Classical and quantum massive string

NUCLEAR PHYSICS B ELSEVIER

Nuclear Physics B 464 (1996) 85-116

Classical and quantum massive string Z b i g n i e w H a s i e w i c z a,1, Z b i g n i e w J a s k 6 1 s k i b,2 Institute o f Theoretical Physics, Wroclaw University, pl. Maxa Borna 9, 50-204 Wroclaw, Poland h lnstitut de Recherche sur la Mathdmatique Avancde, UniversiM Louis Pasteur et CRNS, Strasbourg, France a

Received 7 April 1995; revised 5 January 1996; accepted 9 January 1996

Abstract The classical and the quantum massive string model based on a modified BDHP action is analyzed in the range of dimensions l < d < 25. The discussion concerning classical theory includes a formulation of the geometrical variational principle, a phase-space description of the two-dimensional dynamics, and a detailed analysis of the target space geometry of classical solutions. The model is quantized using the "old" covariant method. In particular an appropriate construction of DDF operators is given and the no-ghost theorem is proved. For a critical value of one of the free parameters of the model the quantum theory acquires an extra symmetry not present on the classical level. In this case the quantum model is equivalent to the non-critical Polyakov string and to the old Faidie-Chodos-Thom massive string.

1. Introduction The first quantum model of massive string was proposed twenty years ago by Chodos and Thorn [ 1 ]. It was originally formulated as the tensor product of d-copies of the standard free field representation of the Virasoro algebra and one copy of the free field Fairlie representation with the central charge c = 26 - d. Although satisfying all consistency conditions of a free string quantum model it had some important drawbacks. First of all it contained a non-physical continuous internal quantum number. Secondly because o f the lack of a path integral representation it was not clear how to implement the idea of "joining-splitting" interaction. The attempts to construct tree amplitudes by means of the operator formalism known from the critical dual models led to a continuous J E-mail: [email protected]. 2 Permanent address: Institute of Theoretical Physics, Wroclaw University, Poland. E-mail: jaskolsk @ifi.uni.wroc.pl. 0550-3213/96/$15.00 (E) 1996 Elsevier Science B.V. All rights reserved PII S 0 5 5 0 - 3 2 1 3 ( 9 6 ) 0 0 0 1 4 - 4

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range of intercepts. Finally the rather ad hoc phase space formulation made the physical interpretation of the classical system obscure. For these reasons the FCT string did not receive much attention at that time. Soon after the Polyakov paper [2] on conformal anomaly in the BDH string [3], the Fairlie realization of the Virasoro algebra reappeared in the quantization of Liouville theory [4]. At about the same time the Fairlie realization was used in the rigorous proof [5] of the Kac determinant formula [6]. One of the early attempts to clarify the relation between non-critical BDH string and the Liouville theory suggested by Polyakov's work, was made by Marnelius [7]. He considered a modified string model determined by the world-sheet action

°/

S[ M, g, ~, x] = - ~

~

dZz gabO,x"Obx~u~

M

13 f 2~" M

,-2~d2z (gaOOaq~Ob~+2Rg~o)_l.~f ~Z~d2zexp(~o) " (1) M

The analysis of [7] was devoted to the case of non-vanishing cosmological constant. As a side remark it was pointed out that for ~ = 0 and/3 = ~ the canonically quantized model should be equivalent with the FCT string [ 1 ]. The equivalence of FCT and the non-critical Nambu-Goto string was first observed in the paper [ 8 ] and subsequently used in a BRST reformulation of the old subcritical dual model [9]. It was shown that the dual model prescription for amplitudes is consistent only for d = 25 where the "Liouville field" plays the role of an extra dimension. In another approach [ 10] the non-physical cut singularity of the one-loop non-critical string amplitude was interpreted in terms of the closed FCT string with a continuous mass spectrum. In both approaches in order to achieve the unit intercept of the leading Regge trajectory one assumes a non-vanishing imaginary value of the zero mode of the "Liouville momentum". The action ( 1) regarded as a two-dimensional conformal field theory action describes a special case of the induced (Liouville) 2-dim gravity coupled to the conformal matter. This system has been extensively studied over the last few years, both as a conformal field theory input of the modern dual model construction [ 11 ] (this application being restricted by the famous c = 1 barrier) and as a dilaton gravity toy model for analyzing formation and evaporation of black holes [ 12]. (For a review with references to the literature see [ 13] .) The basic idea of the present paper is to consider the action (1) as a world-sheet action of a relativistic one-dimensional extended object. In contrast to the previous attempts to relate this action with non-critical string we assume that the bulk and the boundary cosmological constants are zero, i.e. there is no Liouville exponential interaction. Another important novelty of our approach is the reparameterization invariant formulation of the classical variational problem. Such formulation implies an additional constraint on the zero mode of the "Liouville momenta" removing the unwanted continuous degree of freedom present in the conventional field theoretical analysis [ 14] of

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the action (1). The interpretation of classical solutions of the system as trajectories of a onedimensional relativistic extended object depends on the causality principle one imposes on possible motions of such systems. According to the commonly used "micro-causal" principle a string trajectory is causal if there exists a parameterization such that each point of the string moves with a speed less or equal to the speed of light [ 15,16]. An interesting feature of the system determined by the action (1) is that the space of its micro-causal solutions coincides with the space of micro-causal solutions in the NambuGoto and the BDHP classical string models. It turns out however that the "stringy" interpretation of the system can be maintained with an essentially weaker notion of causality. This leads to a new classical model of a relativistic extended object which we called the massive string. The covariant quantization of this system yields in the range of dimensions 1 < d < 25 a consistent quantum theory of free relativistic string with properties depending on the value of the parameter fl and the intercept a ( 0 ) = a0 - 2ft. Our considerations concerning the quantum massive string theory and its relations to other non-critical string models can be summarized in the following diagram.

!

massive s ~ n g

~

FCT siting, ¢0o= 0 0 < # < _ - -24-d ~ao
string too = 0

o < ~ < 2~s'~-~, ao =1

llI

=~, ~o=~.,(~

Ill

F'C~ strlng ta0 =0, fl = - ~ , a0=l crilical nmsive slaing

d-1

N-G~,~ 1 < d< 2s, ~(o)= ~ ~

d-2

h_

FCT s~dngwo =0

Nambu-Golo string

classical

~

,

BDHP si~hng

~

Nmnhl-Goto stl~ag 1 < d < 25, or{O) =1

Narab~Goto siring

d =~, ,~(o)=I quantum

The number of functional degrees of freedom varies along the vertical axis. All arrows from classical to quantum models stand for the covariant operator quantization [ 17]. The diagram contains non-equivalent quantum models obtained for all admissible values of parameters t , a0 in the FCT string and d, ce(0) in the Nambu-Goto string.

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The part of the diagram concerning the Nambu-Goto string displays the no-ghost theorem proved long time ago by Brower [20] and by Goddard and Thorn [21]. The no-ghost theorem for the FCT string can be deduced using methods of Ref. [21] further developed in [22,23]. A proof along these lines was given in [24] for the special case of fl = ~ , a0 = 1. In the present paper we follow an independent line of arguments based on a suitably generalized method of spectrum generating algebra [20]. This together with the results of [25,26] yields a general "if and only if" version of the FCT string no-ghost theorem presented on the diagram. Let us note that for all admissible values of the parameter fl the quantization of the massive string leads to the quantum FCT model with the zero mode of the "Liouville momenta" equal zero. In consequence in all these models a ( 0 ) = a o - 2 f l < 1 and the physical states on the first excited level are massive which justifies the name of the model. For fl in the range 0 < fl < ~ and a0 < 1 the number of physical "functional" degrees of freedom in the classical and in the quantum model is the same and equals d. For 0 < fl < ~ and a0 = 1 and for the discrete series

tim =

24-d 1 4--~ ÷ 8 m ( m + 1) '

ars(m) = 1 -

((m+l)r-ms) 2-1 4 m ( m + 1)

