- Email: [email protected]
Particle theories with minimum observable length and open string theory
Volume 245, number 1
PHYSICS LETTERS B
2 August 1990
Particle theories with minimum observable length and open string theory Mitsuhiro K a t o
1,2
Theoretical Physics Group, Physics Division, Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94 720, USA Received 16 February 1990
We investigate a class of particle theories that have m i n i m u m observable length. Among them we find the theory equivalent to the open bosonic string theory. For this theory, starting from a reparameterization invariant and non-local action with respect to a world-line parameter, we derive the Virasoro constraint and the Veneziano amplitude. The possible relation to other attempts to introduce a fundamental length scale into particle theory is also discussed.
Various recent investigations [ 1-6 ] suggest the existence of a minimum observable length in fundamental string theory. This might be an essential observation for understanding the underlying new principles of physics that would differ fundamentally from those of conventional local field theories. A rough derivation of the existence (non-existence) of minimum observable length in string (particle) theory is as follows: Taking a discretization of the world-sheet (world-line) parameter as a regularization, we consider the path integral (we fix the reparameterization degree of freedom)
2
X[(~f)
2
-F ( ~ - ~ ) ]} ,
(1)
where ~ is an interval of discretization, 2 is a dimenThis work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy physics of the US Department of Energy under Contract DE-AC03-76SF00098. On leave of absence from National Laboratory for High Energy Physics ( K E K ) , Tsukuba, Ibaraki 305, Japan. 2 E-mailaddress: [email protected]
sional constant and 8x is a finite difference in accord with the discretization. Note that the { factors in ( 1 ) cancel out while they do not in (2). This difference leads to different { dependences of ( ( S x ) 2): ((aX)2)
string ~ 2 ,
(3)
((8X)2)
particle .w {)2 .
(4)
Taking {--+0 we find that the space-time distance of any two points on a string world-sheet cannot be smaller than 2 even in the continuum limit (see ref. [ 6 ] for a more precise argument and for implications of a new uncertainty relation). It is an idea of Lee [ 7 ] that regarding a discretized theory as fundamental leads to a minimum observable length or limitation of successive observation in a small volume in particle theory. The minimum length, however, depends on { in this case. He has introduced a fundamental length scale I and fixed the number of discrete points by hand so that the average of { coincides with 1 ~. There is another way to achieve a minimum observable length in particle theory. If we define an alternative action of O ( 1 ) instead of Ze (Sx/{) 2 ~ 0 ( 1 / {) keeping it quadratic in x, we can observe a mini~1 In the faithful application of his formulation to a relativistic particle, e can differ from point to point and its distribution is also determined by the action regarding the world-sheet parameter as a dynamical variable.
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
43
Volume 245, number 1
PHYSICS LETTERS B
mum length in this theory as well. This is indeed possible. For example, let us take an action like f dr ~fAx, where Ax means a finite difference. Then its discretized version becomes Ee(ax/~)Ax~O(l) as desired. It seems, at first sight, that we have introduced an unwelcome scale in the definition of Av, i.e. an interval on which the difference is taken. This unpleasant situation will be an illusion if this action is derived from a reparameterization invariant action by gauge fixing, because in this case the interval of A is gauge variant and physics does not depend on it. In this paper we investigate a class of particle theories that have the feature stated above. In particular, we will construct the theory that is equivalent to open bosonic string theory. Actually any theory containing a finite difference can be written in terms of an infinite number of derivatives and such a theory is equivalent to the conventional first-order derivative theory with infinitely many degrees of freedom [8]. Therefore it is not so unnatural that a stringequivalent theory is a theory of this kind. Furthermore, since the Virasoro algebra, which is essential for string theory, comes from a one-dimensional mapping, it seems possible that we can obtain it from the reparameterization invariance of the world-line provided there are infinite degrees of freedom ~2 We start our discussion from the following action, denoting particle coordinates and the einbein field on the world-line by xU(r) and v(z) respectively ~3. S= ~ dzL,
5)
2 August 1990 02"
v ( r ) - , v ' ( r ' ) = dr' v(r).
