Nonperturbative weak-coupling analysis of the quantum Liouville field theory

ANNALS

OF PHYSICS

153. 147-201

(1984)

Nonperturbative Weak-Coupling Analysis Quantum Liouville Field Theory*

of the

E. BRAATEN, T. CURTRIGHT, G. GHANDOUR,+ AND C. THORN Physics

Department. Gainesville. Received

Unicersit.v of Florida, Florida 326 I I June 3. 1983

Two independent weak-coupling expansions are developed for the Liouville quantum tield theory on a circle. In the first. the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Blcklund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g’” using both approaches, yielding identical results.

I. INTRODUCTION The Liouville field theory [ 1] is a two-dimensional subject to the nonlinear wave equation

model of a scalar field, @,

(1.1) Several problems in theoretical physics can be reduced either to this model or to simple generalizations which are similar in their dynamics. Our interest in the Liouville field theory stems from Polyakov’s work [2] on the problem of covariantly quantizing the relativistic string. That work has motivated many studies of the Liouville theory (3-101. In Polyakov’s approach to the string problem, it is necessary to solve the quantum field theory associated with (1. l), and then to obtain correlation functions for products of exponentials of the Liouville field. As a first step in solving the Liouville quantum field theory, Curtright and Thorn 13 ] presented a conformally covariant quantization of the model on a finite

* Research supported in part by the Department of Energy, Contract by the Research Council of the University of Kuwait, Project SP012. ’ On leave from the Department of Physics, University of Kuwait.

DE-AS-05-81ER40008,

and

147 0003-49

16/84 $7.50

Copyright ‘P 1984 by Academx Press. Inc. All rights of reproduction m any form reserved.

148

BRAATEN

ET

AL.

spatial interval, 0
QUANTUM LIOUVILLE

FIELD THEORY WEAK-COUPLING

ANALYSIS

149

Liouville theory starting from the BCT exact operator solution. We review the connection, due to the existence of a Backlund transformation, between the Liouville field # and a free masslesspseudoscalar v. We discuss the explicit free field functionals for a,#, :exp( g4):, and :exp(2g#):, and we rearrange these expressionsto facilitate carrying out the weak-coupling expansion. We calculate the lowest order matrix elementsof these operators between free field energy eigenstates.These results immediately lead to the necessity of restricting the free field states to those which are odd with respect to a discrete symmetry, the so-called “zero-mode parity.” We present the results of computing corrections relative to these lowest order matrix elements. For a,# and :exp( g#):, the corrections are given to O(g”), while for :exp(2gd):, the corrections are given to 0( g6). In order to reveal the relevant features of this alternate weak-coupling expansion, we discussin great detail the calculation of corrections for :exp(g#): to O(g’). Higher orders can be done without introducing any essentially new techniques. We also discuss to this order the consistency of restricting the free field state space. In Section IV, we assessthe validity of the operator manipulations used to obtain the BCT exact operator solution. We explain how the presence of a previously unnoticed zero eigenvalue, for an operator appearing in the explicit free field expression for a,#, invalidates certain operator manipulations on the even zero-mode parity sector of the free pseudoscalartheory. We go through a careful examination of the Liouville field equation to show how it fails on this even parity sector. We attribute this failure, and other difficulties on the even parity sector, to the presenceof a singular operator distribution, 6(P), where P = 1’:” da @(a). This operator distribution does not contribute in the odd zero-mode parity sector, and we conclude that the operator manipulations of BCT remain valid in that sector. Finally, in three appendixes, we collect several technical results which are of use in evaluating the terms in the weak-coupling expansions for matrix elementsof Liouville operators.

II. LIOUVILLE

PERTURBATION

THEORY

Our first method of analyzing the weak-coupling limit of the quantized Liouville field theory on a circle is to use continuum perturbation theory. We shall formulate the exact eigenstatesof the Liouville Hamiltonian as seriesinvolving known “unperturbed” continuum states, and then use these series to obtain weak-coupling results for exact eigenstate matrix elements of various local Liouville operators. We shall also discussthe use of the perturbation seriesto check the conformal transformation properties of the exact energy eigenstates. To implement this program, we shall find an unperturbed Hamiltonian, Ho, whose spectrum and level degeneraciescoincide with those of the Liouville Hamiltonian, H. To begin, then, we define H and discuss its spectrum. H was previously defined in [3] using canonical quantization methods. These methods shall also be used here, so let us briefly recall someof the results of ]3 ]. The

150

BRAATEN

canonical r=O by

Liouville

ET

AL.

field on a circle, 4, and the conjugate variable,

71, are given at (2.la)

(2.2a) Z*(U)

= &

$,

(u*ne*inu + b*,e*i”uJ

(2.2b)

where [Q(u), $a’)]

= id(u - a’)

I%. Qkl = 4,k,O and where q independent of of#* and rr*. The quantum and (2.2), was appropriate for

(2.3) Ian, bk] = 03

= lb,, bkL

(2.4)

and p are canonically conjugate zero-mode operators which are the nonzero-mode operators appearing in the Fourier decompositions Liouville Hamiltonian, expressed in terms of the operators in (2.1) defined in 13] to have the same form as the classical Hamiltonian Eq. (1. 1), with the mode operators normal-ordered. That is,

where 4’ and TC+ annihilate the usual Fock vacuum, IO). In 131, we showed that the definition (2.5) was consistent invariance in the sense that the energy momentum density operators defined in such a way that their Fourier components .2r

L*k E

dueiiko

['OO:

with conformal Too f To, can be

'011

0

satisfy the Virasoro

algebra, 2

IL:,L;l=(k-m)L:+,

+&(k3-")6,T-,,+;

k’6

k.-m

(2.6) IL:,L;]

=o.

QUANTUM

It follows

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSIS

151

from this algebra and the relations H=L,+

+L,

9=L,-L,t that the energy and momentum eigenstates can be put in the form L +,\I . . . L+;‘LI;’ -I

... L:;qE,O)

(2.7)

where (A), {p) are sets of nonnegative integers and where the state lE, 0) is any energy and momentum eigenstate satisfying HjE,O)=ElE,O) L: JE, 0) = 0,

(2.8)

k > 0.

For a fixed state IE, 0) the states (2.7) are linearly independent [ 111. In [3] we used a variational argument to show that there is at least one such state 1E, 0) for each E > 0. and that it also satisfies .P IE, 0) = (L, - L,+)lE, 0) = 0. We believe it is a strong working assumption that there are no other states satisfying (2.8). In the following we shall make this assumption, which we shall see is consistent with all of our weak-coupling calculations. Then it is not hard to see from the linear independence of the states (2.7) that the degeneracies of H are given by the formula

god(n)x”=~ fi l 1 --s

&,

(2.9)

(1 -xq*

where d(n) is the number of states at a given energy E with n < E < n + 1. Our next task is to introduce our choice for H, and discuss some properties of the unperturbed system. Our H, is obtained by discarding the coupling between zero- and nonzero-modes of the Liouville field. We define (2.10)

H,=h+R where h=&2+Te

&ml2

2gq

(2.1 la)

g 2%

R=2

I

0

da z-(a)

7c+(u) = f (a-,a, n=l

+ b,,b,).

(2.1 lb)

152

BRAATEN

ET AL.

Thus, the unperturbed nonzero-modes are free oscillators, governed by Liouville quantum mechanics (LQM). The unperturbed energy eigenstates are then of the form

but the zero-modes

are

(2.12) where (A} = (A, ,..., A,) and (p) = (p ,,..., p,} are two sets of nonnegative integers, and where IE) denotes a pure zero-mode eigenstate; i.e., it is an eigenstate of h and acts as a nonzero-mode “vacuum” for each E: hJE)=ElE)

(2.13)

a,jE)=O=bJE), The unperturbed

energy spectrum

n = 1, 2,....

(2.14)

is thus given by

The spectrum of h is 0 < E < co and is nondegenerate. It therefore follows that the level degeneracy and the spectrum of H, are identical to that of H as given in (2.9). We next discuss how one obtains the zero-mode energy eigenstates and other relevant features of the LQM. The spectrum of h and several other useful properties of the zero-mode states IE) can be establish by working directly in the “9” representation. The zero-mode Schrodinger equation is (2.16) and can be readily solved upon changing the variable to exp(gq). (q 1E) = i

(2k sinh rrk)“’

Ki, (F

eKq)

One obtains 19 ] (2.17)

where k = (4zE/g*)‘/=,

O
(2.18)

and where K, is a modified Bessel function. (Our conventions for Ki, agree with those of [ 12I.) It follows from (2.17) that the E = 0 wavefunction is excluded from the space of bounded wavefunctions 191. This property is easily understood upon considering the purely repulsive character of the potential, exp(2gq). It is also clear from the potential

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSIS

153

that h does not have any degenerate levels. Rather, one state exists for every positive E (or k). This of course is due to the transmissionless nature of exp(2gq), so that any q-space wave packet approaching from q = --oo will eventually be totally reflected. All this is conveniently expressed in terms of the asymptotic q behavior of the eigenstates in (2.17)

x e-K@/2

1 -

&l

+4k2)ernX4

+ cp”))

(2.19a)

(

(4 I E),zi

W7c)“’

(2.19b)

sin( gqk + 6(k))

where the phase shift is given by (2.20)

One can also use (2.17) to establish the continuum

normalization

of the h eigenstates

(E”~E’)=$B(k”-k’)

(2.21)

which is consistent with (2.19). Next we give some results for LQM matrix elements of various zero-mode operators. Later we will use these results extensively in carrying out the perturbative evaluation of energy eigenstate matrix elements for Liouville field operators. We will only briefly indicate derivations of these results here (see Appendixes A and B, however). Rather, we will emphasize features of the matrix elements of importance for the perturbative analysis to follow. First, consider the LQM matrix element of an arbitrary exponential of q. (~“1 cog9 IE’) = 4rZjr(n) x /r(a/2

(&In

(k”k’

+ i(k” + k’)/2)

sinh rk”

sinh rk’)“’

I-(a/2 + i(k” - k’)/2)1’

(2.22)

where u > 0. This may be obtained in the q representation by using (2.17) and by consulting the literature on K-transforms (e.g., [ 12, Chap. 10, Eq. (10.3.49)].) Note the behavior of this matrix element as a -+ 0 is as expected from (2.21). Also note that if k’ and k” are held fixed, all g dependence is explicitly displayed on the RHS of (2.22) as an overall scale factor gzu-‘. Similarly, K-transforms can be used to evaluate matrix elements of exponentials of

154

BRAATEN

ET AL.

both p and q. One need only note the equivalence of either shifting q or resealing m in (2.17). Thus we have (E”j e o&Tei4Plg =

JE’)

m -cc

ewqe(llgMWdq)

dq(E”Iq)

(q

1 E’)

= -.$ (k”k’ sinh nk” sinh nk’)“’ x

00 dqe

= -$ (k”,’ sinh nk” sinh r&‘)‘/2 X [” =

(Elf

dqeQgqK-,,,, (F cl2 1 eaK9

IE’)

ei4k’

egq) K,, ($

(me4) eKq)

(2.23)

2F,(u/2 + i(k’ - k”)/2, a/2 + i(k’ + k”)/2; a; 1 - e2’)

where in the last step, we again used Eq. (10.3.49) from [ 121. Using the series expansion of the Gauss hypergeometric function in (2.23) we have IE’) (E” / e4hvei4pIR (E” 1eay91E’) eiDk’I’(a) = f(a/2 + i(k’ - k”)/2) T(a/2 + i(k’ + k”)/2) xl

(2.24)

? f(n + a/2 + i(k’ - k”)/2) r(n + a/2 + i(k’ + k”)/2) n! f(n + a) IT=0

(1 _ eZbJn

Using I-( 1 + s) = XT(X), this immediately shows that matrix elements involving exponentials of q multiplied by polynomials in p will always reduce to matrix elements of exponentials in q multiplied by polynomials in k’ and k”. The simplest example, eugqIE’)ig

(E” / eag9p IE’) = (,“I

(~+

k”22ak”),

(2.25)

follows from the terms linear in ,f3in (2.24). Finally, we point out that canonical commutator algebra can be used to reduce many other matrix elements to the above forms. As an illustration, we evaluate a particular matrix element containing one zero-mode propagator. We first observe that e2K9

-

-&

IPYhl

(2.26)

QUANTUM

follows

LIOUVILLE

immediately

(E”IeagqP

i

FIELD

THEORY

WEAK-COUPLING

155

ANALYSIS

from [q,p] = i. So we have

&

1

ezgqlE’)=~(E”le”pqP(~~[p,hjlE’)

= -+$

(,“I

eagq P (A)

= &

(E”(

eagqp (E’).