'

where m = 2,3 . . . . ;1 < r < m - 1; 1 < s < r, due to the presence of the zero norm states the number of physical degrees of freedom in the quantum theory is smaller than in the classical one. The largest space of null states appears in the critical massive string model corresponding to the critical values/3c = / ~ 2 = " ~ , ac = a l l ( 2 ) = 1. In this case the number of physical functional degrees of freedom reduces to d - 1. This phenomenon can be interpreted as a manifestation of a new symmetry on the quantum level or "anti-anomaly". Let us stress the equivalence of the critical massive string and the non-critical NambuGoto string with the intercept .~_.2'. On the classical level the Nambu-Goto string has d - 2 physical functional degrees of freedom. Due to anomalous commutation relations one of the classical symmetries is broken in the quantum theory. In consequence the covariant quantization technique leads in the range 1 < d < 25 to the quantum model with d - 1 physical functional degrees of freedom. It seems as if it is impossible to invent a classical counterpart of the critical massive string with the right number of degrees o f freedom. The model can be obtained either by anti-anomalous quantization of the classical massive string or as an anomalous quantization of the Nambu-Goto or BDHP string. There exists yet another way of constructing this quantum theory. It was recently shown that the Polyakov sum over random surfaces leads in the range of dimensions 1 < d < 25 to the critical massive string model [24]. These three different interpretations of the same quantum model indicate a reach internal structure of non-critical string and may shed new light on the problem of interactions. The organization of the paper is as follows. In Section 2 we formulate a geometrical variational principle for the classical massive string. In order to emphasize the geometrical character of the theory we use reparameterization invariant boundary conditions. This clarifies the geometrical origin of the constraint removing the continuous internal degree

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of freedom from the model. It also considerably simplifies the phase-space analysis of the two-dimensional dynamics of the system which is given in Section 3. The target space interpretation of the classical model as well as its relation to the notion of classical causality is analyzed in Section 4. It is shown in particular that if one assumes the strong classical causality condition then the massive string coincides with the Nambu-Goto and the BDHP classical models. It turns out however that the model admits "stringy" interpretation with a weaker notion of causality, which allows for an essentially wider space of classical solutions, In this respect the massive string can be seen as an extension of the classical string models. One of important features of the classical massive string is that it does not admit the target space light-cone gauge. Due to the non-zero central term in the classical (Poisson bracket) algebra

{Lm, L,} = i(m - n)Ln+m - 4flim3t~m,-n the constraints are of the second kind. Thus, in contrast to the BDHP and the NambuGoto string models, solving constraints before quantization is a prohibitively difficult task, In Section 5 we use the covariant operator method [ 17-19] to quantize the model. Following Brower's ideas we construct the DDF operators and show that they generate the full space of physical states. The metric structure on this space is then completely determined by general results concerning unitary irreducible highest weight representations [27] of the Virasoro algebra. The corresponding no-ghost theorem is formulated for the whole range of free parameters of the model. Section 6 contains conclusions and a brief discussion of some open problems.

2. Variation principle A classical open CTP string trajectory will be described by a set (M,g, ~o,x) where M is a rectangle-like 2-dim oriented manifold with distinguished "initial" aiM and "final" afM opposite boundary components, g is a pseudo-Riemannian metric on M, ~p : M ~ ~ is a scalar function on M and x : M ~ R d is a map from M into d-dim Minkowski space. It is assumed that the metric g on M has a trivial causal structure such that the "initial" and "final" boundary components are space-like while the other components aM \ (aiM U afM) are time-like. On the space of classical trajectories we consider the action functional

°fv

S[ M, g, ~o, x] = - ~---~

d2 z gaOaax~ OoXUrluu

M

fl f x / - ~ d 2 z (gabOa~OOO~O+ 2Rg~o) , 27r M

(2)

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where Rg is the scalar curvature of g, and r/u~ = diag(-1, +1 . . . . . +1). The constant fl is dimension-less in units of h, and the constant ot is conventionally expressed in terms of the slope parameter d with dimension of length-squared a = 2~" The group of global symmetries of the action functional (2) consists of Poincar6 transformations in the target space and constant rescalings of the internal metric g. The action is invariant with respect to general diffeomorphisms f : M - . M ~ preserving the initial and final boundary components and their orientations. The transformation rule for the scalar curvature Reeg = e-O( Rg + [-lgO) ,

implies that it is also invariant with respect to the rescalings of metrics g ~ e o g with a conformal factor p satisfying the equation l--]gLO=--

1 v~Oa(

Vt-~gabObo) = O.

Before formulating variational principle it is convenient to restrict this huge gauge invariance by introducing some partial gauge fixings. This will be done in two steps, first partially restricting reparameterization invariance and then introducing gauge fixing for the rescaling symmetry. Let us fix a model manifold M and a normal direction n along the boundary aM. For any string trajectory (N,g,q~,x) there exists a diffeomorphism f : M ~ N such that the normal direction nf.g of the metric f*g is proportional to n, and the boundary components meet orthogonally at corners with respect to f*g. This means that fixing M and imposing these conditions on metrics provides a good partial gauge fixing for the reparameterization invariance, which we shall call the (M, n)-gauge. In this gauge the space of classical trajectories is the Cartesian product .A4]t x WM x g d where .M~t is the space of metrics with the normal direction ng (x n and with right angles at the corners, 14;M is the space of real-valued scalar functions on M, and gd is the space of maps from M into d-dim Minkowski target space. The reparameterizations form the group D~t of diffeomorphisms of M preserving comers and the normal direction n. The action of D~t on .M~t x 14;u x gdu is given by (g,~o,x)

f E ./)nu

, (f*g,~oo f, x o f ) .

(3)

In the (M,n)-gauge the rescalings also form a group which can be identified with the abelian group /CoM of 1-forms on aM satisfying f ~ = 0. For any g E .M~t and E/C0u there exists a unique solution Q[g,~] to the boundary problem E]sQ = O,

(n~gOaQ)e8 =10u K,

satisfying the normalization condition - ~ d 2 z P = O. M

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91

e g in the boundary condition above denotes the einbein (or the volume form) of the 1-dim metric induced on OM. The action of/CoM on the space of trajectories .A4~ x ~VM × grim is given by

, (eolg,,dg,~,x).

(g, ~p,x)

(4)

It follows that in the (M, n)-gauge the group of gauge transformations is a semisimple product of D~t and /COM. Let us observe that the transformation rule for the geodesic curvature t
implies that for a fixed k E/CoM the condition Kgeg = K is a good gauge fixing for the gauge symmetry (4). Choosing ~ = 0 one gets the ~D~t-invariant gauge-fixing condition Kg = 0,

(5)

which we shall call the geodesic gauge. It can be thought of as a counterpart of the constant curvature gauge in the BDHP model. In this partial gauge the space of trajectories is given by the product . M ~ × WM x g~ where .A4~ is the space of metrics from A4~ with zero geodesic curvature. Let us note that the same space of metric has been obtained in the Polyakov sum over bordered surfaces from the requirement of a proper structure of boundary conditions for the Faddeev-Popov operator [29]. In the geodesic gauge the group of gauge transformations reduces to D~t and its action on the space of trajectories is simply given by the restriction of (3) to .A4~ x WM × g~. Calculating the full variation of (2) on .A4~ x WM x g~t one gets

°fM

-SS[M,g,~o,x] = -

V,_2_~d2z(Vlsx~)Sx~ + ol

dsn.Oax~Sx ~

1rj OM

+ fl-- f v / ' ~ d 2 z (I:]g~°+ Rg) 6~° + 1r fl--f

m

c~M

M

-'~8 (OOa'°~br"o-g a!gabgCd 2 b o~c~OOd~O-7 2 l-]g 1 "gab-" 2"a'b~O)] +fl---fdstalbOgabnCOcq~--ff-f d s t a t o S g a b n c o c ~ ° ' ~7rr P,Ms OM~

(6)

where 0MI, OMs denote the time-like and the space-like boundary components, respectively. Let us note that in the geodesic gauge variations in the metric sector satisfy the boundary conditions

natbSgab = O,

naVa(gbCt~bc) -- naVl~Sgab = O,

(7)

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along all boundary components. The requirement of the vanishing bulk variation leads to the following equations of motion: []g x t* = 0 ,

(8)

[~g ~ + Rg = 0 ,

(9)

Tab =

(10)

0,

where Tab is the energy-momentum tensor: cr ( 1 ca "~ Tab = --Tr \OaxlZObx~ -- -~gabg 3cX~ C3dXl.~) "-[-~-'~"IT Oa~OOb~O-- ~gabg Cgc~OOd~O-- 2 r-lg ¢,Ogab-- 2VaVb~O

.