(8)
Let us take the v= 1 gauge. Then the lagrangian becomes L=-
f~f
dz
(9)
which gives the equation of motion ~4
10, In order to recover the reparameterization invariance we use some constraints that are given by the equation of motion with respect to v(z) at v= 1. Here we derive these in a different way. Consider an infinitesimal reparameterization 6x= ¢:~. Before fixing a gauge we had a relation a L = ( d / d r ) ( e L ) with 6v= ( d / d r ) (cv). Now this time we have an extra term proportional to E in 8L besides the total derivative term and we require that it does not affect the action:
aL= ~ (~L)-½L,
e
.
Taking a Fourier mode ¢---eaexp(iar) (a is arbitrary), we get the set of constraints
1 qo,~=-~.t,,F
( a'dz d)
.f;'=O,
(12)
where
F(a,d)=f(d+ia)(d+ia) L=-
~x"
xuf
(1 d ) l f u
"~ v
'
6)
\T~/T,
(7)
From now on we concentrate our argument on the case
f(z)=
,2 This, on the other hand, may show that constructing a particle theory equivalent to closed string is difficult. We might need a bi-particle theory. ,3 We use a space-time metric qu,=diag( + , - , ...,- ).
sinh(nz)
rcz cosh[ ( ~ - s ) z ]
(O
f~(d/dr) can be understood as a sum of differences. In particular we have
(d) X=--
fo ~
44
f(d~d -
(13)
where .fc=dx/dz and x is a constant with dimensmn (length) 2. We require f ( - z ) = f ( z ) so that the theory possesses time reversal invariance. Note that the substitution o f f ( z ) = 1 gives a conventional (massless) particle theory. This action is invariant under a reparameterization z--, r' with the transformation law
x(r)~x'(,') =x(r),
(ll)
~n=l
(--1)n-lAZmr X
and t'4 Eqs. (9) and (10) were first analyzed by Naka [9].
(14)
Volume 245, number 1
(d) ./'~
1 2= ~A~x,
PHYSICS LETTERS B
(15)
where A a x ( r ) = x ( r + 6 ) - x ( r - 6 ) . It will turn out later that s = 0 theory is equivalent to the open bosonic string theory. It should be stressed again that the peculiar interval 2~ is an artifact due to our choice of the gauge v= 1. If we take the V=Vo gauge then 2~ in the above will be replaced by 2gVo and the physics will not depend on it. Before going on, it may be instructive to consider the classical meaning of our equation of motion. For s = ~ theory we have A,2=O
(16a)
2 August 1990
XoU(r)=f
s(d)
-~ x U ( r ) ,
(d/d,)2 ff d)
x.U(r)= (d/dr)2+n2
\drjX'(r)
(20b)
and where we neglected the total derivative term in (19) [8,9]. This is single derivative theory with infinitely many degrees of freedom which are just simple harmonic oscillators. For special values of s there occurs an absence of the particular modes, but we will not consider that case for simplicity though it is straightforward. Defining conjugate momenta po"= - ( 1/X)2o ~ and p fl'= - ( 2 / x ) cos ( ns ) 2 f for the variables Xou and x . . respectively, we obtain the hamiltonian
or r+~z
(20a)
(
H=-½~cP°~'P°U- n=l ~ 4 cos(ns) p"upn
f dr'2(r')=0.