(E’ - h)p 1.E’)

(2.27)

Here P denotes the principal part of the singular operator (E’ - h)) ‘. If the singularity of the operator (E’ - h)-’ is removed by an i& prescription instead of using P, an imaginary part is obtained for the matrix element. This may be seen by inserting a complete set of LQM energy eigenstates. From the state normalization in (2.21), we have (2.28) Thus 6(E’ - h) e 2gq (E’)

= 1 6(E’ - h) eznq IE’)

= IE’)(E’I

eznq jE’)-$

= ,E’)& where in the last step we have used the diagonal (k’ = k”) case of (2.22) for (r = 2. Combining (2.29), (2.27), and (2.25), we obtain (,!r(

eWq

’

E’-h+ie

x (E”( eagq IE’).

(2.30)

Note that the net effect of inserting (E’ - h + is)- ’ exp(2gq) into the matrix element is simply to multiply (E” 1eagq(E’) by a simple polynomial in k’, k”. This effect generalizes to more complicated matrix elementsconsidered below (cf. Appendix A). Relations of the type given by (2.30) are crucial for an evaluation of perturbations on the LQM. Many additional such matrix elements can be efficiently obtained by further operator manipulations using [q,p] = i, reducing the matrix element in question down to (2.30), for example. We describe some of these manipulations and tabulate the ensuing relations in Appendix A. Alternative derivations using contour integrals are described in Appendix B.

BRAATEN ETAL.

156

Next we discuss the particular perturbation on the LQM which arises from the coupling between zero- and nonzero-modes in the Liouville quantum field theory. Comparing (2.5) and (2.1 I), we may write the LQFT Hamiltonian as H=H,$H,

(2.31)

where H,=-e

2m2 zgq 2n da(e2gm-(de2gm+b) J 0 g2

(2.32)

- 1).

H, describesthe coupling between the zero- and nonzero-modes of the Liouville field. This perturbation is weak as g + 0 in the sensethat 2lc

1

_ 1) do(e 28m-(de2gm+w

-0

=

-2g2

g

4

n=, n

(anbn

+ aenhn

-

a_,a,

-

b&b")

+ O(g3)

(2.33)

and therefore H, is 0( g’) relative to Ho. Becausewe have chosen an Ho whose continuous spectrum and level degeneracies coincide with those of H, we may directly relate the exact and unperturbed energy eigenstates using the Lippmann-Schwinger scattering formalism. Denoting an exact eigenstate of H by III?), we have IlE) = IE) + (E-H

+ ie)-’ H, lE)

We choose the ic prescription in (2.34) as a convention. Letting E+ --E does not give an independent set of states. For E < 1, where the energy levels are nondegenerate,it is clear that a different prescription can only produce an overall change in phase of the state defined by (2.34). This point is manifest in the explicit results we obtain below for matrix elements of certain Liouville operators (cf. (2.50)). We now consider matrix elementsof the operator ze”gm. .-_ eagqe”weagm+

(2.35)

between exact energy eigenstates as given by the perturbation series (2.34). For simplicity, we consider matrix elements involving states IlE”) and IlE’) with 0 < E’, E” < 1, since for this energy range the states are nondegenerate and therefore completely specified by their energy labels. One should keep in mind, however, that higher energy matrix elements can also be evaluated by our methods.

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

157

ANALYSIS

Using (2.34) we have (E’j

(2.36)

:eag4: l/E’) n”

1 ’ El’--Ho-i&

We may evaluate the terms in this double sum eigenstates. An equivalent, but somewhat more eliminate the nonzero-mode operators in the (E -Ho)-’ H, factors by normal-ordering the full IE’) and IE”) are nonzero-mode vacua. We shall illustrate this last remark in detail for of (2.36). We have

(En 1 :e”Km’o’:

1 E’-H,+iE

eaKqp‘4-eaK4+

H,

” 1E’).

by inserting complete sets of H, systematic, alternative is to first product of :exp(ag#): and the expression, exploiting the fact that the n”, n’ = 0, 1 term on the RHS

E’-~,+jFH1lE’) do (E”I e weab?b +(o)

- 27~(E” I eagq

1

1 Et-h-R+&

eh’qe2K@-(U) IE’)

ezRqIE’)[.

E’-h+ie

(2.37)

We further reduce the first matrix element on the RHS to (Et!

1 Et-h--R+&

I &%‘qeW@+(0)

jJ’

=--I

dr (Et’1

e2gqeb’~-(U) IE’)

eagqeag@+(0)eir(E’-h-R+iE)e2g@~(”’e2gq

IE’ >

eagqeago+(0)e2g~R(a.-T)eiT(E’-h+iE)eZqg

I >

0

=

---I

‘1” dt (E/J

Er

0

where we define 4,’ to be nonzero-mode HI-

operators

(2.38)

time-evolved

according to R (not

(si (u, r) = eirRf#i* (u) eeisR _

* ’ \/47I

(2.39)

G ,~,

leTinr(a

n

dine+

Leiino).

Next we reorder the $’ and #- factors in (2.38) using e - L%m,kJ“,r”)eagmR+(o’.r’~eOgmRoe-“g”R+‘”~”~’ =(1-e

i(r”-r’),i(u”-n’)

-a5g2/47c(l >

_

ei(7”-T’)ei(u’-u”))-nDgZ/4z.

(2.40)

158

BRAATEN

ET AL.

This allows us to eliminate all the nonzero-mode operators from (2.38) in favor of a c-number function of o and r. That c-number function may then be easily integrated over (T, as called for in (2.37), by using the binomial expansion of each of the two multiplicative factors on the RHS of (2.40). The result for (2.37) is a series of zeromode operator matrix elements summed over the nonzero integer eigenvalues of R. The r integration is then trivially done for each term in the series to obtain our final result. (En

/ :e”Rm’o’: E’

4?z??t2 y, = -T--g ,,

-ho

+

%

ie

H1

+

%*j2’)

IE’)

(,n

/ eaR9

(n!)” P(ag2/27c)

E, _ h 1 2n + ie ezgq IE’). (2.41)

Terms in (2.36) involving more factors of (E - H,))’ H, may be similarly expressed, with all nonzero-mode operators reduced to sums over integers. The results are complicated, but completely straightforward to obtain. We shall present explicit results for the higher n”, n’ terms in (2.36) only as they are needed to a definite order of g in the following (cf. (2.47), (2.48), and (2.49) below). Next we discuss the order in g of the result in (2.41) and of the higher IZ”, n’ terms in (2.36), to explain more precisely what is meant by the weak-coupling of the nonzero-modes as g -+ 0. This discussion is facilitated by an additional simplifying assumption, which still leads to nontrivial results for the matrix elements under consideration. We assume that g+o

with E”, E’ = O( g’)

(2.42)

in this limit. That is, g + 0 with k”, k’ both of order 1. We then rewrite the zero-mode matrix element in (2.41) using the completeness relation (2.28).

(,U 1eagq

l

+

,E,) =gi:; & ““i;;g:i;k”“,x,’

E’ - h - 2n + ie The matrix which are dominated (2.41), we (,I,

c;; lE’) .

(2.43)

k

elements in the numerator of (2.43) exponentially suppress values of E, not O(g*), as is clear from (2.22), and so the integration in (2.43) is by E’ -E, = O(g*) = E” -E,. Since n = 0 is excluded from the sum in may therefore expand the matrix element as I eWq

’

E’-hh22n+ie

eZg9 IE’) ezgq (E’)

=--

2y

f NY0

g,fm dk @“I eaR91Ek)(Ek( e2g9IE’)[ 0

and we see that the Nth term in this sum is of order

g x g3 x gZN

E’iE”)

N,

(2.44)

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSIS

159

relative to the matrix element @“I exp(agq)jE’), the factor of g3 being due to (Ekj exp(2gq)lE’). Taking into account the explicit factors of g in (2.41), we conclude that the n”, n’ = 0, 1 term in (2.36) is a series in powers of g2, starting with a lowest order behavior of order g6, given the conditions in (2.42). We give the order g6, g*, and g” terms in (2.41) explicitly, for later use.