The trace part of (10) yields Qg ~o=0,

(11)

which, together with the equation (9) implies Re = 0.

(12)

In order to obtain a well-posed variational problem one has to impose boundary conditions in the x- and ~p-sector for which the boundary terms in the expression (6) vanish. Since the classical system under consideration is supposed to describe a free open string with some additional internal structure the variations 6x t" and 6q~ should be arbitrary at the "ends". This leads to the following boundary conditions along aMt: n~O~x~ = 0,

naOa~O= O.

(13)

On the space-like boundary components one needs Dirichlet type boundary conditions describing initial and final configurations of the system. D~-invariant boundary conditions of this type were first introduced in [28]. More recently these boundary conditions have been analyzed in the context of the covariant path integral quantization of non-critical Polyakov string [24]. The derivation of the gauge invariant boundary condition in the x- and ~- sector of the present model is essentially the same. The basic idea is to impose the Dirichlet boundary conditions in the parameterization in which the einbeins induced on the space-like boundary components are constant. For every classical trajectory ( g , ~ , x ) E 2¢1~ x WM x E~ we define the initial configuration as a set (e/g, ~Oi,Xi) where (e/g) 2 = gabtatb( dt)~OM, ,

q~i = ~PlaM,,

All possible initial configurations form the space

X~ = X~M' .

(14)

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93

where ./~i consists of all einbeins o n a g i , ),Vi is the space of real-valued functions on aMi, and Cd is the space of maps xi : aMi --~ •d. D u e to the boundary conditions (13) the functions from W i and C/~ satisfy Neumann boundary conditions at the ends of aMi. The initial configuration defined by (14) is ~D~-covariant, i.e. for f E D~4 one has ( e : *g, (~o o f ) i , ( x o f ) i ) = ( ~ / i*e ig, ~ i o y i ,

XiOyi),

y i = fl~Mi •

Due to this property one can define the space Ci of gauge invariant initial configurations as the quotient M i X ~/~i X e d

Ci =-

~q- x ~Di

where Di is the group of orientation preserving diffeomorphisms of the initial boundary aMi, and the action of R+ x Di on .A4i x Vii x C/d is given by

component

(a',~') ER+ x ~i

( ei, ~Pi,xi )

" ("~'Z* ei, ~i o )1, xi o "y) .

(15)

The group ~+ of constant rescalings of einbeins has been introduced in order to avoid over complete boundary conditions. In fact using the Gauss-Bonnet theorem one can easily show that for the zero-curvature metrics from .A4,~ with fixed geodesic curvature of aM the internal length of the boundary cannot be arbitrary. The space of gauge invariant final configurations Cf is defined in a similar way. It is convenient to use a common description for both spaces introducing a model interval L and the quotient space CL ~-

M L x WL × ~ d R+xDL

canonically isomorphic with Ci and Cf. If we assume the induced orientations o n aMi and cgMf, the isomorphisms are given by

cL ~ [(~,,~,~)] ~

[(~,*~,,~o ~,i,~o ~,;)] E c;,

where '~i : c~Mi ~ L is an arbitrary orientation preserving diffeomorphism, "yf : cgMf L is an arbitrary orientation reversing diffeomorphism, and the square brackets denote the gauge orbits of corresponding elements of 79i, Pf, and 7~z. The gauge invariant boundary conditions for classical trajectory (g, q~,x) E .M~ x )'V~t × £ d are then defined by [(ei, q~i, x i ) ] = c i ,

[(ef,~f,xf)]

=cf,

(16)

where ci, c f E CL. We shall check whether the variational problem (6) is well posed with the boundary conditions (16). For this purpose it is convenient to introduce a more tractable description of the space of gauge invariant configurations. Let ~" be an einbein on the model

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interval L. One can easily check that for every (R+ × DL)-orbit c E CL there exists a unique (if, Y) E ~'VL × C~ such that c = [(~',ff,2)]. In fact the condition ~'= ~ defines a smooth gauge slice for the action (15), and provides a 1-1 parameterization of CL

, [(~',~,Y)] ~ eL.

WL × E~ ~ (,~,~)

Let ci = [(~',~/,Y/)] and cf = [(~',fff, Yf)]. Then the boundary conditions (16) take the form Xi = "Xi O ~ i [ g ] ,

~i = ~i O ~i[g] ,

xf = Yfo~/f[g],

~of = fffo,~/[g],

(17)

where the diffeomorphisms yi[g] : aMi

, L,

~/f[g] : a M f

,L

are uniquely determined by the equations ~i[g]*~'o¢ el,

~/[gl*~'cx e / ,

(18)

and the convention concerning orientation. The analysis of the boundary conditions (17) is the same for both space-like boundary components and we restrict our considerations to the initial boundary. Let s E [0, 1 ] be an arbitrary parameterization of OMi. The variation of (17) yields 6Xi(S) = (OsXi)(S)6"~i[g](s) ,

6~Oi(S) = (Os~Oi)(S)~'~i[g](s)

(19)

where

6yi[g] (s) 6"yi[g] (s) - dsyi[g] (s) " In the parameterization ~ E [0, 1] of L such that ~" = const • d~" the solution to the equation (18) can be easily calculated:

l:

~/i[g] (s) = ~

ei(s') ds',

0 1

where el(s) = x/g(Os, Os) and li = f el(s) ds. Using the equation above one gets 0

( s) "}- ~ 6"~i[g](s) = -- 61"~i[g] ei(s'--'~)

1

JI 6 e i ( s t ) dst

and

~g(Os, 0~)

= 2O~(ei6~/i[g] ) + 2 -," ei.

Inserting (19), and (20) into the OMi-boundary terms in (6) one obtains

(20)

Z Hasiewicz, Z Jaskdlski/NuclearPhysics B 464 (1996) 85-116 -

-

7r OMi

95

ei ds [anaOax~'6xu + fln"Oa~OS~o+ fltatb6gaonCOc~O]

1/ I

- -

e~ ds [an"O,x~O, xu + fln"O.~oO,~o - 2BO~(nao.~o)] ~ i [ g ]

o 1

f

li ~ j e i ( s ) 0

ds na aa~p .

Note that for the metric variations 6g satisfying the conditions (7) the variations ~ i [ g ] (S), ~li are arbitrary and independent. It follows that the 0Mi-boundary terms vanish if and only if the following conditions are satisfied:

otnatbOaX~ObXl~ + flnatboacpab~O + 2flnatbVaVbtp f eidsnaOaq~ = O,

= 0,

(21) (22)

OMi

where in the derivation of (21) the identity natbVaVb~o = KgtaOa~o -- taOa(nbOb~O) and the condition us = 0 have been used. The conditions (21), (22) should be regarded as (off-shell) constraints on possible boundary values of dynamical variables. The first condition implies that a part of the Euler-Lagrange equations (10) - the Tnt component of the energy-momentum tensor along the space-like boundary - should be regarded as an off-shell condition. The second condition means that the zero mode of the momenta of ~0 should be zero off-shell. This removes the unwanted continuous internal degree of freedom and is very important for the "stringy" interpretation of the model. Since both conditions are consistent with the equations of motions the variational problem is well posed.