(16b)
n2
+ - - cos(ns) xnux./' This means that the particle is free on the average over the interval 2re, while there are fluctuations in the scale smaller than 2re (notice that 2 is an acceleration and so is proportional to a "force" ). We can make a Fourier expansion of 2(T) on the interval [-z~, tel and find a spiral motion of the particle in each mode, like a charged particle in a constant magnetic field. For an s ¢ n theory, the qualitative nature is the same. This point will be discussed again later. Now we rewrite the lagrangian in a form appropriate for quantization. Using the relation
1 fs(Z) - 1 +
~ z2 ~=1 2 cos(ns) z2.k_n 2
K
= _ ½~PouPou _ ... G n . ~ ( a . u a " n=l
+a.ua.U), (21)
where x.~=½ ~ c ~ × [a.. exp( - i n r ) + a * . . e x p ( i n r ) ] P.u=iG
(17)
),
,
(22)
/nG
~x
X [a.~ e x p ( - i n r ) - a ~ u exp(inr) ]
(23)
we obtain expressions xU(r)
=Xol'(z)+2 ~ cos(ns) x.U(r)
(18)
and
L=- 1 ()?o,:to" + 2 .=,~ c o s ( n s ) ( 2 . 1 , 2 . " - n 2 x . u x # ' ) ) , where
(19)
and c. = Icos (ns) l, e. = sign (cos (ns)). We quantize the theory using the canonical commutation relations [Xou, Po.] =iqu. and [x.u, Pm~,] = iqu.3 ..... or equivalently [ a.u , a ~. ] = - e.qu.fi.,m. The hamiltonian in the quantized theory is defined in normal ordered form. As we shall soon see, we must use constraints like ( H - a o ) l > =0, where ao is a constant. Thus we can easily see that these theories have the same spectrum as the open bosonic string theory taking K = 2 a ' where a ' is a slope parameter, though there exist many negative norm states besides the time component excitations unless s = 0. 45
V o l u m e 245, n u m b e r 1
PHYSICS LETTERS B
We now turn to the discussion of constraints. Making use of the general formula (12), we can calculate the constraint for the theory with f ( d / d r ) for generic s. We, however, only present the results for two special cases s = 0 and 7tjust for simplicity. We define
p'(~/=- !~"(r/=po'(r/+ Z p~(rt K n=l
2 A u g u s t 1990
the equivalence of the Hilbert space to that of the open bosonic string. To confirm the equivalence of the s = 0 theory and open bosonic string theory, let us derive the Veneziano amplitude. We again go back to the generic s theory and assume the form of the vertex operator (tachyon or ground state particle emission) as
=po~,+i ~ en n~cn n=l
~]
]~
X [a. ~ e x p ( - i n r ) - a ~ ~ exp (inr) ] .
(24)
× exp( i 2x//~d7 n=, ~
fl2kuat"U) '
(27)
(i) s=Tr case: ~a--
i 2~rxsin(Tra)
f drexp(iar) p~,(r)P~(r+~) • (25)
where k ~' is the momentum of the emitting particle, qU=XoU(O) and x=2ee' is taken. With standard technique we find that the four-point amplitude 1
Since a is arbitrary, we use more conveniently
f ~ (O,-k4lV(k3)xH-~°V(k2)lO, kl) ,
pu(r)pU(r+rO (r arbitrary) or alternatively its
0
Fourier mode
K~= ~
X
where the k, (i--1, 2, 3, 4) are external momenta, gives the correct Veneziano amplitude if and only if
drexp(inr)pu(r)p~(r+~)
1
+ -E~flnfln = - - and ~ o = 1
(n an integer) as constraints ~5. They coincide with what Naka [ 9 ] imposed by hand. (ii) s = 0 case: i q~ = - - - A 27rx
sin(~a) cos(Tra)
× ~ dr exp(iar)
p~,(r)p~'(z),
(26)
where A is a regularization dependent constant. As in the previous case we use Pu( r)p u (r) or L~= ~
dr exp(inr)
pu(r)p~(r)
as constraints. Using the mode expansion of pSi(r) and commutation relations, we can easily find that the L~'s satisfy the Virasoro algebra! Therefore by imposing the physical state condition ( L ~ - Olod~,o) Iphys ) = 0 for non-negative integer n we can quantize the s = 0 theory consistently and so find ~5 N o t e t h a t , a l t h o u g h ~ v a n i s h e s for i n t e g e r a d u e to a coeffic i e n t sin ( h a ) , we d o h a v e Kn as c o n s t r a i n t s . B e c a u s e for a r b i t r a r y a o t h e r t h a n i n t e g e r we still h a v e c o n s t r a i n t s . T h e s a m e is t r u e for the s = 0 case.