-2&f&25,

=am2 +

~~~2-4(51,-2S,~,)(~~3~(E”]e’at”g’~E’)

i( (

-*~~-4(2i,-hC)(~)*)~E”le~gq[e2g9,~]lE’)

(2.45)

+ o(p).

In computing (2.45), we have used the results of Appendix C to replace sums involving inverse powers of integers with Riemann zeta functions, and we have replaced powers of (E’ - h) with appropriate commutators in the zero-mode matrix elements on the RHS of (2.44). Also, each LQM matrix element appearing in (2.45) may be reduced to a polynomial in k’ and k” times (E”J exp(agq))E’), using operator manipulations as discussed in Appendix A. Before giving the results of that reduction, however, we complete our discussion of the order in g of the general n”, IZ’ terms in (2.36). From our experience with the n”, n’ = 0, I term above, we can anticipate that the general n”, n’ term in (2.36) will be a series in g2 multiplied by (E”I exp(agq)lE’). We now establish the minimum power of g2 in this series, given the conditions in (2.42). This can be done using four simple rules which are easily verified. (i) For every factor of (E-H,))’ H,, there results an intermediate integral g J‘dk, which arises from using (2.28). This gives an overall power of g”“+“‘. (ii) For every H, there results a factor (E, I exp(2gq)(E,) (2.22) and (2.42). This gives an overall power of g3@“+#‘).

which

is 0( g”) given

(iii) For every propagator, (E -Ho)-‘, there results a minimum power of go, if nonzero-mode energies are carried by the propagator, while there results a minimum power of l/g2, if only zero-mode energies are carried. (iv) For every nonzero-mode of g.

created or annihilated by :exp(ag#):, there is a power

One can now determine the overall minimum power of g for the general n”, n’ term by inserting between the H, factors intermediate Ho eigenstates in such a way as to increase the number of propagators carrying only zero-mode energies while simultaneously decreasing the number of nonzero-modes created or annihilated by :exp(ag#):. In general, for every 2N or 2N + 1 propagators to the left (or to the right)

595/153/l-11

160

BRAATEN

ET AL.

of :exp(ag#):, the optimum arrangement of intermediate H, eigenstates is such that there are N propagators to the left (right) of :exp(ag#): carrying only zero-mode energies, one of which is the propagator immediately to the left (right) of :exp(ag#):. The only feature of the rules not revealed by considering the n”, n’ = 0, 1 term is the second part of rule (iii). This feature is exemplified by the n”, IZ’ = 0,2 term in (2.36). By inserting intermediate H, eigenstates into that matrix element, it is possible to arrange for the left-most (E - H,))’ to carry only zero-mode energies of order g2 and in that case, the :exp(ag#): factor must necessarily reduce to exp(agq). Thus we see that the minimum power of g for the n”, 12’ = 0, 2 term is g6, relative to @“I exp(agq)lE’). This is the same order as the n”, n’ = 0, 1 term discussed above. Similarly, by applying rules (ib(iv) to the general n”, n’ term in (2.36), one can straightforwardly establish the following lower bound for the order of the term relative to (E”I exp(agq)lE’), given the conditions in (2.42).

O(g IO(g

3(n"+n')

n”, n’ term is at least

1

lt3(n"tn') >

if n” + n’ is even if n” + n’ is odd.

(2.46)

Note that different terms corresponding to the interchange of n” and n’ are obviously of the same order. The result in (2.46) shows that the calculation of the matrix element in (2.36) can be completed up to and including contributions of order g”, given (2.42), by considering those terms in the perturbation series with n” + 11’ < 3. In fact, as one readily establishes, using rules (i)-(iv), the only terms contributing with n” + n’ = 3 are those with n”, n’ = 0, 3 or 3. 0. We shall not go through all the steps needed to evaluate the terms with n” + n’ = 2 or 3 in (2.36). Suffice it to say that the analysis involves straightforward embellishments on the steps used to arrived at (2.45), again using identities from Appendix C. We simply state the results. (,“I

:eag*:

’ E’-H,+i&

=m4

1 (-81,-

1614&64(5L

x@"Ieagq

+ (-41,

H, E’-~o+ie

E,A\+iE

H”E”

-2i2r3,($)2)

e4gqI E’)

- 32(2& - [2C13)(&))(E”i

eagq E, -i

+ ie ezgq[ezgq, h]lE’)

- 21;, (E” I engq E, _ : + iE ezgq[ (ezgq, h], h] IE’) + 81;,a2 6

(,!!I e(n+4)R4 IE’) 1 + W2)

(2.47)

QUANTUM

(E”I

H,

E”-H,-ie

LIOUVILLE

l

‘e

a&-, . ’ E’

4 1 (4[,+8((a’

=m

FIELD

THEORY

-Lo

+

ie

WEAK-COUPLING

H,

ANALYSIS

I61

IE’)

+8)1,-41,~,)&)~E”!e’“i4)gq~E’)

+ 21, (E”I eca+2)gq [e2gq, h](E’) + 2& (E”I [h, e2gq] e(n+2)gq IE’) 1 + %+) @“I zag*:

l E’-H,+ie

H, E, -;

= 32m6C, @“I eagq E, -i

0

(2.48)

+ ie H, E, -ho

+ ie H, IE’)

+ ie e6gq IE’) + O(g”)

(2.49)

The terms in (2.36) with n”, IZ’ = 2, 0 or 3, 0 may be obtained from (2.47) and (2.49), respectively, by complex conjugating and interchanging E” and E’. The results in (2.45), (2.47), (2.48), and (2.49) may be further simplified using the material in Appendix A to reduce the zero-mode, LQM matrix elements. Each individual LQM matrix element reduces to (,“I exp(agq)lE’) multiplied by a polynomial in k” and k’. We next give the result of this reduction, after combining all the contributions up to and including U(g”) to obtain (E”ll :exp(og#): l[E’) to this order. (,U /I :eW@: IIE’) = e-iX(E”) (,,, ( eWq (E’) eiX(E’) x /l+

($3C3[+(2-a)(4+2a+3a2)

+ 2a(2 - a)(k”2

+ k’2) - (k”’

+(-g)‘&[+ (2 - a)(2 - 2a2(2 -a)’

where k = (4nE/g2)“2 x(E)=+

I

- 2a + a2)(4 + 6a + 5a2)

(k”2 + k’2) - 4a(2 - a)(k”2

- 2(2 - a(2 - a))(k”2 + 2(k”’

- k’2)2

+ k’2)(krr2

+ k’2)2

- k’2)2

-k”)‘]

+ o(g12)/

(2.50)

and

(

$

1

3~3k(l+k2)+3(g)si,k(l-k4)+O(g’2).

Several features of this result deserve comment.

(2.5 1)

162

BRAATEN

ET AL.

First, note that the matrix element has an imaginary part which is taken completely into account by giving phases, x(E”) and x(E’), to the external states. If we had chosen a different ic prescription in our original relation between llE> and IE), (2.34), it would have changed only these phases. Second, the behavior of the coefficients in (2.50) as a -+ 0 and a -+ 2, is such that the matrix elements satisfy the Liouville equations of motion (1.1). To see this, first note that

(E’/ll(~-~jOllE’)=-(E.ll[H,

[fJ~4llllE’) = - (E” -E’)*

XC-

(,“I1 4 IlE’)

* (k”* - k’*)*

;

(2.52) f

(E”I/ :P@: ‘E’)i

. a=0

Using (2.50) we then express the derivative of this last matrix element as --& (E”II :eag*: IIE’) laze = (1 - (6)’

x

[3(k”2 -k’*)*

,-i,&F")ti~(E')

(2.53) CX=O

where we have used (2.21) and (k”’ - k’*)* d(k” - k’) = 0. Using (2.22) we next obtain the identities -~(k”2-k’2)2$~E~‘,e”g*,E’)l a=0

= - &

(k”* - k’*)*

sinh zk” sinh zk’)“*

(k”k’

2

= - &

sinh nk” sinh nk’)‘/*

(krr2 - k”)(k”k’

-I sinh+(k”+k’)sinh+(k”-k’))

-

:-* (k”*-kt*)(

k”

tanh nk”/2

x sinh nk”/2 sinh nk’/2 cash rk” - cash nk’

k’

tanh nk’/2

) I” (2.54)

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSIS

163

and

Comparing (2.53) with (2.50) evaluated at a = 2, using (2.55), we then explicitly find to O(g”) that

This is the quantum Liouville equation for this matrix element. Finally, to close this section, we briefly discuss the conformal transformation properties of the exact eigenstates, as defined by the perturbation series (2.34). It follows from (2.6) that the conformal charges commuted with H give [H,LZ]

=--nL,i,

(2.57)

]iE) = (E - n) L; II@.

(2.58)

from which HL;

On the one hand, as is clear from the definition (2.5), H is a positive operator, and so it follows from (2.58) that L;

l/E) = 0

for all n > 0, if E < 1.

(2.59)

On the other hand, it is not directly obvious from the series in (2.34), without using the commutator (2.57), that the energy eigenstates obey the constraints given in (2.59).

As a check on the consistency of our perturbation methods with conformal invariance, we have directly evaluated the inner product of Lz ]]E’) with :exp(ag#): ]E”), where a is arbitrary. The calculation was carried out to O(g”) for arbitrary n > 0 and to 0( g”) for 12= 1. To that order, this inner product vanishes as expected from (2.59). The details of this calculation are straightforward and will not be presented. However, it is important to note that the particular value of the conformal improvement term given in [3] is required for the inner product to vanish. This completes our presentation of weak-coupling results for the Liouville system using continuum perturbation theory. In the next section, we shall compare the results obtained using continuum perturbation theory with those obtained from the weakcoupling limit of the BCT exact operator solution.

164

BRAATEN ETAL.

III. WEAK-COUPLING

ANALYSIS

FOR THE QUANTUM

B~~CKLUND TRANSFORMATION

We turn now to a weak-coupling study of the exact operator solution for the Liouville theory obtained earlier by Braaten, Curtright, and Thorn [5]. We begin by recalling the results of that work. Let ~(a, r) be a free masslessquantum field defined on a circle, 0 < cr < 271,with Fourier decomposition: + Bke-ik(7--(7))

(3.1)

and with Q, P, A,, and B, satisfying

[Q,PJ =i;

[Ak,AIj

= [Bk,BIj =k&-,

(3.2)

and with all other combinations vanishing. It is also useful to define a related field 13/ by (qff, r) = - 2;

zQ + (Ake-ik(“+o) - Bke-‘k”-“)).

(3.3)

(Note that a@/& = awl& - P/2n.) The energy momentum tensor operator for I, which was shown in [5] to coincide with that for the Liouville field 4, is given by

where co1= 1, the symbol N denotes free field normal ordering, and the last term is required by the Backlund transformation. In this section we shall denote normal ordering with respect to the 4 field by the usual colons. Reference [5] presented 4 f #‘, :egg:, and :e2gm:as explicit functions of the field w. We quote those results here for easeof reference (we shall take r = 0): :eg@w:_

1 eg”‘“‘-qo) mi

ekwd+

(3.5)

N{F(P) egPd(u’--o)cosh(gv(a’)) eg*‘O’)}.

(3.6)

where x(a)-’

= ,f:n f(od:‘o,,

In this formula and in the following ones we use the following definitions: f(a) = (4 sin’ p)“li“

(3.7) -I/2 2 g2

