The cluster expansion for wormholes

Volume 223, number 3,4

PHYSICS LETTERS B

15 June 1989

THE CLUSTER EXPANSION FOR WORMHOLES R. B R U S T E I N i and S.P. DE ALWIS 1,2

Theory Group, Departmentof Physics, The Universityof Texas, Austin, TX 78712, USA Received 23 February 1989

We discuss the consistency of Coleman's solution to the cosmological constant problem by taking into account wormhole (end) interactions. The effects of the interactions are estimated by using the cluster expansion for a gas of wormhole ends.

I. Introduction

Wormhole configurations [ 1-3 ] induce drastic effects on low energy physics. In particular it has been argued by Coleman [4 ] that wormholes drive the cosmological constant to zero and determine all other coupling constants of low energy physics as well. Coleman's theory is based on the following postulates: (a) Physics below the Planck scale is determined by a quasi-local gauge invariant action SMo (g, -..), where M8 -1 > M ~J is the short distance cutoff of the theory. (b) Expectation values of observables may be evaluated using the euclidean path integral (EPI) [5], ((O)) = N - 1 f [ dg ] exp ( - SMo) O (only the integration over metrics is shown explicitly ). (c) Wormholes make an appreciable contribution to the EPI at a characteristic scale b > Mbq ~. Below this scale the EPI is dominated by smooth geometries. Postulate (b) has been criticized by several authors [ 6 ]. The source of criticism is the fact that the euclidean action for gravity is unbounded so that the EPI is ill-defined. These problems may perhaps be avoided by a Minkowski-space formulation of Coleman's theory [ 7 ]. Above the scale b the effects of wormholes can be represented by the insertion of local operators [ 2,8 ], and they shift the coupling constants 2i in Smo: 2 , - - , 2 , + f ( a ) . The wormhole parameters a are integrated over with a gaussian measure. The resulting expression for ((O)) is ((O))=~

,f

dap(a)

(O)),+f(,~).

(1)

The expectation value ( O ) is computed on a closed connected manifold. The distribution p ( a ) m a y be written a s p ( a ) = P ( a ) Q ( a ) , where

P( a ) = e x p ( - ½aiAiT~aj)

(2)

and

Q(a)=exp(f[dg] exp(-S~(,~)))~exp[exp(3

G2(al)A(a))l.

(3)

Research supported in part by the Robert A. Welch Foundation and NSF Grant PHY 8605978. 2 Address afier March 1, 1989: School ofNatural Sciences, Institute for Advanced Study, Princeton, NJ08540, USA. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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The wormhole propagator A is a numerical matrix. The values of Newton's coupling constant and the cosmological constant at cosmological scales are used in eq. (3). The last approximate equality depends in particular on postulate (c). The distribution Q(a) overwhelmingly favors A ( a ) = 0, but also drives G - ~ ( a ) to its m a x i m u m on the surface A(a) = 0 [9,10]. For obvious reasons one needs the low energy G-~ ( a ) to be bounded. This requires Gho~( a ) to be bounded [ 10 ]. If indeed G-~ is b o u n d e d then its maximization may determine all other couplings as well [ 10]. (The fact that the other couplings should be bounded functions o f a was first noted in ref. [ 11 ].) It has also been argued that the factor Q ( a ) overwhelms any suppression o f large wormholes coming from the classical action [ 12,13 ]. It then seems that postulate (c) is violated and wormholes at all scales are dense. This would have unacceptable effects on low energy physics. In order to discuss these questions in a systematic fashion, we follow Preskill [ 10 ] and develop a theory of wormhole (end) interactions using the cluster expansion [ 14 ]. These are the interactions which induce nonlinear corrections in a to the action SMo.The so-called wormhole interactions merely change P ( a ) and are less important. For further discussion of this point we refer the reader to ref. [ 10 ]. We consider configurations o f the Misner-Hawking type. We find a power law two-body (repulsive) interaction between the wormhole ends ("instantons"). We introduce a mean field description for the long range effects of this interaction [ 15 ]. (See also ref. [ 16 ]. ) The scalar field that arises in this way is very similar to the axion field considered by some other authors [3,17,18 ]. The remaining interaction is a short range interaction. This interaction induces non-linear shifts in the a's. We analyze the convergence properties o f the cluster expansion for instantons, and conclude that the vanishing of the cosmological constant is consistent with the instanton gas picture. We find, however, that the question of the boundedness o f G - ~( a ) is sensitive to the details of the physics at the fundamental scale. We also discuss the problem of large wormholes.

