covariant loops and strings in a positive definite hilbert space

IL NUOVO CIMENTO VOL, 37 A, N. 3 1 Febbra io 1977

Covariant Loops and Strings in a Positive Definite Hilbert Space (*)

F. ]~ 0 I:I~LICtt

Department o/ Physics, Syracuse Uqdve~'sity - Syracuse, N. Y. 13210

(ricevuto il 5 0 t t o b r e 1976)

S u m m a r y . - - Relat ivist ic loops and strings are defined in the conven- t ional way as solutions of a one-dimensional wave equation with certain boundary conditions and satisfying the orthogonal gauge conditions. Conventional pseudo-Cartesian co-ordinates (rather than null-plane co-ordinates) are used. The creation and annihilat ion operator four- vector ~ and a~ are required to be spacelike (orthogonal to the to ta l momentum P~), so tha t the resulting Fock space J~+ is posit ive definite. This requirement is shown to be mathemat ica l ly consistent with Poincar6 invariance and to impose no addi t ional physical constraints on the system. I t can be implemented in a canonical realization of the Poincar6 algebra as a condition on state vectors, or in a noncanonical realization as an operator equation, as is done here. The space JC+ is further res t r ic ted by the Virasoro conditions to a physical subspace ~ which is of course also posit ive definite. In this way there arises no crit ical-dimension problem and Poincar6 invariancc holds also in 3 § 1 dimensions. The energy and spin spectra are the same as usual, leading to l inear Regge trajectories, except tha t there are no tachyons and no zero-mass states. The leading Regge t ra jec tory has negative intercept.

1. - B a c k g r o u n d .

T h e r e is s t r o n g e v i d e n c e a t p r e s e n t t h a t n o n - A b e l i a n g a u g e fields p r o v i d e

t h e (( g lue )~ for q u a r k s to c o m b i n e i n to h a d r o n s . A s i m p l e m o d e l l eads to p r o m -

i s ing r e su l t s on SUa h a d r o n d y n a m i c s ( c h r o m o d y n a m i c s ) ( 1 ) . B u t r e a l i s t i c

m o d e l s in 3 + 1 d i m e n s i o n s a r e v e r y diff icul t to c a r r y t h r o u g h .

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (1) C. G. CALLAN, N. COO~ and D. G~OSS: Phys. Rev. D, 13, 1649 (1976).

242

COVARIA.NT L O O P S A N D S T R I N G S IN A P O S I T I V E D E F I N I T E I I I L B E R T S P A C E 2 ~

I t is therefore suggest ive to resor t to s tr ing models. These have for some

t ime been known to arise as l imit ing cases of gauge field theories (2.4), and

mos t recent ly have iu fact been shown to be equivalent to ch romodynamies in I -'- I d i m e n s i o n s (5).

However , realist ic s t r ing models encounter the difficully of the free rel-

a.tivistie s tr ing: thei r q u a n t u m dyn'mfics seems lo ca r ry through ~nly in 25 + 1 dimensions (e) or in 1 + 1 dimensions (7). Str ing models in 3 + 1 di-

mensions thus awai t the solution of this difficulty.

This solution was accomplished recent ly , as has been briefly repor ted for

a noncowtr iant formula t ion (8), and in somewhat grea ter detai l for the con-

vent iona l covar ian t and canonical formulat ion (9). Bu t the la t ter involves

an indefini te-metr ic t I i lbe r t space JG• which mus t be res t r ic ted to a t)ositL:e -

met r ic physical subspace + by suitable conditions. [n the presen t pape r we shall p resen t the relat ivist ic s i r ing and loop in

3-~-1 dimensions in a formulat ion which is also mani fes t ly co~ariant , bu t which leads directly to a posi t ive definite Hi lber t spaee. While this formulat ion

is ma themat i ca l ly equivaleni to the one previously (9) given, it seems physical ly

considerably more attracti~'e.

Our t r e a t m e n t uses a <, t imel ike gauge )b i.e. uses the convent ional Cartesian

co-ordinat izat ion of Minkowski space ins tead of null-plane co-ordinates ((< null

gauge >>), which is more commonly used for this problem.

Beyond tha t , however , our descript ion differs in the following essential

way f rom previous work: we observe t ha t the relative posit ions ~, and m o m e n t a

zt, of the points on the s t r ing are spacelike vectors. (, Rela t ive )> here means relat ive to the center-of-mass posit ion and to ta l m o m e n t u m of the string.

This is in tu i t ive ly obvious and will be shown to be ma themat ica l ly correct. I f these spacelike vectors ~, and ~, have no ~0 and ~ - e o m p o n e n t s in the c.m. f r ame then it follows tha t ~, and ~# are or thogonal to the to ta l m o m e n t u m / ) ~ .

This fact is sufficient to ensure a posi t ive definite Hi lbe r t space for these operators, i.e. the Pock space can be cons t ruc ted f rom creat ion opera tor four- vectors (a:) which are spacelike. This idea is lhe subject of sect. 2.

(2) Y . NA)iBLT: Phys. Rev. D, 10, 4262 (1974). (a) Cx. T'HOOFT: ,Yucl. Phys., 72 B, 461 (1974). (4) A. P. BALACHANDRAN, R. ]~AMACIIANDRAN, J. SCIlECHT]~R, K. C. WALI and H. RUPER:rSBERGER: Phys. :Rev. D, 13, 361 (1976). (5) I. BARS: Phys. Rev. Lett., 36, 1521 (1976). (s) P. GODDARD, J. GOLDSTONE, C. REBBI and C. B. TIIOR.~-: 5~ucl. Phys., 56 B, 109 (1973); C. R~.~BI: Phys. Rep., 12 C, 1 (1974); J. SCHER~: Rev. Mod. Phys., 47, 123 (1975), ~nd other papers quoted there. (7) W. A. BARDEEN, I. BARS, A. J. HANSON a]ld R. D. P>'CC;EI: Phys. Rec. D, 13, 2364 (1976); A. PATRASCIOIU: NUC[. Phys., g l B , 525 (1974). (s) F. ROIIRLICH: Phys. Rev. Lett., 34, 842 (1975). (~) F. ROItRLICn: Nucl. Phys., 112 B, 177 (1976).

