LETT]~RE AL NUOVO CIMENTO VOT.. 10, N. 15 10 Agosto 1974

Quantum Dynamics of a Massless Relativistic String.

~ . ]3ATRASCIOIU

The Institute /or Advanced Study - t)rineeton, N . J .

(ricevuto il 2 Maggio 1974)

Not long ago K o s l s i (1) first and shortly thereafter GODDARD, C-OLD8TONE, ]~EBBI and THORN (~) (GGRT) developed the classical and quantum mechanics of a massless relativistic string. The problem is interesting both because of its relevance to the dual- resonance model (a) and because it became apparent that one can write the quantum mechanics for the relativistic string only in 26 dimensions. I t is this lat ter aspect of the problem that is discussed in the present letter. We find that , besides the purely transverse modes of oscillation foUnd by GGRT, the string has also longitudinal modes. Hence there are nontr ivial motions of the string in 2 dimensions. The longitudinal modes restore the Lorentz covariance of the system in any number of dimensions. Our presentation follows the one of GGRT.

The dynamics of the string is contained in its action which is taken proportional to the area swept out by the string:

Vf ~ T t

(1) s - ~ h c ~ j j [ ~ ~ \ ~ / \ & / j = ~:~ 0 ~'~ 0

In the above expression x~(a, ~) are the co-ordinate vectors in physical space of points on the surface; a and z are parameters used to label points on the surface; ~, h and c ~ are dimensional constants which we shall choose to be 1. The principle of least action applied to (1) yields the following equation of motion and boundary conditions:

(2a) ~ ~ + ea ~x~ -- 0 ,

~s ~Lf (2b) ~ (0, 3) = ~ (. , ~) = o,

where ~g ~ ~x~,/~v and x~, ~ ~x~/~a.

(1) G. KO~CISI: Progr. Theor. P h y s . , 45, 2008 (1972). (~) P. GODDARD, J . GOLDSTONE, C. REBBI and C. B. THORN: Nucl . Phys . , $6 B, 109 (1972). (s) Y. NAYmU: ProCeedings o! the In ternat ional Con/erence on S y m m e t r y and Quark Models (Detroi t , Mich., 1969).

676

Q U A N T U M D Y N A M I C S OF A MASSLESS R E L A T I V I S T I C S T R I N G 677

The action being invariant under the Poincar~ group, its generators P~ dud M ~ are constants of motion. The generators are given by

0

(3b) M ~ = f d ~ (x ~ ~ - - x ~ ") , 0

where ~ -= ~ s The action is also invar iant under an arbitrary nonsingular reparametrization of

the surface which leaves the lines x~(a ~ 0, 3) and x~(a = ~r, ~) unchanged (gauge freedom). One can show that one can always use this freedom of parametrization to choose a and ~ such that

(4a) ~ .x '= 0,

(4b) ~ z + x ' 2 = 0 .

Such a gauge will be called orthonormal. The equations of motion become

(Sb) x't'(0, 3) = x'~(~r, v) = 0.

The most general solution of (5a) and (5b) is

(6) x~(a, v ) = q ' + V 2 [ a ~ + ] ' ( v q - a ) + ] ~ ( 3 - - a ) ] ,

where ]~'(u) is periodic

]'(u)=f'(u+2=),

obeys the gauge conditions (4a) and (4b)

(7) [% + 2f(u)]2 = 0

and is otherwise arbitrary. Therefore ]~'(u) can be expanded in a Fourier series:

[ +~ ] (8) x~(a, z) = q, + V~ a ~ + i ~ ~ cos na exp [-- i n r ] .

The gauge condition (8) gives

(9) ~a~+ noa_ l = 0 .

678 A. PATRASCIOIU

Notice that in an orthonormal gauge ~, x" and P" obey the following relation:

(lO) f d a x a = vr(q~ + 2P~r).

0

We observe that requiring a gauge to be orthonormal does not determine it uniquely. The most general reparametrization which preserves orthonormality is given by

(11)

where g(u) is periodic

{ ~ = 3 o + r + g ( r + a ) + g ( r - - a ) , = a + g(r + a) - - g ( r - - a) ,

g(u) = g(u + 2~),

leads to nonsingular reparametrizations 8(~, ~)/8(r, a) ~ 0

(12) l + 2 d ( u ) r

and is otherwise arbitrary. Expanding g(u) in a Fourier series gives

(13)

= v o + T + ~ An cos n a exp [-- i n v ] ,

= a - - i ~ A n sin na exp [-- inT] . riCo

In their paper GGRT use this gauge freedom to choose ~ such that

(14) 1 0 x t ~ ~-~ (x + x 3) = 2 P % .

That is, suppose we are in some orthonormal gauge (r, o) where x~(v, o ) # 2 P C r . Then, they say, from (6) and (11) it is apparent that we can always go to another ortho- normal gauge (f, 8) where (14) holds true. In fact, if we are willing to look at the set of all possible motions of the string, the statement is false as we will show now.