46
(28/
n
(29)
are satisfied. Only the s = 0 theory naturally realizes (29) with V(k) = :exp[ikuxU(0) ] :. We may impose (29) for the other theories by hand. It is, however, unnatural in the sense that V(k) is not expressed in terms of a fundamental variable xU(r). Thus we have established the equivalence of the s = 0 theory and the open bosonic string theory by showing the coincidence of the Hilbert space including Virasoro constraints as well as a natural derivation of the Veneziano amplitude. Sibling theories of s ~ 0 share qualitatively many properties with the s = 0 theory. Let us return to the discussion of the behaviour of the particle in this kind of theory. As mentioned before, the normal motion of the particle is a spiral movement like that of a charged particle in a constant magnetic field. So its quantized state is analogous to the Landau level. Notice that the Landau level in the magnetic field B in the z-direction can be obtained by using either H = (1/2m)(p-eA) 2 with [Px,Py] = 0 or H = ( 1/2m)p2 with [Px,Py] = ieB. This is suggestive in connection with the argument of Townsend [ 10 ]. He considered the de Sitter group as
Volume 245, number I
PHYSICS LETTERS B
a natural extension of the Poincar6 group by introducing a fundamental length scale. The main difference of the former from the latter is a non-trivial commutation relation [Pu, P~ ] = ( - i/). 2)Ju~ for the translation generator P~, where J ~ is a Lorentz generator. He took the Planck length as a constant 2, which should have the dimension of length. As a consequence, a particle in the space-time with this symmetry group has large fluctuations at the Planck scale in its motion. What we can observe is an average motion on a larger scale. This is again very similar to our particle system. Another possible hint from this direction is obtained by the eigenvalue of p2. Let us simplify our argument by considering three-dimensional euclidean space-time whose symmetry group is a semi-direct product of O ( 3 ) and the translation group:
[Ji, Jj ] =iEakJk,
(30a)
[8, Pj] =ieakPk,
(30b)
[Pi, Pj] =0.
(30c)
According to Townsend we replace (30c) by
[e~,Pjl=iE,jkJk.
(31)
Then we have O ( 4 ) , - - O ( 3 ) × O ( 3 ) as a symmetry group. The unitary representations are characterized by two Casimir invariants corresponding to each O ( 3 ) and we denote them by 1~(ll + 1 ) and 12(12+ 1 ). The quadratic Casimir o f O (4) c = j2 + p2, then becomes 2ll (1~ + 1 ) +212(12+ 1 ). Since we observe the particle only in its average motion, we will classify particles with eigenvalues p2 and j2. Decomposing the representation characterized by l~ and 12 into representations of Ji, i.e., diagonal O ( 3 ) , we get the relation
p2=c-j2=21l(l~+l)+212(12+l)-j(j+l),
(32)
2 August 1990
where j(j+ 1 ) is an eigenvalue of j2 and j=ll+12, I~+ 12- 1, ..., 1Ii - 12I, or alternatively p 2 = ( 2 n + 1 )j+n(n+2) +,~2,
(33)
where n=ll+12-j, ~=1~-12. Remarkably eq. (33) shows a linear dependence o f P 2 o n j analogous to the Regge behaviour (!), although its slope depends on the quantum number n. What does this mean? These arguments may suggest that a modification of the space-time structure at small scales is incorporated into our particle theories (and the string theory) in a yet unnoticed way.
Acknowledgement I would like to thank A.A. Tseytlin for drawing my attention to ref. [ 10 ] and R. Cahn for correcting the English of the manuscript.
References [ 1] G. Veneziano,Europhys. Lett. 2 ( 1986) 199;CERN preprint CERN-TH5316 (1989). [2]T. Yoneya, in: Wandering in the fields, eds. K. Kawarabayashi and A. Ukawa, (World Scientific, Singapore, 1987) p. 419; in Quantum string theory, eds. N. Kawamoto and T. Kugo (Springer, Berlin, 1988) p. 23; Mod. Phys. Lett. A4 (1989) 1587. [ 3 ] D. Amati, M, Ciafaloni and G. Veneziano, Phys. Lett. B 197 (1987) 81; Intern. J. Mod. Phys. A 3 (1988) 1615. [4] D.J. Gross and P.F. Mende, Phys. Lett. B 197 (1987) 129; Nucl. Phys. B 303 (1988) 407. [ 5 ] M. Karliner, I. Klebanov and L. Susskind, Intern. J. Mod. Phys. 3A (1988) 1981. [6] K. Konishi, G. Paffuti and P. Provero, University of Pisa preprint IFUP-TH46/89 GEF-TH89/9 (1989). [7] T.D. Lee, Phys. Lett. B 122 (1983) 217. [8] A. Pais and G.E. Uhlenbeck, Phys. Rev. 79 (1950) 145. [9] S. Naka, Prog. Theor. Phys. 48 (1972) 1024. [ 10] P.K. Townsend, Phys. Rev. D 10 (1977) 2795.