3. Two-dimensional dynamics In string models one has two conceptually different notions of evolution: the inner time evolution, and the evolution in the target space. Although in some cases (e.g. NambuGoto string in the light-cone gauge) the inner and the target times can be identified, the interpretation of classical trajectories of two-dimensional system as histories of a one-dimensional extended object in the target space, is a difficult problem and has to be analyzed in each model separately. In this section we shall concentrate on the twodimensional dynamics of the massive string model, leaving the discussion of the target space evolution to the next section. In general the space of states and the time evolution of a classical system are given in terms of Cauchy data for the Euler-Lagrange equations of the corresponding variational principle. This formulation of dynamics assumes the existence of an evolution parameter for which the Cauchy problem is well posed. In the case of systems with

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reparameterization invariance there is in general no gauge independent notion of (inner) time. The standard method of dealing with this problem is to formulate the variational principle with some fixed choice of time parameter. In this formulation the initial and final boundary conditions are gauge dependent. The structure of the Cauchy data for the resulting Euler-Lagrange equations can be analyzed by the phase space Dirac method. If the Hamiltonian of the classical system obtained in this way weakly vanishes one gets a consistent formulation independent of the choice of inner time. The classical system determined by the action functional (1) has been analyzed within this framework in a number of papers [ 14]. In contrast to the scheme above the formulation of the variational problem given in the previous section is gauge invariant. In consequence the problem is well defined on the quotient space x w,,,, x

(23)

For a given choice of boundary conditions the solution (g, ~p,x) of the Euler-Lagrange equations is determined up to the iDa-action and can be seen as a point in the space (23). The interpretation of the Euler-Lagrange equations as dynamical equations requires an introduction of an evolution parameter. The uniformization of metrics from .A4~ allows for a D~t-invariant definition of the inner time as the corresponding Teichmiiller parameter. As we shall see this leads to a consistent phase space formulation. The advantage of the gauge invariant description of the 2-dim dynamics is that it minimizes the number of dynamical variables and therefore the number of constraints in the corresponding phase space formulation. Moreover the dynamical variables, the constraints, and the equations involved have a clear geometrical interpretation. In order to analyze the dynamical content of the model one needs some parameterization of the quotient space (23). With our choice of the inner time the conformal gauge is especially convenient: (24)

g=eO~t ,

( , @ , , ~ t ) : ( [ 0 , t]x[0,Tr],(-01 0)), ,

tC•+.

In this gauge each point in the quotient (23) is uniquely represented by a set (t, •, ~,, x) ~ R + x WM X WM x £ ~ . In the conformal gauge the action functional (2) takes the form t

0

'n"

0

where dot and prime stand for the partial derivatives with respect to the parameters ~and o- respectively. By simple calculations one gets the equations of motions (8), (11 ) and (12) - 2 ~ + x ~n = 0,

- ~ + ~o" = 0,

-~J + Q" = 0,

(26)

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and the energy momentum tensor

'~

:)

~

:1

---n0'~,'

= a ( : + x '2) + fl (~b2+ ~p,2) +

qr

+ ---flO'qd - 2--fl~p'' , ¢/"

T,,, =T,,~ = a--±x' + ~((o~o' + 0~o' + a'~ - 2~b'). 7/"

(27)

cr

(28)

7/"

The boundary conditions along the time-like boundary components are given by x u' = O,

~p' = O,

d = O.

(29)

Using natural identification of aMi and aMf with the model interval [0, 7r] the boundary conditions (5), (16) along the space-like boundary components can be written as follows: O/ = Ol~Mi=

~S = ~I~Ms=

O,

0,

(30)

x: =~fo~f[es],

Xi = "Xi o ~i[ai] , ~oi = ~i o ~ ; [ a ; ] ,

where the diffeomorphisms ~i[Qi]

: [0,7/']

:'

[0,7/'],

~f[~gf]

: [O,'rr]

)

[0,7/']

are uniquely determined by the equations Qi

01o-~i[~i] Of. e x p , - ,

~9i = ~lOMi,

(31) -~-,

Pf = ClaMs.

l_~t us note that with the boundary conditions (29), (30) there is no dynamical degree of freedom in the metric sector. Indeed the only classical solutions to the equation (26) satisfying (29), (30) are &l = const. Inserting these solutions into (27), (28) and (31) one can easily check that Pd completely decouples. The resulting system is determined by the equations of motions

2---~ (~2 + x , 2 ) +

-Yc ~ + x "~ = O,

(32)

- ~ + ~p" = 0 ,

(33)

(~2+~o'2) - 2 ~ o " = 0 , ~

(34)

and the constraints --a.tx' + fl---~b~p'- 2 ~ b ' = 0 ,

(35)

do-(o=O.

(36)

7/"

71"

7"/"

0

98

Z.

Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

The phase-space analysis of the system above is straightforward. The non-vanishing Poisson bracket relations are {p~' (o-), x ~ ( o ' ) } = ~ " ~ ( 0 . - 0 . ' ) ,

{to(0.), qff0.')} = 3(o" - 0.').

(37)

The Hamiltonian generating the inner time evolution of the system

0

leads to Hamilton's equations

f = {ffI, f } , "rrp~ (0.) , o~

~t~(0.) =

7r to(o') fi '

¢(0.) =

p~(o) = - ~ x ¢ ' ( o ) ,

~(o) = -~"(o').

7/"

7/"

The phase-space constraints are given by

2

x'2+ to 71"

2

H ( o ' ) -- ~ a p + 2Ir

2zr

~"

V(o-) - - p • x + to~pl _ 2to I = 0,

,/

too - --

do.to = O.

7/"

0

For any canonical variable Z = expansions

x~,p ~, ~, to and for H ( 0 . ) , V(0.) we define the mode

~X)

O0

Z(o') = ~ Zncosno',

O0

H(o') = ~ Hncosno',

n--'O

V(tr) = ~

n=O

Vn sinno'.

n=l

In order to simplify the analysis of the algebra of constraints Hk, I4, too we introduce

L o - - ~rI-lo = F-I, 71"

- -

It

po, 7r

L+k =-- ~ ( Hk -4- iVk) , Ct~k =-- ~

k > 0,

Pk q: ikx/'~x

21(~

ikx/Cfi )

,

k>0, k>0.

By straightforward calculations one gets

(38) where

Z Hasiewicz, Z Jask61ski/Nuclear Physics B 464 (1996) 85-116

99

+oo

1 +oo

1

--00

- 2v/-dik/3

.

--00

The Poisson brackets (37) imply the relations /z

u

_'

x

v

_

/zv

{am, an} - tm~7 tSm,-n , { Lm, a n } - -ina~m+,,, X L nX } ___i ( m _ {Lm,

x n, n) Lm+

{tim, fin } = im6m,_n , { L~, fin} = -in/3m+n - 2m2 v/--fiSm,-n , ~o ~ _ . {L.,, L. } -t(m

- n)L~+ n - 4/3im3t3m_n.

Calculating the algebra of constraints one gets { Lm , Ln } = i ( m - n ) Ln+m - 4 flim3 ~m,-n ,

{Lm,/30} = 0 .

(39)

An important property of the algebra above is that {Lm}m>O is a family of second kind constraints. We close this section by the formulae for the conserved charges related to the global Poincar6 symmetry. Using the Noether method one gets the conserved currents: •a

a

b

J l~"a = _~ V / - ~ gab Obx[l.tX~,l

In the phase space the total energy momentum of the string is given by 1r

F

0

where F is an arbitrary curve connecting opposite sides of the strip of parameters o-, ~'. The total angular momentum of the string reads fr

=

n~ju~, F

(p~(o.)x~'(o.) - p P ( o ' ) x U ( o ' ) )

do',

0

= P~"x~) -- P~ x~ + i Z n=l

a_.a.

n

4. Target space dynamics In this subsection we shall discuss the space-time interpretation of classical solutions of the massive string model. For that purpose let us briefly recall the assumptions

Z Hasiewicz, Z Jask61ski/Nuclear Physics B 464 (1996) 85-116

100

concerning the target space behavior of classical solutions in the Nambu-Goto and the BDHP string models. The Nambu-Goto string in the orthogonal gauge and the BDHP string in the conformal gauge are determined by the same set of equations -

= 0,

(40)

±2 + x t2 = 0 ,

(41)

±. x I = 0,

(42)

Yd' + x u "

with the boundary conditions x'~(O, 7") = x'~(#,~ ") = O.

(43)

A simple consequence of (41) and (43) is that the ends of string move with the speed of light. Another one is that the string world-sheet is time-like, i.e. detaaXgabX ~ = _(jr2)2 <~_0.

