ASPEcrs RELATED O:" STRI:"G TlIEORY REVISTA MEXICANA DE FÍSICA 49 SUPI.EMFXro 1,53-60

Bosonic string theory with constraints linear in the momenta

JUNIO 2003

M. MontesinosDepartamento de Física, Centro de Investigación y de Estudios Avanzados de/Instituto Politécnico Nacional,

Al'. Instiruto I'olir~cnico Nacional No. 2508,07000 Ciudad de M~xico, D,F., Mexicoe-mail: me rced@jis. cinvesta ~~m.x

J.D. Vergara!nslitulo de Ciencias Nucleares, Uni~'ersidQdNacional Autónoma de México,

Aparrado poscal 70-543, México, D.r:, Mexicoe-mail: vagara@nuclecu.lInam.flL'(

Recibido elIde abril de '2001; aceptado el 1dc agosto de '2001

The Hamiltonian analysis or Polyakov aclion is rcvicwcd pUlling cmphasis in two topics: Dime observables and gauge conditions. In (hecase 01'the dosel! slting il is computcd Ihe changc 01'its aclion induccd by Ihe gaugc lransform¡¡lion coming fmm Ihe first class constraints.As expccted, Ihe lIamiltonian action is nol gaugc-invariant duc lo tbe lIami1tonian constmint quadr.llic in tbe momenta. However, il ispossible to add a bounJary tenn to Ihe original aetion to build a fully gauge-inv<lrianl ac(ion at firs( order. In addition, two relatives of stringIheory whose aclions are fully gaugc.invariant under the gaugc symmetry involvcd when the spatial slice is closed are built. The first oneis pure dilTeomorphism in the sensc it has no Hami1tonian constraint ami tbus oosonic string theory bccomes a sub-sector of its space ofsolutions. The secoml one is associated with the tensionless bosonic slring, its houndary term induces a canonical transfonnation and tbefully gaugc-invariant action written in terms al' tbe new canonica! variahles becomes linear in Ihe momenla.

Keywords: Canonical quanlizalion, string Iheory

Se revisa la acción hamiltoniana de Polyakov con énfasis en dos tópicos: observables de Dimc y condiciones de nonna. En el caso de la cuerdacerrada se calcula el c~tmbio er, la acción inducido por la transformación de Ilonna de las constricciones de primera clase. Como se esperaba,la acción hamiltoniana no es invariante de nonna debido a los términos cuadráticos en los momentos en la constricción hamilloniana. Sinembargo, es posible agregar un término de fmnlera a la acción original para construir una acción completamente invariante de norma antetransformaciones de primer orden. Adcmác;, se construyen dos parientes de leona de cuerdas cuyas acciones son completamenle invariantesde nonna bajo la simelna de norma involucrada cuando la rebanada espacial es cerrada. La primera de ellas es difeomorfismo puro en elsentido de que no tiene constricciones hamiltonianas y por tanto, el sector bosónico dt.::la tcona de cuerdas viene a ser un subespacio de esteespacio de soluciones. El segundo está asociado con la cuenla bosónica sin tensión, sus ténninos de frontera inducen una transformacióncanónica y la acción completamente invariante de norma escrita en lérminos de las nuevas variables canónicas es lineal en los momentos.

Descriptores: Cuanliz.ación canónica, teona de cuerdas

PAes: (14.60.Ds; (14.20.Cv

1. Canonical analysis

Rclalivislie free strings propagaling in an arbitrary D-dimensional fixed background spacctime with metric 9 =9"v(X) dX" dXv; J.l, J) = O, 1, ... ,D -1, can be deseribcd,for inslanee, wilh lhe Polyakov aelion [1J

The varialion of SiTa', X"] wilh respeel 10 Ihe backgroundcoordinates X 1J and lhe melric l'ab yields the cqualions ofmOlion

Q cda X"a Xv a x"a x-v - Q cdh I - OTab:= 21'abl' e d 91J1/ - Q' a b 9JJI/ - 21'abl' cd - a lab - 1 (2)

respccliyely. Bere, hab = DaX1J8bXI/ 91J1/ is the induced mClric olllhe world sheet, 'iJ is the covariant dcrivative a'iSOCialed wilhthe Levi-Civila connection of I'ab. r~1ilare the Chrisloffcl symbols ,i'isocialcd with the background metric g1Jl/' The Lagrangianformalism is more common lhan lhe Hamiltonian one for slring peoplc cOlTIlTIullity. Howcver. lhe Hamiltonian framework isa necessary step lo perfonn the quantization of the theory using Dirac's rncthod 12). Also the Hamiltonian framework is lhenatural arena lo analyze lhe issues of observables and gaugc conditions for the lheory which have relevance bOlh in ils cJassicaland quantum dynamics. Thal is why here lhe canonical analysis is reviewcd pulting cmphasis in lhese two topics. To go lo the