-4

(3-S)

QUANTUM

LIOUVILLE

4u) =

FIELD

THEORY

WEAK-COUPLING

u - ?r&(U) 2n

E(U) = 2n + 1

165

ANALYSIS

(3.9)

for n < u/27z < n + I (3.10)

< = (2 sin( g’/2)/g’)“‘.

Also, the N symbol includes the following prescription for ordering P’s and Q’s: N{ eaQg(P)} E eaQ”g(P) euQ”

(3.11)

and @(u)+/tj?(u)- are the annihilation/creation components of @(a). The operator :ezg@(@: is obtained from a short distance limit: :e*gO(“):

5

lim

f

@ _

0)

:eg@(“‘:

:eg@(“):

0-n

_

(3.12)

* 1 2 e2ga(o'-~(u)t~(u)

e2g$(o'

+

Cm where

+ L ei&Wu)/2N{F(p)

enPd(o’-u)eg(~(o’)-e(o’))

(3.13)

2

Finally, we quote the result for the derivatives of the Liouville field: ,&,)

* q(u)=wt(u>

f q(u)

T g;2

eg(8(~)-*~(o)~*Q/*)y*(u)eg(8(o)+*~(~)+*QI*)

(3.14)

where

+ +N(F-(P)

egPdCo’-o)eg($b’)

Tcb(o’))

1)

(3.15)

and -1

(3.16)

where P denotes the principal value prescription.

166

BRAATEN

ET AL.

Before turning to the weak-couphng Iimit of matrix elements of these operators, it is very helpful to rearrange the expressions somewhat. First of all, it is convenient to use the identity

COW &9 N (P’ + g*/4y2 to reexpress

(tan:$

)’

cosh(gQ)/

tan15

);1

=

:egm: and :e*@: as follows. (3.18)

where V and p are obtained from X and X? by replacing F(P) by F(P) = $

P2 + g*/4 P sinh 2g+ sin 2 -g* 2 4

(3.20)

As we shall see, the weak-coupling limits of V and P are easier to calculate than those of X and X. It is more important to give a revised form for 4 + #‘, since the weak-coupling expression for Y;’ as it stands possesses a zero eigenvalue. By formally rearranging the Q and P dependence, we can express (d f q%’in terms of the inverse of a positive operator W, for which this zero eigenvalue problem is absent. Thus

112 X

(3.21)

where

F+(p) egPd(o’~~)N~g(B(o’)-tc”(u’)) + eTgQP- (P) e gPd(u’-~)N~g(~(o’)TD,(o’)),rgQ

(3.22)

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

167

ANALYSIS

with F t (p)=izg

47c

p sinh !!2

(3.23) 112

PZ+g2

F- (P)A 47c

(3.24)

sinh2 !!? + sin* s' 2 2

and the symbol {A, B} E AB + BA. Also, v,(o) = w(a) - Q is w(a) with zero-mode operators deleted. The expressions (3.2 I)-(3.24) follow from (3.14)-(3.17) by a series of formal manipulations. As we shall see in the weak-coupling analysis, (3.19) and (3.21) satisfy the Liouville equation in the odd zero-mode parity sector of the free field state space, but not in the even sector. It is useful at this point to set up a basis of energy eigenstates of the free field state space P which is particularly useful in the comparison of the results of this section with those of Section II. The energy operator for the free field theory is just H"=

5

(3.25)

(A-,a,+~-,B,)+;

n=l

=Lo++L,

(3.26)

where L,i are two of the conformal generators L,i , n = 0, k 1, *2,..., defined by 2lr

L;+J

da e*'""(T;o

f zg.

(3.27)

0

As discussed in Ref. [5] a basis of eigenstates of (3.25) may be taken to be IP 1>.**,&I, {P l,...,p,},P')=(Lt,)'l

a.. (L+,)'x(LI,)~~

.a. (L:,J'mlO,P')

(3.28)

where (A}, (p} are sets of positive integers and the state 10, P') is the eigenstate of P with eigenvalue P' annihilated by A, and B, for k > 0: IO,P')==

eiQp' IO)

with 10) the unique zero energy eigenstate of H". The states (0, P') continuum orthogonality relation (O,P"IO,P')=

S(PU -P').

Finally, we introduce the zero-mode parity energy eigenstates

(3.29) satisfy the (3.30)

168

BRAATEN ET AL.

where E is just the energy eigenvalue of (3.31). Clearly an equally good basis of eigenstatesof (3.25) would be (3.28) with 10,P’) replaced by the states IP’*/4n, +). We therefore define the even/odd zero-mode parity sectors R(i)/R(P’ of the free field state space to be those generated by states of the form (3.28) with 10,P’) replaced by 1P/*/4x, k). Obviously the full state space is Z = R(+’ +X’-‘. As we shall see, it is the subspace of states R(-) which is to be identified with the state spacefor the Liouville theory. In Section II we considered a weak coupling expansion for matrix elements between states with energies of order g *. The analogous expansion for the operators of this section would be for matrix elements between the states 10,P’) with P’ of order g. To lowest order in g with k” = P/‘/g, k’ = P’/g held fixed, the matrix element of :egg: is (O,P”l:egm:IO,P’)

=-&

ttan;g

1”’ ttan+

i”

@P”l cos;gQ IOyPf)[l +o(g6)1

=&

ttanhk+

)‘;’

i’;’

cash .i:.2k”

ttan;ik+

” +“(g6)‘~13~32~

From (3.32) it is easy to work out the matrix elements between states IE, *) with E = g2k2/4n, k > 0: (E”, + I :egm : (E’, +) = 2:rn [tanhk,i’;;i

X

(E”, - 1:eg*: IE’, -) = --&

nk ‘I nk’ cash cash2 2 cash nk” + cash nk’

271rn[tanhki+f2

X

,,,,x,:

1

I”*

(1 + OWN

(3.33)

~tan~~~‘~2

izk” nk’ sinh sinh 2 2 cash nk” + cash nk’

+O(d)). 1(1 (3.34)

(Of course (E”, + ( :egg: IE’, - ) = 0 by parity conservation.) Comparison of these lowest order results with Eq. (2.22), with a = 1 reveals that the Liouville energy eigenstatesllE> should be identified with the free field eigenstatesIE, -), and not with IE, +>.

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

Before turning to corrections to these formulas, results for 4 f 4’ and :ezg*:. These are (vq(~

ANALYSIS

169

we continue with the lowest order

* fqlO,P’)

?rP” 2g + tanh z

x

(

tanh--

nP ” 2&T

X 4g sinh

[

+ tanh 7cp’ 2g ii

1 I + e-Q

103P’Xl

+ OWN

(1 + O(g6N

fgP”-P’)TiE]

=&tran:k

X

(0. P” 1

i’:‘(tan~~]“2~tanh~+tanh~]

1

(3.35)

(1 + W)). I

Note that Eq. (3.35) implies that (0, P” I 4’ IO, P’) = 0 to this order. The matrix elements of 4 i 4’ between the states /E, k) are easily shown to be

(E”, + I(4 f $‘)IE’, +>

=i (,,hxl+ i’;;(tan;&)“2 nk” 7sk’ 1 cash 7zk”/2 cash cash - 2 2 2 cash nk’/2 X cash nk” - cash nk’

1 cash nk’/2 2 cash nk”/2 (1 + OWN

(3.36)

170

BRAATEN

ET AL.

Note that (E”, +I 4 IE’, -) = 0 by parity conservation but one must show (E”, + 14’ IE’ -) = 0 to this order by direct calculation. It is now straightforward, using (2.22) to verify that (3.37) agrees with the LQM matrix element

Finally, the lowest order matrix element of :e*@: is given by (cf. Eq. (3.19)) (0, P” I :e2gm: IO, P’) = 167r2m2 g2

x (0, P”I

= 128n*m* g3 X

l l IO,P’)(l cash gQ tanh x cash gQ 2 112 I/* k’ [ t ;.-‘i

nk I’ tanh T+

[ tanh F

+ O(g4))

1

k112 _ k’* tanh$ sinh+(k”-k’)

+

k”‘+k’*+2 nk, nk ” cash cash 2 2

1

(3.38)

(1 + O(g4)).

And between parity eigenstates these become ((E”, +I :e2gmIE’, -) = 0 by parity): (E”,

+ I :e2gm:1E’, +) = 16n2m2 g3 itanhk$i’;‘itan:;Fi; nk” 1 cash nk”/2 nk’ cash 2 cash 2 - 2 cash zk’/2 cash nk” - cash ak’ +

kr’* + k’* + 2 nk” 4 cash 2 cash F

1 cash nk’/2 2 cash nk”/2

(1 + O(g4N 1

(3.39)

QUANTUM

LIOIJVILLE

FIELD

THEORY

WEAK-COUPLING

I71

ANALYSIS

280..I E’ , _ )

(E”,--/:e

7tk ” rk’ sinh ~ sinh 2 2 X cash nk” - cash nk’

+Wg4N* 1(1

(3.40)

And we again see that only the Z(-) matrix element, Eq. (3.40), agrees with Eq. (2.22), with a = 2. These lowest order calculations make clear the necessity of restricting the free field state space to GY’-’ in order to identify the explicit operator solution of Ref. [ 5 ] with the quantum Liouville field theory. Not only do the matrix elements between Z(+’ states disagree with the weak-coupling analysis of Section II, but also the Liouville field equation fails on Z(+) as can be seen by comp’aring Eqs. (3.36) and (3.39). Indeed :

= ($$)

(,“,

+ i
+) - F

=-i&qtanhk+ )‘;1( ,,,:, y2 X

(E”, + / :e2ga: p,

+>

(3.41)

k”‘,;,k’2 + ik, (1 + O(g4)). cash cash --22

Note that the k” and k’ dependencefactorizes in the term which violates the Liouville equation: since the term is even under k” + -k” (or k’ + -k’), the RHS does indeed vanish for XC-’ matrix elements. We shall discuss what goes wrong with the derivation of the Liouville equation on Z(+) in Section IV. We turn next to a discussion of the corrections to the above lowest order matrix elements. Our plan is first to quote the results of our calculations and then to give a sketch of the crucial stepsin their derivation. The matrix elementsof :eg*: and 4 f 4’ have been calculated up to and including terms of order g” relative to the lowest order :

172

BRAATEN

ET AL.

(0, P” / :egm:IO, P’) = (0, P” 1N, egmIO,P’), 5 [,(,“*

- 2(k”*

I

1 +[

- k’*)*

+

(,“*

+ k’*) + 61 + O(g’*)

[-(k”Z

4-Q’ kr2)

_

- k’*)*

6&r/4

+

+ 2(/Y* + k”2) + 31 _ 4k”*k’*

k/4)

(3.42)

(

and

(O,P”I(j f $‘)lO,P’) = (0, P”I(qd f @‘)lO, P’),, 11 - c3 ($1’

(k”* -k’*)*

(k”’

- k’*)*

(k”* + k’* - 2) + O(g’*)

i

where the matrix elements with subscript zero are just the lowest order results given in Eqs. (3.32) and (3.35). Note that these lowest order matrix elements are multiplied by polynomials in k’* and k”*. Thus matrix elements between the states IE, k) are obtained by simply replacing the lowest order matrix elements by their definite parity counterparts (see Eqs. (3.33), (3.34), (3.36), and (3.37)). Comparison with the weakcoupling results of Section II confirms that the two calculations agree on R’-‘. As we shall discuss more fully later, the presence of the operator P/tanh(nP/2g) between pt and P in Eq. (3.19) makes the calculation of matrix elements of :eZRm: more difficult than those of :eg*: and 4 + 4’. We have only completed the calculation for x*~@: up to and including terms of relative order g6: (0, P” I :ezg*: )0, P’) = (0, P’l I :e2g’:J0,P’),[1-~3

+ 128L*

($)3(kfr2-kf2)2]

(tanhkL+)‘;;i

,,,:,

i’-

(3.44) -

cash -

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSIS

173