244 F. ROHRLICII

A comparison with e lec t rodynamics is instruct ive. In e lec t rodynamies the Loren tz condition k . A ( k ) - = 0 or A ~ k . A / k ~ is used as a condit ion which physical s ta te vectors mus t ()bey (in a weaker sense only, k.A(-)(k)iq~} = O,

but t ha t is i r re levant here). Thus the t ime componen t of A~' is e l iminated by

cancellat ion with the longi tudinal componen t k . A . The ghost s ta tes (which

are due to A ~ are thus el iminated and ~ is res t r ic ted to a posit ive definite Hi lbe r t space.

I n the s t r ing the analogous condition is _P.a, = 0 or a ~ ~-- P . a , / 1 ~, where

Pa is the to ta l m o m e n t u m a.nd a~ is the annihi la t ion opera tor for the n-th mode.

W h e t h e r this condit ion holds as an opera tor condition or as a condit ion on

the physical s ta te vectors is i r re levant a t the momen t , bu t will be the topic of sect. 3. I n ei ther case, this condition ensures the el imination of the t ime

componen t s of a~ in favor of tile space components a . which obey the (, r ight ,> commuta t i on relations, thus ensur ing a posi t ive-metr ic t I i l be i t space.

Bu t there is an impor t an t difference be tween k . A (-)-- 0 and P . a . - = 0: k,

is a null vector , while P~ is (by choice of the representa t ion of the Poincar~

group) a t imcl ike vec tor (*). This has the consequence tha t A ~ can be climinat(~d

only b y cancellation, while a~ is ac tual ly zero in the c.m. f r ame (where P - ~ 0),

and appears in other f rames only via Lorentz t rans format ions : a~ is a.ssured

to be spacelike with three independent components , while A~ has only two.

I~ow it is also clear why a t imel ike gauge is preferred over a null gauge: = r with only in the null ga.uge the condition a~+ 0 yields a spaeelike vec tor a ,

two independen t components , while in the t imel ike gauge the condition an~ 0 leaves a , with three independent components .

The ghost el imination in the s t r ing is thus carr ied out by the condition P . a , = O. But the resul t ing posi t ive definite Hi lbe r t space J~+ c ~ • is not

the physica l subspace (/). The la t te r is obta ined by fu r the r res t r ic t ing JC+ b y means of the Virasoro conditions (~o).

I n the execut ion of this p rog ram one can proceed in two a l te rna t ive but equivalent ways depending on the par t icular realization of the Poincar6 algebra

which one wishes to adopt , as will be explained in sect. 3. Ou the basis of these

p re l iminary studies, the loop (sect. 4) a.nd the s t r ing (sect. 5) will then be dis-

cussed in detail, and some of thei r physical s ta tes will be exhibi ted explicit ly

(sect. 6). A s u m m a r y concludes this pape r (sect. 7).

(*) As is customary, wc characterize the string and the loop as free relativistic systems with P~ pointing into the future light-cone. 0 o) Here and in the following we shall ignore the presence of states of zero norm, because their elimination is a technical point that would only detract from the main argument: the physical space is actually the quotient space r where J~o is the subspace of JG+ which has all zero-norm states.

COVAIr LOOPS AND S T R I N G S IN A P O S I T I V E D E F I N I T E I I I L B E R T SPACE 245

2 . - G h o s t e l i m i n a t i o n .

In the convent ional t r ea tmen t of the string the ghost elimination is carried through as follows (11). One star ts with the Virasoro conditions on the physical s tate vectors [~} and one asks for creat ion operators A, which a) are spa cc- like, i.e. satisfy the ((right >> commuta t ion relations, and b) generate the ~ec- tors I~} when act ing on the vacuum. Thus it is required tha t the A~ produce exactly all the [~}, not more, and not fewer states. This leads to a conflict be tween the Virasoro algebra and the Poinear6 algebra which can be resolved

only by the special choice of D ---- 25 + 1 dimensions and by a uni t in tercept of the leading Regge t ra jec tory .

As was already poin ted out in ref. (~), this way of el iminating ghosts involves too strong a requirement . A weaker requ i rement is to restr ict the indefinite-metric space JC+ first to J~+ by means of the conditions / ) . a , = 0 and then to res t r ic t it fu r the r to ~ by means of the Virasoro conditions. In this way JC+ is genera ted by the a~ in the c.m. f rame and by spacelike a~ ~ in any other frame. Bu t there "~re no creation operators which generate exact ly 4). The scheme is ~b c ;E+ c JC~.

In order to be able to impose the conditions P . a ~ = 0 in an appropria te manner in the theory of free strings and loops, the following will be shown:

a) these conditions are physical ly eminent ly reasonable,

b) they ,~re consistent with Poincar~ invarlance,

c) they are consis tent with the commuta t ion relations,

d) t hey ~re consis tent with the Virasoro conditions.

The remainder of the present section will be devoted to i tem a), the following sect. 3 to i tems b) and c), and sect. 4 and 5 to i t em d) for the loop and the string, respect ively.

Le t x2 be a vector f rom some origin 0 to an a rb i t r a ry point P of an ex- t ended classical physical object. Le t Q~ be a vector f rom 0 to some reference point of t ha t object, for example its center of mass (c.m.). Then we denote by ~ the position of P relat ive to tha t reference point:

(2.1) ~ = x~-- Q~.

I f all these vectors are t ime dependent due to the mot ion of the object, (2.1) must be refer red to <~ equal t ime ~> of all th ree vectors. Such a specification

(11) R.C. BROW~R: Phys. Rev. D, 6, 1655 (1972); P. GODDARD and C. B. THORN: Phys. Zett., 40 B, 235 (1972).

2 4 6 F. ROtIRLICH

of (( equal t ime >> is of course f rame dependent in special relat ivi ty. But , if the object is a free object, far removed from all external influence (a closed system), then the f rame in which its c.m. is at rest is a preferred frame. The t ime meas- ured in tha t f rame is the proper t ime ~. If the system has c.m. velocity Vg, t hen the motion of the c.m. position Qt, is

(2.2) Q ~ ( v ) = Q , ( O ) + V " z

according to Galileo's law of inert ia. I t follows that , for every point P in the sys tem and every 3,

(2.3) x . ( r ) = Q . ( 0 ) + V .~+~ . (~ ) .

I t is obvious tha t formally the same equation will also hold in relativistic quan tum mechanics with x,, Q,, V, and ~, being now four-vector operators.