54 M. MONTESINOS AND J.D. VERCiARA

Hamiltonian fonnaiism, il is mandatory lo choosc a time co-ordinatc €o = T and a spacc coordinatc (1 = a ami 3SSUIllCthal !he world sheet M has lhe topology M = R x E. Themelrie 1ab is pUl in me ADM [onn

and so'¡=::¡:= J-dethab) = ,N,jX with, = +1 ifN > o and ~ = -1 if N < O. Duc 10 the faet T is timc-likcand a is spaee-like _N2 + ,\2X < Oand X > O.Taking inloaeeounl (3) Sbab, X"] aequires lhe fonn

where C,,\ =£,\'-,\£', Co£=£g' - .f£', K. = lGo2(gt,' -M'),- -amI B.T. stand s fm boundary tcrms. Therefore, H and D are

tirsl elass amI the theory ha, D - 2 physieal degrees of free-dom 12(D - 2) in lhe phase spacel per spaee poinl.

{H(g), H(W = D(K.) + B.T.,

{D(£),H(g)} = H(C, g) + BT.,

{D(£), D('\)} = D(C,'\) + B.T., (11)

Then a straightforward computation yields the Poisson brack-eLsbeLween Lheconstraints

Even though lhe Hamiltonian and difTeomorphism con-slraint'i have an evident mcaning, it is intcresting lo sec whatthe consLraint surface means from the pcrspcctive of the gc-ometry of the induccd mctric hab. By using the equation ofmotion for X p, ami Lhedefinitions of the induced mctric com-ponenL' lhe eonslrainL' (6) bccome

Geomelric perspeclú'e

(5)

(3 )

X'\)X '

,\ JN2.!._~ 'X N°

wilh

where !he dependeney of the phase spaee variables and La-grange multiplices in lcems of lhc Lagrangian variables is

- _ 2a,,jX X" 2o,,\,jX X'"PIJ - - ----¡¡- glJII + N gllll!

N.A=----,- 4m,jX

the Hamiltonian 3ml diffcomorphism conslraiTlls, rcspcc-lively. Here X'" = aX" laa. The standard varialional prin-cipie is fonned with the aClion (4) and with lhe houndaryconditions

- JIJ-D = X p", (6)Thcrcforc, the consLraint surfacc (9) just means

(13)

-;::;;(Jó - - 8 4>whcrc Y = pp.PIIg'hlgÓIl - 40-2X' X' . To computethe algcbra of constraints (he Hami1tonian and diffcomor-phism constraims are smeared with arbitrary fields '(T,O")and é{T, a)

(15)

(14)

( o 2 2)hiT= -lG.ja +.\. hUU1

h,u = '\'huu.

::::::2~j5tJX'P, + .\.X'¡!l XJI' gJ.1ol'1

h X'"X'"0"(1 = gll1'1

These relalionships among !he world sheet metrie eompo-nenL'i amI Lhe Lagrange multipliers JUSl carne a'i a consc-queTlCCof describing string dynamics from a canonical pcr-spcctivc. Also it is possiblc to writc down the induced mctrichab in tcrms of Lhephase spacc variables and Lagrange mul-lipliers

From the second cquaLion h,u can be plugged jnto the firstal\(l then lile constrainL surface looks Iike

hiT:::::: XJ1.Xl'gtJl'

::::::4.j2Pp,Pl'gJ1.1'+ 4.j.\.PJ1.X'¡!l +.\.2 X,¡!l X,I' g¡!l1'1

I X.," X',"l,O" :::::; g¡!ll'

(9)

(8)

(7)

(10)D(£) = ( do El).lE

H = O, D =0,

X"(T;,a) = x;(a), i = 1,2,

and the constraints

where xna) are lhe iniliai (al TI) amI final (al TO) slring eon-figurations.The varialion of S[X", p",t,,'\J under (7) all(llhe condi-

tion ÓS = Oyield lhe equations of motion

X" = 2~}"g""+ '\X'",'-- :::;."ag." ( o '" - )'p" =~y ax" + ~80 X g""+'\P"