47
PHYSICS LETTERS B
2 August 1990
Particle theories with minimum observable length and open string theory Mitsuhiro K a t o
1,2
Theoretical Physics Group, Physics Division, Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, CA 94 720, USA Received 16 February 1990
We investigate a class of particle theories that have m i n i m u m observable length. Among them we find the theory equivalent to the open bosonic string theory. For this theory, starting from a reparameterization invariant and non-local action with respect to a world-line parameter, we derive the Virasoro constraint and the Veneziano amplitude. The possible relation to other attempts to introduce a fundamental length scale into particle theory is also discussed.
Various recent investigations [ 1-6 ] suggest the existence of a minimum observable length in fundamental string theory. This might be an essential observation for understanding the underlying new principles of physics that would differ fundamentally from those of conventional local field theories. A rough derivation of the existence (non-existence) of minimum observable length in string (particle) theory is as follows: Taking a discretization of the world-sheet (world-line) parameter as a regularization, we consider the path integral (we fix the reparameterization degree of freedom)
2
X[(~f)
2
-F ( ~ - ~ ) ]} ,
(1)
where ~ is an interval of discretization, 2 is a dimenThis work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy physics of the US Department of Energy under Contract DE-AC03-76SF00098. On leave of absence from National Laboratory for High Energy Physics ( K E K ) , Tsukuba, Ibaraki 305, Japan. 2 E-mailaddress: [email protected]
sional constant and 8x is a finite difference in accord with the discretization. Note that the { factors in ( 1 ) cancel out while they do not in (2). This difference leads to different { dependences of ( ( S x ) 2): ((aX)2)
string ~ 2 ,
(3)
((8X)2)
particle .w {)2 .
(4)
Taking {--+0 we find that the space-time distance of any two points on a string world-sheet cannot be smaller than 2 even in the continuum limit (see ref. [ 6 ] for a more precise argument and for implications of a new uncertainty relation). It is an idea of Lee [ 7 ] that regarding a discretized theory as fundamental leads to a minimum observable length or limitation of successive observation in a small volume in particle theory. The minimum length, however, depends on { in this case. He has introduced a fundamental length scale I and fixed the number of discrete points by hand so that the average of { coincides with 1 ~. There is another way to achieve a minimum observable length in particle theory. If we define an alternative action of O ( 1 ) instead of Ze (Sx/{) 2 ~ 0 ( 1 / {) keeping it quadratic in x, we can observe a mini~1 In the faithful application of his formulation to a relativistic particle, e can differ from point to point and its distribution is also determined by the action regarding the world-sheet parameter as a dynamical variable.
0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
43
Volume 245, number 1
PHYSICS LETTERS B
mum length in this theory as well. This is indeed possible. For example, let us take an action like f dr ~fAx, where Ax means a finite difference. Then its discretized version becomes Ee(ax/~)Ax~O(l) as desired. It seems, at first sight, that we have introduced an unwelcome scale in the definition of Av, i.e. an interval on which the difference is taken. This unpleasant situation will be an illusion if this action is derived from a reparameterization invariant action by gauge fixing, because in this case the interval of A is gauge variant and physics does not depend on it. In this paper we investigate a class of particle theories that have the feature stated above. In particular, we will construct the theory that is equivalent to open bosonic string theory. Actually any theory containing a finite difference can be written in terms of an infinite number of derivatives and such a theory is equivalent to the conventional first-order derivative theory with infinitely many degrees of freedom [8]. Therefore it is not so unnatural that a stringequivalent theory is a theory of this kind. Furthermore, since the Virasoro algebra, which is essential for string theory, comes from a one-dimensional mapping, it seems possible that we can obtain it from the reparameterization invariance of the world-line provided there are infinite degrees of freedom ~2 We start our discussion from the following action, denoting particle coordinates and the einbein field on the world-line by xU(r) and v(z) respectively ~3. S= ~ dzL,
5)
2 August 1990 02"
v ( r ) - , v ' ( r ' ) = dr' v(r).