Notice that unlike the previous two cases, this result is not simply the lowest order result multiplied by a polynomial in k2; also, the extra piece is a correction of order g4 (as opposed to g”) relative to the lowest order. Since this extra piece is even under k” --) -k” (or k’ --) -k’), it will not, of course, contribute in G?‘-‘. Therefore Eq. (3.44) is consistent with our conclusion that to the order calculated, Nmeg@, N,eZg”, and d f 4’ agree with the weak-coupling results of Section II provided the free field state space is restricted to Z(-). For the restriction to Z(--) to be consistent, the operators 4 f $‘, :egm:, and :ezgO: must not communicate between R(+) and Xc-‘. As discussed in Ref. [S 1, any matrix element of :e**:, or :ezg*:, 4 + #‘, between states of the form (3.28) with (0, P) replaced by IP’*/4?t, i->, may be algebraically reduced to matrix elements between states of the form IE, k). Furthermore, if the operator v = (1/2n) 1:” da d’(a) is identically zero, none of the operators :eg*:, :e”*:, $ f d’ communicate between Z’+’ and X(-j. Thus if v = 0, we may consistently restrict our state space to &P-j. It is precisely for states in Z(-) that our weak-coupling calculations of :eRO:, :e2gm:, and 4 f 4’ agree with the weak-coupling analysis of Section II. Note that Eq. (3.43) confirms the vanishing of v to the order calculated, i.e., up to and including terms of order g”. We do not wish to present complete detailed derivations of the results summarized in Eqs. (3.42), (3.43) and (3.44). However, since the weak-coupling analysis involves some unfamiliar features, and since some of the steps are not completely straightforward, a brief sketch of the derivations is perhaps worthwhile. Consider first the operator :eg@:. Referring back to Eqs. (3.18), (3.20), and (3.6), we see that the operator V-’ is given by (take u = 0 for simplicity)

Next, it is convenient to do a Fourier analysis of the part of the integrand that contains nonzero modes:

where 2x da

Ek’=&

eg(~(u)im(~))-eg(~(u)*~(u))+e-ihu

(3.47)

0

Notice that E$ = 1 + O( g’) and E: = O(g) for k # 0, and that Et depends on the A,+, only and E; on the B,., only.

174

BRAATEN

Inserting

ET

AL.

(3.46) into (3.45), the integral can then be done with the result

V(u)~‘=~N{(E,+egQ+E;e~ga)Z(P)}

+

+ E;epga) (3.48)

where I-(1-&)

1’2 Ir(l+$ifJ

Z(P) =

(3.49) i

T(l+&)

1

lp-(1-$-i$)i.

Note that if P = O(g), Z(P) = 1 + 0( g”) and V(a)-’ = cash gQ + 0( g’). Thus, we can invert V-’ to obtain V in a weak-coupling expansion (VP i = cash gQ - A):

v=

(3.50)

Note that the terms in (3.48) involving Et for k # 0 have an explicit factor of order gz so these terms are of order g3; thus A is at least of order g*. For our weak-coupling calculation we need the matrix element of (3.50) between states ) 0, P): (0, P”] vjo, P’) = c (0,P”J n=o

cos;gQ

(A cos;gQ)n

IQP~)

(3.51)

with k” = P”/g and k’ = P’/g fixed. From Eq. (3.32) we see that the matrix element (0, P, I l/cash gQ IO, P, > is exponentially suppressed for IP, - P, I B g, so it is safe to take the P’s occurring in the A’s to be of order g. Thus to calculate to a given order, one truncates the sum in (3.50) at the appropriate n and expands A as a power series in g and P truncated at the appropriate order. One then has a finite number of matrix elements to evaluate involving the operators Q, P, and Et. The operators Et can be removed by the standard procedure of moving positive indexed A’s and B’s to the right and negative indexed A’s and B’s to the left using the algebra (3.2) and the fact that A, IO, P) = B, (0, P) = 0 for k > 0. This part of the calculation is completely straightforward, and it suffices to say that it leads to sums over reciprocals of powers of integers which can in turn be expressed in terms of the Riemann zeta function (cf. Appendix C).

QUANTUM

LlOUVILLE

FIELD

After removing the operators elements of the form

THEORY

WEAK-COUPLING

17.5

ANALYSIS

E,* from the matrix element, one is left with matrix

(0, P”I,,,ltgQ fi (WWY e’lrgQ) cosi gQ jI0,P’> I

1

to evaluate, where q, = f 1 and G,(P) is a polynomial in P and g which is at least of order g*. We now show that a matrix element of this form can always be reduced to (0, P" 1l/cash gQ 10, P') multiplied by a polynomial in P", P', and g. To see this, consider the operator 1 cash gQ

N(Pke*RQ)

= ,,,;

gQ (Wk

1 cash gQ cash gQ) f

N(Pk sinh gQ))

I cash gQ ’

Now

N(Pk cash gQ) = f {Pk, cash gQ} + N((Polynomia1 of order
N(Pk sinh gQ) = $ [Poly of order k + 1, cash gQ] where {A, B} = AB + BA and [A, B] = AB - BA. Thus we have 1 cash gQ

N(Pke*gQ)

1 = Poly of order cash gQ !

k,

1 cash gQ

Polyoforderkt

(3.53)

i

1,

1 coshgQ

1 ’

Using (3.53) in (3.52) we can express (3.52) in terms of matrix elements of the same form with only n - 1 factors of polynomials in P and g multiplied by l/cash gQ. Continuing this reduction procedure, we eventually arrive at the desired result:

(3.54)

(O,P"l ViO,P') = (0,P"l

cos;gQ

I",")il

+ (Power series in P", P', and g)).

It should now be clear how to obtain the result (3.42). In practice the calculation can of course be made more efficient by following these steps in a judicious order. The matrix elements (0, P"I qdf 4' IO,P') can be computed in a parallel fashion. First the operator W,(u) is expanded perturbatively around l/( 1 t erzgQ). Then the nonzero-mode operators A, and B, are commuted out to the left or right leaving a matrix element of zero-mode operators only. This is finally reduced to

176

BRAATEN

(0, P” I l/( 1 + e**gQ)I 0, P’ ) multiplied of identities of the form 1 Pk 1 + 1 + efZgQ

ET AL.

by a polynomial

;TLcQ

1 1 + e’2gQ 1+

1

Poly of order k + 1, 1 +

1 1 + er2gQ

N(pke*gQ)

ei2gQ

in P’ and P” by making use

Polyoforderk+

&Q

1

(3.55)

1,

The matrix element (0, P” 1:ezgo : 10, P’) presents a new problem because of the factor P/tanh(rrP/2g) between fit and P in Eq. (3.19). After expanding ft and p perturbatively, the nonzero-modes can be removed as usual and then, using operator identities like Eq. (3..53), the matrix element can be reduced to polynomials in P’ and P” multiplying matrix elements of the form (0,P’I

l p” cash gQ tanh(lrP/2g)

l (0,P”). cash gQ

We next show that this can be reduced to a matrix element of the form (0, P”l =

G(P)tanh

-$-,

tanh gQ] IO, P’)

71P‘I G(P”) tanh - G(P’) tanh z 2g

il% i

sinh i

(3.56)

(P’ - P”)

where G(P) is a polynomial of order n + 1. The operator in the above matrix element can be manipulated as follows: [G(P) tanh g,

tanh gQ ]

1 = coshgQ

cash gQG(P) tanh ?!- sinh gQ 2g 1 - sinh gQG(P) tanh ?P- cash gQ ~ cash gQ 1 2g 1

1 = coshgQ

y i

1 = coshgQ

G(P - ig) tanh 6 (P + is)

P”

1

tanh g cash 2g

gQ.

--

1 2

G(P + ig) tanh c (P - i&)

(3.57) 1 cash gQ

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

177

ANALYSIS

For the last equality to hold, G(P) must satisfy G(-ig)

= G(+ig) = 0 (3.58)

*G(P - ig) - iG(P + ig) = P”.

The first equation enables us to ignore the i& prescriptions on the operators l/tanh(nP/2g) in Eq. (3.57). The solution for G(P) is most conveniently written down in the following form: I . exp-cosgx 1 (eXP- 1) cash gQ tanh E cash gQ = r [( sin gx 2g

--g

P

tanhE,

2g

tanhgQ

I

. (3.59)

Equating powers of x yields the desired identities. Using these identities, one can eventually reduce the matrix element (0, P”I v+

to a linear combination

p

tanh(rcP/2g)

BlO, P’)

of

(0, P” 1tanh z

tanh gQ /0, P’ ) and (0, P” 1tanh gQ tanh 5

/ 0, P’ ),

the coefficients being power series in P”, P’, and g. Since these two matrix elements are independent, (0, P” ( :e 2g**. 10 , P’) will not be given simply by the lowest order expression multiplied by a power series in P”, P’, and g. However, in the odd parity sector, one linear combination of the two matrix elements vanishes, so (E”, -I XZ~~*:IE”, -) d oes h ave a simple factorized form. To illustrate the weak-coupling calculations, let us carry out in detail the calculation of (0, P” ( V (0, P’) through sixth order. One might think it necessary to keep terms up to n = 3 in (3.50), but in fact n = 2 suffices since the O(g2) term in A annihilates a zero-mode state standing next to it. Thus the “outside” d’s are each of order g3, so the nth term in (3.45) is of order ( g2y+ ‘. The n = 1 term is (0, P’/

’

cash gQ

A

cos: gQ lo’ “)

= (0, P” I ,,,h’ gQ Wcosh gQ( - W>>)

= (03p”l,,,~ + WO)

,,,;

gQ IO, P’)

gQ N(coshgQ(-2r,~~(~)2+(~)2] (3.60)

178

BRAATEN

ET AL.

For the n = 2 term, each A contributes of order g3, so we may take the lowest order form for d (the O(g*) term vanishes in the matrix element):

and for Et we may take its lowest order form -E: = -The nonzero-modes

ig kfi

A

-k

ig kGBk

may then be im mediately rem01 Jed, yielding

(O,P”Icashl gQ Acos;gQ A cos;gQ lO,P’) =g

(j!,

$)x

X cos;gQ + (Q, P +

cowl

cos:gQ

+-“’

A7 (e-8, ($+g)) -Q,

(g-g))

cos;gQ

IO,PJ) (3.61)

g>

To reduce the matrix element in Eq. (3.61), note first that N (emgQ (g--g))

= (g-g)

N (,-gQ

= coshgQ

Substituting (O,P”l

($+g))

coshgQ+N($sinhgQ) ($+g)

(3.62)

-N(gsinhgQ).

(3.63)

(3.62) for the left factor in (3.61) gives for that matrix element coslgQ + (0, P”( X

($f)

N (CgQ ($+$$))

l N ZsinhgQ ;* cash gQ (

)

cosigQ

10,Pf)

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

= (0, P”I ,,,L gQ [ (g-g) +N

(EsinhgQ)

179

ANALYSIS

eegQ g

(&+g)]

cos’gQ

IO,P’)

+ (0,P”l

1 N g. sinhgQ cash gQ ( 1 1 -igP N 2n sinh gQ X cash gQ cosi gQ lo’ “)

where we used (3.63) in the last step. Inserting the term with (Q, P) + (-Q, -P) yield (0,P”l

(3.64)

(3.64) in (3.61) and combining

with

’ A ’ A ’ 10,P’) cash gQ cash gQ cash gQ

=S,$12 (&-)‘(O~p”l cosigQ WsinhgQ) X

1 cash gQ

+ (0, P”l

N(P sinh gQ)

co,: gQ lo’ “)

1 N(coshgQ[Z($)*++&)*]) cash gQ (3.65)

Now, use the commutator lcash gQ

identity

,,,L gQ NV’ sinh,,,L gQ NV sinh II=-8g2 gQ)

+ 4g2

1

sQ> cash gQ

1 1 N(P’ cash gQ) cash gQ cash gQ

to reduce (3.65) to

co,p”i cash’ gQ A ,,,; gQ A ,,,; =i,g

1-f

+ (0, P”I X

cos;gQ

($)‘(Pff2-P’*)*

gQ

lo,

p'>

(0,P”l

cos;gQ

IO,P’)

1 NjcoshgQ[3(%)*+4/$)*]j cash gQ ‘“‘P’)!’

(3.66)

180

BRAATENETAL..

Finally, combine (3.60) and (3.66) in (3.50) to obtain (0,P”l VlO,P’)=(O,P”J

coslfgQ ]O,PO(l-r,

($&P-k’?)‘)

+oshgQ[ X

cos;gQ

($)z+2($)2]/

(3.67)

I",p')'

The anticommutator identity 1 ’ N 1 ip2 -;) cashgQ I = coshgQ

coshgQ/ cos;gQ

can now be used to reduce the last term in (3.67) to obtain, at last, (O.P”j lqO,P’)=(O,P”I

VjO,P’),

;

1 +&

(9’

x [ - (k”2 - Icy2 + 2(/C’* + P)

+ 31 + O(g8)/