Hel'./Hex. Fú. 49 SI (2(XJ3l53 60

BOSONIC STRING THEORY WITH CONSTRAINTS LINEAR IN THE MOMENTA 55where the equation of motion for X~ wa~ used. These ex-pre,,,;sionsmean that in arder lo compute the induccd metricon the world sheet is necessary 1) fix the gauge and thus todetermine the Lagrange multipliers ~, and .\, 2) wilh eachparticular choice for!he Lagrange multipliers solve the equa-tions of motion, 3) plug these solutions into the constraintsand into !he gauge fixing conditions to drop the gauge free-dom completely, finally, 4) insert!he final expression for X~toge!her wi!h !he Lagrange multip!iers into the RHS of themeLric componente;;.

By using (6) !he metric components can be written interms of !he constraint~, the Lagrange mullip!iers and justDnemetric component

class constraints, il is interesting lO compute Ibefinite trans-formation of these variables due lo bOlh Poincaré and Weylinvariance, ¡.e., it is assumed in Ibis part of the paper that thebackground metric g~" is the Minkowski one TI"".

i) Poincaré invariance is X~(T, a) = A~X"(T,a) + a~,7ab(T,a) = ,ab(T,a) wiIb A~ a Lorentz transforma-tion amI a" a translation. Using Ibe explicit form of,ab in (3) and Ibe definition of Ibe momenlum (5)jinitePoincaré invariancc meaos, in the Hamiltonian frarnc-work, that the phase space variables and Lagrange mul-tiplicrs transfonn as

hu = <Q,2H + <Q,.\15 + (-16~2a2 + .\2) ha•.

hra = 2~15 + '\haa. (16)

Thus, on the constraint surface

( 2 2 2)hrr= -16~a +.\ haa, (17)

X~(T,a) = A~X"(T,a) +a",

j5~(T,a) = A~P"(T,a),

~'(T,a) =~(T,a),

.\'(T,a) = .\(T,a). (19)

due to the fact X, and N transform as X(T,a) =e2w(r,a)X(T,a), Ñ(T,a) = E'ew(r,a)N(T,a). Never-thelcss, on!y E' = 1 leaves action (4) invariant underIbe transformalion (20).

ii) Two-dimensional Weyl invariance is X~(T,a) =X~(T,a), 7ab(T,a) = e2w(r,a),ab(T,a) for arbitraryW(T, a). Using the explicit form of ,ab in (3) and thedefinilion of the momentum (5)finite two-dimensionalWeyl invariancc mean s, in the Hamiltonian framework,that Ibe phase space variables and Lagrange mulliplierstransform al)

as expected [sce Eq. (14)].In addilion, it is interesting 10 compule the energy-

momentum tensor (2) in terms of the phase space variablesand !he Lagrange multipliers. By doing !his the componentsof T = Tabd(ad{b become

Tu = _2a.\2 (1+ ~) H - 4a.\~15,- 16a2~

.\ '" -Tra = --H - 2a.\D,8a -1 '"Taa = - 8a H, (18)

and !hus the energy-momentum tensor vanishes on the con-straint surface. However, !he trace of the energy-momentum

zero vanishes identically, Ta a = Tabgab = OH + 015 = O,and 001 jusI 00 the constraint surface.

X~(T,a) = X"(T,a),

~'(T,a) = E'MT,a),

ti = ::f:l,

- 1 -P~(T,a) = -P"(T,a),

E'

.\'(T,a) = .\(T,a);

(20)

Gauge transformations

Before compUling !he gauge transformation on the phasespace variables and Lagrange multipliers induced by the first

I

Let us go back to general ca~e, namely, when the back-ground mctric g~v is left arbitrary. In this ca'iC, it is easy 10compute the infinitesimal gaugc transformation induccd bythe constraints (9) on the phase space variables [2]

X"(T,a) = X~(T,a) + {X"(T,a),H(f)} + {X"(T,a),D(f)},

= X~(T,a) + 2 (f9~" p") (T, a) + £,X~(T, a),

j5~(T, a) = P~(T, a) + {P"(T, a), H(,¡;)} + {P~(T, a), D(E)},

-O<l8g= P~(T,a) + fY 8;~(T,a) - 8a2f(T,y)J(y,a)X'"(T,y)g~"(T,y) I~';:~:

+ (8a2,¡;X'"g"")'(T,a) -£(T,y)J(y,a)P~(T,y) I~~~:+£,P"(T,a), (21)

£,X~ = £X'~, £,P~ = (EP~)'. The gauge symmetry (21) is associated with the two-dimensional diffeomorphism invarianceof Ibe lheory. Por closed strings no boundary terms appear in the constraints algcbra (1 1) and the transformation law for the

RCI: Mex. Frs. 49 SI (2003) 53-60

56

phase spaee variables simplifies aeeordingly

:vt. MONTESINOS AND J.D. VERGARA

X.(T,a) = X.(T,a) + 2 (fO.v Pv) (T,a) + C,X.(T,a),

- _ - :::::JJI/Jf)gor:p 2 /1/ I -P.(T,a) - P.(T,a) + ,y ex.(T,a) + (8<> ,x O.v) (T,a) + C,P.(T,a),

while the Lagrange multipliers transform as

g(T,a) = ~(T,a) + t. + C,~ - C,,',A' (T, a) = A(T,a) + i + C,A - ••.

Taking inlo aeeounl (22) and (23) lhe gauge transformalion induces a lransformaliou in lhe aClion for the closed slring

with

H = ["' da (dí +ói5) = H(f) + D(E;).J.,After a direCl computation

S[X", j5.,~', A'J = S[X., p.,~, AJ+ ["' da [, (p.Pvo"V - 4<>2 X'. X'" o.v)] T:T' ,} 0'\ T-T1

= S[X.,P.,~,AJ + ["' da(,VV

g"Vr:".la] T-n

(22)

(23)

(24)

(25)

(26)

Thercfore S[X., p.,~,AJ is not gauge-invariant and behaves in lhe same way as lhe aClion for Ihe relalivislic free particle (e!ReL 3). The rcason why S[X", p.,~,AJ fails 10 be gauge invariant is beeause lhe Hamiltonian eonstraint is quadralie in lhemomenla like in syslems wilh finite degrees of freedom 14,51. In spile of lhis, fully gauge-invarianl aelions under finile gaugesymmetries for systems with finite degrees of frcedom were built in ReL 3. Now, lhose ideas are here eXlended 10 ficld theory.In the particular case whcn the background metric 9/1>11 is constmIl, fOf instancc whcn 911:11 is lhe Minkowski rnctric 1]/1>11. lheaetion for lhe closed string

¡s. al first oeder, fully gaugc-invariant.

(27)

Observables

Dirae observables or observables for shor! are funetions de-fined on lhe redueed phase spaee of lhe lheory. They are eon-slanl nloog lhe gaugc orbits of lhe cOTlstraintsurface and thusthey llave wcakly vanishing PoisSOTlbrackets with lhe Hamil.toniao and diffeomorphism constraints. Al infinitcsimallcvclthis meaos observables must be gaugc invariant undcr thegauge traosformation (22). From lhe transformation law forlhe phase spaee variables (22) il is clear Ihal for closed slrillgspropagaling in a constant hackground gj.lY !he linear ami an.guiar momenlUm

1"'p.= daP"(T,a),u,

M"V= ("' da [X"(T,a)pV(T,a)- XV(T,a)p.(T, a)] , (2R)Ju,

respeelivcly, are observables; and lhus !he values of p" alldM. are independelll of any particular choice for the gaugeconditions. From their own definitions apparently thc linearmomenlum P¡¡. and thc angular momcntum j\,f¡J.Y dcpcml onT, However, by computing their derivative with respect to T

and using the equations of molion ?¡J. = O = M¡J.Y. There-fore, P¡J. and Mj.lV are indeed indepcndcnt of the lime coor-dinate T and, of course, of the space coordinate u. Noticethat two string configurations having same Pj.l and Mj.lV donot represent the same physical string configuration bccausePj.l ami l\1j.lV do not lahcl the fuI! reduced phase space ofstring theory. Morcovcr, it is possible to build other ohserv-ahles fmm comhinations of the prcvious ones as, foc exam.pie, lhe square mass M2 of lhe elosed string M2 = - P" p•.Up lO herc, it has been showll Pj.l and Af¡J.v arc ohserv-ables because lhey are invariant under the gaugc transforma-tion gCTleratedby the firsl class constraints (22). Howcvcr,

