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Transversity of a massless relativistic membrane
Volume 198, number 4
PHYSICS LETTERS B
3 December 1987
TRANSVERSITY OF A M A S S L E S S RELATIVISTIC MEMBRANE M.E. LAZIEV ~ a n d G K SAVVIDY b a
Yere~an Phvsws lnsntute, Yerevan 375036, 4rmema, USSR b TheoryDtvlston, CERN, CH-1211 Geneva 23, Swuzerland Received 15 June 1987
We consider the relatlvlsUctheory of the open two-d~mensmnalsurface which is characterized by an action proportional to the volume swept out by the surface in spacenme 11~sshown that classlcall~there are onl~ D - 3 dynamicallyindependent transverse components among the D funchons A ~ ( a j, a_.. z) which represent "theworld volume (D ~sthe dzmensmnof spacettme) and that the surface boundary moves with the velocity ofhght
At present, the idea that all elementary particles correspond to different excitation modes of a single f u n d a m e n t a l s t u n g has been intensely developed In the framework of th-ts -tdeology one can naturally explain such difficult problems of G r a n d U n i f i c a t i o n theory as the choice of the group and representations, Yukawa couplings, unlficatton of gravity with other forces Presumably, there ex-tst underlying principles in the basis of superstrlng dynamics that are as simple and economic as the gauge principle The string theory exists and -ts cons-tstent with q u a n t u m mechanics a n d relatiwty for qmte definite groups a n d d I m e n s m n s of spacetime The question arises whether there exist consistent theories for objects with a large n u m b e r of d i m e n s i o n s 9 If not, then the string is indeed a u n i q u e object, if yes, then the string is nothing but a h m l t of more general objects A concrete questton we want to answer is Is the m o t i o n of a massless relanvlStlC surface transverse like m the string case of not9 W-tth thts ram. we consider a two-dimensional open surface M whose action is p r o p o m o n a l to the volume swept out by its mot-ton in the D - d i m e n s m n a l spacet-tme with coordinates X" [ 1-4]
j dr daj da2 ,
Mk~=0g"/0~z ,
g~z=MX~'g,o34~
(2)
Introducing the m o m e n t u m current density P.~= -05~/0 X " = Tgl/2g"~X./~ with c o m p o n e n t s P~ and P.~' we can write the equaUon of mot-ton as
O~P~+"0~, P~~, = 0 .
(3)
and the constraint equations as P ~ X } = - L P ~ } ~, P~,~P'¢=T2gg~I~=T2A ~ , A ~/~ is the algebrmc adj u n c t of the element g~,/j If the equations of m o t i o n (3) hold, then the act-ton v a r t a n o n has the form
8S= f f da , da2
--fd f
S = - T f d z d a t da~ Detl/2g =
f l = 0 , 1, 2, X"=X~*(~) is a parametric representation of the covered volume ~ e ~ × [z,.. zf~.] I f r is identified to time X ° = t = r , then, at a given instant of time t, the surface is gtven by the D - 1 f u n c n o n X ' ( z , al, a2) The volume swept out by the surface must be n m e - h k e the points move with a velocity less or equal than the veloc-tty of light X~ = 1-X{~>0. X ~ = - X ~ < 0 , so Detg~>0 The action (1) -ts m v a r l a n t u n d e r the reparametrlzatlon ~ =
0.,( P~,'S X u )
(do'2P~'-P:~-do.l)~X ~
(4)
0~2
(1)
where g,~z=X~X1,/~, Xf~=OX/O~, 4~(T. a~. a2), a.