in agreementwith Eq. (3.42). Of course, the absenceof the g8 term is not obvious but can be shown by extending the calculation to higher order. We feel that this sample calculation adequately illustrates the essential features of the weak-coupling analysis of the results of Ref. IS]. The higher order calculations are more involved but require no new techniques or tricks.

IV. CONSISTENCY

OF THE OPERATOR

BACKLUND

TRANSFORMATION

In this section we reexamine the arguments of Ref. [5] which led to the exact operator solution discussed in weak coupling in the previous section. This is necessary becauseRef. [5] did not make clear the need for the restriction of the free field state space to X’--). In the derivation of the Liouville equation in Ref. [5], we noted certain caveats associatedwith manipulations involving the operator 4 f 4’. In retrospect, we seethat the original definition of 4 f 4’ in terms of Y-’ is ambiguous because Y-’ possesses a zero eigenvalue in A?‘+) (at least in the weak-coupling limit). The new definition of 4 f 4’ in this paper (see Eq. (3.21)) does not suffer from this problem. When we repeat the manipulations of Ref. [5] using this new definition, we shall seethat they are valid in Z’-’ but not in X(+), in agreement with the weak-coupling results of Section III. Our general conclusion is that the manipulations used in Ref. [S ] go through intact on X ( - ‘, but require modification on XC+‘, and the quantum

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSIS

181

Backlund transformation is consistent if and only if the free lield state space is restricted to SF-‘. The simplest situation in which a problem associated with the operator 4 f 4’ arises is the Liouville equation itself:

(4-l) or, algebraically,

(4.2) We shall find that Eq. (4.2) is satisfied by (3.21) and (3.19) only on X’-‘. We can rewrite the desired commutators of the remaining elements of the Virasoro algebra with (d * 4’ in the form

[L:&p]=e*‘k”

{[Lo’, fj f 4’1 + k(4 f 4’) + ik2CJ

[L:, 4 * 4’1 = eriko[Li,

d * qi’]

(4.3a) (4.3b) (4.3c)

wo’, 4 f 4’1 = [L,r, 4 f 4’1 T [Y”, 4 f 4’1.

We find these equations to be valid as written on the full free field state space 3. It is only the requirement that the first term on the right side of Eq. (4.3~) be given by (4.2) that demands the restriction to A?-‘. To understand these conclusions, let us examine the derivation of Eq. (4.2), in some detail. Referring to Eqs. (3.21), we see that tanh $-,

X W,[L,i,

WI’]

Wieg(+**)+

eg($*@‘)

182

BRAATEN

ET

AL.

Now from Eqs. (3.22) and (3.24), the commutator

of WI’

~e8ww) da’ e &?Pd(o’-0) [L T 03

X

The integral can be simplified integrating by parts.

by writing

becomes

WJU’))

the commutator

le *gQ *

(4.5)

as a derivative

and then

do’ egPd(o’-o) g

egPd(u’-u)

2n _ 2 sinh q

a(al _ 0)

do’

Neg($(O’) FOn(o’))

eH’db--o)~eg($(o’)

TOnto’))

The first term cancels the {P, WI’} term in Eq. (4.5). After multiplying and right by W,, Eq. (4.5) then becomes W

f

[LT

0’

W-11 It

W

*

=

+irZg

W

2rL X

eTgQeg(BWn)m

(4.6)

*

P2 +g2 sinh2 gP/2 + sin2 g2/2

l/2

sinh $

Now consider the inverse of the last three factors. (e

go**“)+ e’gWw =

,>-’

wIle’gQe-g(WO~)+

1 f(u

on the left

_

u,)2

F+(P)e

+ ergQFp (P) e gPd(o’--o)Neg(B(o’)~FB,(o’))

gPd(u’-o)Neg(B(u’)i$.(o’))eigQ

eg(jr TO”)+e *gQ

W,

QUANTUM

LIOUVILLE

FIELD

THEORY

where we have used Eqs. (3.22)-(3.24). furthermore be written 2n

Ce-

I0 f(o’

g($fi,)+

+

WEAK-COUPLING

Using

ANALYSIS

Eqs. (3.20)

and (3.13),

183

this can

da’ -1’

efih’(o’-d/2~{~(p)

egPd(o’-o)eg(~(u’)fo(o’))

P F ig sinh( g/2)(P T ig)

= 2[e-m7’ r&(o))+P(u)- ’ P sinhgP/2

112

sinh gP/2 P f ig P sinh( g/2)(P f ig)

SW g/W F 8 P f ig

112

“2 peg(+ ?@,I+ - ’ 1 .

We now replace the last three operators on the RHS of Eq. (4.6) with this expression and the first three operators with its hermitian conjugate, and obtain (4.7) Inserting

(4.7) into (4.4) then yields the result

tanh x I

2g ’

e*g~-ptp~e2g6+

(4.8)

184

BRAATEN

According

to the Liouville

ET AL.

equation (4.2), this should equal 112

2mzi p;e2g*:=ig 87r2

g

P

x e2gB-pt

112

f++

(4.9)

tanh z 2g Comparing

(4.8) with (4.9), we see that (4.2) will be satisfied if and only if tanhg

P 2g ’

P*

(4.10)

tanh s

2g

However, (4.10) is false as an operator statement, as can be seen in weak coupling by replacing p by l/cash gQ. First note that one can move tanh nP/2g through l/cash gQ as follows: (P” 1tanh $

,,,~ gQ P 1P’ )

= PI

l

cash gQ

p

7cp,

IP’)-

tanh z 2g

sinh -

(4.11)

2g

where we have used the explicit form for the matrix element (P” / l/cash gQ 1P’). We may also write Eq. (4.11) as an operator identity: 1 tanh E 2g coshgQ

P=

1 cash gQ

P

* 2ig

tanh E 2g

1 W) cash gQ

e+gQ

cash gQ

P (4.12)

where either + or - may be taken. Using (4.12) and its adjoint we may write the LHS of (4.10) in lowest order as 1 1 1 P 1 tanh x P i =2 2g ’ cash gQ cash gQ cash gQ tanh np cash gQ 2g eigQ 1 1 f 2ig P WV cash gQ cash gQ cash gQ 1 P cash gQ co:;Q ‘(‘) & 1 P 1 cash gQ tanh g co& gQ ’

T 2ig f2

2g

(4.13)

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

185

ANALYSIS

However, it is also clear that the offending terms in Eq. (4.13) will not contribute to matrix elements between states in &” (-I. For there are no P = 0 states in R’-‘, so that (4.14) if /ly,-)ELZ+‘. W)I w, -) = 0 Clearly l/Gosh gQ /v, -) E R’-’ so if (4.13) is evaluated between states in J?‘-‘, the offending terms disappear by (4.14). This argument may be easily extended to all orders in perturbation theory. Write

p;‘= qc ,to

1

“1 A-t cash gQ 1 cash gQ

(4.15)

where at=coshgQ-

pt-’ (4.16)

E cash gQ - eRQLIL - ePgQLJR where 0, and Q, do not depend on Q. Then 1 it tanh z 2g ’ cash gQ L

,& (egQQL +epgQf4d] 1 L 2g =-

tanhE

(4.17)

But in the same way (4.12) was derived, it can easily be shown that efgQ 1 tanh -!? = *2ig 2g ’ coshgQ J cash gQ w [

(4.18)

cashl gQ *

Thus (4.17) becomes 1 tanh g 2 = 2ig ..,; 2g ’ cash gQ L 1

‘tp)

gQ

,,,;

gQ

ia, - ‘,I’

Using Eqs. (4.15), (4.18), and (4.19) it is now easy to show that tanh z 2g

ptP - ft

P

= 2igF+cS(P) cos;gQ

PL-QRI

Jo

tanh g 2g e*gQ

f 2ig fits(P) = 2igP+d(P) x {Ll, -Q,

cash gQ

P

’

cash gQ f e*gQ(egQRL + e-gQf2,)}

PtP

(4.19)

186 tanh

BRAATEN

c

pt

_

pt

&

P

ET AL.

= 2igT+ts(P) 2 ( :l:rzR)

p+P

(4.20)

tanh g 2g = 2igti+tg(P)(egQQL - CgQflR)

P+P

(4.21)

where in the last line we averaged the two alternative forms. We would like to show that the RHS of (4.20) or (4.21) is zero if a state of the form (w, -] e2g@, with / w, -) E XC-), stands on the left. The argument is very simple. First note that d(P) :egm:Iv,->=0 by (4.14). (Recall that :e g*: does not communicate rewrite (4.22) as &6(P)

(4.22) between XC+)

and Z’-).)

(4.23)

Vega+ / v, -) = 0

and since eg*- contains only raising operators,

(4.23) implies

6(P) Veg*+ 1v/, -) = 0 now multiply

We

(4.24)

(4.24) by egg+(O) to obtain 6(P) eg*(0)+V(o) eg*(0)+ 1w, -) = a(P) f(o) e2g*(o)+ 1W, -) = 0

(4.25)

which was to be proved. Thus the results (4.21), (4.25) and their adjoints can be used to show that (4.10) holds between all states in Z(-) in spite of the fact that it fails on Z(+‘. The extent to which the manipulations of Ref. [5] were formal is now quite transparent. At various points in certain of the derivations, matrix elements of the operator J(P) were inadvertently dropped. It is also clear from the argument surrounding Eqs. (4.22~(4.25) why it is that such matrix elements are zero between states in X(-j. Thus the results of Ref. [5] stand, provided one restricts the free field state space to GF-). The consistency of this restriction depends, of course, on the vanishing on the full state space of the operator v = Ii” da #‘(a). We have not yet provided an all-orders proof of the vanishing of V, but the calculations of Section III confirm its vanishing at least up to and including terms of order g”. We close this section with a survey of those results of Ref. [5] that hold on the full state space 3 and those results which require the restriction to Z(-l. It is very important that the conformal properties of the Liouville operators in the form of Eqs. (4.3) and (a = 1,2) IL;,

:ew3b):]

= ,*i”c([~;,

:eagm(o):] + qag)

:ewm(o):)

(4.26)

where J(g) = f + g2/8z, J(2g) = 1, hold on the full state space Z. A careful examination of the derivations of (4.3) and (4.26) along the lines of Ref. [5] but with

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSIS

187

the new definitions (3.18) and (3.19) reveals that at no point does one encounter the problems with the operator 6(P). (Recall that for conformal invariance the form of the functions F(P), F,(P) is completely irrelevant.) Thus we conclude that Eqs. (4.3) and (4.26) hold on the full state space. We have also confirmed this through order g6 by direct calculation. It is important that this be so because the proof that :ezg*:, 4 do not communicate between 828’+ ) and Z(--) and the proof that 4’ can a priori communicate between 2”) and Z(-’ only through v rely on the conformal properties displayed in Eqs. (4.3) and (4.26). Thus the statement that one can consistently restrict the free field state space to &P(-) if and only if v = 0 remains valid. On the other hand many of the important results in Ref. [5] hold only on Z(-‘. We have already noted that the Liouville equation itself (Eq. (4.2)) is one of these. Another is the equality between the Liouville energy momentum density and the free field energy momentum density: (p;puville

_

q3

v, ->

=

0

(4.27)

for 1w, -) in PC-). The equality fails on aC(+) (as one should expect from the failure of the Liouville equation itself). It is clear that the derivation of this relationship involves a reordering of factors tanh zP/2g which produces matrix elements of 6(P). Finally, we mention that matrix elements of 6(P) are also encountered in the proofs of locality of the operators :e@:, :ezgm:, and d f 4’. They can be ignored on Z’-‘, but they lead to a violation of locality on 3 (+). Their contribution to the matrix element (E”, + I[ :egm’“‘:, :eg”‘P’: ]IE’, +) for E, E’ of order gz has been computed in the weak-coupling approximation, and it is nonvanishing at order gs relative to the matrix element of the anticommutator. V.

