QCD1+1 with massless quarks and gauge covariant Sugawara construction

29 December 1994 PHYSICS LETTERS B ELSEVIER

Physics Letters B 341 (1994) 195-204

QCD+n with massless quarks and gauge covariant Sugawara construction Edwin Langmann a.l, Gordon W. Semenoff b'2 a Physics Department, Royal Institute of Technology, I00 44 Stockholm, Sweden b Department of Physics, The University of British Columbia, Vancouver, B.C., V6T IZ1, Canada

Received 29 June 1994; revised manuscript received 30 July 1994 Editor: EV. Landshoff

Abstract We use the Hamiltonian framework to study massless QCD1+~, i.e. Yang-Mills gauge theories with massless Dirac fermions on a cylinder (= (1 +1 ) dimensional spacetime S l x R) and make explicit the full, non-perturbative structure of these quantum field theory models. We consider Ne fermion flavors and gauge group either U(Nc), SU(Nc) or another Lie subgroup of U(Nc). In this approach, anomalies are traced back to kinematical requirements such as positivity of the Hamiltonian, gauge invariance, and the condition that all observables are represented by well-defined operators on a Hilbert space. We also give equal time commutators of the energy-momentum tensor and find a gauge-covariant form of the (affine-) Sugawara construction. This allows us to represent massless QCDI+t as a gauge theory of Kac-Moody currents and prove its equivalence to a gauged Wess-Zumino-Witten model with a dynamical Yang-Mills field.

1. Introduction The path integral formalism provides a very convenient starting point for perturbative calculations in quantum field theory. Alternatively, an algebraic framework in the spirit of the Hamiltonian formalism can be very useful for understanding the non-perturbative structure of interacting quantum field theories, especially quantum gauge theories. In this letter we wish to illustrate this idea in the simple context of ( 1 + 1 ) dimensions. We study massless QCDI+t, i.e. massless Dirac fermions on spacetime S I x R coupled to a Yang-Mills ( Y M ) field [ I - 3 ] , and we demonstrate that in this case all the mathematical tools and results required to complete such an algebraic approach do exist (they have been mostly developed in the context conformal quantum field theory and the representation theory o f the affine Kac-Moody algebras (= current algebras on S 1) and the Virasoro algebra (for references close to the spirit of the present paper see [ 4 - 6 ] , for a recent discussion of the history of the subject we refer to [7] ). Using the latter, we outline of a simple, non-perturbative and rigorous construction of these quantum gauge theory models in terms of operators on a Hilbert space. t Supported in part by the 'Osterreichiscbe Forschungsgemeinschaft'under contract 09/0019. 2 Supported in part by the Natural Sciences and Engineering Research Council of Canada. 0370-2693/94/$07.00 (~) 1994 Elsevier Science B.V. All fights reserved SSDI 0370-2693 (94)01282-2

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E. Langmann, G. W. Semenoff l Physics Letters B 341 (1994) 195-204

We restrict ourselves to the massless case for simplicity, mainly because representations of the affine KacMoody algebras in Fock spaces of fermions with mass m > 0 are more complicated and less understood than for m = 0 [5]. In general, we allow for Np fermion flavors and a gauge group H = U ( N c ) , S U ( N c ) , or another Lie subgroup o f U ( N c ) . Put differently, we use fermions transforming in the fundamental representation o f the group G = U ( N c ) x U ( N F ) and gauge the Lie subgroup H of G. To simplify our notation, we first concentrate on the special case NF = 1 and H = G = U ( N c ), and then discuss the modifications required for the general case in Section 8. We note that a rigorous construction of 2 dimensional QCD has also been given by Klimek and Kondracki [8]. In contrast to our approach, they study the model in 2 Euclidean dimensions and use methods from constructive field theory (see e.g. [9] ) which are close in spirit to the path integral formalism. We believe that our results here provide an example that Hamiltonian methods can provide a powerful alternative to these methods ( o f course, at the moment the latter are much further developed than the former [9] ).