Therefore at the boundary 0f2 of the parameter space O.ef~ the b o u n d a r y conditions should hold" do2 P;{' --dal P~,~-Ion = 0
(5) 451
V o l u m e 198, n u m b e r 4
PHYSICS LETTERS B
These condmons may be written in different forms depending on the way the boundary O~ ~s given if parametrically, e , = e , ( z ) , then in the form of t o'l ¢2P. - ~ , Pt . ¢r2 = 0 and so on Under varmtlons, the mmal and final posmons of the surface are fixed. 8 X # ( T , n , O') = S X U ( T f i n , O ' ) = 0 , whereas 8XU(r, Of~) ~s arbitrary Denote by 0M the boundary of the surface M which is given by the equation X ' = X " ( r , Of~) In the case of an open surface, the surface boundary 0M must be lnvarxant under the reparamemzatlon (2), OM(Of~)=0M(Of~(O(~)), the action (1) to be changed as
3 D e c e m b e r 1987
d a = x / X .2, X ~2 2- (X~,X~:)-"~da, do"2 Is the differential element of area on surface M. However from here it does not yet follow that the motion of the M surface Is transverse. Let us construct such a parametrlzatlon of the surface M in which the energy density P~ will be constant over the whole surface. For that we choose the transformation matrix M~ in (2) m the form
M~=
1
,
L, F=[,
~1 = a l ,
~2 =f(t, a~, a2)
(10)
S = - T [ d~ Det'/2g(~) T f d ~ Det'/2g(~)
(6)
and the boundary condmon (5) as doe P ~ ' - - d a l
Then from (7) we have P~ =Uo/f~, and if we put f~_ = pto, then P~ = const over the whole surface. The funcnon f c a n be constructed explicitly
EdifY1' r~2' t)=~z "i P6(a~, a;, O d e ; ,
P~'-IoD=O
(11)
0
From the definmon of momentum current density Pg it follows that they transform as
where E = f P ~ d o " 1 do'2 = f ( E / ~ 2 ) d r r I dry2
MB P"~ -
~'~" "P~'
(7)
det M
The energy-momentum current through the arbitrary surface S inside the swept out volume is
P.=fPLda, da2+fdT(da2P~'-da, P~~)
(8)
is the total energy of surface M, and/~6 =EDC For definiteness, the lnmal parametrlzatmn region f2 ~s chosen as a square a , ~ [ 0 , rt], a2e[0, re], while the boundary conditions (5) have the form Pg'(a, = 0, n) =Pg~(a2 =0, n ) = 0 S i n c e / ~ has a qmte certain form as a function of Xt and Xo, then
IfS is a closed surface, then Pu=0, xn virtue ofeq (3), therefore P~ = fP/~ da, da2 for the surfaces S covenng the boundary If S belongs to a surface swept out by the boundary 8M, Pu=0, due to condluons (5), 1 e the energy-momentum current through the surface boundary is zero There are similar relaUons for the angular momentum One can readily be convinced that the acUon (1) depends on the normal velocity component X, ~ of the surface Indeed, let X ° = r , then X~=(1, X,), X~, = (0, Xo,), and the acuon will be rewritten as
The new region boundary O~ can be found from (11 ), it has the form of a "rectangle" of which three sides are straight, while the fourth side is a 2 = f(6~, re, t), its area being zt2 The boundary condition is unchanged on the three sides, while on the last one it has the form
S=-TJx/1-X?X,
L, P;~'-P,~2 }
~ dAdt,
where
IX? I =x,.x~, ×x,._/Ix., ×x,~ I, 452
(9)
T[X~,X~2 - (X~,Xo2)2] X[(l - - X 2l )(XGIXG2--(X~IX~2) 2 2 2) "~-X~I(XIXrT2)'Ji -
]-1/2=E/7~2
=0.