(8)
Let us take the v= 1 gauge. Then the lagrangian becomes L=-
f~f
dz
(9)
which gives the equation of motion ~4
10, In order to recover the reparameterization invariance we use some constraints that are given by the equation of motion with respect to v(z) at v= 1. Here we derive these in a different way. Consider an infinitesimal reparameterization 6x= ¢:~. Before fixing a gauge we had a relation a L = ( d / d r ) ( e L ) with 6v= ( d / d r ) (cv). Now this time we have an extra term proportional to E in 8L besides the total derivative term and we require that it does not affect the action:
aL= ~ (~L)-½L,
e
.
Taking a Fourier mode ¢---eaexp(iar) (a is arbitrary), we get the set of constraints
1 qo,~=-~.t,,F
( a'dz d)
.f;'=O,
(12)
where
F(a,d)=f(d+ia)(d+ia) L=-
~x"
xuf
(1 d ) l f u
"~ v
'
6)
\T~/T,
(7)
From now on we concentrate our argument on the case
f(z)=
,2 This, on the other hand, may show that constructing a particle theory equivalent to closed string is difficult. We might need a bi-particle theory. ,3 We use a space-time metric qu,=diag( + , - , ...,- ).
sinh(nz)
rcz cosh[ ( ~ - s ) z ]
(O
f~(d/dr) can be understood as a sum of differences. In particular we have
(d) X=--
fo ~
44
f(d~d -
(13)
where .fc=dx/dz and x is a constant with dimensmn (length) 2. We require f ( - z ) = f ( z ) so that the theory possesses time reversal invariance. Note that the substitution o f f ( z ) = 1 gives a conventional (massless) particle theory. This action is invariant under a reparameterization z--, r' with the transformation law
x(r)~x'(,') =x(r),
(ll)
~n=l
(--1)n-lAZmr X
and t'4 Eqs. (9) and (10) were first analyzed by Naka [9].
(14)
Volume 245, number 1
(d) ./'~
1 2= ~A~x,
PHYSICS LETTERS B
(15)
where A a x ( r ) = x ( r + 6 ) - x ( r - 6 ) . It will turn out later that s = 0 theory is equivalent to the open bosonic string theory. It should be stressed again that the peculiar interval 2~ is an artifact due to our choice of the gauge v= 1. If we take the V=Vo gauge then 2~ in the above will be replaced by 2gVo and the physics will not depend on it. Before going on, it may be instructive to consider the classical meaning of our equation of motion. For s = ~ theory we have A,2=O
(16a)
2 August 1990
XoU(r)=f
s(d)
-~ x U ( r ) ,
(d/d,)2 ff d)
x.U(r)= (d/dr)2+n2
\drjX'(r)
(20b)
and where we neglected the total derivative term in (19) [8,9]. This is single derivative theory with infinitely many degrees of freedom which are just simple harmonic oscillators. For special values of s there occurs an absence of the particular modes, but we will not consider that case for simplicity though it is straightforward. Defining conjugate momenta po"= - ( 1/X)2o ~ and p fl'= - ( 2 / x ) cos ( ns ) 2 f for the variables Xou and x . . respectively, we obtain the hamiltonian
or r+~z
(20a)
(
H=-½~cP°~'P°U- n=l ~ 4 cos(ns) p"upn
f dr'2(r')=0.