2. Preliminaries To fix our notation and summarize the algebraic structure o f the model, we first recall the canonical formalism for massless QCDI+I on the semiclassical level (= on the unphysical Hilbert space, no filled Dirac sea). Let G = U ( N c ) be the structure group of the YM-fieid and T a the generators o f the Lie algebra g o f G in the fundamental representation of U ( N c ) obeying 3

[T a, Tb] -- i'~abcT~,

(T ~)* = T a,

tr(TaT b) = "rab

(1)

with ('r ab) an invertible matrix ('metric tensor in color space'). Denoting as (tab) the inverse matrix o f ( : b ) , we also introduce Ta ~- rabT b, and for X E g we write X = )(aT a = XaTa where Xa = tr(XTa) = r~bX b etc. Then tr(XY) = XaY ~ for X, Y E g. With A~ =_ A~Ta the YM-field and ~b, ff -- ~p.y0 the fermion fields, the Lagrangian density for massless QCD1+I is

1 E.QCD = ---~-~e2tr (FulFil') - i~',/' D~,~p

(2)

with D ~ p -- (0~ + iA~)~p, e the coupling constant, a~ - ,9/e~x~, x ° = t E R time, x r = x E A ~ [ - L / 2 , L / 2 ] the spatial coordinate, and F~,~ = auA~ -a,,A~, + i [ A u, A~] the YM field strength 4 . More explicitly, ~ = "~,~,, ,/,(*) cr, o J E {1,2} and A , B E {1,2 . . . . N} are spin and color indices, respectively. T a =- T~B, and ~p(*) - "gtr,A' TO be specific, we choose the Dirac matrices as yo = o'r, yl = i0-2, Y5 ---- -Y°Yl = 0"3 (0-i the Pauli matrices as usual). With the usual canonical procedure [10] we get the momenta II~ conjugate to A~, viz. 5 Fl°a(x) ~ 0 and II~(x) = ~ ( F 0 1 ) a ( X ) , and the following canonical (anti-)commutator relations ( C ( A ) C R ) 6 [ I I ~ ( x ) , A b ( y ) ] = it~ 8~bd~(x -- y)

{~,.A ( x ) , tk~,,B ( y ) } = 8c~o~8,tBS(X -- y)

(3)

3 [., .j is the commutator, * and tr(.) the adjoint and the trace of N x N-matrices, respectively; repeated indices are summed over throughout unless stated otherwise. ,t/z, v E {0, 1} are space-time indices; our metric is g~ = diag( 1, - 1), and the anti-symmetric tensor ~ ' with ~01 = 1. 5 The symbol "" means weak equality (constraint). 6 {., .} is the anti-commutator.

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197

etc. as usual. Moreover, the resulting Hamiltonian is 7

f

HQcD=H(°)+

dxtr ( --~Fll(X)2 + Al(x)j(x) ez

- Ao(x)G(x) )

(4)

A

where we introduced the free fermion Hamiltonian H~0) = f dx ~* ( x ) y 5 ( - i a l ) ~ b ( x ) ,

(5)

A

the fermion currents

pa(x) =_ ~b*(x)Ta~b(x) jO(x) - ¢* ( x ) y s T ° O ( x ) and j

=

(6)

ja Ta etc., and the Gauss law operators

G(x) - -DiIIl(x) + p(x),

(7)

where ( DuX)a =~ OuXa -t-i[A~, X] a = O~Xo - Aa~A~X c. The primary constraint II ° _~ 0 implies the secondary constraint [ l I ° ( x ) , Hoct)] = - i G a ( x ) ~- 0 (Gauss' law). One also gets [Ga(x), HQCD] = -iAat~A~(x)GC(x) ~- O, hence there are no tertiary constraints. From (2) we deduce that the vector- and the axial fermion currents, J~ = ~y~Tade' and (J~)o = ~y~?/sTo~b, obey on the semi-classical level the equations of motion D~J ~ = D~J~ = O. We also obtain the symmetric and gauge invariant energy-momentum tensor

I

l

~ ~

i-

®u~"= ~-~tr (~gU F #F,~# - FU'~F~,)) - ~ p [ruO" + 3 ? 0 u] ~p

(8)

which is derived on-shell, i.e. by taking into account the equations of motion DuFU~' + e 2 j " = 0 and yUDu~b = O. We also use light-cone coordinates, i.e.

J±(x) = ½(p(x) q:j(x))

(9)

and similarly for ®. We write L + = :FO ±± = - i ~ * ½ ( 1 q : y s ) O , ¢ M-

/e2 0+-

= L~o + t r ( A , J +)

= ¼tr (Fl~)= e~O -+,

(lOa) (lOb)

where again we used the equation of motion for ~p.