(12)
03)
rYi = 0 r~ a 2 = / ( a l tr t)
hence the energy density P6 is constant on surface M The next step m gauge fixing (eqmvalent to par-
Volume 198. number 4
PHYSICS LETTERSB
ametnzatlon fixing) is to choose such a paramemzatlon of surface M, #j and ~2, in which the densmes of the energy currents m different dlrecuons are equal Po~1 = P o0"2
(14)
In order to realize (14), one has to clarify the remaining arbitrariness in the choice of paramemzatlon To preserve the previous conditions X ° = z , P'o=E/n 2, it is necessary that d e t M = l , and the M~ matrix upper line should have the form of (1, 0, 0), then t~ t=t. fi6=P~o=E/n 2 and
(EIn2k~ /
\ Pff: /
f 1 \h,
0
0 ~/EI.2~ libel
/
go,
g ' ll
ha,
ha.l \ Pg~- I
8t = g ( # t , 02. t),
o t,
82 =h(d~, 62, t)
(15)
The condmon (14) and equahty d e t M = 1 are now eqmalent to equations for the g and h functions E
--~ ( g - h ), + P~'(g-h )~, + P~'-(g-h )e2 = 0 , g~,h0.2-g~he , =1
(16)
The first of them is solved by the characterlstm method d61 n2/5~, d~-- E '
dO2 7~2 dt - E P ~ - ' "
(17)
3 December 1987
The first integral of the system (21) can readily be found, it is G(a, b)=c3 Solving it with respect to 62 we find that #2 =~,(6t, t, c3) and GO..= - Ge, / ?0., The latter being subsntuted to (21 ) we find the second mtegral
h = f 7e, ddl _[_c4 ~G~-~16~=G(ab)
(22)
Hence the general solution has the form
h=j
da, +H(G(a, b)) ,
where H is an arbitrary function Thus the sought after transformaUons (15) are found, the arbitrary functions G and H determine the boundary 0 ~ which can be chosen wlthm these frames So, m this gauge we have
Uo=E/n 2,
X°=t=r,
PoO't =Poo'2 ,
(23)
where again the mmal notations are introduced Let us make use of the zero component of the equation of motion (3) (1 e further results are true on the mass shell only)
OoP'o+Oo,Ud' =ooeg'= (0o, +0~2)eg' = 0 ,
(24)
whose solution has the form ofPg' = Z ( a : - a2), and the boundary condmon (5) implies that Z = 0 Therefore on the whole surface M Pg' =Pg-~ = 0
(25)
Let c,=a(#~,~2, t),
c2=b(~,,#2, t),
(18)
be two independent first integrals of the system (17), whmh exist under definite assumptmns with respect to Pg', Pg-" functions, then the general solution has the form
g=h+G(a(6,. 62. t). b(#:, ~2, t)) .
(19)
where G is an arbitrary function Subsmutang (19) m the second equation of (17), we get an equanon for h Ga, h<. - Ge~ h~, = 1 ,
(20)
On the other hand, using explicit expressions for the Pg' and Pgecomponents through X,, Xo, we find that
( X,X~, )( X0.,X~ ) -X~, ( X,X~. )=0 ,
d~l = dh
Ga,
G,~:
(26)
whence one can see that the transverse condition holds (XtX~,) = 0.
l=1,2
(27)
In the transverse gauge (27) the expressions for the current densmes are particularly simple p,=
whose charactenstlc equation is do"2 __
XG2) -X22 ( XgXal ) = O,
(XgXo 2 )(Xol
P~'=
_ (E/n2)X,
n~-T 2 E
,
{ X . X ~ + , - ( X . , X o . )Xo,+, } t
~
(28)
the equations of motaon have the form 453
Volume 198, number 4
x,,=
(+)2 7r-T
PHYSICS LETTERS B
{o.,(x~,x~-x..(xo,x~))
+O~.(X~_X~., - X ~ , (X.,X~. )) } .