(44)

Note that this is the property one has to assume for all trajectories in the NambuGoto string model. Indeed only for time-like trajectories the Nambu-Goto action is a real-valued functional and the orthogonal gauge is well defined [30]. In the BDHP model the action is real and well defined on trajectories (g, x) with g Lorentzian and non-degenerate on the whole strip of parameters (including boundaries). Only for such trajectories can one prove the validity of the conformal gauge. In this model the property (44) is a consequence of the constraint equations (41) and (42). In the relativistic theory of classical point-like particles the causality principle is formulated as the condition for the energy-momentum of the particle to be time-like. In the case of relativistic one-dimensional extended objects the notion of causal motion is less obvious and depends on the way such objects may interact with themselves and other classical systems. In the commonly accepted formulation of causality in the classical Nambu-Goto and BDHP models the string is regarded as a collection of points which may individually interact. One says that a string trajectory is causal if there exists a parameterization such that all points of the string move with a speed less or equal to the speed of light [ 15,16]. Such property of string trajectory we shall call microcausality. If we assume that classical strings may interact only as a whole the causality principle is much less restrictive - it requires the spectral condition

p2 ~ 0 , where P~' is the total energy-momentum of the string. Trajectories satisfying the spectral condition above will be called macro-causal. The notion of micro-causality plays an important role in the classical Nambu-Goto and BDHP string models. First of all for micro-causal trajectories one can show the validity of the light-cone gauge which in order allows to find all micro-causal solutions of the system ( 4 0 ) - ( 4 3 ) [ 15,16]. Secondly, the micro-causality implies the spectral condition for classical solutions [ 15] . The inverse implication does not seem to be true

Z Hasiewicz, Z Jask61ski/Nuclear Physics B 464 (1996) 85-116

I01

but we do not know any macro-causal solution of the Nambu-Goto model which is not micro-causal. What can be easily shown is that the system ( 4 0 ) - ( 4 3 ) admits tachyonic motions. A simple example in the three-dimensional target space is given by t(o-, r) = cos(r + o-) + cos(r - o-), x(o-, r) = ½(sin2(r + o') + sin2(r - o') ) ,

y(cr, r) =~" - ¼(sin2(r + o') + sin 2 ( r - o-)). To conclude this brief discussion of causality in the classical Nambu-Goto model let us mention that it is related to a rather subtle structure of the phase space which has no impact on quantum string models obtained by the covariant quantization techniques. As far as the system ( 4 0 ) - ( 4 3 ) is regarded as a two-dimensional o'-model its reduced phase space can be identified with the space of all classical solutions. If we however consider the same system as a classical string model the corresponding phase space is "smaller" and consists of solutions satisfying some causality assumptions. In the socalled old covariant approach the "big" phase space is quantized. It turns out however that the quantum spectral condition automatically appears due to the properties of the Fock space holomorphic representation (the tachyonic ground state appearing there is of a different origin and has nothing to do with classical tachyonic motions). As we shall see the same phenomenon takes place in the covariant quantization of the massive string model. Let us now turn to the classical solutions of the model derived in the previous subsections. The general solution to the equations of motion (32), (33) satisfying the boundary conditions (29) can be written in the following standard form

xU(~r,r) = ½ ( f U ( r + ~) + f U ( r - (r)), ~(~r, r) = ½( h ( r + ~r) + h(r - or)), where fU(z ), h(z ) are arbitrary functions such that

f~(z

2rr +27r) = f ~ ' ( z ) + ~ p u ,

2¢r h(z +2zr) = h ( z ) +--x-to0, P

and P~', too are the total energy-momentum of the string and the zero mode of the Liouville momentum, respectively. Inserting the general solution to the constraint equations ( 3 4 ) - ( 3 6 ) one gets the equation for the functions f~', h Ol

~ ft2 Jr

h '2 - 2flh" = 0,

(45)

and the periodicity condition h (z -I-2~') = h(z ). The equation above can be seen as the Hill equation [ 16] 8/3H" = - - ot ~ f 12H

(46)

Z Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

10:2

with respect to the function H = e x p ( - ¼ h ) . The problem is to find conditions for the function f a under which the equation above admits strictly positive periodic solutions. Since we are not aware of any simple method to solve this problem we shall restrict ourselves to a simple ansatz giving a subclass of solutions, large enough to exhibit peculiar features of the system at hand. We start with the discussion of micro-causal solutions to the equation (45). For such solutions the ends of string cannot move with a speed greater than the speed of light and one has f a ( z ) <_ O. Then by the constraint equation (46) H " >_ 0, and the function H ~ is monotonic. Since H ~ is periodic this is possible only for h = const. One obtains a rather surprising conclusion that the micro-causality does not allow for the Liouville excitations in the classical solutions of the massive string model. Moreover the micro-causal solutions of this model precisely coincide with the micro-causal solutions of the Nambu-Goto and BDHP models. It means that with the micro-causality principle imposed all three classical string models are identical. Let us now consider macro-causal solutions. A large subclass of such solution can be obtained by the following light-cone ansatz. In the Minkowski target space we introduce the light-cone coordinates x ± =-~2(x °+x d-l) i ~d-2 in its in a two-dimensional time-like subspace and the transverse coordinates ti x trJi=l orthogonal complement. In these coordinates the Lorentzian scalar product reads 2

x . y = - x + y - - x - y + + Xtr.

For f + of the form p+ f+(z)

= --z

(47)

+ c+ ,

the constraint equation (45) can be solved with respect to f z

f-(z)=~--Wi('¢"eh"-2.h")dz'+ck2 Jtrq- 2

(48,

0

It follows that the collection ( f + , f - , fir, h) where f t . h are arbitrary functions satisfying the periodicity conditions f~r( Z + 2"rrl = fitr( Z ) + 2"n'P~r , Ol

h( z + 2"n') = h( z ) ,

and f + , f - are given by the formulae (47), (48), is a classical solution of the massive string. Solutions obtained in this way are macro-causal. Indeed, calculating 2 P + P - one gets

Z Hasiewicz, Z Jaskrlski/Nuclear Physics B 464 (1996) 85-116

--

-

71" ~

+

103

dz,

"rd"

which implies the spectral condition p2 = _ 2 p + p - + p~ _< 0. Let us note that what we really need to linearize the constraint equation (45) in the light-cone coordinates is the condition for the function (f+)~ to be strictly positive. With this condition satisfied one can calculate f - in terms of f + , ftr, and h: f-(z)

__aF/2 2~h'2 - 2 t 8 h " 2Jtr+

=

a f+/

dz' + c-.

0

It is convenient to formulate the ansatz above in a slightly different way. Let us first observe that any function f + with strictly positive and periodic derivative can be expressed in the form f+(z)

p+ = --y(z) o~

+ c +,

where y is an orientation preserving diffeomorphism of the real line such that y(2kTr) =2k~-

for

k E Z,

(49)

and y ' ( z + 27r) = y ' ( z ) . (50) p+ Let ( f + --- --~-z+ c +, f - , ftr, h) be an arbitrary light-cone solution, y - a diffeomorphism satisfying (49) and (50), and

~ ( z ) = -b-Z o One can show by explicit calculations that the collection (f+ oy, f- or +~,ft roy, hoy+

21ogy')

(51)

is a new solution of (45) with required periodicity properties. Note that except of the non-homogeneous part ~ in the expression for the new f - , the formula (51) is the transformation rule for classical variables with respect to the conformal transformations. The appearance of the non-homogeneous term ~ results from the fact that the constraints of classical massive string are not conformally invariant. Calculating 2 P + P - for the deformed light-cone solution (51) one gets 2p+p_=p~+

ot -g

--

~r

+

h ~2

dz - -FZ

-~,,3 - dz • --

,ge

It follows that the transformation (51) may lead to tachyonic solutions. Indeed a simple example of a tachyonic string motion is given by f t r = h = 0 and y #: id.

Z Hasiewicz, Z Jaskrlski/Nuclear Physics B 464 (1996) 85-116

104

In order to illustrate some features of the macro-causal solutions let us consider a particular case of the light-cone solution in 3-dimensional Minkowski space time given by the functions p+

f+(z) = --z ,

f~r(z) = acosmz,

OL

h(z) =bcosnz.