I t is obviously possible to describe the evolution of the string by taking pictures of it at a succession of times. That is to say, we can certainly choose

(15) x~ a) = 2P~

Then from (10) follows that if we want an orthonormal gauge, a must be chosen such that

(16)

a

p o a = : r f d a ~ ~ .

0 f i x e d ~r

In this gauge the orthonormality condition (4b) requires the ends of the string to move with the speed of light.

QUANTUM DYNAMICS OF A_ MASSLESS R E L A T I V I S T I C STRING 6 7 ~

The question is whether we can use the remaining gauge freedom to choose an ortho- normal gauge where (14) is true. The answer is no because in general this would imply performing a singular change of variables. Indeed, the Jaeobian of a transformation which made x ~ proportional to ~ is proportional to

~(~' d) oc [a~ + 2p(~ ~- a)] [a~ ~- 2p(T-- a)].

Without violating the gauge condition (7), one can have a motion of the string such that

a ~ ~- 2]#(u) = O

for some finite interval in u. This condition means that for some finite time interval the end of the string, which is always moving with the speed of light, has zero velocity in the x 1 and x 2 directions (recall x#= (1/V/2)(x~ x3)). An example of such a motion is provided by a string in 2 dimensions, or equivalently by a string moving radially in any number of dimensions.

Next we turn our at tent ion to the quantum mechanics of the string. As we have already seen, the gauge condition (4) yields an infinite system of nonlinear coupled equations (9). They cannot be solved for some normal modes in terms of the other; this is the crucial step in applying the noncovariant quantization.

The covariant quantization retains all degrees of freedom as independent and enforces the constraints on the physical states. In our case the operators are q0, ao and a, which obey the following commutat ion relations:

(17) [a~, a~] = g"~ ~._~ [qo", a~] = i V~ g"~.

The gauge conditions (9) require all possible states in our theory to obey

(18a) L.v IV> = O,

(18b) Lo IV}---- ~o [~>,

where co

(19) L~v = � 8 9 :az~+z'a_t: �9 -r

~Y>O,

The solutions to (18a) and (18b) have been studied (4). The results are that for ~o = 1 and 1 ~ D ~< 26 and for ~o < 1 and 1 ~< D <~ 25 there are no negative-norm solu- tions to (18a) and (18b) (D is the dimensionality of space-time). However, in general the solutions include null states (states of zero norm orthogonal to all the other solu- tions of (18a) and (18b)), which are undesirable, as they introduce an arbitrariness in the definition of physical states. Thus if ]p> has positive norm and In> is null, then

IP '> - tP>-4- ~ln>

has the same norm and couplings as [p>.

(~) 1~. C. BREWER: Phys. Rev. D, 6, 1655 (1972); P . GODDARD and_ C. B. THOR~: Phys. Left., 40 B, 235 (1972).

6 8 0 A.. P&TRASCIOIU

The stipulation made by GGRT was that the existence of null states reflects the arbitrariness in choosing some orthonormal gauge from the many possible ones. Indeed in enforcing (18a) and (18b) we require only that the gauge be orthonormal, but do not specify it otherwise�9

If the existence of null states were a gauge effect, then for any positive-norm states IPl> and [p~> and null state In> obeying (18a) and (18b) and for every operator O, there must exist some gauge transformation (13) such that

(20) <pl[O[[p~> -[- I n > ] = (p~lOIp2>,

where 0 is the operator obtained from O by performing this gauge transformation. To disprove this assertion we only need one counter-example to (20).

From (8) and (13) we can compute the change in q and an under an infinitesimal gauge transformation. We find

(21a)

(21b)

(21c)

n # o

ao" = 4 ,

�9 d ~ a z~ �9 n u ,u .u

$ - - ~ ' t - - - - a o A n - a n T o - - ~ _ " a n + m A _ m . Tt, q'b m~O

Now in (20) take O to be some a~, n # 0 , and choose IPl> and [p~> such that <pl[a~lp~>= 0 for all m (for example [Pl> and 1202> have the same mass but different spins). Then the r.h.s, of (20) is clearly zero�9 The 1.h.s. can always be made nonzero by choosing the null state such that

This example shows that null states are not gauge effects, which could be gauged away by a clever choice of gauge.

However they are innocuous, since they can be eliminated by hand in a Lorentz- covariant manner. This is a corollary of the following theorem: if ]p> and In> satisfy (18a) and (18b) and In> is null, and if O is a gauge-invariant operator ([O,L~v] = O for all _At), then <p[OIn > = O.

The proof is trivial once one notes that , O being gauge invariant , OIp > also satisfies (18a) and (18b). Then In> being a null state, the theorem follows�9 The corollary is that if O is gauge invariant and In> a null state, then OIn > is also null. Now the Lorentz group generators M~ are gauge invariant , so tha t null states can be eliminated from the theory by hand, in a Lorentz-covariant manner.

I t is a pleasure to thank R. DASH~, A. N~v~u, T. REGGE and C. B. THORN for useful discussions.