(16b)
n2
+ - - cos(ns) xnux./' This means that the particle is free on the average over the interval 2re, while there are fluctuations in the scale smaller than 2re (notice that 2 is an acceleration and so is proportional to a "force" ). We can make a Fourier expansion of 2(T) on the interval [-z~, tel and find a spiral motion of the particle in each mode, like a charged particle in a constant magnetic field. For an s ¢ n theory, the qualitative nature is the same. This point will be discussed again later. Now we rewrite the lagrangian in a form appropriate for quantization. Using the relation
1 fs(Z) - 1 +
~ z2 ~=1 2 cos(ns) z2.k_n 2
K
= _ ½~PouPou _ ... G n . ~ ( a . u a " n=l
+a.ua.U), (21)
where x.~=½ ~ c ~ × [a.. exp( - i n r ) + a * . . e x p ( i n r ) ] P.u=iG
(17)
),
,
(22)
/nG
~x
X [a.~ e x p ( - i n r ) - a ~ u exp(inr) ]
(23)
we obtain expressions xU(r)
=Xol'(z)+2 ~ cos(ns) x.U(r)
(18)
and
L=- 1 ()?o,:to" + 2 .=,~ c o s ( n s ) ( 2 . 1 , 2 . " - n 2 x . u x # ' ) ) , where
(19)
and c. = Icos (ns) l, e. = sign (cos (ns)). We quantize the theory using the canonical commutation relations [Xou, Po.] =iqu. and [x.u, Pm~,] = iqu.3 ..... or equivalently [ a.u , a ~. ] = - e.qu.fi.,m. The hamiltonian in the quantized theory is defined in normal ordered form. As we shall soon see, we must use constraints like ( H - a o ) l > =0, where ao is a constant. Thus we can easily see that these theories have the same spectrum as the open bosonic string theory taking K = 2 a ' where a ' is a slope parameter, though there exist many negative norm states besides the time component excitations unless s = 0. 45
V o l u m e 245, n u m b e r 1
PHYSICS LETTERS B
We now turn to the discussion of constraints. Making use of the general formula (12), we can calculate the constraint for the theory with f ( d / d r ) for generic s. We, however, only present the results for two special cases s = 0 and 7tjust for simplicity. We define
p'(~/=- !~"(r/=po'(r/+ Z p~(rt K n=l
2 A u g u s t 1990
the equivalence of the Hilbert space to that of the open bosonic string. To confirm the equivalence of the s = 0 theory and open bosonic string theory, let us derive the Veneziano amplitude. We again go back to the generic s theory and assume the form of the vertex operator (tachyon or ground state particle emission) as
=po~,+i ~ en n~cn n=l
~]
]~
X [a. ~ e x p ( - i n r ) - a ~ ~ exp (inr) ] .
(24)
× exp( i 2x//~d7 n=, ~
fl2kuat"U) '
(27)
(i) s=Tr case: ~a--
i 2~rxsin(Tra)
f drexp(iar) p~,(r)P~(r+~) • (25)
where k ~' is the momentum of the emitting particle, qU=XoU(O) and x=2ee' is taken. With standard technique we find that the four-point amplitude 1
Since a is arbitrary, we use more conveniently
f ~ (O,-k4lV(k3)xH-~°V(k2)lO, kl) ,
pu(r)pU(r+rO (r arbitrary) or alternatively its
0
Fourier mode
K~= ~
X
where the k, (i--1, 2, 3, 4) are external momenta, gives the correct Veneziano amplitude if and only if
drexp(inr)pu(r)p~(r+~)
1
+ -E~flnfln = - - and ~ o = 1
(n an integer) as constraints ~5. They coincide with what Naka [ 9 ] imposed by hand. (ii) s = 0 case: i q~ = - - - A 27rx
sin(~a) cos(Tra)
× ~ dr exp(iar)
p~,(r)p~'(z),
(26)
where A is a regularization dependent constant. As in the previous case we use Pu( r)p u (r) or L~= ~
dr exp(inr)
pu(r)p~(r)
as constraints. Using the mode expansion of pSi(r) and commutation relations, we can easily find that the L~'s satisfy the Virasoro algebra! Therefore by imposing the physical state condition ( L ~ - Olod~,o) Iphys ) = 0 for non-negative integer n we can quantize the s = 0 theory consistently and so find ~5 N o t e t h a t , a l t h o u g h ~ v a n i s h e s for i n t e g e r a d u e to a coeffic i e n t sin ( h a ) , we d o h a v e Kn as c o n s t r a i n t s . B e c a u s e for a r b i t r a r y a o t h e r t h a n i n t e g e r we still h a v e c o n s t r a i n t s . T h e s a m e is t r u e for the s = 0 case.
