Decemb er ITPSB HIDDEN GRAVITY IN OPENSTRING FIELD THEORY 1 W Siegel Institute for Theoretical Physics State University of New York Stony Brook NY ABSTRACT We clarify the nature of the graviton as a b ound state in op enstring eld theory The at metric in the action app ears as the vacuum value of an open string eld The b ound state app ears as a comp osite eld in the free eld theory hep-th/9312117 14 Dec 1993 1 Internet address siegelinstiphysicssunysbedu INTRODUCTION Fundamental strings as opp osed to hadronic strings have b een prop osed as the solution to two problems unied theory and quantum gravity The use of strings as a solution to the former problem hinges on its use as a solution to the latter Since the compactications of string theories to four dimensions are so numerous and the compactication itself do es not seem to b e predictable it is not clear that the predictability of string theory for low energy physics is much greater than that of renormalizable or lowenergy phenomenological fourdimensional quantum eld theory of particles The greater predictability of string theory is exp ected from the hop efullynotto omuch higherenergy corrections arising from the direct or indirect eects of gravity including the eects of sup ersymmetry whose breaking is b est explained through the use of sup ergravity and the sup erHiggs eect In other words the unication p owers of string theory although originally thought to b e great b ecause of the few mo dels available in ten dimensions b efore compactication have b een reduced to a solution to the unication of gravity with sup er grand unied theories Therefore string theory is basically a solution only to the problem of quantum gravity but it is the only known solution Ordinarily that might b e sucient and p erhaps even desirable but until the exp erimental situation improves it would b e useful to have an alternative theory of quantum gravity for purp oses of compari son The only prop osed mechanism free of ghosts and the resultant eectively non renormalizable ambiguities is the app earance of the graviton as a b ound state in a renormalizable eld theory One advantage this might have over string theory is the correct prediction of the dimensionality of space time sup erstring theory has criti cal dimension while renormalizable eld theory with a nite numb er of elds and b ounded p otential has critical maximum dimension However the only known theory in which the graviton has b een demonstrated to app ear as a b oundstate p ole in the Smatrix is op enstring theory Therefore the phenomenon of b oundstate gravity in op enstring theory warrants a closer study with an eye toward isolating those asp ects that are essential to this phenomenon but might not require string theory In this pap er we p oint out two curious features of this mechanism in op enstring theory that may b e crucial to the understanding if not the formulation of b ound state gravity in a more general setting In the following section we discuss the nature of the vacuum value of the op enstring eld Originally this was intro duced as an analogy to gravity where unlike most other eld theories there is no kinetic quadratic term in the action until after expanding ab out the at metric However in classical op enstring eld theory as opp osed to classical closedstring eld theory there is no gravity We discuss the origin of this phenomenon from the vacuum values of massive spintwo elds the relation of their Stueckelb erg elds to the Skyrme mo del and the relation of these massive elds to the massless graviton in the quantum theory In section we study the generation of the graviton at the quantum level In op enstring eld theory the graviton app ears in a onelo op diagram rather than through an innite sum of oneparticleirreducible graphs as in most other known theories with b ound states such as QED and QCD We p oint out that this implies that the b oundstate graviton actually app ears in free op enstring eld theory We discuss the analogy to the theory of a free twodimensional massless spinor which has a massless scalar as a b ound state The b oundstate graviton in op enstring eld theory is thus a higherdimensional analog of b osonization Besides this analogy b etween free theories we also describ e the analogy b etween interacting op enstring eld theory and the Schwinger mo del which shows how these b ound states manifest themselves as new p oles in the fundamental elds It is useful to analyze this phenomenon from the p oint of view of an eective the ory where a new redundant eld is intro duced for the b ound state and we therefore discuss the gaugeinvariant eld theory of coupled op en and closed strings in section In the nal section we give a more