As ment ioned before, ~ is the t ime in the c.m. frame. Therefore, with V~ we can wri te

(2.4) o o 0 x ~ = q .... ( ) + ~

and o Q .... (0) character izes the origin of the t ime axis. This means tha t we can choose

(2.5) ~2m.=0.

In the c.m. ]rame the relative position ~ has no time component. I t follows tha t ~, is a spacelike vector and satisfies

(2.6) V.~ ~ 0 or /~.~ ---- 0 ,

since V, and P~ must be proport ional to one another . I t is obvious tha t , if

P~ and $~ commute , this or thogonMity also holds in quan tum mechanics.

Le t p~ be the momen tum of the point P of the object. Then we can define t h e re la t ive momen tum

(2.7) ze~ =-- p~-- _P~t/V ,

where V is the invar iant volume of the object. By definition of the to ta l momentum,

(2.s) Y ~" = 0 , P

where the sum extends over all points P of the sys tem (possibly an integral).

C O V A R I A N T L O O P S A N D S T R I N G S I N A P O S I T I V E D E F I N I T E I I I L B E R T S P A C E 2 4 7

Let us fu r ther assume a uni]orm mass density so that in suitable units ps, and x, are related by

(2.9) p~ = dx#/dv = :~'.

Then (2.7) follows from (2.3) with

(2.10) :~" = ~",

(2.11) ~9'---- P~/V.

But, since (2.5) holds for 'ill 3, it follows tha t

(2.12) :t~ ---- 0 ,

i.e. t ha t :r~ is a spaeelike four-vector and satisfies

(2.13) P ' : t : O.

This follows also f rom (2.6) if we note t h a t / ~ is a constant of motion. I t is obvious tha t , if P~ and ~r~ commute , then this or thogonal i ty will also hold in quan tum mechanic.s.

If the relativistic str ing is paramet r ized by a e [0, vr] and v, it is de-

scribed by (~)

(2.1.1) x,,(a, ~)=Ql,+ P~T+~r ~/:r,,~o ~176

(We sha.ll use Q~ f rom here on to designate Q~(O).) This expression is exact ly o the form (2.4), if we use V= zz and (2.10). With (2.9) one finds (e)

(2.15) p a ( a , ~ ) = + -~= ~ a ~ . c o s n a e x p [ - - i n v ] . %/:r nr

F r o m a comparison of these two expressions with (2.3) and (2.7) it follows tha t the last t e rm in (2.14) is to be identified with ~"(a, 3), while the last t e rm of (2.15) is :r,(a, 3). The or thogonal i ty relations (2.6) and (2.13) are then equivalent to

(2.16) P . a~ ---- O.

In the quant ized form the ~ (n > 0) are re la ted to the annihilat ion operators a. ~ by

(2.,7) ,~ = v ~ a ~ .

2 4 8 r . ROHRLICH

3 . - A l t e r n a t i v e d e s c r i p t i o n s .

I t is obvious from (2.14) tha t the relativistic quantum mechanics of the string has to do with position operators. This notion becomes more specific when we consider the generators of the Poincar6 algebra

(3.1)

(3.2)

n

pt, -~ f W ( a , T) da , 0

$$

M ~ =__ f (xl' p~-- x~ p~') da = Q#_P, - Q,_p, -4- s ~'~ , 0

where S~ v depends explicitly only on ~ and not on Q~ or zoo. In the derivation of (3.2) one observes tha t both ~ and ~ vanish when summed over V (see, e.g., (2.8)), because we assumed a uniform mass density. The interpretat ion of this result is a separation of the to ta l angular momentum into an (orbital) angular momen tum due to the c.m. motion and a (spin) angular momentum due to the internal motion characterized by ~ and ~r~:

(3.3) S~ --f(~l,~r,-- ~'vr~) do'. 0

The M ~ components of (3.2) show an analogous separation for the boost operators.

Any four-vector operator Q~ which yields the Lorentz generators by (3.2) will be called a <~ position operator ~>. Given P~, the Q~ are of course not unique. For each possible Q~ there exists an appropriate S~ ~, so tha t M~ ~ and zP, will satisfy the Poincar6 algebra ft. The algebra of the operators Q% zP~ and S~ ~ will be called the Q P S algebra. I t is a realization of the Poincar4 algebra ff if (3.2) leads to ft.

The most commonly used Q P S algebra is the covariant canonical algebra. I t shows independence of the c.m. variables Q~ a n d / ~ from the internal va-

riables S ~

(3.4) [Q,, S ~] = 0, [P~, ~ ' ] = 0.

The c.m. variables satisfy the canonical commutat ion relations (CCR)

(3.5) [Q~, Q~] = 0, [P~, ~p~] -- 0, [Q~,/~] : ig,~

COVARIANT LOOPS AND STRINGS IN A POSITIVE D E F I N I T E HILBERT SPACE ~ 9

and Sa~ satisfies the Lorentz algebra:

(3.6) [S ~'', S ~ = i(g aq S "~ + g'" S ~'q -- g~ ~o _ g,~ S~,,,) .

We note tha t Q, and St, do not belong to the enveloping algebra of ft. I f we want to impose the condition

(3.7) P , , S ~" = 0 ,

we see tha t it would lead to a contradict ion. Therefore (3.7) can only be im-

posed as a condit ion on the state vectors:

(3.8) P~ s~'l~> -- o.

Therefore, the covar iant canonical QS_P algebra (3.4)-(3.6) is not compatible with the or thogonal i ty relations (2.6) and (2.13) as operator equations, since they would imply (3.7) via (3.3). Only the weaker or thogonal i ty relations

(3.9) P'}[q~> = 0 and P.~[q~> = 0

are consistent with this Q P S algebra. And (3.9) was in fact used in ref. (~).

I t is clear tha t commuta t ion relations such as (3.5) will necessarily lead to an indefinite-metric t I i lber t space and tha t (3.9) will restr ict it to one of positive metric. I t is, however, a reasonable question to ask whether this indefinite- metr ic space needs to arise in the first place, or whether it can be avoided completely.

This will indeed be possible if a Q P S algebr.~ can be found in which (3.8) need not be imposed, but holds ~s an operator equat ion (3.7). This would be somewhat analogous to replacing the Gupta-Bleuler conditions (Lorentz gauge) by the Coulomb gauge condit ion V .A = 0, which holds as operator equation. However , in our case we will not lose manifest covariance.