2. The cluster expansion for wormholes The EPI for gravity is Z = f [ dg ] exp [ - S (g) ]. One way of performing the EPI is to separate the metrics into conformal equivalence classes [ 5 ]. The metrics are written as gu,=£22~u~, where £2 is the conformal factor and is a background metric. The EPI then becomes Z = f [d£2] [d~] exp[ - S ( £ 2 , ~) ]. S(g2, ~) is some quasi-local action, determined by the fundamental theory at the Planck scale. Coleman's assumption is that the integral over £2 is saturated by wormhole configurations. To study the implications of this assumption in detail we need some explicit form for wormhole configurations. The general framework developed in this section is independent of the particular configuration chosen. However, the details studied in the next section may depend on that choice. We choose to study the MisnerHawking configurations [1,2]. Without additional degrees of freedom (such as axions), such wormholes are not saddle points o f the classical action [ 17,19,20 ]. Nevertheless the sum over such configurations has a drastic effect on low energy physics ~. One Misner-Hawking wormhole (for ~u~ = fi,~) o f size 2b is represented by g2= 1 + b2/Ix-xol 2. This conformal factor is singular. However, it is possible to define two coordinate patches x, x' and see that this metric actually describes two asymptotically fiat spaces with a spacetime wormhole connecting them. The wormhole ends (the instantons ) occupy the regions Ix - Xo [ 2 = b 2 and Ix ' - x612 = b 2. It is also possible, in principle, to find wormhole configuration with non-fiat background metrics. In this case, however, the analysis is more complicated. If the other ends o f the wormholes are attached to the same asymptotic region there are some corrections ~ We would like to stress that even with axions, for instance, multi-wormhole configurations are not solutions. They are close to being solutions when they are far apart, but the Coleman mechanism determines the a's to be at a point where wormhole ends are closely packed. 306

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to the above conformal factor [ 1 ]. We do not expect these corrections to change the qualitative conclusions. The action of one instanton [2] is $1 = (3n/G)b 2. To study the interactions of instantons we have to write down a configuration describing two wormholes and compute its action S> Then Sire = $ 2 - 2S~. The two wormhole configuration (for wormholes connecting three distinct asymptotic regions ) is g2= 1 + b 2/Ixl 2+ c 2/Ix-Xo 12. The instantons are at x = 0 and x = Xo. Note that we have implicitly assumed that I x - x o l > b + c. The resulting Sint yields a two-body bare interaction potential. The action for a given N-instanton configuration of size 2b at points x~ is

S N= N~ b2+

E v(r~j), l <~i
(4)

where r~j= Ix~- xj I. The two-body potential V is 3re (bi2j4 1 V(r~j) = -~+ ~

b6

)

(r~_b2) 2

(5)

for r~j> 2b. For ro~<2b we take V(r~j) =oo to simulate the fact that the instantons cannot sit on top of each other. Since the last term in the potential turns out to be unimportant we neglect it in what follows. This potential is essentially a repulsive Coulomb potential (at long distances). The system of instantons is therefore closely analogous to a repulsive Coulomb gas. The "force" that holds this system together is Coleman's factor Q (or) which overwhelmingly favors non-zero density of instantons. We expect the long range potential to be screened in this system. We can rewrite [ 15 ] V~jas V~j= Vs + VL. The long range potential VLis given by 3nb 4 ( 1 VL= ~ ~2

exp(--ro/lD) ~ r~ J"

(6)

The short range potential Vs is given by

Vs(ro) =

3nb 4 e x p ( - r J l o )

G

r~

=~ ,

' r°>b' r~j<~b.

(7)

The split is made such that VL is non-singular at the origin so that we can use the sine-Gordon transformation o n i t [15]. The long range potential may be accounted for by a scalar field 0. It represents the mean-field effects of the interaction. Define the inner product (f, g) = f d4x d4yf(x)v(x, y)g(y). Milnor's theorem ~2 asserts that [for a sufficiently regular v(x,y)] there exists a gaussian measure [d0]v such that e x p [ - ½ ( f , f ) ] = f [dO] vexp [i f d4x O(x)f(x)]. If we choose f ( x ) = Y~N=~ qO4(X__Xi)with q2= (3rr/G)b 4 we can write

exp(-~
O(x~)):=~[dO]v~= exp(iqxi).