(29)
and c o n d l t m n (12) wtll be rewritten as
1-X~ =
{X~,X~..-(X~,X~.) 2 )
(30)
da2(X~, X~.. - (X., Xo. )X~ 2) (31)
from these after m u l n p l y m g by X., a n d X.2 we get
X ~ , X ~ - (X~,X,._)2 l a,2 = 0
ber o f transverse d y n a m i c a l degrees o f freedom ~s D-3 The c a n o m c a l q u a n n z a t l o n o f the theory will be p u b h s h e d elsewhere The a u t h o r would hke to t h a n k S G M a t l n y a n and A B Z a m o l o d c h l k o v for interesting discussions
The b o u n d a r y c o n d m o n s (5) are now
-da~(X~.X~.,-(Xo,X.2)X~,)[on-=O,
3 December 1987
Note added C.N P o p e r e f o r m e d one o f us ( G K S ) that sxmflar p r o b l e m s have been discussed m refs [ 5 - 7 ] One of us ( G K S ) would like to t h a n . . M Jacob and the T h e o r e n c a l Physics Dxvlslon for the h o s p l t a h t y extended to hxm and for n u m e r o u s discussions
(32)
References Together with (30) it m e a n s that the b o u n d a r y o f the surface M moves wtth the velocity of hght X, 10n = 1
(33)
Hence the a c u o n (1) has the form S - - - - 7~2T2 f
---if-
{X~,,X~,~-(X~,X~_)2}da, da2 dt (34)
It can be readily checked that all mlttal constraints are equivalent to (27) and (30) and hence the num-
454
[1] P A M Dlrac, Proc R Soc 1268 (1962) 57 [2] P A Colhnsand R W Tucker, Nucl Phys B 112 (1976) 150 [3] P Goddard. T Goldstone. C Rebbl and C B Thorn, Nucl Phys B 56 (1973) 109 [4] A Sugamoto, Nucl Ph~s B 215 (1983) 381 [5 ] J Hoppe. Quantum theo~ of a relatlvlsnc surface, ~achen preprlnt PITH~ 86/24 (1986) [6] K Klkkawa and M Yamasakl, Prog Theor Plays 76 (1986) 1379 [ 7 ] M J Duff T Inaml. CN Pope, E Sezgm andKS Stelle. Semi-classical quannzatlon of the supermembrane, CERN preprmt TH 4731/87 (1987)
PHYSICS LETTERS B
3 December 1987
TRANSVERSITY OF A M A S S L E S S RELATIVISTIC MEMBRANE M.E. LAZIEV ~ a n d G K SAVVIDY b a
Yere~an Phvsws lnsntute, Yerevan 375036, 4rmema, USSR b TheoryDtvlston, CERN, CH-1211 Geneva 23, Swuzerland Received 15 June 1987
We consider the relatlvlsUctheory of the open two-d~mensmnalsurface which is characterized by an action proportional to the volume swept out by the surface in spacenme 11~sshown that classlcall~there are onl~ D - 3 dynamicallyindependent transverse components among the D funchons A ~ ( a j, a_.. z) which represent "theworld volume (D ~sthe dzmensmnof spacettme) and that the surface boundary moves with the velocity ofhght
At present, the idea that all elementary particles correspond to different excitation modes of a single f u n d a m e n t a l s t u n g has been intensely developed In the framework of th-ts -tdeology one can naturally explain such difficult problems of G r a n d U n i f i c a t i o n theory as the choice of the group and representations, Yukawa couplings, unlficatton of gravity with other forces Presumably, there ex-tst underlying principles in the basis of superstrlng dynamics that are as simple and economic as the gauge principle The string theory exists and -ts cons-tstent with q u a n t u m mechanics a n d relatiwty for qmte definite groups a n d d I m e n s m n s of spacetime The question arises whether there exist consistent theories for objects with a large n u m b e r of d i m e n s i o n s 9 If not, then the string is indeed a u n i q u e object, if yes, then the string is nothing but a h m l t of more general objects A concrete questton we want to answer is Is the m o t i o n of a massless relanvlStlC surface transverse like m the string case of not9 W-tth thts ram. we consider a two-dimensional open surface M whose action is p r o p o m o n a l to the volume swept out by its mot-ton in the D - d i m e n s m n a l spacet-tme with coordinates X" [ 1-4]
j dr daj da2 ,
Mk~=0g"/0~z ,
g~z=MX~'g,o34~
(2)
Introducing the m o m e n t u m current density P.~= -05~/0 X " = Tgl/2g"~X./~ with c o m p o n e n t s P~ and P.~' we can write the equaUon of mot-ton as
O~P~+"0~, P~~, = 0 .