3. Fourier transformation In the following we find it convenient to work in Fourier space. Having in mind the thermodynamic limit L ~ cx~ at the end of our construction, we use the following suggestive notation: Fourier space is A* = { k = ~ n I n C g } , and for functions f on A* we write f A . d k f ( k ) = ~-'~-teA"~ f ( k ) so that # ( x - y) = for k, q E A* ~J A" ~dkeiktx-y)" Then the appropriate 6-function on A* is ~( k - q) = #k.qL/2zr = fA dxe-ix(k-q) 2~ 7 We assume periodic boundary conditions for the YM fields.

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For the Fourier transformed operators we use the following conventions (k E A*), ~(*)(k) = [

dx ~b(.)(x)e¢~-,ikx

(lla)

A

(to simplify our notation, we use periodic boundary conditions for the fermions; it is trivial to modify our equations so as to allow for anti-periodic boundary conditions),

~ A v ( x ) e -ikx, j LTr

(lib)

av(k) = /

A

and in all other cases

2(k) = f

dxX(x)e-i~'

for X= IIv,G,p,j,J",J~, and 0.

(llc)

A

For convenience of the reader, we write down the non-trivial C(A)CR in Fourier space, [fI~(k), , ~ ( q ) ] = ig~,,rat'~(k + q)

{~t#,a(k),~,,B(q) } =

6o.o.,6ts~(k

--

q)

Vk, q E A*.

(12)

4. Filling the Dirac sea

Our approach is in the spirit of the algebraic approach to quantum field theory [ 11 ] where the non-trivial aspects of quantum field theory (as compared to quantum mechanics) arise due to the existence of unitarily inequivalent representations of quantum field algebras [ 12]. The essential physical requirement selecting the appropriate representation is positivity of the HarailtonMn on the physical states. The crucial simplification in (1+1) (and not possible in higher) dimensions is that one can use a quasi-flee representation [5] for the fermion field operators corresponding to "filling up the Dirac sea" associated with the free fermion Hamiltonian H~°), and for the YM operators one can use the naive Schr&linger representation. At this point, we have to take this as an assumption to be checked at the end of the construction. However, this assumption is plausible due to the facts that, (i) quasi-free representations for fermion fields in different external YM field are unitarily equivalent in (1+1) dimensions [13], (ii) the YM field on a cylinder has only a finite number of physical degrees of freedom (namely the eigenvalues of the parallel transporter Pexp (i fA dxAl (x)), see e.g. [3] ), and as all representations of a finite number of quantum degrees of freedom are unitarily equivalent (von Neumann's theorem), the simplest representation for the YM field algebra should do. We therefore construct a representation of the C(A)CR algebra given above on a Hilbert space 7"/which is a tensor product of a YM and a fermion Hilbert space, 7-/= ~l'~ym @ ~'~F, with 7-/yM the usual Hilbert space of functionals of Av(k ^a ) with I]aV(k) = ~ i 0 / O A a ( - k ) , and "h/F the Fermion Fock space with vacuum I'~F such that ½(l+ys)~(k)lqF = ½(1--yS)~'(k)flF=O

Vk > 0

½(1 + y s ) ~ * ( k ) l ~ F = I ( 1 - T s ) ~ ( k ) ~ F = O

Vk
(13)

(by abuse of notation, we do not distinguish the quantities introduced on the semi-classical level in the last Section from the well-defined operators representing them on 7"/). It is well-known that the presence of the Dirac sea requires normal-ordering:...: of the fermion bilinears, hence/~0) = ~A.dq: q~.(q)y5~t(q) : (which is

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199

positive by construction [6] ), and similarly for Ja~(k) and L~(k), where the tilde indicates normal ordering with respect to the free fermion vacuum IIF. This modifies their naive commutator relations following from the CAR (12) as Schwinger terms show up [4,6] (for a mathematical rigorous discussion of this construction of fermion bilinears in the presence of a Dirac sea and how normal ordering leads to Schwinger terms, see [5] ). In our case [4,6], [ J ~ ( k ) , Jbi(q) ] = iAa~J~(k + q) :F k~(k + q)rab

(14a)

and

[L~o(k),Lo~(q)]

(k-q)L~o(k+q):t: ~

k2 -

8(k+q)