(8)

We can now express the EPI for gravity as

Z = f dotP(a)I[dg]

exp(-S[~])£o

~.. ZN,

(9)

where ~2 As quoted in ref. [ 15 ].

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ZN=

PHYSICS LETTERS B

~ [Vs(ro)+VL(ro) ] ×

l-ld4x/exp

: f [d¢]v f l-Id4x, e x p ( - ~

: f [d¢]v

Vs(r,j)X/__I~

15 June 1989

(a.O)(x;)

(6~.O)(x;)exp[iq¢(x;)]

f I~ d 4 x i e x p ( - ~j rs(rij) )Xi~=l (~'O)(xi)

(10)

with c~= exp ( - q 2/b 2 ) a and 0 = O exp (iq¢). Since Vs is a short range potential it makes sense to perform the cluster expansion• Thus we write

exp(-- f Vs(ro))=WN(Xl .... ,XN) t
=U(XI ) ••• bl(XN) 21-E U2(X;' Xj) H ~l(xk) Jt-•••'~-bIN(Xl' •••' XN)'

(11)

with u i (x,) = 1, u2 (x;, xj) = exp [ - Vs (r;j) ] - 1, etc. being the standard cluster expansion functions• Clearly we may write our integrand in an analogous fashion: exp(-~

i
V s ( r ; j ) ) f i (~•O)(Xi)=J~fN(Xl ..... XN)

i= 1

= a(x, ) ... a(XN) + ~, a2(X+,Xj) I] a(Xk) + ... + au(X~ ..... XN), with ak(Xi,,..., Xik) =Uk 1-[ (&'O).

(12)

Define the (modified) cluster coefficients by

,f ~;(x,,...,x;) l-Id4xi.

~=- ~.

(13)

These cluster coefficients are approximately independent of the volume V= f ,f~ d 4x for large l. The partition function ZN is expressed in terms of the cluster coefficients: ~.. ZN= ~ U ~ {

•

m ,

}

(V/~)m;'

(14)

•

where mr is the number of clusters of/instantons and Y~m/=N. Let us introduce a chemical potential term exp(/tN) and write the free energy fgas exp ( - f#) =

exp(pN) ZN= {m;} ~ I~I/--~fl[V~exp(lz')]'m=exp(V~exp(Id)~) " . N].

(15,

The mean instanton density is

V :-

V0# u = o : ~ l/~

1 cc -V,=~,

I)-----~ (/-1 ~al .•. (ffialf

I a = ~ 0 -1- E = ~ a#O V

308

U/(Xl,

.... Xl) Oal (Xl) ..• Oal(Xl) I--[ d4xi

fOax/~d4x+o(~2) •

(16)

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The contribution of the instanton gas to the action is given by the free energy ff a t / l = 0; 8 S ( a ) = Y~~ 8S (~) ( a ) , where 8S (/) ( a ) = - ( 1/l!) Z ........ , ~a,... 6~aJ 1~ d4xi ul( x, ..... xt)O~, ( x~ ) ... Oo,( xt). Since the cluster coefficients (defined with the short range potential Vs) are small for large separations, it makes sense to replace the product of local operators by a single local operator by using the operator product expansion. This expansion is defined by the background field action: Q~, (x,) 0,2 (x2) ~ C b, ~ (x, - x2 ) 0b (Xz). Note that we now have to include in the set of operators also operators constructed out of all powers and derivatives of 0 as well. Hence we may write g S ( l ) = - ~a~...~ b ~ l . . , d~'f~,...~,

,

f

Oh(X) d4x,

(17)

w h e r e f ~...~, = f ]-[ d4z~ ul(zl, .... z/_t)C~,~:(zl b~ b, ~ (zz) ... )Cb~ The wormhole shifted action takes the form

C bbl - 2 al

for/>_- 2 a n d f = 1 for l= 1.