Re\~Mex. Fís. 49 S 1 <2(03) 53-60

BOSOi'IC STRINe; TIlEORY WITIl CONSTRAINTS LINEAR IN TIlE \10MENTA 57

are observables r see Ref. 6 for an altemalive approaeh in lhecase of p-branes]. 81H, ",hat about if baekgrounds ha" noisornctrics? This simply would mean that thcrc w01l1<1 be 110

observables a~sociatcd wilh isornctrics, howcver, still lherewould be observables, i.e., invarianl enlities under lhe trans-formalion (22) assoeiated wilh lhe lrue physieal degrees offrecdorn of slrings.

Other quanlities used iTlslring theory are lhe 'center ofma~s' coordinales of lhe string

whal ahout Poinear6 and \Veyl invarianee? Noliee lhat un-dcr Poincaré invariancc (19) the linear rnotl1cntutll Pp. amilhe angular morncntum .\1JlV transform as PIJ = A~PII'M'JlII = A~A~Mat1 + (all A~ - a" A~)pa whilc undcrtwo-dirncnsional Wcyl invariancc (20) thcy are fully guugc-invarianl (laking" = 1) p~ = p~,M'~v = M~v he-cause P~(T, a) = j5~(T, a) and X~(T, a) = X~(T, a). Thiswill be very important in a momcnl. Duc lo lhe faet theobservables Pp. amI MJJII in a Minkowski target arc asso-c¡atOO with il'i isornClrics (Killing vector ficlds), it is nat-ural 10 cxpcct 1h31 lhe analogous of PJJ and M¡J.II in arbi.trary backgrounds, whcrc Poincaré invarinncc is lost. wouldbe assoeiatcd wilh lheir Killing vector fields too. ¡u facl, ifv = v~(X){)/{)X~ is a Killíng vector field oflhe haekgroundspaeetime 9 = g~v(X)dX~dXV , lhen a slraighlforward ap-plíeation of Noclher's theorem to (1) implíes

However, lhe 'centcr of mass' coordinalcs of a closed slringare flot observables under the transformation (22). This miglllbe souree of confusion wilh intuition. Certainly, X¡'¡(T) aremeasurable quantities, hut measurahle quantities me not,in general, observables of lhe theory. In additioll, underPoincaré invariance (19) the 'cenler of mass' cooniinalestransform as X~(T) = A~X"(T) + a~(a2 - a¡) whileundcr Weyl invariance (20) they are rully gaugc-invarianlX~(T) = X~(T) beeause X~(T,a) = X~(T,a).

So far, it has becll cxhibited the transfonnation laws fOl'p~ and M~v under i) Poinear6 invarianee (19), ii) lwo-dimensional \Veyl invarianee (20), and iii) the tmnsforma-tion law associated with the first c1ass eonstraints (22). Lelus compare with gravity. Here, gravity is nol string gravily,cather, it is Einstein's general relativity. In fOUTdimensionalgeneral relativity is neittler the symmelry of the kind asso-ciatcd with global Lorentz invariance (19) nOf lhe symrne.try of the kind associated with two-dimensional \Veyl invari-anee (20) (lhis lypc of symmelry is also nOI presenl in lheDirae-Nambu-Goto aelÍon (71), mlher, the gauge symmetrypresent in genera) relativity is or lhe same kind that the Orlecoming fram lhe firsl c1ass eonslraints (6), (22). Indce<l, fromla~t cornputations imponant nOlions can been drawn whichmake shapc lo lhe meaning of observables in generally eo-variant theories, in particular, roe general relativity. The first