(14b)

with the second terms on the r.h.s, of (14a) and (14b) the Kac-Moody and Virasoro cocycles, respectively [ 4 ]. Moreover,

[ L~(k),Ja~(q) ] = -qJai(k + q)

(14c)

with no Schwinger term arising here. Note that these relations are exactly the ones of the semi-direct product of an affine Kac-Moody algebra and the Virasoro algebra playing a prominent role in conformal field theory. We can now write the Gauss' law as t~a(k) = - ( D - - ~ l ) a ( k ) + ~a(k) ~- 0 where (DlII1)a(k) = ik(I~Ii )~(k) - Aabcf^.dq,4bl(k + q)[ICl(--q), SO Eqs. (14a) imply that [Ga(k), J ~ ( q ) ] = iAa~J~ (k + q) ~ k~(k + q)'l'ab. Thus the presence of the Schwinger terms implies that these fermion currents no longer have the classical commutator relations with the Gauss' law generators and therefore do not transform covariantly under gauge transformations. To restore gauge covariance and obtain fermion currents having canonical transformation properties (without Schwinger terms), we note that [t~a(k), (,41)b(q) ] = -k~(k + q)rab + iAabc.4~(k + q), hence the operators

Y±(k) -J±(k) T,~t(k)

(15)

obey the desired relations [~a(k), f~b (q) ] = i'~abCJ'c:1:(k + q).

(16)

Similarly, the naive energy-momentum components L ± (k) = L0~ (k) + J'A"dq tr (AI (k + q) J± ( - q ) ) are not gauge invariant, [ t~~( k ) , / , ~ (q) ] = :Fk,4~(k + q), but obviously there are unique polynomials in ,41 which can be added to make them gauge invariant,

L±(k) = L~o(k) + f J

dqtr(,~t(k+q)J+(-q) q: ½Al(k+q),~l(-q)).

(17)

A*

Recalling that normal ordering is only unique up to finite terms, it is natural to regard the J ~ ( k ) and L±(k) as the currents and energy-momentum components obtained by a gauge covariant normal ordering preserving the transformation properties under gauge transformations. To construct the full energy-momentum tensor - - especially the Hamiltonian - - we also need the operators ^

~l(kl = f A.dq ~--fftr(fll(k+q)fI,(--q) ) .

(15/

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At this point a technical difficulty arises: the operators L+(k) and /l~/(k) do not have a common, dense invariant domain of definition in 7"/ (we recall that the sum of two unbounded Hilbert space operators can be defined directly only if these operators have such a domain). It is, however, possible to define the operators

!L+(k) + e2/~t(k)! - - and therefore the components ~°°(k) and O°l(k) of the energy-momentum tensor - by normal ordering i...i the YM field operators with respect to the YM vacuum IIyM obeying (~iHl(k)+fi,

l(k))f~VM=0

VkEA*.

(16)

Note that firm is just the ground state of the free YM Hamiltonian

,,<0> = ! h.dktr nVM

( e2

~--~fi, (k)fll(-k)

+ Al(k)Al(-k)

)

!.

(17)

Especially, we get the gauge invariant Hamiltonian HQCD = O°°(0) - G(A0), or equivalently

Hoco= ,q~o). - - -,_ v,< Mo>+ ? d

dktr(,~,l(k)~(-k)

do(k)t~(-k)).

(18)

A•

We see that, similarly as in the Schwinger model [ 14], there is an additional 'gluon mass term' resulting from gauge invariant normal ordering.

5. Observable algebra relations For the covariant currents we get the following commutators, [ J'f~(k), f~b (q)] =

iaa{3~(k + q) q: Sab(k, q)

(19)

with the Schwinger term Sab(k,

q) = k~(k + q)rab

--

iAabcACl(k+ q).

(20)

It is natural to regard the latter as the gauge covariantform of the Kac-Moody cocycle, and we can represent it in explicitly covariant form as S~b(k,q) = -[l~l°(k), (DiA-"~)b(q)]. Moreover, we get

[L+(k),L±(q) ] = ( k _ q ) L ± ( k + q ) q: NCbk

_

&k+q)

(21)

and [ L + ( k ) , d~a(q) ] = --qJa~( k + q) -- [ A~'J~ ]a (k + q)"

(22)

Remark: For some readers it might appear that there is a contradiction between the existence of Virasoro and current algebras and the model not being conformally invariant. However, as the Hamiltonian of the model is not equal to - L + (0) + L - ( 0 ) , these algebras do not correspond to symmetries of the model which could be used to answer dynamical questions, but they only provide interesting (non-conserved!) observables. It might be useful to recall an analogous situation in quantum mechanics: the harmonic oscillator operators a and a* obeying [a, a*] -- 1 exist for every QM model, but only for the harmonic oscillator with the Hamiltonian cx a*a their algebraic properties become powerful for answering dynamical questions.