S = S [ g l + ~ S ( a ) = Y~ [, ) d4x [,~b_f b(~ exp(iq0))] Oh(x) T

= f d4xx/~ [ A o - f ° ( ~ e x p ( i q O ) ) ] - I

d4x [ (M~,,)o +f~(c~ exp(iq~))] x/~/~(x) +...,

(18)

X~oo ~ , a l . . . OLal. w i t h f b ( a ) = ~l= i J~cb...... t~ We now consider the linear approximation. We simplify the discussion by assuming that only the operators and R are relevant for wormhole physics. We note in passing for future reference that the Debye length (screening length) is obtained from the action for the field 0. In the linear approximation

S-- f

X/~ d4x { [Ao - f ° ( a

exp (iq0)) ] - [

(MZl)o+ f t ( a

exp (iq0)) ] R ( x ) +...},

(19)

where f o ( a ) = b - 4~ o + o (c~ 2 ) and f t ( a ) -- b - 2t~ 1+ 0 ( ~ 2 ). The shifted cosmological constant at the wormhole scale then is

A~'( a ) =Ab-b-46~° ( exp( iqO ) ),

(20)

and the shifted Mp~ at the wormhole scale is [mbp~(a) ]2= [mbp~(a) ] 2 _ b -zt2, (exp(iq0) ).

(21)

There is no value of 0 for which A = 0 and the Planck mass is bounded. We also note that if exp (iq0) = 0 (for 0~) all the a dependence disappears and the Coleman effect is washed out completely at any scale! Of course there is no particular reason why 0 should be infinite. In fact, since A ( a ) ~ A o - C~o( exp (iq0)), the Coleman factor Q ( a ) is maximized for ( 0 ) =0. So henceforth we take ( 0 ) to be fixed at (q~) =0. The dilute gas approximation is actually inconsistent. We are forced to go beyond this approximation. The value of the density of instantons at which the cosmological constant vanishes is b-4. This means that the instantons are densely packed. Furthermore in the dilute gas approximation G - ~is unbounded.

3. The cosmological constant and GNewton

We would like to gain some qualitative understanding of the effects of instanton interactions on the preferred values of low energy coupling constants. For this purpose we obviously need to discuss the convergence properties of the cluster expansion, and obtain some information on the sum of the series. 309

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It is useful to consider first only the cosmological constant and only one wormhole parameter ~3 ao - corresponding to the operator ~. The effective cosmological constant at the wormhole scale is

A(ao) =Ao - ~ btato =Ao - 8A(ao),

(22)

and the mean instanton density is

(N)

= ~ lbtalo = ao ~

V

8A(ao)

(23)

The cluster coefficients bz are computed with the short range potential Vs = ~ b 4 exp( --r/lD)r 2 Since this is a repulsive potential we have the following bounds [ 21 ]: bt( - 1 )/-1 > 0 , 1

I btl

7~ ~

(24)

l I-2

~ z~-.'

(25)

where B = - b2 > 0. The radius o f convergence ~ o f these series therefore obeys 1/eB<~ ~ <~1/B. Note that the convergence properties o f the series are determined solely by the second coefficient, - b2 = B, and that there is a singularity on the real axis at a o = - ~ . We can estimate B for the potential Vs:

B=-J

[exp(-Vs)-l]

d4r~l 4.

(26)

From eqs. (24) and (25 ) one obtains [ 21 ] bounds on the mean instanton density and 811: ao (N) +Ba-----~ 1 ~< ~ ~
(27)

B - l log( 1 + B a o ) ~< gA~
(28)

These bounds are valid for 0 < ao < B - l , i.e. for values o f ao inside the region of convergence o f the cluster expansion. The m a x i m u m value o f the mean instanton density inside the region o f convergence is o f the order o f B - ~ and the m a x i m u m value of SA inside this region is 6A ~ B - i. Assuming that ( N ) / V ~ B - l we can estimate lo. The Debye screening length has to satisfy a self-consistency condition arising from the identification of the mass o f the mean field ~ and the inverse screening length: l-2 D ~ q 2 T( N ) ~q2154'

(29)

with q2 = (3rt/G) b 4. Therefore l~ ~ ( 3zc/G)b 4 and B ~ [ ( 3n/G )b 2 ] 2b4. Let us check now the consistency of the instanton gas picture. The Coleman distribution function Q ( a ) , eq. (3), is peaked at A ( a ) = A - 8,/1= 0. Here A is the cosmological constant at long scales (without the additive wormhole contribution). We see that as long as 0 < AB < 1 there is a solution with ao positive and the cosmological constant indeed vanishes. Since A is expected to be of the order o f M 4 this b o u n d is expected to be obeyed for M2m/M2w not too large. Otherwise the value of ao for which the cosmological constant vanishes would be an unphysical value. The conclusion is that the wormhole scale cannot be much below the Planck scale if the wormholes are to drive the cosmological constant to zero. ~3 We replace ~ by a in this section for simplicity of notation. 310