The Q P S algebra in which (3.7) hohts is the covar iant bu t noncanonical algebra

(3.10)

(3.11)

(3.12)

where

(3.13)

[q. , Q.] = - iM,,l~'~, [•., P,] = O, [q,, i' ,] = i i ' [ ~,

[ Q., s Q~] = i[ ~ s " ~ - 1 ~ s~,J I P . , [ P", s ~] = o ,

P~"-- 9~ , - - P . P ~ / P ' .

Thus, this algebra involves the project ion operator P ~ onto the space per- pendicular to P~. I t is not a Lie algebra., and it is not canonical, as is obvious

17 - l l Nuovo Gimenlo A.

250 ~. ~OH~L~CH

from (3.10). Bu t (3.7) is obviously consistent with it. And, of cours% it is a realization of ft.

In the following sections the above covariant noncanonical algebra will be used to describe the string and the loop (closed string), so tha t one never encounters ghost states.

I t remains to implement this algebra by equal-time commutat ion relations (ETCt~) between Q~, P~, ~ and n~. In the canonical case,

(3.14) [~,(a), n~(a')] -~ ig~(5(a-- a') -- V) ,

while the components of ~ commute with each other, and so do the components of ~z~. The 1IV t e rm in (3.14) is necessary to be consistent with the vanishing integral of ~(a) and of ~z~(a') over the volume V. The internal variables ~ , ~z~ also commute with the external ones Q~, 2~. The QPS algebra (3.4)-(3.6) is then satisfied. Upon subst i tut ion into (2.1) and (2.7) one finds the CCI~

(3.15) [x~(~), x~(~')] = 0, [x,(a), p~(~')] -~ ig,~ ~(~-- ~'), [pz((r), p~(a')] : 0.

For the noncanonical algebra (3.10)-(3.12) the si tuation is different. The internal variables must satisfy the ETCI~

(3.16)

[~.(~), ~(a')] = 0 ,

[~(~), ~(~')] = iioi~ (~(~ - ~') - - ~ ) ,

[~.(~),~(~')] = 0,

which differs from (3.14) in the replacement of g~' by -P~,'. But, in addition to this difference, the internal and external variables no longer commute. While the internal variables are t ranslat ion invariant (they are relative po-

sitions and momenta) ,

(3 . ,7 ) [r P~] = o , [z , , P~] = o ,

their CI~ with Q, are nontrivial :

(3.18) [Q", ~(o)] = - i~(o) P~/P~, [Q,, n~(a)] = - i~,(a) p,/p2.

One verifies tha t the noncovariant QPS algebra indeed follows from this algebra. And in part icular it is consistent with

(3.19) ~ ' P ---- 0 , ~r-P ~- 0

as operator equations.

COVARIANT LOOPS AND STRINGS IN k POSITIVE DEFINIT'E HILB'~.RT SPACE 251

Finally, i t is interest ing to observe tha t the variables x , and p~ no longer satisfy canonical, or even in tui t ively obvious, commuta t ion relations. One finds

[x~(a), x~(a')] = -- i( M,,,' + $~,(a') P ~ - $,'((r) P~,) /P ~ ,

(3.20) [p~(a), p~(a')] = 0,

[x~(a), p~(a')] = iP~ ~ ~(a-- a ' ) - iTt~'(a) P~/P'~.

In a noncanonical theory , these variables have thus lost their p r imary importance as well as their intui t ive quali ty. The fundamenta l variables are the external

variables and the in ternal ones. And tha t is consistent with our physical description. A closed relativistic sys tem is described by two completely dif-

fe rent sets of variables: a) the (~ external ~) ones which account for the tr ivial uniform motion and for the constant orbital angular momen tum dependent on the arbi t rar i ly chosen origin, and b) the in ternal ones which describe the interest ing s t ructure and dynamics of the system relative to its c.m. in a. t rans- la t ion-invariant way, its energy spectrum and its spin spectrum.

4. - The covar iant loop.

The loop is defined classically as a closed one-dimensional line t racing out a t imelike cylindrical 2-surface in M3+1 which is minimal in the sense of Iqambu.

In the orthogonal gauge its defining equations are (if we assume a physical r~nge for a as [0, 2~] such tha t V----2u)

(4.1) p" = x"t',

(4.2) x,(a + 2Jr) =: x , (a) , p"(a + 2Jr) = p"(a) ,

(4.3) (p ~ x') -~ = O.

Ttle classical equat ion of motion and the periodici ty condit ion can be taken

over into quan tum mechanics formally unchanged. The or thogonal gauge

condition (4.3) is not l inear and, therefore, not well defined, since ~ and ~a are mathemat ica l ly speaking operator-valued distributions. Since we shall

be able to express the in ternal variables in te rms of Fock-space operators, eq. (4.3) can be defined in te rms of a normal ordered product (Wick product)

(4.4) :(p -- x')'2: + (mo/V)~--= 0 (at least weakly) .

The addit ive constant is so far undetermined.

2 5 2 ~. ROHRLIOH

In an at tempt to apply the noncanonical realization of if, one could apply the ETCR (3.20) and find a suitable Hamiltonian which will yield (4.1). How- ever, in the spirit of treating the trivial c.m. motion separate from the internal motion, we proceed differently, although the result will be the same. We make the separations (2.3), (2.7), in the spirit of (2.9) and (2.10), since we will have a (~ uniform mass density ~) and a Hamiltonian that depends only quadratically on ~:

(4.5) x~(a, .) = Qu + .pl, z/2~ + ~.(a, ~),

(4.6) p,'(a, ~) = .e./2# + :~,,(,~, ~:).