CONCLUSIONS

We have developed two independent weak-coupling approaches to the quantum Liouville field theory (LQFT) on a circle. In the first approach, perturbation theory was applied directly to the canonically quantized theory. The unperturbed system consisted of the zero-modes, obeying Liouville quantum mechanics (LQM), and the nonzero-modes, which were free harmonic oscillators. The exact energy eigenstates were expressed as infinite series in terms of the unperturbed eigenstates, using the Lippman-Schwinger formalism. The computation of matrix elements in LQFT was then reduced to the evaluation of complicated matrix elements in LQM. As an application, matrix elements of the operator :eagO: between energy eigenstates )]E) with E = 0( g*) were computed to order g” relative to the lowest order term. The second weak-coupling approach was based on the exact BCT operator solution to the LQFT, in which the operators a,#, :eg@:,and :e2gm:were given as functions of a free massless field v. These expressions contain the inverses of complicated operators. Weak-coupling expansions for the inverses of these operators were developed and used to compute matrix elements of LQFT operators between free field

188

BRAATEN

ET

AL.

energy eigenstates ]E, +) and ]E, -) with E = O(g’). The results were identical to those of the first weak-coupling approach, provided that the Liouville eigenstates l[E) were identified with the odd zero-mode parity eigenstites ]E, -) of the free field. The necessity of restricting the free field state space to either the even or the odd zero-mode parity sectors was recognized in Ref. [5]. The weak-coupling expansion showed that the states must in fact be restricted to the odd sector, X(-). Reexamination of some of the formal manipulations used in the BCT construction revealed that it was indeed valid in Xc--). The failure of the construction in Z(‘) was traced to the presence in certain expressions of the operator 6(P), which does not contribute to matrix elements in Z(-). The fact that the Liouville states must be identified with the odd sector X(-j of the free field implies that this theory has no E = 0 state. This is in accord with some arguments of D’Hoker and Jackiw [9] that the LQFT on the infinite interval has no “normaliiable translationally invariant ground state.” They further suggest the possibility of a translationally noninvariant vacuum. In view of the developments in the present paper, the BCT construction shows that, even on a finite interval with periodic boundary conditions, there is neither a normalizable nor a continuum-normalized zero-energy state. Nevertheless, there is still not the slightest evidence that translational symmetry must be broken. It should be emphasized in this regard that the absence of a zero-energy state does not prevent a consistent physical interpretation of the Liouville theory, since the Liouville energy spectrum is continuous and E = 0 is only a point at the edge of that continuum. There is an important question of principle in the BCT construction which has not been completely resolved. The consistency of the construction requires that the Liouville operators do not communicate between the Z(+) and Z(-’ sectors of the free theory. As shown in Ref. [5], this will be the case if and only if v = J‘i” do #‘(a) vanishes as an operator function of w. We have shown by direct calculation that v= 0 to order g” in the weak-coupling expansion, but we have not succeeded in proving it to all orders. A weak-coupling expansion for the LQFT had previously been developed by Leznov [lo]. His treatment differs from our first approach in that, in the unperturbed system, the zero-modes are governed by the free particle Hamiltonian h =p2/4n instead of by LQM. Thus the unperturbed system has an energy degeneracy exactly twice that of the complete Liouville system. In addition, we feel that Leznov’s approach fails to take properly into account the nonpertubative effects contained in the exact solution to Liouville quantum mechanics. Gervais and Neveu [S] have suggested an alternative to the BCT solution for the LQFT on the circle. In Polyakov’s approach to the quantum relativistic string, the Liouville coupling constant is g* = 12x/(26 - D), where D is the dimension of spacetime [2]. Now the Virasoro algebra of the conformal generators (2.6) contains a term k’ 6 k.-m with a coefficient 7t

=f 1+& g (

2

2

)

+&.

QUANTUM

LIOUVILLE

FIELD THEORY WEAK-COUPLING

189

ANALYSIS

Since this has a singularity at D = 26, Gervais and Neveu claim that this solution cannot be appropriate for the string theory. They suggest an alternative solution for the LQFT in which the coefficient in the conformal algebra is n/g” + l/12, and they give an operator expression for :eAgO:, which has the proper commutation relations with the conformal charges L: , namely, (4.26) with an appropriate value for J(-g). However, we strongly emphasize that the conformal properties alone are not enough to ensure the self-consistency of the solution. Recall that, in the BCT construction, the conformal properties were satisfied in the Z(+’ sector, but that other important properties, including locality and the quantum Liouville equation, failed to hold in this sector. Therefore, further consistency checks must be carried out before the Gervais-Neveu solution can be accepted as a consistent quantization of the Liouville field theory. APPENDIX

A: ZERO-MODE

OPERATOR

ALGEBRA

In this appendix, we use canonical algebra to evaluate LQM matrix elements of zero-mode operators. We have already introduced this method in Section II in the course of evaluating (E” 1exp(agq) (E’ - h + ie)-’ exp(2gq) IE’), Eq. (2.30). Here we generalize to include all the LQM matrix elements needed to obtain the O(g’*) results given for @“/I :exp(ag#): j[E’) in the text, Eq. (2.50). We accomplish this by deriving zero-mode operator identities. The basic commutation relation

[wl = i

(A.11

and the definition of the LQM Hamiltonian h=&p2+Te

immediately

4nm2 zgq g

(A-2)

lead to e*gq --&

hhl

(A.3)

as previously observed in the text. In addition we have

[h, eWq] = -5

{p,eagq}.

(A-4)

This in turn gives eagqp= + {eagq,p} + + [eogq,p]

(A.5) = 5

[h, eagq]+ + iageagq

which could be used to quickly obtain (2.25) directly without recourse to (2.24).

190 Conjugating

BRAATEN

(AS)

ET AL.

also gives 644

pe

This relation immediately

Using (A.4) and (A.l),

allows us to compute

we then obtain

A useful limit of this identity follows

upon taking p -+ -a.

[h, eaRq]e-a~q=u2&if!Cp,

(A-9)

More complicated products of multiple commutators and exponentials of q may be similarly reduced. Consider [h, [h, exp(agq)]] exp(J?gq). We may reduce this by following several routes. For example, use 1 = exp(-agq) exp(agq) and [A, BC] = [A, B] C + B [A, C] to obtain

= [h,((A,eagq] epaRq)] e(at4)aq [h,[h,eagql 1e4tYq - [h, eagq]([h, e-nRq] eugq) esgq. The previous identities may now be employed to reduce this further. The first term on the RHS simplifies immediately using (A.9):

(A. 10)

QUANTUM

LIOUVILLE

FIELD

THEORY

The second term on the RHS of (A.10)

WEAK-COUPLING

ANALYSIS

requires more work.

191

We have

[h, eag4]([h, epagq] eagq) e4gq = [h, eagq] (ai $

+ 2~)

e4gq 2

= (h, eagq]eSgq a(u+2P)5+i$ !

1

(A.12) Next we reduce

(A.13) 2

= -yz c eY”“p- 2iyg eygq 471 Finally, combining (A.lOt(A.13)

and using (A.5) we obtain

[h, [k eagq11e4gq =

4m2a(l

+ a)e(a+4+2kq

-2 a’(a+W) g2 a+P 4Kh e(0+4)gq ---e

2a3

a+P47c

g2

(at5)gq

(A.14)

h.

This result is fairly compact, but not always usable in evaluating LQM matrix elements because of the first term, e(“+412)gq. It is convenient to eliminate the ezgq factor in this term. This may be done through the use of the graded Jacobi identity

[A P,C)I + p, [C,A]lf

{C, [B,A])=O

(A.15)

192

BRAATEN ET AL.

in conjunction with (A.3) and (A.4). We have etY+2b7-

8

ieygq,

[P,

h])

-TGz

(A.16) 2

?x2 2 peygqot~ = 32rc2m

+&

h-Te it

4nm2

2gq

)

g

,e Y8q

i

[h, [h,eygqll.

Next we easily evaluate peygqp= (p + igy) peygq eyRq+ igyp eygq (A. 17) =

4;rreYKq

h -!.!!$e2Kq i

-

igyeYh"Jp

1

g

Substituting this into (A. 16), we obtain 2 4m2(1

+y)e(Y+2)gq=y3

eYKq+ 2y-$

$ i

{h, eyRq]

)

(A.18) + $ [h, [h, eygq]]. We now use this result to replace the first term on the RHS of (A.14). [h, [h. eagq]] eBBq= +

2ap@ - a(a + P)) & he(a+hq (a + P)(l i- a + PI 471

+ 2ab(P + 2a + a(a +P))&++dtRqh (a+P)(l +a+@ 471 a(1 + a)

(A. 19)

QUANTUM LIOLJVILLE FIELD THEORY WEAK-COUPLING

ANALYSIS

193

These last two operator identities are very useful in evaluating the matrix elements encountered in the O(g”) calculation in the text. To see this, we apply (A. 18) to obtain a recursion relation for (A.20)

Z“+, = (E”I eagq E, -L + iE e2c”+‘)EqIE’).

We use (A.18) with y = 2n, and substitute for the right-most operator in this matrix element. Note that the identity in (A. 18) acts to reduce the power of exp( gq) by 2. Also, a single commutator with h in Eq. (A. 18) will eliminate the denominator in (A.20). A second commutator with h can be reduced using (A.8). The net result is 4m2(1 + 2n)Z,+, e(a+2nm

IE’).

a + 2n

(A.21)

Through this recursion relation, we reduce all the Z’s down to expressions containing I, and matrix elements (E” 1exp(ygq)l E’). I,, however, directly evaluates to (A.22)

Z,=--&-$-~~.+2ik’+~(k”~-k’~)~(E”~e~@~E’).

a

This is shown in the text, in (2.26) through (2.30), or by a different method in Appendix B. Also note that naively setting n = 0 in (A.21), dropping the first term on the RHS, gives the correct real part of I,. Thus we can reduce all the matrix elements defined in (A.20) to simple matrix elements of exponentials. Using (2.22) in the text, it is then clear that Z,,+, becomes@“I exp(agq)lE’) times a polynomial in k” and k’ of degree 2(2n + 1). Applying both (A.19) and (A.21), along with (2.22) in the text, it is now straightforward to complete the evaluation of the matrix elements occurring in expressions (2.45), (2.47) (2.48), and (2.49). The results are simple polynomials in k” and k’ multiplying (E”I exp(agq)[E’). Summing these leads to the O(g”) result in (2.50).