The corresponding string world-sheet is described by

( P+ t(O',T) =

~

ota2m2+_fl__b2n2"~ +

4v~P +

j 7-

~a2m - 1 6 v ~ P + (sin2m(7- + o') + sin2m(7- - o-))

flb2n

16v/~p + (sin 2 n ( r + tr) + s i n 2 n ( r - or))

flbn

(sin n(7" + o-) + sin n ( r - o') ) ,

a x(o-, 7-) = ~ (cos m(7" + o-) + cos m(7- - o-) ) ,

p+ y(o',r) = V'~Ot

ola2m2 + flb2n2~ 4V'2P +

] 7-

ota2m -+ 16v~P-------~ (sin 2m(7- + o-) + sin 2m(7- - o') )

flb2n

-F 16x/~p---------~ (sin 2n(7- + tr) + sin2n(7" - tr))

flbn

x/2P+ ( s i n n ( 7 + tr) + s i n n ( r - tr)) .

(52)

For the parameters a = 1 , fl = ~11, a = 0.6, b = 1, P+ = 0.8, n = 3, m = 1 the parametric plot of the string history is presented in Fig. 1. We shall analyze more closely this particular string motion. First of all, as one could expect from the discussion above, some parts of the string oscillate with a speed greater than the speed of light. In consequence for the amplitudes of this oscillations high enough one has several disjoint pieces of the string on a fixed target time hyperplane. The configuration of the string at the time t = T corresponding to the upper bound of the plot in Fig. 1 is shown in Fig. 2a. This strange behavior of string trajectory is in apparent contradiction even with the weaker principle of macro-causality. What saves the day is that all classical macro-observables of the string are concentrated on its one "material" piece. For all other "ghost" pieces the total energy-momentum and the angular momentum are zero. This can be easily seen from the structure of the equal time lines in the space of parameters, i.e. curves y ( s ) = ( t r ( s ) , 7 ( s ) ) determined by the implicit equation

t(o'(s),7-(s) ) = T.

Z Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

105

6

-'

-0

Y X

1 Fig. 1.

For the configuration of Fig. 2a these lines are drawn in Fig. 2b. It is clear that calculating appropriate line integrals of conserved currents one gets non-zero result only for the curve m connecting opposite sides of the strip of parameters. The only "material" piece of the string is that corresponding to the curve m ( the line M in Fig. 2a). Due to this property the strange time evolution of "ghost" pieces of the string (they appear, disappear, split and join with each other and with the "material" piece) is in a perfect agreement with the macro-causality principle and the global conservation laws.

Z Hasiewicz, Z Jask6lski/Nuclear Physics B 464 (1996) 85-116

106

Fig.2.a

Y

"~

Fig.2.b

0

pC)

-i

-1

-0.5

0 X

0.5

o:s ~ i s

pC ~ 215 ~

Fig. 2.

Another important feature of the solution (52), closely related to the structure of equal time lines, is that the "ghosts" pieces are inessential from the point of view of the Cauchy problem for the target space dynamic. Indeed in order to determine the evolution of the system it is enough to know "positions" and "velocities" of all classical variables at each point of the "material" piece of the string at a fixed target time. Solving the equations of motions in both time directions one obtains all "future ghosts" (lines F in Fig. 2a) and all "past ghosts" (lines P in Fig. 2a). It means that the macro-causality principle allows for a consistent formulation of target space dynamics, and therefore, for a "stringy" interpretation of the solution (52). The question arises whether the discussion of one particular macro-causal solution given above is valid in general. The main property of the equal time lines required for the "stringy" interpretation is that for each moment of the target space time one has only one "material" piece of the string. Analyzing the geometry of levels of the time component x°(o -, ~') of a general macro-causal solution one can easily show that this property holds except instants corresponding to saddle points of x°(o ", r). At these moments "ghost" pieces join or separate from the "material" one so the identification of the "material" piece is ambiguous. This ambiguity however does not lead to any ambiguity in the Cauchy problem - all possible choices of "material" piece at an "interaction time" lead to the same string trajectory and the same values of macro-observables. It follows from the considerations above that one can provide a consistent "stringy" interpretation for all macro-causal solutions of the massive string model. The space of these solutions can be regarded as the reduced phase space of the massive string. It is essentially bigger that the reduced phase space of the Nambu-Goto and the BDHP string models. It seems that the generalized light cone ansatz provides a good local

Z Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

107

parameterization of the reduced phase space but the analysis of singularities of this parameterizations is a difficult open problem. Since solving constraints before quantization seems to be a very difficult task the only available method is to represent the algebra of constraints on the quantum level. In the next section we shall discuss the covariant quantization of the "big" phase space. AS in the case of the Nambu-Goto string the quantum spectral condition automatically appears.

5. First quantized massive string In these section we discuss the covariant operator quantization of the massive string. Following standard prescriptions of string theory [ 17-19] we start with the CCR algebra [ a~, ot~ ]

=

mr/aVt~m,_n,

[ fin,, fin ] = m S m . - , ,

m, n v~ 0,

(53)

[ P~,, x~ ] = - i 8 ~ .

supplemented by the conjugation properties =

+ =

+

,

= S-m,

(Bm) + = B-re.

The space of states is a direct sum of the Fock spaces F, along d-dimensional spectrum of momentum operators: 7"[ = f

dap Fp.

In each Fp there is a unique vacuum state lip satisfying a ~ t l p = flmflp = O,

m > 0;

PuIlp = p u l ) p .

In order to define the constraints operators (38) we introduce the standard normal ordering of quadratic expressions: i.t v :area .:=

OlmO[n , ~ ~

[ anam ,

m < 0 m >_ 0

(54)

with similar rules for l~mt~n and mixed products. With this definition the action of the normally ordered constraints operators 1 +oo

+oo

tn = 2 ~ : ° / - m "°in+m: ..}_1 ~--~ : ~ - m ~ n + m : "I- 2 V / - ~ i k f l k + 2flrn,O, m=- cx~ m=- oo

(55)

on the vacuum state is well defined. The action on states from 7"[ is then uniquely determined by the commutation relations with excitation operators: = --motto+ n ,

[ L . , flm] = -mflm+n q- 2 i n 2 v / ~ Bn,-m •

(56)

108

Z

Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

Due to the normal ordering the central term of the algebra of constraints gets shifted by ~ ( d + l) [ L . , Lm] = (n - m)L.+rn + ~ ( d + 1 + 48fl) (n 3 - n)6n,-m.

The physical subspace 7-/oh C 7-[ is defined as the set of all vectors ~O satisfying ( Ln - 6n,oao ) ~9 = O .

The parameter a0 in the conditions above is left arbitrary at the moment. Its value will be restricted by the no-ghost theorem. In this representation of the quantum theory the algebra of Poincar6 charges is realized by the translation operators pu and the Lorentz generators: P

M u" = ( P ~ x ~ - P " x ~ ) + i Z

~, _ a v

a-nan

n>0

n

a/z

-n n

Following the DDF approach [ 31 ] we introduce the formal operator series i

X " ( O ) =q~ + a~O + ~ --man'l~e-im° mq:O ~ ( 0 ) = Z t r i m e-in'O, m m4=O oo

p~'(O)=(XU)'(O)

= ~

a ~ e -ira°,

n,=--O0

I](0)

=qbt(0)

=

Z

flm e -imO ,

m~0

where q~ = Vrdx~ is the operator canonically conjugate to a0 and 0 is a real parameter. From (53) one gets [XU(0), X~ ( O') ] = -27ri~7u~ e( O - 0 ' ) , [ ~ ( 0 ) , ~ ( 0 ' ) ] =-2~-i(e(0 - 0') + (0' - 0 ) ) , [ Pu ( O) , X~(0') ] = -2~ri6~ ~( 0 - 0 ' ) ,

[FI(0) , ~ ( 0 ' ) ] =-27ri(6(0 - 0') - 1), [P~,(0), P~ ( O' ) ] = - 2 ~rirlt,~6' ( O - 0 ' ) , [l-I(0), II(0') ] =-2~ri6'(0 - 0 ' ) , Commutation relations with constraint operators follow from (56) [ Lm, X u ( 0 ) ] = - iP ~ (O) e ira° , [ Lm, ~ ( 0) ] = - i l I ( O)e in'° + 2 m v / - ~ e ira° , .d [ Lm, Pu( O) ] = -t-S-z( Pt~( O)eim°) , U[7

Z Hasiewicz, Z Jaskdlski/NuclearPhysics B 464 (1996) 85-116

109

•d [ Lm,II( O) ] = -t~-~(II( O)e imO) + 2im2 v/-fle ira°.