Now the equation of motion (4.1) becomes

(4.7) ~ . = ~-.;

the periodic boundary conditions (4.2), (4.3) simply impose periodicity in a on ~a and ga, and the orthogonal gauge condition (4.4) becomes

(4.8) P~/2g + 2n:(n 4- ~')~: + m~/2:~ = 0 (at least weakly),

where we have used (3.19). The ETCR (3.16) will yield the equations of internal motion (4.7) if we

take as (internal) Hamiltonian

(4.9) 2 ~

H = �89 :(~* + ~'*):da @

because one finds (2.10) and (4.7). Since ~,(a, ~) is periodic and must satisfy (4.7), it can be expanded into

modes:

(4.10) i ~[a~ a)]+~exp[--in(v--a)])

The restriction n r 0 is necessary because ~, and ~z, have vanishing average over the volume V-----2z. The averages of xa and p , are just responsible for the external variables Qg and P , which we have separated out. Now the ~ and ~, can be replaced by the sets a~ and fl~ (n---- 4-1, =k 2, ...). From (3.16) it follows that their CR are

(4 .n) [ ~ , < ] = mi~ ' O,~,.o = hS~, &q, [ ~ , ~ ] = o.

C O V A R I A N T LOOPS AND S T R I N G S IN A P O S I T I V E D E F I N I T F . HILB~ .RT SPACE 2 5 ~

Furthermore, the self-adjointness of ~a and zr~ requires

(4.12) ~,_. = ~ * , ~ = ~.~

and the orthogonali ty relation (3.19) becomes

(4.13) a , , 'P = 0, fl,~'P = O.

Thus ~ and fl~ are spaeelike four-vectors. ~ o w we are in a position to construct our Hilbert space. The operators

a~ and fl~ are convenient ly replaced by

(4.14) { ~.~ - < / C 4 , ~ - D."/v~ (n > o),

a~ * - ~ * / v ~ , ~* - ~*/V-~ (n > o),

so tha t these vectors satisfy the algebra of creation and annihilation operators:

(4.15)

Ea~, a:*] = P~," 5~.,, [a~, a~] = 0 (m, n > 0),

b ~ _ ~ [ ~, b:*] = P~" tt~,,, [b~, b~'] 0 (m, n > 0),

[a~, b:] = 0, [a~, b:*] = 0 (m, n > 0).

The vector operators a~ and b~ arc the operators obtained when one boosts the rest f rame operators (0, am) and (0, b~) by means of a Lorentz transfor- mation. The Fock space JC+ constructed in the rest f rame from the vacuum is clearly positive definite. This proper ty is not changed by a boost.

Now we turn to the constraints (4.8). When expressed in terms of the a~ and fl~ these become the conditions

(4.16) _ (p2 + m~) =

= 4a ~ (A(. ~) cxp [-- in(': + (0] + A ~ exp [-- in(,:-- a)]) n

where

(at least weakly) ,

{a) = A c ~ ) - - ~ �9 (4.17) A. _ �89 Z :a.-a._.~: Z :fl,~'fl,,-., ~Jtn = 2 m~O m~O

They Call be satisfied with

(4.18) A(~ ~, 19> = 0, A~' 19> = 0 (n > 0)

254

and

(4.19) (p2 + M 2) u = 0,

where the opera to r M ~ is defined b y

(4.20) M 2 - - mo 2 + 8~Ao,

(4.21) Ao �89 +

F . R O H R L I C H

The s ta tes I~0) which sat isfy (4.18) define a subspace of J~+ which we call the

phys ica l space r Equat ions (4.18)-(4.20) toge ther with (4.13) are equivalent

to the Virasoro conditions.

The mass opera tor M 2 mus t be s t r ic t ly positive~ because we choose a rep-

resen ta t ion of ff for which zo~ is timelike~ corresponding to a mass ive free

phys ica l sys tem. F r o m (4.21), i.e.

r

2 4~ ~ t (4.22) M 2 mo -t- = n(a, .a,, + b*,, "b.), 1

t hen it follows tha t

(4.23) m o > 0

is a necessary and sufficient condit ion to ensure P-" < 0. Thus no tachyonic

s ta tes are possible. The spin tensor (3.3) can be expressed in t e rms of the a~ and fl~ b y substi-

t u t ion of the mode expansions (4.10) into (3.3) b y using (2.10). One finds

' +" " +" " - i - , r (4.24) S"' = -- ~ t:r a , - a , ~ . . . . . n # 0

This pe rmi t s one to ver i fy t h a t

(4.25) [S, ,v, A.] = 0 .

A special case of this relat ion is of course [S~ ~, M ~] = 0 which holds in the non-

canonical represen ta t ion of the Poincar6 algebra, as well as in the canonical one.

The A~ sat isfy the Virasoro a lgebra

= -, ~,m(m--- 1) ~5,,, t ,.o (all in, n ) . (4.26) [A,~, A,] (m- - n)A ..... + ~ ">

F r o m this algebra, it immedia t e ly follows t h a t

(4.27) [M ~, A,] u = 0 (n > 0).

COVARIANT LOOPS AND STRINGS IN A POSlTI~TE D]~FINIT~, I tILB]~RT SPACE r

Moreover, one verifies tha t

(4.28) [a,,'.P, A,] = m ~ + , . / ) = 0

and the same for the fl~. These relations establish the internal consistency of the theory: the con-

straints are pr imary constraints and first-class ones, commuting with each other, with (4.13) and with the I tamil tonian on the physical subspace ~b.

Final ly, f rom (3.18) and (4.10),

(4.29) [Qz, ~:] = _ i ~ p~lp2, [Q~, An] = 0, [P", An] = 0

can be ascertained. Equations (4.25) and (4.29) ensure tha t the generators of the Poincar6 algebra M, ~ and P , all commute with the constraints on r There- fore, the Poincar6 transformations produce automorphisms of ~b and do not

lead out o/ the physical subspace.

This ensures the Poincar6 invariance of the solution which we have obtained: manifest covariance must be supplemented by the invariance of the physical subspacc.

The defining equations of the classical covariant loop are exactly (4.1) through (4.3), and the internal t Iamil tonian is (4.9) (of course without the normal ordering). The theory carries through exactly as above (but with commutators replaced by i t imes the Poisson brackets) up to (4.16). The constraints are satisfied by

(4.30) A: ~) = o = A~' (n # 0)

and

(4.3~) M~ = 4~(A~ ) + A~)) .