APPENDIX

B:A CONTOUR INTEGRAL

In Section II of the text, we used operator methods to evaluate the matrix element e2gq JE’). Here we briefly

discuss the evaluation

of this matrix

(B-1)

element

using contour

194

BRAATEN

integration methods. Inserting ezgq, we obtain

ET

AL.

a complete set of LQM states (cf. (2.28)) to the left of

I, = g fca dk (E”( eagq (Ek)

E,-k,+ia

-0

@kle2”qlE’).

(B-2)

Using Eq. (2.22) and the special case (Ekl eZgqIE’) = b

(kk’ sinh nk sinh nk’)“’ E,-E’

X

(B.3)

sinh + (k + k’) sinh q (k - k’) we find that the integrand in (B.2) is even in k. We therefore extend the integral to the entire real line. Then we may cancel the propagator in (B.2) with the energy difference appearing in the numerator of (B.3), and take into account the i& in (B.2) by choosing a contour G? in the complex k plane. I, then becomes (k”k’

sinh 7tk” sinh zk’)“’ z sinh 7cz

sinh : x

n

(z + k’) sinh 1 (z - k’)

T(a/2 + ri(k” + 5z)/2).

(B.4)

The contour GFruns along the real z axis from -co to +co, with small detours above the pole at z = -k’ and below the pole at z = +k’. We now consider the integral (sinh nz) eeizx

4x1 =J/z sinh t

(z + k’) sinh f

x n T(a/2 + vi@” + &)/2) PI=* c=*

(z - k’) (B-51

where G?’is the contour previously defined. Clearly the determination of I, only requires dl(x)/dx evaluated at x = 0. This we obtain by finding a differential equation for J(x), which we solve for small x by the method of series. A differential equation for J(x) follows by comparing (B.5) with the same

QUANTUM

LI~UVILLE

FIELD THEORY WEAK-COUPLING

ANALYSIS

expression integrated along a shifted contour %?= SY + 2i. This comparison that d dx

a-ik”+-e-.x

a + ik” + x

a - ik” - - d dx

= -4Gi(a

+ i(k”

+ k’))(a

195 shows

d

a + ik” - --$ - i(k”

- kr))(exfik’x

J(x) + e-x-ik’x)

03.6)

where G = jT(a/2 + i(k” + k/)/2) T(a/2 + i(k” - k’)/2)1*.

03.7)

The terms on the RHS of (B.6) are essentially the residua of the integrand in (B.5) at the poles z = +k’ and z = -k’ + 2i. The solution of interest for the differential equation in (B.6) is uniquely determined by the method of series. We immediately obtain

J(x) = - $z

x(a + i(k” + k’))(a - i(k” - k’)) + 0(x3).

Evaluating aYJ/dx at x = 0, we then obtain (a + i(k” + k’))(a - i(k” - k’))(E” 1engqIE’)

03.9)

where G has been combined with other factors in (B.4) to give (E”I exp(agq)lE’). This is precisely the result for I, obtained by operator methods in the text, (2.30). Other LQM matrix elements can also be evaluated using contour integration. For example, we have obtained I, of Appendix A by these methods. Our results again agree with those obtained by manipulating the zero-mode operators.

APPENDIX Calculating vacuum expectation leads to various sums over integers, of inverse powers of integers. In this We tabulate specific results used in First, we define a canonical form S(N,P)=

C: SOME SUMS

values of products of nonzero-mode operators where usually the summand consists of products appendix, we discuss how to evaluate such sums. the weak-coupling analyses of the text. for the sums in question. Let

co x n,,n*,.,., n,.+=, n1n2

1

1 **. ItN (n, + n2 + -.. + ‘ZNY~

(C.1)

196

BRAATEN

ET AL.

This may also be expressed as a multivariable (x1xz

. . . dxp

integral. nfi+n*+ ***

‘.f +n,&-1

XP)

n,n2 me*nN =

’ -dx, . . . -dx, 0 Xl XIJ

I

= (-l)“jb

2

N

-F (XI-5 -.. xp)k/k k=l

... 2ln”(l

-x,x2

1

(C.2)

..- xp).

By performing a “triangular” change of variables, all but one of the integrations in (C.2) may now be carried out. That is, let ym= fi xi.

(C.3)

i=l

The Y integrations are now “nested” for the canonical form.

Beginning with y, and continuing through yp-, , the first P - 1 integrations trivially done through the use of integration by parts. We obtain

are

(C.5)

An additional integration by parts reveals a simple symmetry relation for S(N, P) under the interchange of N and P. P! S(N, P) = N! S(P, N).

(C.6)

A generating function for the canonical sums in (C.l) may easily be obtained by comparing (C.5) with the integral representation of the standard beta function. This comparison gives -?

c~~~~S(N,P)/N!=1-Z-(l-a)~(l-/?)/~(l-a-~)

(C.7)

N$f=l

through which the symmetry relation in (C.6) is immediately apparent. It is now in principle a simple exercise to express any of the canonical sums, S(N, P), in terms of the Riemann zeta function [,, = F k=l

l/k”

(C.8)

QUANTUM

LIOUVILLE

FIELD

THEORY

TABLE S(N,P)

P=

1

P=2

WEAK-COUPLING

197

ANALYSIS

I

P=4

P=3

P=5

through the use of the series expansions for the gamma functions appearing in (C.7). Since ln r( 1 - z) = zyEuler + 2

zY,/n

cc.91

II=2

we have aN/lPS(N, P)/N! N,P=

I

= 1 - exp ( -7 [a” + P” - (a + P)“] i/n).

cc. 10)

n=2

It is straightforward to expand the RHS of this equation and produce Table I. Comparing (C.l) and (C.8), we obviously have S(l,P) = cp+,, as displayed in the table, and in view of (C.6), we also have S(N, 1) ==N![N+I. Other sums may be reduced to the canonical form (C.l) through the use of elementary methods. Consider “nested” sums, which are frequently encountered in the weak-coupling calculations of the text. These may often be reduced by interchanging the order of summation and shifting the summation variable. Thus

=rp+l+ k;;;i= F , k(n+1klP 1 kn(n + k)P--l

=cp+, ++s(2,zJ-

1)

(C.11)

198

BRAATEN

ET AL.

where we used l/k + l/n = (k + n)/kn in the next-to-last step. Such manipulations may also be applied to more general sums. For example,

which is not in canonical may also write

form. Nevertheless,

following

simple

the first step in (C.12)

we

(C*13)

So, we conclude that 1 -+nPkq

1 nqkP

(C.14)

= I;,& + i,+,

or using (C.12)

kj_,

(kP(k:

n)4

+

k’(k!+

n,P)

=

‘qiP

-

‘P+q’

(C.15)

Another very useful elementary identity is

f += f (&L) k=l

Used in conjunction sums. For example,

with

k=l

(C.16)

1 = (k + n)/(k + n), (C.16)

=

,gl

(rP'$+n)'+

allows

us to evaluate many

nP-'k;k+n12)

(C.17) = y j=1 +F

( f k,n=l

P-j(kyny+l) ’

1 k,;;i= 1 kn(k + n)‘-’

*

QUANTUM

LIOUVILLE

FIELD

THEORY

WEAK-COUPLING

ANALYSlS

199

The second term on the RHS of the last equality is a canonical sum, S(2, P - l), while the first term on the RHS can be reduced to Cs using (C. 15). P-2

a,

(C.18)

If we now use (C.18) and compare (C.17) with the last equality in (C.l I), we obtain P-2

s(2Yp-

l)=pCp+l

-

x

(C. 19)

Cp-jlj+l*

.i= 1

This provides a simple closed-form Alternate forms of this result are

expression

for the second

row

of Table I.

P-2

-F l-

n=1

np

f:

k2,

+=+(P+z)i,+,-+

;

[p-j~j+,

(C.20)

CP-j(j+]

(C.21)

J-1

and 03

&

P-2

1 k(k

+

n)

P=+(pF2)CP+*-+

T J-I

as is clear from (C.11). As a final illustration of the use of elementary manipulations canonical forms, we consider

to reduce a sum to

(C.22) First, note that

(C.23) so we have a=2

-v

klJc2.n (kl
200

BRAATEN

ET AL.

The first sum on the RHS is obtained as follows: Fk,,kz,n

-- 1 &n3

1

m k,(k,

- x,.&i=,

+ W(k,

+ k, + 4’

(k,
1 (k,+Mk+k,+n)3

(C.25)

=:

, 1

k,gnzl

k, k,(k,

+’ k, + n)3

k,k,n(k,

: k, + 4’

=:, k,,gd =+3,

2) = 21;, -(*&.

The second sum on the RHS of (C.24) is given by (C.20) (C.26) The last sum on the RHS of (C.24) is obtained as follows:

~l$~l$=‘5+k~,

k’(k:n)’

(C.27)

=i5-

k,gl

kn(k:n)‘++

,,;,

k’n’(k+n)

However, 1 k2n2(k + n) so k;l

k2n2(: + n)

(C.28) =

x2

c3 -

355

QUANTUM LIOUVILLE

FIELD THEORY WEAK-COUPLING

201

ANALYSIS

and therefore (C.29) Combining (C.25), (C.26), and (C.29), we finally Riemann zeta functions.

reduce (C.24) completely

to

The results we have explicitly obtained above are sufficient to carry through the evaluation of nonzero-mode contributions to the matrix elements discussed in Sections II and III of the text. In principle, our methods may also be used to calculate any higher order nonzero-mode terms in the weak-coupling expansion of the Liouville theory.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12.

J. A. T. E. E. E. 0.

LIOLJVILLE, J. Math. Pures Appl. 18 (1853), 71. M. POLYAKOV, Phys. Left. B 103 (1981), 207. L. CURTRIGHT AND C. B. THORN, Phys. Rev. Lett. 48 (1982), 1309. BRAATEN. T. CURTRIGHT, AND C. THORN, Phys. Lett. B 118 (1982), 115. BRAATEN, T. CURTRIGHT, AND C. THORN, Ann. Phys. (N.Y.) 147 (1983), 365. BRAATEN, T. CURTRIGHT, G. GHANDOUR. AND C. THORN, Phys. Rev. Lett. 51 (1983), 19. ALVAREZ, preprint CLNS 82/539; B. M. BARBASHOV AND V. V. NESTERENKO, preprint JINR-E2-82-922; G. BHATTACHARYA AND H. BOHR, preprint ICTP/81/82-13; H. BOHR AND H. B. NIELSEN, preprint ICTP/8 l/82-7; B. DURHUUS, H. B. NIELSEN. P. OLESEN, AND J. L. PETERSEN, Nucl. Phys. B 196 (1982), 498; B. DURHUUS, P. OLESEN, AND J. L. PETERSEN, Nucl. Phys. B 198 (1982), 157, 201, 176 (1982); D. FRIEDAN, preprint EFI-82-50-CHICAGO; L. JOHANSSON, A. KIHLBERG, GOTEBORG-82-34; A. KIHLBERG AND AND R. MARNELIUS, preprint R. MARNELIUS, preprint GOTEBORG-82-2; P. MANSFIELD, preprints DAMTP 82/7, DAMTP 82/23; R.MARNELIUS, Nud. Phys. B 211 (1983), 14; Phys. Lett. B 123 (1983), 237; preprint GOTEBORG-82-34; E. ONOFRI AND M. VIRASORO, Nucl. Phys. B 201 (1982), 159. J. L. GERVAIS AND A. NEVEU, Nucl. Phys. B 199 (1982), 59. 209 (1982), 125: Ph.vs. Letf. B 123 (1983), 86; preprint LPTENS 83/10. E. D’HOKER AND R. JACKIW, Phys. Rev. D 26 (1982), 3517; preprint MIT-CTP-1057; R. JACKIW. preprint MIT-CTP-1049. A. LEZNOV, Serpukhov preprint IHEP-82-169 (1982). V. KAC, in “Group Theoretical Methods in Physics” (W. BeiglbGck, A. BGhm. and E. Takasugi. Eds.), p. 441, Springer-Verlag, New York, 1979. H. BATEMAN. “Tables of Integral Transforms” (A. Erdilyi. Ed.), Vol. II. McGraw-Hill, New York, 1954.