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Consider now wormholes at two different scales bs < bL. Using similar arguments to the previous ones, we expect the density of small wormholes to be bounded by 1/bs and the density of large wormholes to be bounded by 1/bE. This bound is a very weak one because it does not take into account the interactions between small and large wormholes and does not assume any a priori preference for either size. But even this weak bound shows that the density of large wormholes is suppressed: nL/ns ,~ (bs/bL) 8, or on their own scale by a factor of [ (3n/G)b~_]-2. For instance the density of l fm wormholes is approximately 10-7° fm-4 in euclidean four-space. Therefore it might seem that the effects of wormholes on physics at the Fermi scale are negligible. However, the mean field q~interacts with the other fields. Thus, for example, it shifts the mass of low energy fermions. The shift in the fermion mass is 5 m f = M w f ( a exp(iq0)). So therefore there are interactions of fermions and the field 0, of the form On(tV, of strength aq n. Recall that q = M p i / M 2 ~ ID whereas a ~ l ~ 4 . Thus for large wormholes we apparently have very strong interactions of this type, unless the density of large wormholes is strongly suppressed. A mechanism for such a suppression of large wormholes (actually to zero density) has been proposed by Preskill [ 10 ] and further developed by Coleman and Lee [ 20 ]. The details of the argument depend on the linear approximation in which both A and M~,~ are linear in the density. However, as we have seen all terms in the cluster expansion contribute and the expressions for A and M2p~bear no simple relation to the density. Thus it is not clear whether the constraint on the sum of instanton densities Z ni< 1 for instantons of different sizes is sufficient to suppress large wormholes when MZl is maximized. For a detailed discussion of the suppression mechanism in a simplified picture ofinstanton interactions see ref. [22]. Is G ( a ) at the wormhole scale bounded away from zero? To answer this question we must include all the wormhole parameters corresponding to the complete set of operators in the theory. The general expression for Mzj ( a ) - 1/ 8nG at the wormhole scale is 1

M 2 , ( a ) = ( M ~ , ) o - ft. X

aa' . . . r~'att"l . dal...al"

a 1. . . a l

The superscript 1 corresponds to the operator R. For example f 1o = O, f ~ = b2, f I ~ ~ - CnRR log Mp~b and f Io...o=bt, eq. (25). Unfortunately, it is hard to analyse in detail the convergence properties of the general cluster expansion for M ~ l ( a ) . We can estimate the magnitude of the various terms appearing in the expansion o f f using bounds similar to those obtained for bb However, the signs of the OPE coefficients vary and this makes the analysis hard. The signs of the OPE coefficients determine if the effective interaction between instantons is repulsive or attractive. We know that if the interaction is attractive we can expect a phase transition. In this case the whole picture of instanton gas breaks down. In general we would expect the maximum of G to occur near the boundary of the domain of convergence of the cluster expansion. What is the possible functional dependence of Mpi on a? ( 1 ) Mpl has a singularity for some real value of the ai's. This means that MpI will be driven to this point in a space, which is of course a phenomenological disaster. (2) MpL has no singularity on the real axis and is therefore a bounded function of the a ' s (as a real function), and (a) The maximum occurs at a point (on the surface A = 0) which is on the boundary of the region of convergence of the cluster expansion. (b) The maximum on the real axis occurs inside the region of convergence. This means that the cluster expansion is a good approximation. In this case we can approximate the a dependence of the various coupling constants by polynomials in the a's. The last alternative is the only situation in which some estimates about values of low energy coupling constants can be obtained. Deciding between these alternatives is, however, a delicate question involving detailed knowledge of the operator product expansion at the fundamental scale. Of course we can always take the opti311

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mistic p o i n t o f view [ n a m e l y ( 2 b ) ], c o m p u t e G - 1 to s o m e finite o r d e r in the ~ ' s , a n d a s s u m e that the o~'s are fixed by the m a x i m u m o f the resulting p o l y n o m i a l i n s i d e the region o f convergence.