The spin tensor has the same form (4.24). In order to exhibit Regge behavior in this classical case, one can consider

single-mode excitation. Le t n > 0 be a particular fixed integer and assume tha t all cr and fl~ vanish except ~. , and f l~. Le t us fur thermore specialize to the center-of-mass f rame where a~.. = (0, a• etc., and take [a.[ = [~,]. Then

(4.32)

(4.33)

~ = S~l~4~ = s~l~nl" ,

i *

The only nontriviM AM is (in this reference frame)

(4.34) A2n----L~. 1 ~ 1 2

256 F. ROHRLICH

The solution of (4.32) and (4.34) is

(4.35) a , = 4 ~ ( . ~ , § i~,), M ~. = ~ - = (~" + ~ ' ) ,

where e,, ev and ~,', ey are each a p~ir of orthonormal vectors. Substi tution

of this result in (4.33) leads to

i 2 (4.36) S = 8 ~ (8, -q- g,,),

where e~ • ~ ---- e, and e~ • e, = e,. The vectors e, and e. are of course in general not parallel. Equat ion (4.36) is the classical form of the linearly rising Regge trajectories of the covariant loop.

The quantum mechanical analogue of this result is more easily carried out af ter the Regge trajectories of the string are derived. These will be discussed in sect. 6.

5. - T h e c o v a r i a n t s tr ing .

The covariant string can be regarded as a l imiting case of the covariant loop: as a collapsed loop. I t emerges from the loop by a pointwise identification of its two halves [0, z~] and [n, 2hi:

(5.1) { ~(a) = x~(2z- a),

p ~ , ( a ) = p . (2~- a).

When this condition is added to the defining equations of the loop (4.1), (4.2) and (4.4), the defining equations for the relativistic string result.

Equivalently, instead of (5.1) one can add to the defining equations of the

loop the symmet ry conditions

(5.2) { x.(a) = x~(- a),

p.((~) = p . ( - a).

One can easily see that , in view of the periodicity (4.2), eqs. (5.1) and (5.2)

arc equivalent.

C O V A R I A N T LOOPS AND STI{.II~GS 1N A P O S I T I V E I)]!IFINIT~] I t I L B E R T 8PAC]~ 2 5 7

An immediate consequence of these defining equations is the p roper ty of the end points of the str ing ~ = 0 and ~----~z:

(5.3) x'.(o) = o x' . (z) = o .

in (4.23) for the loop which will also be required The posi t ivi ty condition of m o

for the quan tum string ensures tha t the end points will never move with light velocity. Tiffs is not so in the cla.ssical theory, in which the choice m~ = 0 does

not seem to be prohibi ted: there is no nontr ivial classical vacuum state of the string.

The range of a now is [0, ~]; consequent ly the volume V of (2.7) is V----~z and the loop t Iamil tonian (4.9) is replaced by the string t tamil tonian

(5.4) H = �89 (n~ + ~") da . 0

Heurist ical ly one can imugme a loop of constant mass densi ty ~)t == r and a

range [0, 2z] to be collapsed to a str ing of mass densi ty 2~ and a range [0, hi, so tha t the total mass rema.ins unchanged:

2~

0 0

The Hamil tonian densi ty . ~ = :(~z 2 + $,2):/2 is proport ional to the respective densities p (loop) and Q (string).

The equations of motion (4.7) remain formally unchang(d , and so do the general solutions (4.10) except tha t the s y m m e t r y requirements {5.2) at �9 = 0 rest r ic t these solutions to

(5.6) a~ = fl,~ (all n :/: 0),

result ing in the solutions (2.14). There is therefore only one set of mcde para- meters {a~, n = :[: 1, ~ 2, ...}.

This simplification permits an easy formal derivation of all the r(:sults

for the covar iant string from the eovar iant loop. These results are of course identical with the results obtained from the defining equations and the string Hamil tonian (5.4). We summarize them as follows.

The solution of the covar iant string is defined in terms of a Fcek repre- sentat ion of the set of spacelike four-vectors {a~, n > 0} related to the a~

by (4.14) and satisfying the a lgebr~ (4.15) of creation and annihilation operators.

258 F . R O H R L I C H

These operators determine the mass operator

co

(5.7) ~ - ,no ~ + 2~/1. = ~ + 2~ Z ~ C . . 1

and the spin operator

i i (a~" a: '~ "" = - - a n ~ n ) �9 (5.s) s ~ ~

These two operators characterize the internal motion, the energy spectrum and the angular-momentum spectrum of the string. They commute with each other, so that there exist simultaneous eigenstates. These are defined on the positive Fock space ;E+, whose positivity is ensured by

(5.9) g~.P ---- 0,

which follows from (2.14) just as (2.16) did. The definition of the physical subspace ~b of JC+ is given by the Virasoro

constraints

(5.10) A,,]~> = 0 (n> 0, ]~>e r

(with A. defined as in (4.17)), and by the condition that the total mass (5.7) of the free covariant string is due to its internal energy, (4.19). This ensures that the constraint (4.4) is satisfied weakly.

The Poincar6 generators M, ' and pu commute with the Virasoro generators A, (n> 0) on ~b. Poincar6 invariance of r is thereby ensured.

As a consequence of these results one concludes as in the covariant loop:

1) M s is strictly positive for m~ > 0 and this choice ensures the absence of zero-muss states, which must be excluded for algebraic consistency of the Poincar6-Mgebra representation with timelike P~ in terms of the QPS algebra; the operator 1/P a would not be defined otherwise. There are no tachyons.

2) The spaeelike nature of the a~ ensures the absence of ghosts: ;E+ and in particular r are spaces of positive definite metric.

3) Poinear6 invariance is manifest and ensured by

(5.11) Jig% A0] Iw> = 0 , [P,, A.] I~> = 0 ( n > o).

4) All vibrations are transverse because (4.4) implies p.x'l~0 } = 0, which in turn implies that ~ . ~ ' q - ~ " ~ has vanishing matrix elements on ~b in the

c.m. frame.

C O V A R I A N T LOOPS AND S T R I N G S IBT A P O S I T I V E D E F I N I T E HILBF, RT SPACE 259

6. - The covariant string in the c e n t e r - o f - m a ~ frame.