46
(28/
n
(29)
are satisfied. Only the s = 0 theory naturally realizes (29) with V(k) = :exp[ikuxU(0) ] :. We may impose (29) for the other theories by hand. It is, however, unnatural in the sense that V(k) is not expressed in terms of a fundamental variable xU(r). Thus we have established the equivalence of the s = 0 theory and the open bosonic string theory by showing the coincidence of the Hilbert space including Virasoro constraints as well as a natural derivation of the Veneziano amplitude. Sibling theories of s ~ 0 share qualitatively many properties with the s = 0 theory. Let us return to the discussion of the behaviour of the particle in this kind of theory. As mentioned before, the normal motion of the particle is a spiral movement like that of a charged particle in a constant magnetic field. So its quantized state is analogous to the Landau level. Notice that the Landau level in the magnetic field B in the z-direction can be obtained by using either H = (1/2m)(p-eA) 2 with [Px,Py] = 0 or H = ( 1/2m)p2 with [Px,Py] = ieB. This is suggestive in connection with the argument of Townsend [ 10 ]. He considered the de Sitter group as
Volume 245, number I
PHYSICS LETTERS B
a natural extension of the Poincar6 group by introducing a fundamental length scale. The main difference of the former from the latter is a non-trivial commutation relation [Pu, P~ ] = ( - i/). 2)Ju~ for the translation generator P~, where J ~ is a Lorentz generator. He took the Planck length as a constant 2, which should have the dimension of length. As a consequence, a particle in the space-time with this symmetry group has large fluctuations at the Planck scale in its motion. What we can observe is an average motion on a larger scale. This is again very similar to our particle system. Another possible hint from this direction is obtained by the eigenvalue of p2. Let us simplify our argument by considering three-dimensional euclidean space-time whose symmetry group is a semi-direct product of O ( 3 ) and the translation group:
[Ji, Jj ] =iEakJk,
(30a)
[8, Pj] =ieakPk,
(30b)
[Pi, Pj] =0.
(30c)
According to Townsend we replace (30c) by
[e~,Pjl=iE,jkJk.
(31)
Then we have O ( 4 ) , - - O ( 3 ) × O ( 3 ) as a symmetry group. The unitary representations are characterized by two Casimir invariants corresponding to each O ( 3 ) and we denote them by 1~(ll + 1 ) and 12(12+ 1 ). The quadratic Casimir o f O (4) c = j2 + p2, then becomes 2ll (1~ + 1 ) +212(12+ 1 ). Since we observe the particle only in its average motion, we will classify particles with eigenvalues p2 and j2. Decomposing the representation characterized by l~ and 12 into representations of Ji, i.e., diagonal O ( 3 ) , we get the relation
p2=c-j2=21l(l~+l)+212(12+l)-j(j+l),
(32)
2 August 1990
where j(j+ 1 ) is an eigenvalue of j2 and j=ll+12, I~+ 12- 1, ..., 1Ii - 12I, or alternatively p 2 = ( 2 n + 1 )j+n(n+2) +,~2,
(33)
where n=ll+12-j, ~=1~-12. Remarkably eq. (33) shows a linear dependence o f P 2 o n j analogous to the Regge behaviour (!), although its slope depends on the quantum number n. What does this mean? These arguments may suggest that a modification of the space-time structure at small scales is incorporated into our particle theories (and the string theory) in a yet unnoticed way.
Acknowledgement I would like to thank A.A. Tseytlin for drawing my attention to ref. [ 10 ] and R. Cahn for correcting the English of the manuscript.
References [ 1] G. Veneziano,Europhys. Lett. 2 ( 1986) 199;CERN preprint CERN-TH5316 (1989). [2]T. Yoneya, in: Wandering in the fields, eds. K. Kawarabayashi and A. Ukawa, (World Scientific, Singapore, 1987) p. 419; in Quantum string theory, eds. N. Kawamoto and T. Kugo (Springer, Berlin, 1988) p. 23; Mod. Phys. Lett. A4 (1989) 1587. [ 3 ] D. Amati, M, Ciafaloni and G. Veneziano, Phys. Lett. B 197 (1987) 81; Intern. J. Mod. Phys. A 3 (1988) 1615. [4] D.J. Gross and P.F. Mende, Phys. Lett. B 197 (1987) 129; Nucl. Phys. B 303 (1988) 407. [ 5 ] M. Karliner, I. Klebanov and L. Susskind, Intern. J. Mod. Phys. 3A (1988) 1981. [6] K. Konishi, G. Paffuti and P. Provero, University of Pisa preprint IFUP-TH46/89 GEF-TH89/9 (1989). [7] T.D. Lee, Phys. Lett. B 122 (1983) 217. [8] A. Pais and G.E. Uhlenbeck, Phys. Rev. 79 (1950) 145. [9] S. Naka, Prog. Theor. Phys. 48 (1972) 1024. [ 10] P.K. Townsend, Phys. Rev. D 10 (1977) 2795.
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