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6. Equations of motion Using (18) and the algebraic relations above it is straightforward to work out the equations of motion for all e2 ^ observables of the model. For example, we obtain [ HQCD, J+ (k)] = : F i ( D - - ~ ) ( k ) + [ ~ ] (k) 4-i~l'I (k). e2

Using 00J + +i [ HQCD, J+ ] = 0, and transforming to position space, we can write this as DoJ + +D1 J+ = 4-~-#II I. Noting that 111 = -I11 = -2~-~eU~Fu~ and jo = j+ + j - = _j5~, jI = _ j + + j - = _ ~ , this can be written as

DvJ v = 0

(23a)

D~J~ =~----~e-U~Fuv.

(23b)

The second of these equations shows a covariant axial anomaly. Evaluating 00II I (k) + i [ HQCD, I11 ] = 0 we obtain DoII 1 - j = 0. Together with the Gauss' law - - which we can write as Dil-ll - p ~ 0 - - this comprises the usual equations of motion of the YM field

D~,F~'~ + e2j ~ ~- 0

(24)

identical to those obtained on the semi-classical level. Using this and Eq. (23b) rewritten as Doj - D 1 p = - £i~ ,~ 1, e2

we obtain D~I/i - D~l-Ii ~ --;-rI1, or equivalently e2

D~DVIII + - - I I i ~ 0

(25)

77

generalizing the Klein-Gordon equation one has in the Abelian case [ 14]. Equations (23b) and (25) have also been obtained by by Sorensen and Thomas [ 15] in a path integral approach.

7. Bosonization The celebrated (affine-) Sugawara construction allows to write the free Virasoro generators Lff in terms of the Kac-Moody currents J+,

L~o (k) = q : ½ /

dq ~,tr(J+(k+q)J+(-q)) ~

(26)

J A

(see e.g. [4]; for k = O,+2¢r/L this was already given in [16]) with normal ordering ~Jff(k)./~(q),~ J~(q)Ja~(k) for k <> q and J'~(k)J~(q) otherwise [4] (note that J+(-k)llF = 3'-(k)f~F = 0 Vk > 0). Combining this with Eqs. (17) and (15), we observe that the terms involving AI can be arranged such that ^

L+(k)=q:½/ J

dq~tr(J+(k +q)J+(-q))~.

(27)

A*

Thus the Virasoro generators L+(k) are obtained simply by replacing the non-covariant currents on the r.h.s. of Eq. (26) by the covariant ones! It is natural to regard this as the gauge covariant version of the Sugawara

construction. Especially, we get the Hamiltonian of the model in the following form

H~D = ½i/ a . d k t r ( ~ (Y+(k)Y+(-k) + Y - ( k ) ? - ( - k ) ) ~ , + ~ f-II(k)fI~(-k) - ~,o(k)~(-k) ) ". (28)

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E. Langmann. G.W. Semenoff/ PhysicsLettersB 341 (1994)195-204

It is now manifestly positive definite on the physical Hilbert space where t~(k) ~ 0 (note that J + ( - k ) = at+ (k) *) thus justifying our choice of representation of the field algebra. It is easy to see that this Hamiltonian is identical with the one obtained from a gauged Wess-Zumino-Witten model with dynamical gauge field s (see e.g. [17]) and a gauge group U ( N c ) equal the flavor group and coupling constant g = e/v/-~ (to make this explicit, one has to rescale v/-~A,,(x) ~ A,,(x)). This equivalence has been known from the path integral approach [ 18,19] (see also [20] ).