The simultaneous eigenfunctions of M 2, S ~ and S~ will be denoted by

IN; S, M.>{.~, (6.1)

where

(6 .2 )

(6 .3 )

(6 .4 )

M~]N; SM )(.} = (m~ -t- 2:~N)IN; SM.)(.) ,

S~ ~M.)(.~ = S(S _- a)lN; SM.)(. / ,

k~zl~T ; ,SM|~{n } : J~'[ ]N ; KSMI){n} ,

and the subscript (n} indicates the set of modes which are excited. The operator M 2 is given in the c.m. f rame by

co

(6.5) M ~ m o + 2:r ~ t ~ - - ? i n n �9 a n

1

and the spin vector operator by

co

(6 .6 ) s - - i ~ , , ,~ . • a ~ 1

I ts components ' commuta t ion relations are a special case of (3.12). Since in the c.m. f rame

(6.7) P]> ..... ~ 0 , P~ .... = MI> ...... ,

this relat ion becomes the algebra of On:

(6.8) S • S = i S .

How we must relate the eigenfunctions (6.1) to the :Fock states which are

obta ined by operat ing with the a** and their prcducts on the vacuum. Such ~ general s ta te has the form

(6.9) f i (a~')~" (a~)~"~ --(a~'~)~"~ ]0~ :~ Iv1 v . , . . .~ ,

where the vector v . is defined by

(6.10) v,, = (v: , v . , v : ) ,

260 F. R o m ~ i c n

and has only nonnegat ive integers as components. The quantum number which determines the mass eigenvalues according to (6.2) follows from (6.5) to be

co

(6.11) N = ~ n(v: + u~ + v:) .

Obviously, there are in general many sets {~, v2, ...} which will result in the same mass eigenvalue. This degeneracy is given by the part i t ion T(~) defined implicit ly by

f l co (6.12) ( 1 - x')-~ = 1 + ~ . p ( ~ ) x '~ . ~'==1 N--1

In part icular , for the lowest values of 2r one has

(6.13) p(0) = 1 , p(1) ---- 3 , p(2) = 9, p(3) ---- 22.

Of course, not all of these states will bc in the physical subspace q). This subspace will be character ized by

(6.14) ./~.Iq~ ---- 0 ( n > 0),

where

(6.15) z . - ~ ~ ~ m ~ . - ~ =

: t %/~((n @ m)a,~.a,+m @ �89 ~. ~/m(n--m}am.a,_,~(1--5,,o}(l -- 5,,,) { n > 0 ) .

Now let us consider the lowest levels. The ground state is the vacuum state and is obviously in r

(6.16) M]O} = mo[O}.

The condition m0 :> O ensures tha t M is a s tr ict ly positive operator. The first excited state N = 1 is of course a s tate in which only the n = 1

mode is nonvanishing. Le t us, however, consider the more general case in which

only one mode is excited, bu t let this mode be an a rb i t ra ry one, n, corresponding

to one factor of the product (6.9). The string in this special case is then equi-

valent to the three-dimensional isotropic harmonic oscillator~ a system which has been studied many years ago, especially in connection with the independent- part icle model (shell model) of the nucleus (1~). The states in which only mode n is excited are (in configuration space) the product of three simple harmonic-

(12) See, e.g., H. A. BETHE a.nd R. F. BACHER: Rev. Mod. Phys., 8, 82 (1936), ospecially p. 172.

OOVARIANT LOOPS AND STRINGS IN A P O S I T I V ~ D ~ F I N I T E HILBERT SPACE 261

oscillator wave funct ions:

(6.17) <x],,> = ~,:(x) ~ ( y ) yJ,:(z) with 2~ = n(v~, + v,', + v:).

Let [ZTv~> be a Fock s ta te with e igenvalue /~; then the s ta tes ]iV; SM,>,, are

re la ted to t h e m b y

(6.i8) ]iV; SM,>,, ---- ~ IN; v.> <ZTv.IhT; SM>.. Vn

With the Condon and Short ley phase convent ion, one finds for X----1

(6.]9)

I1; 30>1 = - 11 ; 00:1>1,

1 I1; 1 • i>i =-4- ~ (ll; 10oN-4- i l l ; o ]oN) ,

v ~

which is a tr iplet . The nota t ion used here is

(6.20)

One verifies easily t ha t this t r ip le t is in ~b. Since, according to (6.13), the

s ta tes (6.19) are all the /q ---- 1 s ta tes , we see t h a t all ~ ---- 1 s ta tes are in r

Now, if a pa r t i cu la r s ta te I/q; SM.>,, is (is not) in r t hen all 2S+ ] s ta tes of the mul t ip le t are (are not) in ~. This is an immed ia t e consequence of the

c o m m u t a t i v i t y of S with the L , (n > 0). I n the convent ional s t r ing the first exci ted level (h T ----1) is forced to have zero mass. This permi t s only the two independent degrees of f reedom with 21/. = =1= S, so t ha t the I1; 10}1 s ta te was a rb i t ra r i ly excluded f rom q~.

The t r ip le t (6.19), however , does not exhaus t the n - - - 1 mode. One finds the following 2V = 2 s ta tes also in this mode: a N--= 2 singlet s ta te

(6.21)

and a N = 2 quinte t

12;2o>1 -

(6.22)

1 12; ooN = ~ (12; 200>, d- 12; 020>1 H- 12 ; 002>1)

1 ~/6 (12; 200>, + 12; 0 2 0 > , - 212; 002>,),

1 1 2 ; 2 + 3 N = :L ~ (]2; 3o1>, -_L i12; 011>,),

12; 2 & 2>, = d: �89 200N - 12; 020N T 2i12 ; 110>,).

:From (6.13) one sees t h a t there are th ree more /~----2 states.

2 6 2 F. Ir H

These belong to the n = 2 mode. They form the t r iplet analogous to (6.19):

(6.23)

]2; 10>2 ----- -- ]2; 001>,.,,

, 1 ' i ]2 010>2) . ]2;3L~1>~= VV ~(]2;loo>~z ;

The singlet, t r iplet and quin te t with eigenvalue N = 2 form the total of 9 states which accounts for the degeneracy p(2) ---- 9 of the ~ = 2 level. How- ever, the singlet and t r iplet states do not vanish when acted upon by Z~ and L~. Thus, t he y are not in r Bu t the quin te t is in r as can easily be verified.

As one proceeds to N = 3 and higher levels, one finds tha t more and more states result f rom excitat ions involving more than a single mode. General ly

valid expressions are therefore more difficult to obtain. t towever , one can show tha t all single-mode excitat ions with max imum

spin for a given N are in ~P. For the largest eigenvalues of S,, M := 4- S, one

finds, for any n,

(6.24) I N ; S • , r [ ;S--r,r ,O>, r=O

f rom which all other states of this mult iplet can easily be obta ined by means of the usual step-up and step-down operators S~:]: iSv: For the eigenvalues

of the mass operator one finds for this mult iplet

(6.25) M2[N; S M >. = (.m~ i- 2~nS)[N; ,.%11 >,,,

since 2r is independent of M.. Therefore the maximum spin multiplets satisfy

(6.26) N = nS .