(3)
and the constraint equations as P ~ X } = - L P ~ } ~, P~,~P'¢=T2gg~I~=T2A ~ , A ~/~ is the algebrmc adj u n c t of the element g~,/j If the equations of m o t i o n (3) hold, then the act-ton v a r t a n o n has the form
8S= f f da , da2
--fd f
S = - T f d z d a t da~ Detl/2g =
f l = 0 , 1, 2, X"=X~*(~) is a parametric representation of the covered volume ~ e ~ × [z,.. zf~.] I f r is identified to time X ° = t = r , then, at a given instant of time t, the surface is gtven by the D - 1 f u n c n o n X ' ( z , al, a2) The volume swept out by the surface must be n m e - h k e the points move with a velocity less or equal than the veloc-tty of light X~ = 1-X{~>0. X ~ = - X ~ < 0 , so Detg~>0 The action (1) -ts m v a r l a n t u n d e r the reparametrlzatlon ~ =
0.,( P~,'S X u )
(do'2P~'-P:~-do.l)~X ~
(4)
0~2
(1)
where g,~z=X~X1,/~, Xf~=OX/O~, 4~(T. a~. a2), a.
Therefore at the boundary 0f2 of the parameter space O.ef~ the b o u n d a r y conditions should hold" do2 P;{' --dal P~,~-Ion = 0
(5) 451
V o l u m e 198, n u m b e r 4
PHYSICS LETTERS B
These condmons may be written in different forms depending on the way the boundary O~ ~s given if parametrically, e , = e , ( z ) , then in the form of t o'l ¢2P. - ~ , Pt . ¢r2 = 0 and so on Under varmtlons, the mmal and final posmons of the surface are fixed. 8 X # ( T , n , O') = S X U ( T f i n , O ' ) = 0 , whereas 8XU(r, Of~) ~s arbitrary Denote by 0M the boundary of the surface M which is given by the equation X ' = X " ( r , Of~) In the case of an open surface, the surface boundary 0M must be lnvarxant under the reparamemzatlon (2), OM(Of~)=0M(Of~(O(~)), the action (1) to be changed as
3 D e c e m b e r 1987
d a = x / X .2, X ~2 2- (X~,X~:)-"~da, do"2 Is the differential element of area on surface M. However from here it does not yet follow that the motion of the M surface Is transverse. Let us construct such a parametrlzatlon of the surface M in which the energy density P~ will be constant over the whole surface. For that we choose the transformation matrix M~ in (2) m the form
M~=
1
,
L, F=[,
~1 = a l ,
~2 =f(t, a~, a2)
(10)
S = - T [ d~ Det'/2g(~) T f d ~ Det'/2g(~)
(6)
and the boundary condmon (5) as doe P ~ ' - - d a l
Then from (7) we have P~ =Uo/f~, and if we put f~_ = pto, then P~ = const over the whole surface. The funcnon f c a n be constructed explicitly
EdifY1' r~2' t)=~z "i P6(a~, a;, O d e ; ,
P~'-IoD=O
(11)
0
From the definmon of momentum current density Pg it follows that they transform as
where E = f P ~ d o " 1 do'2 = f ( E / ~ 2 ) d r r I dry2
MB P"~ -
~'~" "P~'
(7)
det M
The energy-momentum current through the arbitrary surface S inside the swept out volume is
P.=fPLda, da2+fdT(da2P~'-da, P~~)
(8)
is the total energy of surface M, and/~6 =EDC For definiteness, the lnmal parametrlzatmn region f2 ~s chosen as a square a , ~ [ 0 , rt], a2e[0, re], while the boundary conditions (5) have the form Pg'(a, = 0, n) =Pg~(a2 =0, n ) = 0 S i n c e / ~ has a qmte certain form as a function of Xt and Xo, then