(57)

For a fixed light-like vector k (k 2 = 0) we define a basis of DDF operators as follows. li--d--2 a basis Let U be a light-like vector satisfying the condition k. U = - 1 and r~tei~i=l in the Euclidean subspace orthogonal to both k and U. We define d - 2 families of the transverse operators [ 31 ] -

2Ir

A,in(k) = ~ f dO:el. P(O)eimk'X(O) :,

(58)

0 and one family of the longitudinal (Brower) vortices [20] 27r

im l o g ' ( k . P(O)))e imk'x(°) : . Bin(k) = ~1 f dO : (k'. P(O) - -~ o

(59)

In addition we introduce the operator corresponding to the Liouville degree of freedom [241

1/

2~-

C,,(k) = ~

dO : ( I I ( 0 ) - 2V/fl l o g ' ( k - P(O)))e imk'x(°) : .

(60)

o The primes over logarithmic terms denote derivatives with respect to 0. The power series present in ( 5 8 ) - ( 6 0 ) describe well-defined operators on the subspaces Fp C 7-/with p satisfying k. p = x/a. Calculating the algebra of DDF operators (58), (59) and (60) one gets [A/~(k), AJ(k) ] = m~iJt~m,_n,

[ C,n( k ), Cn( k) ] =mt~m.-n , [Bm(k),Bn(k)] = (n - m)Bn+m(k) q- 2n3t~m.-n, [Bn(k), Ai,( k) ] = -mAi,+n( k ) , [ B n ( k ) , Cm( k ) ] = -mC.+m( k ) + 2in2 v/-fl S.._m . This algebra can be diagonalized by the following shift of the Brower vertex [20] B,,(k) = Bn(k) - / : n ( k )

+ Bn.0,

where 1

+oo

d-2

£n(k) =~ ~_~ Z A i m ( k ) A i + m ( k ) m=--oo

i=l

1 +~ q-2 Z C-m(k)Cn+m(k) q- 2inv/-flCn(k) + 2flSn,o. m=- oo

In the new basis Aim(k), Bin(k), Cm(k) the only non-zero commutators are

(61)

I 10

Z Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

[A/,(k), AJ,,( k) ] = m~iJc~m,_n , 1

3

[ Bn( k), Bin(k) ] = (n - m)Bn+m(k) + --~(n - n) (25 - d - 48/3)6n,-m, (62)

[Cm(k), Cn(k) ] = m6m,-n.

Using (57) one can find out the commutation relations of the DDF operators with the constraints (58)

[ L , , a ~ , ( k ) ] = [ L n , B m ( k ) ] = [ Ln,Cm(k) ] = 0 , for all m, n E Z. It follows that acting on vacuum states lip such that k . p = the DDF operators A i , ( k ) , B m ( k ) , C m ( k ) generate off-shell physical states, i.e. states satisfying the physical off-shell condition L,~P=O,

n>0.

All states obtained in this way (for different k) we shall call the DDF states. We shall show that the DDF operators together with the family of operators [32]

27r Fm(k)=-~fdO:eimk'X(°):, 0

generate the total Hilbert space of states. The commutation relations read

[Aim(k), Fn(k) ] = [Gin(k), Fn(k) ] = [Fro(k), Fn(k) ] = O, [Bin(k), Vn( k) ] = -nFn+m(k) , In contrast to the DDF operators Fro(k) do not commute with the constraints

[L,n,F,(k) ] = -mF~'(k) , 2tr

1 f F,~ (k) = ~--~

dOei.o : eimk.X(O)

: .

(63)

0

For a vacuum state f~p with

Fn_m(k)Ilp = 0 ,

p • k = v/'J

one has

rn > O,n > m ,

F~m( k )[~ p = ~ p .

(64)

Let us consider the states of the form

~t'(N) ae ( { a l } , { b } , { c } , { f } )

d--2 i = n (k a i - - m , ] ,~a,,,. i=1 •n b " n . . . . ,

i

"'"

. (a/._l)a,

nbl I • C C t l . . . . .

CCl I • F f k k . . . .

• Ffll~'~p+Nk,

(65)

where {ai}, {b}, {c}, { f } are arbitrary finite sequences of non-negative integers such that the eigenvalue

Z Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 8 5 - 1 1 6 d-2

N= ~

mi

n

l

11 l

k

~-~riair, + Y ~ s b s + y ~ t c t + ~ u f u

i=l

ri=l

s=l

t=l

u=i

of the level operator R is fixed. The virtue of the operators Fm (k) is that for N = 0, 1,2 . . . . the states (65) constitute a basis in the space 7-/p. In order to check this property one can look at the structure of the operators Aim(k), B-re(k), C-re(k), F-re(k) in terms of fundamental creation operators:

Aim(k)

-- Oli-- 117 - n

aiok . a_m + "more"

B_.,(k)=k"~-m-

( k ' . a o - ~ m1( m +

l)

) k . a _ m +"more",

C-m ( k ) = fl-m - 2i v/ flmk " ~-m + "more", F_.7 (k) - k- ~_m + "more".

(66)

The terms denoted by "more" are of higher order in the creation operators Or_n,/3-n with n < m. The structure of the "leading terms" in (66) implies that the states (65) are linearly independent. Counting their number level by level one can show that they form a basis of 7"/p. This statement holds if we replace the Brower vortices Bn(k) in the formula (65) by the shifted ones Bn(k). Following Brower's proof of the no-ghost theorem for the Nambu-Goto string [20] one can show that the on-shell DDF states exhaust all physical states. Theorem 1. Let p ~ O. A state rI, E 7-[e is an off-shell physical state if and only if it

is a DDF state. Since p 4~ 0 there exists a light-like vector k such that k. p = ~ and one can use the states (65) as a basis in 7-/p. It follows that any state q' E 7-I can be written as a sum ~I' = ~t)OF + ~t'F, where q~t)I~F E 7"/DDF and ~F contains non-zero excitations F-m. We shall show that the conditions

L,,~

= LmalrF

=

0,

n > 0,

imply q'F = 0. In the expression for ~V in terms of the basis (65) there is a term containing the operator F-re(k) with maximal m and raised to the maximal power fro:

q'F = AFf-~,,( k ) ' . . . " Ff-'i ( k )I~p+N~k +BFf_';,(k) "... • Ff_i (k)~p+l%k + . . . .

(67)

where f., > f~ and A , B contains only DDF creation operators. Due to (63) and (64) one has m

(Ll )f' "..." (Lm) A' ~

= H

f J ! (-j)yj A ~p+NAk '

j=l

and consequently Af~p+,v,,k= 0. Hence the firstterm in the expansion (67) must vanish. Repeating this procedure for next terms we prove ~F = 0.