The relat ion (6.26) shows the desired l inear l~egge trajectories.

7 . - S u m m a r y a n d d i s c u s s i o n .

The present covar iant theory of the quan tum dynamics of relativistic loops and strings differs in two impor tan t ways from previous t r ea tments of these

systems.

a) Tile ghost elimination was previously carried out by seeking a set of t spacelike creation operators A u which will create f rom the vacuum exact ly

all those states which satisfy the Virasoro conditions. Instead, we construct

COVARIANT LOOPS AND STRINGS IN A POSITIVE D E F I N I T E H I L B E R T SPACE 2~3

a positive l t i lber t space /E+ with spacelike creat ion operators first, and then restr ic t it fu r the r to a physical subspace q5 which satisfies the Virasoro conditions. This solves the critical-dimension problem.

b) As a ma t t e r of mathemat ica l technique, we do not use canonical covar ian t commuta t ion relations (which lead to a canonical covar iant realization of the Poincar6 algebra ~); instead, we use noncanonical covar iant CR which lead to a noncanonical (though covariant) realization of F which is not a IAe algebra, v/z. the QP~ algebra (3.10)-(3.12). This has the advantage tha t the

creation and annihilat ion operators are spacelike from the very beginning and no constra int to this effect needs to be imposed. The positive Hi lber t space

Jr arises f rom the very beginning.

Another difference to most previous t r ea tments is the use of the t imelikc ra ther than null gauge. This difference was ah'cady discussed in ~he introduction.

When the relativist ic loop and str ing are defined in the usual way in the ()rthogonal gauge~ the present theory leads to a consistent description: Poincar6 invar iant in (3-I-1)-dimensional Minkowski space (~o) with a positive definite Hi lber t space ;E+ (no ghosts) and a positive definite mass operator (no tachyons). The physics is res t r ic ted (due to the orthogonal gauge condition) to a subspace ~ of Jr For each energy level character ized by 2q there is a. degeneracy p(2q) consisting of a number of spin multiplets. Some of these are in q), some are not. For each there exists a max imum spin mult iplet with quan tum number S~,.(N) which is always in q} and is related to _N by

(7.1) 8 ..... -- N/n

for the str ing (eq. (6.26)) and

(7.2) ~m~ =: 2N/n

for the loop, n being the par t icular mode tha t is excited (for maximum spin only one such mode is excited). This is the leading Reggc t ra jectory.

The equidis tant energy levels thus lead to l inear Regge trajectories whose slope is twice as large in the loop than in the string. For the purpose of com-

parison with exper iment a scale must be in t roduced into the theory which is so far scale independent. With h-----c--= 1 and the quan t i ty V ~ 2 ~ which has the dimensions of length (which we may assunlc to have been chosen until now as uni t length), the dimensionless quanti t ies Q~,/)~, etc. will be replaced by the ratios Q~'/vf2z:~, ~r etc., so tha t Q~, PJ,, etc. now will have dimensions of length, (length) -], etc. The eigen-calues s of M ~ then become mo ~ + hr/a '. The results (7.1) and (7.2) then yield the linear t ra jec tory

(7.3) ~(s) = (~} ~ ' ( s - -~o), n

264 F. ROHRLICH

where the factor 2 is present for the loop, absent for the string. The intercept

~(0) is consequently negative. This shows tha t the identification of (7.3) with

the leading l~egge t ra jectory leads to an intercept of the wrong sign for both the loop and the string.

I t is well known tha t an intercept q- 1 is required to permit the derivation

of the generalized Veneziano amplitude. The string theory presented here

therefore does not permit such a derivation. This is, however, not considered

a serious difficulty. Neither is duali ty an exact symmetry , nor is the generalized

Veneziano ampli tude very satisfactory. For example, it violates unitari ty.

T h e interact ion of string model mesons, on the other h,~nd, will take place

th rough the interaction of their quarks, not of their strings. I t remains to be

seen whether a satisfactory a.mplitude can be derived in this way. The results

for one space dimension are certainly encouraging (6).

�9 R I A S S U N T O (*)

Si definiscono eappi e corde relativistici in rnaniora convenzionale conio soluzioni di un'equazione d'onda unidimensionale con certe condizioni di contorno e che soddisfino condizioni ortogonali di gauge. 5i usano coordinate convenzionali pseudoeartesiane (piuttosto the coordinate del piano nullo). Si richiede the i quadrivettori degli operatori

o % siano di tipo spaziale (ortogonali rispetto all'impulse di ereazione e annichilazione % totale Pg), cosicch6 lo spazio di Fock ~+ risultante ~ positive definite. Si mostra che questa condizione ~ matematicamcntc in accordo con l'invarianza di Poincar~ e the non impone ulteriori costrizioni fisichc al sistema. Essa pub essere effettuata in una realizzazionc canonica dell'algebra di Poinear6 come condizione ai vettori di state, o in una realizzazione non canonica come un'equazione operatoriale, come in questo contesto. Lo spazio ~+ ~ ulteriormente ristretto dalle condizioni di Virasoro al sotto- spazio fisico � 9 naturalmente ~ pure positive definite. In qucsto mode non sorgono problemi di dimensione critica e l'invarianza di Poincard ~ valida aneho in 3 § 1 dimensioni. Gli spettri di spin e di energia sono i soliti che conducono a traicttorie di Reggo lineari, ec~etto per il fatto the non ci sono tachioni c neanche stati a massa nulla. La traiettoria principale di Regge ha intercetta negativa.

(*) Traduzione a cura della Redazione.

g o e a p s a n T n ~ e neTau x CTpyHld n noaoac~rre:tbno onpe~eJtemioM

rH~lb6epTOeOM npocTpaneTae.

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COVA.RIANT LOOF8 AND STRINGS IN A POSITIVE DEFINITE HILBERT SPACE 265

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18 - II Nuovo Gimenlo A.

covariant loops and strings in a positive definite hilbert space

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