IfS is a closed surface, then Pu=0, xn virtue ofeq (3), therefore P~ = fP/~ da, da2 for the surfaces S covenng the boundary If S belongs to a surface swept out by the boundary 8M, Pu=0, due to condluons (5), 1 e the energy-momentum current through the surface boundary is zero There are similar relaUons for the angular momentum One can readily be convinced that the acUon (1) depends on the normal velocity component X, ~ of the surface Indeed, let X ° = r , then X~=(1, X,), X~, = (0, Xo,), and the acuon will be rewritten as
The new region boundary O~ can be found from (11 ), it has the form of a "rectangle" of which three sides are straight, while the fourth side is a 2 = f(6~, re, t), its area being zt2 The boundary condition is unchanged on the three sides, while on the last one it has the form
S=-TJx/1-X?X,
L, P;~'-P,~2 }
~ dAdt,
where
IX? I =x,.x~, ×x,._/Ix., ×x,~ I, 452
(9)
T[X~,X~2 - (X~,Xo2)2] X[(l - - X 2l )(XGIXG2--(X~IX~2) 2 2 2) "~-X~I(XIXrT2)'Ji -
]-1/2=E/7~2
=0.
(12)
03)
rYi = 0 r~ a 2 = / ( a l tr t)
hence the energy density P6 is constant on surface M The next step m gauge fixing (eqmvalent to par-
Volume 198. number 4
PHYSICS LETTERSB
ametnzatlon fixing) is to choose such a paramemzatlon of surface M, #j and ~2, in which the densmes of the energy currents m different dlrecuons are equal Po~1 = P o0"2
(14)
In order to realize (14), one has to clarify the remaining arbitrariness in the choice of paramemzatlon To preserve the previous conditions X ° = z , P'o=E/n 2, it is necessary that d e t M = l , and the M~ matrix upper line should have the form of (1, 0, 0), then t~ t=t. fi6=P~o=E/n 2 and
(EIn2k~ /
\ Pff: /
f 1 \h,
0
0 ~/EI.2~ libel
/
go,
g ' ll
ha,
ha.l \ Pg~- I
8t = g ( # t , 02. t),
o t,
82 =h(d~, 62, t)
(15)
The condmon (14) and equahty d e t M = 1 are now eqmalent to equations for the g and h functions E
--~ ( g - h ), + P~'(g-h )~, + P~'-(g-h )e2 = 0 , g~,h0.2-g~he , =1
(16)
The first of them is solved by the characterlstm method d61 n2/5~, d~-- E '
dO2 7~2 dt - E P ~ - ' "
(17)
3 December 1987
The first integral of the system (21) can readily be found, it is G(a, b)=c3 Solving it with respect to 62 we find that #2 =~,(6t, t, c3) and GO..= - Ge, / ?0., The latter being subsntuted to (21 ) we find the second mtegral
h = f 7e, ddl _[_c4 ~G~-~16~=G(ab)
(22)
Hence the general solution has the form
h=j
da, +H(G(a, b)) ,
where H is an arbitrary function Thus the sought after transformaUons (15) are found, the arbitrary functions G and H determine the boundary 0 ~ which can be chosen wlthm these frames So, m this gauge we have
Uo=E/n 2,
X°=t=r,
PoO't =Poo'2 ,
(23)
where again the mmal notations are introduced Let us make use of the zero component of the equation of motion (3) (1 e further results are true on the mass shell only)
OoP'o+Oo,Ud' =ooeg'= (0o, +0~2)eg' = 0 ,
(24)
whose solution has the form ofPg' = Z ( a : - a2), and the boundary condmon (5) implies that Z = 0 Therefore on the whole surface M Pg' =Pg-~ = 0
(25)
Let c,=a(#~,~2, t),
c2=b(~,,#2, t),
(18)
be two independent first integrals of the system (17), whmh exist under definite assumptmns with respect to Pg', Pg-" functions, then the general solution has the form
g=h+G(a(6,. 62. t). b(#:, ~2, t)) .