Gauge fixing

(31).\= 1,<l=0

lesson from p~and M~v is thal they are gauge-invariant un-<Ier the gauge symmetry a.<.,;sociatedwith the first c1ass con-strainL<.,;(22). The second Icsson is that P¡.¡ and M¡'¡v are in-dependenJ of the time and space coordinates T and 0"; respec-lively, ",hieh labcl lhe poims on lhe world sheet. Thercfore,in any generally covarianllheory having Hamiltonian and dif-feornorphisrn constraints, as general relativity, rnuSl happenthe same phenornenon: observables must be coordinate inde.pendenl emilies too. In string theory, on the olher hand, fieldshave physieal meaning beeause lhey are allaehed to the fixedbackground 1J¡.¡v, ami lhus P¡.¡ (or AfJlV) can be measured inany 'extemai' LoreTllZ reference frarne. The relationship be-lween the values of the linear rnomentum PJl rncasured fmmany two 'extemal' Lorenlz observers is PI-' = A~Pv withA~ a malrix in the Lorentz group. However, lhe presence of'ex lema l' observers plaeed in lhe background manifold is apeculiar fact of string lheory ami it is nOl a general prop-erly of geTlerally covariant theories, for inslance, in generalrelativity 'extcmal' observers are not allowed; there is nota background manifold 'outside' of spacetime whcre 'cxter.na!' observers sit to see how spacctirne propagates, rather,dynamies of the gravilational field must be deseribed froman 'inside' viewpoinl. This is a key conceptual differeTlce ofgeneral relalivily wilh respeello string theory. Nevertheless,as already mentioned it is stiH lrue thal in general relalivityobservables musl be coordinatc independent entities as wcH,ami this faet irnplies a major problern in gravity. In generalrelalivity spacetime coonlinatcs are attached to 'observers'placed in sorne reference frame, so how can an 'observcr'l1leaS1ll'esorne observable, say in his (her) laboralory, if ob-servahles are independent of spaeetime coordinates? In otherwonis, 'local' ohservables in general relativity or in any othergenerally covariant theory are nol allowed because of diffeo-l1lorphism invariance [81.

In any gauge lheory, determinism forces it lo identify gaugerelated phase space variables a<.,;a single point in the reduced¡lhase spaee of the gauge lheory, and lhe lotal number of lheseorhils span its physieal phase spaee. At elassieal level, goodgauge conditions help to single out these physical degrees offrecdom hecause they intersectjust once lhe gauge orbiL<.,;onlhe constraint surface. On the other hand, in quanlum the.ory there are essentially lwo ways to proceed: i) reduce lhenquanlize or ii) quantize then reduce. In i) the relevance of agood gaugc fixing is clcar. Standard quantization of strings isof the kind i) ami so it is important to have good gauge condi.tions to do that. Before going lo {hmpoinl, sorne words aboutother unforlunale choice for the gauge conditions

usually found in the literature. Due lo the fact T is time-likeamia isspaee-likeí'rr = -N'+.\'X < Oand'luu = X > OmUSlhold, whieh means'\' < 16a'<l'. It is elear (31) does

(29)

(30)¡U,X~(T) = daX~(T,a).

u,

¡"'0v = da j5~(T,a)v~[X(T,a)J,u,

Re\'. Mex. Fís. 49 SI (2003) 53-.(>()

ss ;\1. MO:-"'TESINOS A:"'D J.D. VERGI\RI\

nOl satisfy this condilion. Putting it in a diffcrCTlL manlJer. the choice (31) brcaks dOWII lhe causal structurc 011 lhe world sheethccausc w¡lh such a choice T bccomcs spacc-likc and q hccorncs timc-likc.

To fix COTlsistcntly lhe gaugc dcgrccs of [rccdom in lhe action (1). lhe componcnLs orthe invcrsc ofthe world sheet rnctric,,;ab will h: cOllsidcrcd as dynamical variables. In this case, thefe are duce additional constraints, sincc lhe canonical morncnL:'l;l',OC¡;il~d Ln",,/~bare wcakly cqualto zcro

(32)

:::i, appmach. ins(cad of (4), lhe c:ltIonical action is

(33)

ir - T}T/I = 01ir - T}(]'II = 0

1i¡T = T}(]'I = -w, (37)

(42)

(41 )

The a¡gebra of eonstrainLs for string theory allows itlO definea new theory that iooks like the aetion for string theory ex-cept that it has no HamillOnian constraint, being iL"idynamicsanaehed to the diffeomorphism eonstraint only. This theoryis delined by

2.1. Pure dilTl'omnrphism hosonic string tht'oQ'

2. Relativcs or 1Josonic string theory

with jj = X'" p", X'" = (8X")/(8u). The algebra of eon-

strainL'i clases ami thus fJ is firsl c1ass. Thc equations of mo.tion are

This residual gaugc invariance is important in thc quanturntheory of the string 19),