4. Conclusion We have systematically e x a m i n e d the effects o f w o r m h o l e ( e n d ) i n t e r a c t i o n s o n two c o u p l i n g constants, the cosmological c o n s t a n t a n d GNewto,. T h e c o n c l u s i o n is that the v a n i s h i n g o f the cosmological c o n s t a n t is consist e n t with the i n s t a n t o n gas p i c t u r e p r o v i d e d that the i n i t i a l cosmological c o n s t a n t is positive a n d less t h a n M 4 . As for GNew,on we f o u n d that, t h o u g h w o r m h o l e i n t e r a c t i o n s are, i n principle, able to m a k e it b o u n d e d (as expected by Preskill), we c a n n o t m a k e a d e f i n i t e s t a t e m e n t a b o u t this m a t t e r w i t h o u t detailed knowledge o f physics at the P l a n c k scale. W i t h regards to the large w o r m h o l e p r o b l e m o u r results i n d i c a t e that, at least for the M i s n e r - H a w k i n g - t y p e c o n f i g u r a t i o n s , the d e n s i t y o f large w o r m h o l e s is suppressed. T h e r e a s o n is that the relev a n t length scale o f the i n s t a n t o n i n t e r a c t i o n is the screening length a n d n o t the actual size o f the i n s t a n t o n s . H o w e v e r , their effects o n low energy physics seem to be large. F o r the c o n s i s t e n c y o f C o l e m a n ' s theory the d e n s i t y o f large w o r m h o l e s has to be strongly suppressed.

Acknowledgement We wish to t h a n k Willy Fischler, Joe P o l c h i n s k i a n d L e n n y S u s s k i n d for m a n y useful discussions. We also acknowledge d i s c u s s i o n s with Lee Brekke, A h a r o n Chasher, J o h n Preskill, Charles R a d i n a n d B r i a n Warr.

References [ 1 ] C.W. Misner, Ann. Phys. 24 ( 1963 ) 102. [ 2 ] S.W. Hawking, Phys. Lett. B 195 ( 1987 ) 337; Phys. Rev. D 37 ( 1988 ) 904. [3] S.B. Giddings and A. Strominger, Nucl. Phys. B 307 (1988) 854; B 306 (1988) 890. [ 4 ] S. Coleman, Nucl. Phys. B 310 ( 1988 ) 643; B 307 ( 1988 ) 867. [ 5 ] G.W. Gibbons, S.W. Hawking and M..I. Perry, Nucl.Phys. B 138 ( 1978 ) 141. [ 6 ] A. Goncharov and A. Linde, Sov. J. Part. Nucl. 17 ( 1986 ) 369; J. Polchinski, Texas preprint UTTG-31-88. [ 7 ] E. Farhi, MIT preprint CTP # 1661 ( 1988 ). [8] I. Klebanov, L. Susskind and T. Banks, Nucl. Phys. B 317 (1989) 665. [9] B. Grinstein and M.B. Wise, Phys. Lett. B 212 (1988) 407. [ 10] J. Preskill, CALTECH preprint CALT-68-1521 ( 1988 ). [ 11 ] S. Weinberg, Texas preprint UTTG-12-88 (1988). [ 12 ] W. Fischler and L. Susskind, Texas preprint UTTG-26-88 ( 1988 ). [ 13 ] V. Kaplunovsky, as quoted in ref. [ 8 ]. [ 14 ] K. Huang, Statistical mechanics (Wiley, New York, 1987 ) sect. 10.1. [15] D. Brydges, Commun. Math. Phys. 58 (1978) 313; D. Brydges and P. Federbush, Commun. Math. Phys. 73 (1980) 197. [ 16] S.J. Rey, Santa Barbara preprint (1988). [ 17 ] K. Lee, Phys. Rev. Lett. 61 (1988) 263. [ 18 ] A.K. Gupta and M.B. Wise, CALTECH preprint CALT-68-1520 ( 1988 ). [ 19] B. Grinstein, Fermilab preprint FERMILAB-PUB-88/210oT (1988). [20] S. Coleman and K. Lee, Harvard preprint HUTP-89/A002 (1989). [ 21 ] D. Ruelle, Statistical mechanics (Benjamin, New York, 1969 ) sect. 4.5. [22] J. Polchinski, Texas preprint, in preparation.

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