are gh(c{)= i-2j. The scale dimension of all the corresponding antifields c{* is then 1, and the ghost numbers are g h ( c F ) = - i + 2 j - 1 . The scale dimension of)~ is + 1, and of 2* is 0, while the ghost number of 2 is - 1, and of 2* is 0. Plotting the fields on graphs of ghost generation versus ghost number, they organize into the following pyramids:
After elimination of the antifields, the nonminimal terms become L ....
in.
= ( g - 1 ) h e - ( ~ +b-)n{
l 7~ - - ( C~0
~l 2 q-el ~! n21 q - c 2 n l ) q- . . . . q-C3n2
(14)
The gauge choices g - 1 = 0 and 0= 0 fix the classical symmetries. Since ~0=~x, one can only gauge half of 0 away on-shell [3(~0)~p2K=0]. The gauge choice 0 = 0 indeed fixes half of these symmetries on-shell (/k0=0). (On the other hand, 0 = 0 would be too strong a gauge choice, and would not preserve manifest supersymmetry.) The gauge choices cl'°=c,'l = 0 fix the gauge invariances of the usual Faddeev-Popov ghost action cHl~kcl, o and further terms in (13) and (14) fix further ghost-for-ghost symmetries. The complete action after gauge fixing becomes
L~, =p.i;- @ 0 - ½gp2+ ~(j~ + 40c o ) -
~
c~i+1,/~ i ÷ , ~ c0t + l + ( g - - 1 ) ~ - - ( ~ + O - ) ~ l
i=0
i=2j=l
i=0
(15) After defining ~¢ = nij + hi_2, j - 1 and shifting
he-*he+ lp2, ffl ~Ttl +g0--4c°2, ~t]~t] +2c°2-~c° . fori>_-2,
(16)
the action takes on a free-field form i
L=p.;c--½p2+f~+n¢(g--1)--Onl -- ~ • ci_,~i-JTr¢. ~t~
~
~;
~2
el
n~ c 2
i=2j=l
co
c~'
c °"
(17) The essential points for obtaining these results were
286
Volume 224, number 3
PHYSICS LETTERS B
our choice o f gauge-fixing in (12) a n d the basis for the gauge transformations. There are further local invariances, for example,
OO=~b, 6 x = - ( 6 6 ) 7 0 ,
Op=6g=0,
(18)
and we could have also taken these into account by adding ghosts for them. However, for nonvanishing m o m e n t u m this s y m m e t r y can be expressed as the sum o f a x - s y m m e t r y and a field equation, 0= ~
1
[/k(~0) + ¢ ( ~ O ) l
(19)
for arbitrary n. It is therefore redundant, and we believe that it makes no difference whether one includes these extra symmetries or not. The transform a t i o n ( 1 8 ) can also be used to obtain the usual d i f f e o m o r p h i s m s from those given in (3). The effect on the complete L can be obtained by unitary transformations using antibrackets:
S' = e ~ S ,
29 June 1989 5gAB-- ( A , B ) .
(20)
The above results for the superparticle can be ext e n d e d to the superstring. Work along these lines is in progress.
References [ 1] R. Casalbuoni, Phys. Lett. B 62 (1976) 49; L. Brink and J,H. Schwarz, Phys. Lett. B 100 ( 1981 ) 310. [2] W. Siegel, Phys. Lett. B 128 (1983) 397. [ 3 ] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 ( 1981 ) 27; B 120 (1983) 166; Phys. Rev. D 28 (1983) 2567; D 30 (1984) 508; Nucl. Phys. B 234 (1984) 106; J. Math. Phys. 26 (1985) 172. [4] J. Thierry-Mieg,Cont. Rend. Sc. Paris, t. 299, Serie II (1984) p. 1309; E. Nissimov and S. Pacheva, Phys. Lett. B 189 (1987) 57; L. Brink, M. Henneaux and C. Teitelboim, Nucl. Phys. B 293 (1987) 505; R. Kallosh, JETP Left. 45 (1987) 365; Phys. Lett. B 195 (1987) 369.
287