Thcse gauge conditions allow il lo fix thc Lagrange multiplicr~I = 1/ AB, however, they do nol fix eampletely Al, rather,it is left as an arhitrary T-dependent funetion, Al = Al (T).In addition, f = O amI E = E(T). By plugging ( = O amiE = E(T) into (38) 1)' = O, 1)" = E(T). By inserting io (37),those equations set w = O, ami 1)" = E(T) = al T + az, withal, a2 constants, So, Ihe systcm is still invariant under theT-depcndcnt coordinatc transformations

(36)

(34)

(38)

Thc remaining gaugc freedom is associaled to Ihe con formalgroup in Iwo dimcnsions. which is infinile-dirncnsional. TheLagrangian gauge paramelcrs T}a ami lhe Hamiltonian ones(,E) are related by

1)' - .E..- ~l'

it follows that the eooditions (35) do nol lix eomplctely thegauge freedom of the gauge parameters (1)",w). These pa-rameters are only restricted 10 satisfy Ihe diffcrential equa-

tions

where the additional arbitrary parts (p, p) appear from thefaet that the constraints (6) are secoll(hu-y ones in this ap-proach.

Now. the con formal fixing of the world sheet rnetric willbe considcrcd

'YrT = -1, 'Y" = O, 'Y"" = l. (35)

Thcse gaugc conditions set thc Lagrange rnultipliers Aab = O,however, from the inlinitcsimal gauge transfonnation for theintrinsic mctric

This action occorncs lhe action (4) when \Vcyl invariant vari-ables are uscd. al! lhe cOl1straints hcing firsl c1ass. So, lo fixlhe gaugc. fivc gaugc conditions are nccded. Notice that theLagrange mullipliers (~I, Al) are not exaetly the same L1-grange mullipliers of (4). Both sets are related by

Now 10 tlx the additional gauge frcedom associatcd with theconslrainL"i (6), the lighl-cone gaugc conditiolls are chosell (inthe case of the elosed striog propagating in the Minkawskispaeetime)

A. B constanls. ami where the Iight-conc coontinatcs aregivcn by

X" = ~(XO o!o XD-I), P" = ~(Í50 o!o j51J-1). (40)

(43)

The theory defined hy (42) eontains string theory (4) as asub-sector of its spaee of solutions beeause the theory de-fincd by (42) has olle more physical degree of freedom Ihanstring theory (4). This is a general faet, always that a diffeo-morphism constraint appears in the formalism of generallycovariaTll theorics it closes wilh ilself, and thus il is possibleto drop somc of lhe othcr constraints involved in thcir algc-bra and thus to build larger theories whieh will eontaio theformers as sub-seetors, like the ane delined by (42) whieh

(39)Bp+ = -p+2

Hev. Afex. Fú. 49 SI (2(XJ3)53-(i()

BOSOI'\IC STRING TIIEORY \VITII CONSTRAINTS LINEAR IN THE :v10:v1E:\'TA 59emerged rrom (44). lt is prelly obvious lhal a similar con-s1ruclion Itolds for tite oosonic p-brancs where insteild of hav-ing Dne single diffeomorphism constraint lhere will be a linilenumbcr of thcm. Howevcr. al first sight, a supersymmetricver,ion or (42) mighl nol be allowed. A more radical inler-prelation ror lhe lheory defined by (42) and its relalionshipwilh string lheory is lo see (42) as a kind or M-lheory, amIto consider different seetors of this M :!hcory as Orles definedby dirrerem Hamillonian eonstrainl' H's. Notiee lhal in lhecase when lhe spalial surraee is c1osed, lhe aelion (42) is rully

I

gauge-invariant unde~thegaugc transfonnation gencrated byD. Due lo lhe rael H is missing in (42), a deep analysisor (42) can help lO_undersL1ndbeller lhe role lhal lbe Hamil-tonían conSlraint H plays in slring theory bOlh c1assical andquantum mcchanicaIly. Finally. il is important to mcmion thatlhe lheory defined by (42) plays lhe same role wilh respeelloslring lheory (4) as lhe Husain-Kuehar model plays wilh re-speel lo selr-dual gravily ror selr-dual gravily is a sub-secloror lhe space or solulion or lhe Husain-Kuchar model [101.AClually, lo have a beller analogy il would be desirable lohave a Lagrangian rorm ror (42).