general analysis of coupled systems at the free level which allows us to discuss some examples of the mixing that o ccurs b etween states of the op en string and the closed string the Higgs mechanism for the massless elds of the UN string and the coupling of the graviton of the closed string to the lowestmass spin eld of the op en string THE METRIC TENSOR AS AN OPENSTRING FIELD The action for op enstring eld theory has b een rewritten as a single term cubic in the elds which is apparently indep endent of the atspace metric The at space metric app ears through the vacuum exp ectation value resp onsible for generating the kinetic term from the current Q of the BRST op erator Q m 0 1 X C C hi Z Z y y y 1 1 1 S Q 2 3 3 Z Z 2 hi Q I Q d Q Q d Q L L 0 0 1 m 0 1 0 m n mn 0 1 1 1 Q C X P C B C B C X X P P C B C B m 0 1 mn m n 1 0 2 2 where I is the identity element for the pro duct the Spinvariant vacuum up to a m 0 1 BRST transformation and P B B are the momenta conjugate to X C C m 0 1 Strominger has explained the app earance of the atspace metric in terms of the coupling of the op en string to a background closed string in a desire to enlarge the space describ ed by the op enstring eld to include indep endent op en and closed string comp onent elds or even to describ e only closedstring elds Note that neither the atspace nor any other metric app ears in the denition of the Hilb ert R y space inner pro duct functional integration measure h j i The string 1 2 1 2 eld theory formulation is one based on rst quantization and thus uses the Zinn JustinBatalinVilkovisky ZJBV formalism This means that nonvanishing inner pro ducts exist only b etween elds and their ZJBV antields eg a covariant vector n A has a contravariant antield A so the corresp onding inner pro duct involves m n only Kronecker deltas m Here we give a dierent interpretation of Stromingers result Writing the op en string eld as Z 2 d Q I 0 1 m 0 1 0 m n mn 0 1 1 1 Q C X P C B C B C g X X g P P C B C B m 0 1 mn m n 1 0 2 2 hg X i mn mn we interpret the eld g as an open string eld In fact if we compare with the mn usual oscillator expansion of it is clear that this eld minus its vacuum value is a combination of the usual op enstring elds Ie the linearized metric is an innite sum with appropriate co ecients of the massive spin elds that are mass eigenstates in the free theory Equivalently it corresp onds to eliminating the vacuum exp ectation values of all spin tensors except one by simple eld redenitions For any g including traces of higherspin elds indep endent of g that has hg i mn mn mn k replaceg with a new eldg g k g so hg i Thus if we start with mn mn mn mn mn many metrics ie spin elds with nonvanishing vacuum values we can always redene them so only one remains Similar remarks apply to gauge parameters such as those for general co ordinate transformations where vacuum value refers then to the invariances of the vacuum ie the global part of the lo cal transformation The consistency with Stromingers interpretation in terms of closed string elds follows from the fact that in the eld theory of b oth op en and closed strings the closedop en twop oint vertex that follows from quantum corrections implies a direct nonderivative twop oint coupling b etween these op enstring elds and the cor resp onding closedstring elds like the nondiagonal massterm typ e of coupling that can o ccur b etween two spin elds as here for the op enstring eld g and its mn closedstring analog We will discuss this further in later sections However here we are considering a eld theory of op en strings only This is actually more consistent with resp ect to an expansion inh since the relation of the closedstring selfcoupling to the op enstring one ish dep endent Closedstring states still app ear in the theory but only as b ound states so they are not describ ed by fundamental elds but rather by comp osite elds just as eg the hydrogen atom in QED In this interpretation the op enstring eld g intro duced ab ove is the only metric tensor available in the mn theory the only spin eld which couples universally to the energymomentum ten sor of the op en string When onelo op corrections to its propagator are calculated one nds a new massless spin p ole that didnt app ear in the classical treelevel theory ie the graviton At the linearized level it is already clear that the op enstring eld theory has gauge invariances for massless spin elds that resemble those of massless spin In string eld theory expansion in the ghost co ordinates gives Stueckelb erg elds The interacting case requires only a generalization of the Higgs mechanism to gravity This can b e derived in the same way as for massive spin by p erforming a gauge transformation