Z Hasiewicz, Z Jask61ski / N u c l e a r Physics B 464 (1996) 8 5 - 1 1 6

112

It follows from Theorem 1 that the space of physical states ~ph coincides with the space of on-shell DDF states or, which is the same, with the space of DDF states created from physical vacua. Then using the algebra of DDF operators one can analyse the metric structure of the subspace 7"/DDF and deduce the no-ghost theorem for the massive string. Let ~t, with p 4 : 0 be a physical vacuum, i.e. p2 = - 2 a ( 2 f l - a0). Consider the subspace 7-/DDF(P,k) C 7-/ph of states generated from lip by the DDF operators with some k, k. p = x/-d. The metric on ~DDF(P, k) is completely determined by the algebra (62) and the hermiticity properties of the operators A . ( k ) , B n ( k ) , Cn(k). Due to the diagonal form of the algebra (62) the space ~DDF(P, k) is isomorphic to the symmetric tensor product of the spaces 7-/AC(P, k) and 7-/B(p, k) generated from lip by the algebra of A . ( k ) , C . ( k ) and the algebra of Bn(k) operators, respectively. Since the metric on ~AC(P, k) is positive, the metric structure on ~DDF(P, k) depends on the metric structure on ~B(P, k). Calculating the action of the operator Bo(k) (61) on the physical vacuum one gets

Bol~p = ( 1 -

ao)lip

•

It follows that 7-/B(p,k) is the Verma module Vc,h of the Virasoro algebra with the central charge c = 2 5 - d - 48fl, and the weight h=l-a0. In general the metric on Vc,h may be degenerate. Taking the quotient by the subspace of null vectors one gets the irreducible highest weight representation 7-(c,h [5]. It is known [25,26] that the representation 7-[c,h is unitary, i.e. the metric on 7-[c,h is positively defined, if and only if one of the two following conditions is satisfied: c>l,

h>0,

or

c=c,,,

h=hrs(m)

for

m=2,3 .... ;

l
l
where cm - 1

6 m(m + 1) '

hrs(m) -

((m + 1 ) r - ms) 2 - 1 4 r e ( m + 1)

One gets the following: Theorem 2. The space of physical states in the massive string model is ghost free if and only if one of the following two conditions is satisfied: a0 _< 1,

0 < 13 <

24 - d 48

(68)

z Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

113

or

/3=/3m,

ao=ars(m)

for

m=2,3 .... ;

l
l
where fin, ---

24- d 1 4--~ + 8 m ( m + 1) '

ars(m) --- 1 -

((m+ 1)r-ms) 2- 1 4 m ( m + 1)

The "if" part of the theorem above was first proved in [23] by means of different techniques and in a slightly more general setting. The advantage of the method of spectrum generating algebra used in the present paper is the simple proof of the "only if" part of the theorem. This method can be also applied to analyse the structure of the space of null vectors. In particular as a simple consequence of the algebra (62) and the properties of the Fairlie realization of the Virasoro algebra with the central charge c > 1 one gets that for 0 < fl < ~ and a0 < 1 there are no null states. For fl in this range and a0 = 1 all states involving the operator B - l are null. For the discrete series (69) the problem of the null states can be easily reduced to the same problem in the conformal field theory with the central charge 0 < c < 1. Although all technical ingredients required are known a comprehensive analysis of this point is rather involved and we restrict ourselves only to the special case of the critical massive string corresponding to tic = / 3 2 --" ~ and ac = all(2) = 1. The critical massive string has a number of interesting properties. First of all it is equivalent to the old FCT string [ 1 ] and to the non-critical Polyakov string [24] with extra constraint too = 0. It also has the largest subspace of null states. In fact the commutation relations (62) imply that for the critical values /3 = 25-d 48 ' a 0 = 1 all vectors generated by the shifted longitudinal operators Bn are null and decouple from the physical states. The resulting quantum system has effectively one "functional" degree of freedom less than the classical one. This phenomenon can be seen as an extra gauge symmetry (anti-anomaly) of the quantum model. Due to this special structure of null states the "quantum light cone" gauge conditions K,[ph) = k. a , lph) = 0,

n > 0

(70)

can be used to parameterize the genuine physical states. The physical states satisfying (70) are generated by the transverse and the Liouville DDF operators. This subspace provides 1-1 parameterization of the quotient space 7-/ph/7-/nun [ 24]. One may check that an equivalent parameterization of ~ph/7-/nun can be obtained by imposing the following Lorentz invariant quantum gauge conditions fln[ph) = 0,

n > 0.

In this case the solutions are generated by the transverse A/ (58) and the Brower Bm (59) DDF operators. Since the algebra of these operators and the Lorentz generators

114

Z. Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

coincides with the corresponding one in the non-critical Nambu-Goto string with the intercept ----A -~4 both quantum models are equivalent. This gives a simple proof of the observation made in [8]. Let us finally notice that due to the structure of the operator L0 (55) the physical mass spectrum of the massive string is bounded from below. The no-ghost theorem implies that there are no excited tachyonic states in the physical spectrum. Indeed, for all admissible values of fl, ao only the vacuum states with m 2 = 2t~(2fl - a0) may be tachyonic.

6. Conclusions The main result of the present paper is that in the range of dimensions 1 < d < 25 the action functional (2) leads to a new consistent classical and quantum theory of one-dimensional relativistic extended objects. Our derivation of the classical model is based on a new reparameterization invariant formulation of the variational principle. The virtue of this approach is a clear geometrical interpretation of the classical system and a simple phase space formulation of the 2dim dynamics. The most interesting result of this part of the work is the derivation of the constraint o~0 = 0. In the standard formulation [7,14] the origin Of this constraint is not clear and one has to introduce it by hand in order to remove the unphysical internal degree of freedom. In the present approach it appears as a necessary consistency condition for the variational principle to be well posed and diffeomorphism invariant. Let us note that the Euclidean counterpart of the variational problem formulated in Section 2 is interesting by its own. One can easily check that minimal surfaces form a special subclass of solutions to such problem with ~ = 0. It would be interesting to find some local and global geometric characterizations of solutions with q~ ~ 0. The analysis of the classical causality given in Section 4 leads to the following conclusions. First of all if one assumes the micro-causality principle the classical massive string model coincides with the Nambu-Goto and the BDHP models. Secondly if one admits the spectral condition for the total energy-momentum of string as a weaker notion of causality then the space of macro-causal solutions is essentially bigger than the space of micro-causal ones. The important point is that macro-causal solutions still can be given a consistent "stringy" interpretation. Both results mean that the classical massive string is in a way a minimal generalization of the Nambu-Goto and the BDHP models. Our discussion of the target space dynamics leaves some open questions. First of all it is desirable to have a detailed description of the reduced phase space of the massive string defined as a space of all solutions of the constraint equation (45) satisfying the spectral condition. The light cone ansatz which parameterizes almost all microcausal solutions is not a good parameterization for macro-causal ones. It seems that the generalized light-cone ansatz may provide at least a local parameterization of the massive string phase space. Justification of this conjecture and clarification of the global geometric structure are still open problems. Similar questions are also interesting in the

Z Hasiewicz, Z Jaskdlski/Nuclear Physics B 464 (1996) 85-116

115

case of old string models, where the difference between the micro- and macro-causal solutions is unknown. As shown in Section 5 the massive string model can be quantized by the covariant operator technique. The main results of this section are the explicit construction of physical states by suitably modified DDF method and the no-ghost theorem yielding necessary and sufficient conditions for a consistent quantum theory. It was also proved that the critical massive string model corresponding to the critical values of parameters fl = 2~_~..___dd,a0 = 1 coincides with the non-critical Nambu-Goto string with the intercept d--I 24

"

Our results concerning the structure of the quantum massive string were already summarized in the introduction. Probably the most important conclusion is the equivalence of the critical massive string with the non-critical Nambu-Goto string obtained by covariant operator quantization [ 20] and with the Polyakov non-critical string obtained by purely functional techniques [24]. In all three interpretations of the critical massive string the zero mode of the "Liouville momenta" is represented by an operator with a real spectrum and the intercept is never equal 1. On the other hand the tree amplitudes of the old non-critical string theory [34] are ghost free if and only if or(0) = 1 [35]. Thus if the interacting theory exists it cannot be equivalent to the old subcritical dual model. This conclusion suggests that the amplitudes of massive string might not be dual and therefore the conventional duality cannot be used as an organizing principle for the interaction theory. The most interesting question is whether it can be replaced by the principle of joining-splitting interaction.

Acknowledgements We would like to thank Krzysztof Meissner for discussions at early stages of this work and Przemyslaw Siemion for his patience in helping us with "Mathematica". One of us (Z.J.) would like to thank the mathematics department and research institute (I.R.M.A) in Strasbourg, where part of this work was completed, for their hospitality. We are grateful to Prof.R.Marnelius and the referee for pointing to us several references missed in the first version of this paper.

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