2.2. 'lensionll'Ss hosonic string thcory with l'onstnlinls line~lr in lhe Illolllcnla

Slring aclion (4) has another relalive in the case when the background metric g~/I is constanl. say tite Minkowski or Euclideanmelric 1JJJ./I' The Jater is dcfincd by sctting Ct = O in the constraints. namely. it is defincd by the aClion

with

(44)

D - X'" P- ", (45)

where X'" = (DX")j(D,,). Obviously, lhis aClion can nol bc oblained rrom lhe Polyakov aelion (1) beeause ir a were equalzero lben lhe RHS or (1) would vanish loo.

Let liS focus in lhe case whcntltc spatiaJ slice of Lhe 'world sheet' is closed. Thc algebra of constrainls is

{H([), lI(~)} = O,

{D(é), ll([)} = ll(l:., f),

{D(é), D(A)} = D(I:.,A).

Under lbe gauge symmetry generaled by lbe conslrainls, lhe aCliou changes, according lo (26), as

S[X", p",~'YJ = S[X", P",~,AJ+ [' da (di)'="0'1 1"=1"1

(46)

(47)

Tlterefore. the boundary lenn is proportional 10 the Hamillonian conslraint. and thus the action is gauge-invariant on lheconslraim surface. A similar situation appcars in gencral relativity exprcssed interms of Aslttckar variables [11]. In Ref. 1I itwm; no built the fully gaugc-invarianl action associatcd with tIJe sclf-dual action, howevcr, this could be carried out.

Let us come back to thc action (44) and construct, following the stcps of [31. ite;;fully gauge-invariant action. This action isgiven by

(48)

A slraighlforward computation shows that. at firsl onler. Sinv [XI~, PJll,j, AJ is fuIly gaugc-invariant. The boundary tenn in (48)induces the canonicaltransformalion

1 (XO)qO = 2" In Po '

1 (XI)ql = 2" In PI '

I (XfJ)q[) = 2ln PJ) ,

(no Sl1mover D).

-1 -PI =X PI,

(49)

Re\'. Mex. FÚ". 49 S I (2003) 53 -60

60 M. MONTESINOS AND J.D. VERGARA

In tcrms of the ncw phac;e space variables Sinv reads

(50)

wilh

(51)

wilh l~ = (1,1, ... ,1) and il wa, a,"umed a diagonal background metric r¡~v.Notíce thal lhe conslrainl' are linear andhomogeneous in the momenta (and in their derivatives).

1. L. Brink, P. Di Vechia, and P. Howe, Phys. Lel/. B 65 (1976)471; S. Deser and B. Zumino, Phys. Lel/. 65 (1976) 369; A.M.Polyakov, Phys. Len. 103B (1981) 207.

2. P.A.M. Dirae, Lectures on Quantum Mechanics (Belfer Gradu-ate School of Science, New York, 1964).

3. M. Montesinos and J.D. Vergara, Phys. Rev. D 65 (2002)0(>4002.

4. C. Teitelboim, Phys. Re". D 25 (1982) 3159.

5. M. Henneaux and J.D. Vergara, "BRST fonnalism and gaugcinvariant opcrators: Thc example of the free relativistic parti-ele" in Sakharov Memorial Lectures in Physics, edited by L.v.Keldysh and V.Ya. Fainberg (Nova Science Publishcrs Inc.,New York, 1992) p. 111; M. Henneaux, C. Teitelboim, and J.D.Vergara, Nud. Phys. 8387 (1992) 391.

6. B. Carter, in Recen/ Developments in GravitOlion and Math-ematical Pltysics, (Proc. Sccond Mexican School on Gravi.tation and Mathematical Physics, Tlaxcala, 1996); A. GarcÍael al., (Science Nctwork Publishing, Konstanz, 19(7) hep-thI9705172.

7. P.A. M Dirae, Proc. R. Soco úmdoll Ser. A 268 (1962) 57; Y.Nambu, "Lectures al the Copenhagcn Symposium", (1970) un-published; T. Go'o, Prog. Theor. Phys. 46 (1971) 1560.

8. C. Rovclli, Class. Quantum Grav. 8 (1991) 297.

9. L. Briok and M. Benneaux, PrincipIes o/ Slring Theory(Plcnum Press, New York, 1988).

ID. Y. Husain and K.V. Kuchar, Phys. Re". D 42 (1990) 4070.11. M. Montesinos and J.D. Vcrgara, Gen. Re/. Gral'. 33 (2001)

921.

Rev. Mex. Fis. 49 SI (2(x)3) 53-60