8. General case

Our present approach can be immediately generalized to the case with a gauge group H being a Lie subgroup of U ( N c ) and NF fermion flavors. In this case, we have fermions transforming under the fundamental representation of G -- U ( N c ) x U ( N F ) with the gauge group H a subgroup of G. Labeling the generators of the Lie algebra of G such that To for 1 < a < d i m ( H ) span the Lie algebra of H and tr(TQTb) = 0 for _ d i m ( H ) , b > d i m ( H ) , we have AI = z..,a=-~ ~ - - ~ d i m ( H ) A~Ta and similarly for Fu,,, G. With that all equations and a < the discussion from Sections 2-5 essentially remain the same. For the bosonization, it is most convenient to use the G / H coset construction [4], (29) which implies a similar equation for L + = L~. Then obviously L~/n = (L!o)~/n, i.e. the coset Virasoro generators are completely decoupled from the gauge field, and the discussion of gauge covariant normal ordering etc. above applies to L~ only. Especially the Hamiltonian of the model is HQco = Hn + Ho/n where Hc/n = (['I~FO))c/H is completely decoupled from all YM fields, commutes with Hn, and is identical to the one one gets in the free fermion case [4]. The non-trivial interactions of the fermions with the YM field are completely contained in Htt. Thus we can write ~ = 7"[n®7"t6/n with all non-trivial dynamics occurring on ~ n , and 7"16/n provides the superselection sectors of the model (cf. also [ 18] ).

9. Technicalities

To complete the construction of massless QCDI+I and make it mathematically rigorous, one has to establish several technical properties. Firstly (as most of the operators of the model are unbounded), one has to prove that there is a common, dense, invariant domain D C 7"/ for all operators of interest so that the commutator relations given above are well-defined on D. In fact, this can be proven for the Hamiltonian HQCD and all other operators considered above except the energy-momentum components ~11(k) (see also the discussion in Section 4; taking also the latter into account makes things slightly more complicated [24] ). Secondly, one has to prove that all observables of the model, especially the Hamiltonian and the (smeared) Gauss law generators, are represented not only by symmetric but in fact self-adjoint operators on 7"/ [21,22]. Finally, one would like to establish that the thermodynamic limit L ---, cc is well-defined and leads to a relativisticly invariant theory. The proof of these results can be done by using techniques developed in [5,23] in combination with results summarized in [25] and will appear elsewhere [24].

Th~t i~. the ~ g ~ g i ~

has a term - ~

(F~r~)

i. a0di~o. ,o what i~ usu~my referred ~o as g~g~O WZW mod~l I I71.

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203

10. Final comments Recently an interesting reformulation of QCD on spacetime R x • in terms of a gauge invariant, bilocal master field was given and used as a starting point for a systematic semi-classical approximation [26] (see also [27] ). It would be interesting understand this reformulation in our framework (technically this is more complicated due to the presence of the physical YM degrees of freedom on spacetime S I x R). Alternatively one can eliminate the gauge degrees of freedom by 'solving the Gauss' law' [28]. For massless QCDI+I this results in a theory of interacting Kac-Moody currents J+ coupled to a finite number of quantum mechanical degrees of freedom, the latter representing the physical YM degrees of freedom [3]. From a mathematical point of view this gauge fixing procedure is quite delicate, and it would be important to get a deeper understanding, e.g. in the general framework of [29]. Work in this direction is in progress [24]. There are several deep reasons preventing a straightforward extension of our construction to higher dimensions. Most importantly, a YM field there has also an infinite number of physical degrees of freedom, and choosing the appropriate representation of the YM field algebra is a highly non-trivial problem, even for pure YM theory. Moreover, in higher dimensions the physical representations for fermions interacting with different external, static YM-fields are not unitarily equivalent and gauge transformations cannot be implemented by unitary operators in the fermion sector but only by sesquilinear forms [ 30,31 ]. This suggests that the observable algebra of QCD in higher dimensions at fixed, sharp time does not allow for a reasonable Hilbert space representation, hence a standard Hamiltonian formalism might be too narrow a framework for higher dimensional quantum gauge theories. There is, however, a natural generalization of the theory of the affine Kac-Moody algebras to ( 3 + 1 ) dimensions [30,31] which can be expected to provide a first step to a non-perturbative understanding of the fermion sector of QCD3+I.

Acknowledgement One of us (E.L.) would like to thank I. Affleck, A. Carey, A. Hurst, E Lenz, N. Manton, S. Rajeev, M. Salmhofer, E. Seiler, and S. Weber for useful discussions. He would also like to thank A. Carey for an invitation to the University of Adelaide where part of this work was done and the Erwin Schr6dinger International Institute in Vienna for hospitality where this work was finished.

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