(19)
where G is an arbitrary function Subsmutang (19) m the second equation of (17), we get an equanon for h Ga, h<. - Ge~ h~, = 1 ,
(20)
On the other hand, using explicit expressions for the Pg' and Pgecomponents through X,, Xo, we find that
( X,X~, )( X0.,X~ ) -X~, ( X,X~. )=0 ,
d~l = dh
Ga,
G,~:
(26)
whence one can see that the transverse condition holds (XtX~,) = 0.
l=1,2
(27)
In the transverse gauge (27) the expressions for the current densmes are particularly simple p,=
whose charactenstlc equation is do"2 __
XG2) -X22 ( XgXal ) = O,
(XgXo 2 )(Xol
P~'=
_ (E/n2)X,
n~-T 2 E
,
{ X . X ~ + , - ( X . , X o . )Xo,+, } t
~
(28)
the equations of motaon have the form 453
Volume 198, number 4
x,,=
(+)2 7r-T
PHYSICS LETTERS B
{o.,(x~,x~-x..(xo,x~))
+O~.(X~_X~., - X ~ , (X.,X~. )) } .
(29)
and c o n d l t m n (12) wtll be rewritten as
1-X~ =
{X~,X~..-(X~,X~.) 2 )
(30)
da2(X~, X~.. - (X., Xo. )X~ 2) (31)
from these after m u l n p l y m g by X., a n d X.2 we get
X ~ , X ~ - (X~,X,._)2 l a,2 = 0
ber o f transverse d y n a m i c a l degrees o f freedom ~s D-3 The c a n o m c a l q u a n n z a t l o n o f the theory will be p u b h s h e d elsewhere The a u t h o r would hke to t h a n k S G M a t l n y a n and A B Z a m o l o d c h l k o v for interesting discussions
The b o u n d a r y c o n d m o n s (5) are now
-da~(X~.X~.,-(Xo,X.2)X~,)[on-=O,
3 December 1987
Note added C.N P o p e r e f o r m e d one o f us ( G K S ) that sxmflar p r o b l e m s have been discussed m refs [ 5 - 7 ] One of us ( G K S ) would like to t h a n . . M Jacob and the T h e o r e n c a l Physics Dxvlslon for the h o s p l t a h t y extended to hxm and for n u m e r o u s discussions
(32)
References Together with (30) it m e a n s that the b o u n d a r y o f the surface M moves wtth the velocity of hght X, 10n = 1
(33)
Hence the a c u o n (1) has the form S - - - - 7~2T2 f
---if-
{X~,,X~,~-(X~,X~_)2}da, da2 dt (34)
It can be readily checked that all mlttal constraints are equivalent to (27) and (30) and hence the num-
454
[1] P A M Dlrac, Proc R Soc 1268 (1962) 57 [2] P A Colhnsand R W Tucker, Nucl Phys B 112 (1976) 150 [3] P Goddard. T Goldstone. C Rebbl and C B Thorn, Nucl Phys B 56 (1973) 109 [4] A Sugamoto, Nucl Ph~s B 215 (1983) 381 [5 ] J Hoppe. Quantum theo~ of a relatlvlsnc surface, ~achen preprlnt PITH~ 86/24 (1986) [6] K Klkkawa and M Yamasakl, Prog Theor Plays 76 (1986) 1379 [ 7 ] M J Duff T Inaml. CN Pope, E Sezgm andKS Stelle. Semi-classical quannzatlon of the supermembrane, CERN preprmt TH 4731/87 (1987)