Thermal states of the vector meson model in two dimensions

ANNALS

OF PHYSICS

Thermal

102, 71-107 (1976)

States of the Vector

Meson Model

in Two Dimensions*

DANIEL A. DUBIN Faculty of Mathematics, The Open University, Walton Hall, Milton Keynes, MK? 6AA, England Received April 8, 1976

We compute the KMS states for the vector meson model in twodimensional time. We also consider their limit to those of the Thirring model.

space-

1. INTRODUCTION Efforts to analyze quantum field theory in two-dimensional space-time have proved very successful. The most notable achievements are the construction of the 9($), model and the Yukawa model Y, (cf. articles in [l-3]. In the modern approach to statistical mechanics, where there is a field theory model, there is the possibility of computing and analyzing various states on the model algebra of observables, states of particular relevance to statistical mechanics [4-71. Foremost amongst these are the KMS states [4-121, which growing evidence indicates are the thermal equilibrium states [13-201. In an important paper, Hoegh-Krohn [21] has constructed the KMS states for the g(&, theory and found a remarkable symmetry. These Wightman functions for inverse temperature fl are identical with the Schwinger functions for a periodic box of length l/,3 and zero temperature. This paper is of more modest scope. We consider the vector meson model in two dimensions, consisting of a massless spinor field interacting with a massive vector field minimally: the formal interaction Lagrangian density is g&P. This model is exactly solvable: its Wightman functions and the operator solutions to it are known [22-261. But the model is physically trivial as the S-operator is unity. Nonetheless there is good reason to construct the KMS states for it. It is not yet clear as to what the best axioms for a fermion system in the Euclidean f-) Relativistic approach ought to be [27-291. As KMS states deal with complex times in a certain sense, it may well be helpful to have around some states of slightly greater complexity than those of the ideal Fermi gas. * Partially supported by the Overseas Research Travel Fund of the Open University; Institute for Theoretical Physics, The University of Vienna.

71 Copyright All rights

0 1976 by Academic Press, Inc. of reproduction in any form reserved.

and the

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In this paper we compute the KM‘3 states for the vector meson model in the following way. The solution to the model that we use is constructed out of a family of auxiliary free fields of various sorts. If we could compute the KMS states for each of these auxiliary fields, we could then put them together to form states for the interacting fields. We would then have to check that the states so constructed were indeed KMS states for the model. This we do, and everything is favorable. When we come to compute the KMS states for the auxiliary free fields, we do not compute local grand-canonical states and then take the thermodynamical limit [6]. Instead we construct the relativistic free fields on their Fock-Cook spaces (FCspaces hereafter) [3-6, 3&32]. We then modify this construction by replacing the usual one-particle Hamiltonian h by h = h” - p, where p is a chemical potential. This potential is meant to refer to the interacting fields, but it makes sense for the auxiliary fields (mathematical sense, anyway) provided p is restricted to be upper bounded. Eventually we consider p < 0. This p-modzjkation, as we call it, breaks the relativistic covariance of the theory, but that will have to be broken eventually in any event, as the KMS’ condition treats the time and space variables differently. The point of p-modification is that the resulting time translations are associated with the reduced Hamiltonian. The KA4S state will then be a state of prescribed mean density. These two global conditions, KA4S and prescribed mean density, uniquely define these thermal states for TV< 0 [IS, 33-351. As there is no phase transition, each KMS state is a pure phase state in the physical interpretation now usually accepted. The second section considers the KMS states for a real scalar field v of mass m, very well-known material [21,36]. The third section considers the ordered exponential of 1~422,371. Because the KM3 state for 9 is not of FC-type, i.e., ~(&a # 0, we must use the ordering defined through the KMS state [38], and not normal, or Wick, ordering. We then compute the n-point functions for this ordered exponential at inverse temperature /3. In the fourth section we do the same for various other free fields, the auxiliaries which appear in the solution to the model. Section 5 deals with the free spinor field of mass zero, p-modified. Its KMS states are slightly different from those of the preceding boson fields. This difference is the familiar one of using [l - exp(+@)]-l for bosons and [l + exp( +pc)]-’ for fermions. The vector meson model as such is first discussed in Section 6. Following closely the usual solution to the model, as found in [22, 251, we use the material of the first five sections to form the KMS states for the model. In Section 7 we indicate that the limit to the Thirring model [39, 22, 25, 261, known to hold at T = 0, holds for arbitrary p. What we do not do is analyze the representation W*-algebras associated with the KMS states. We also do not compute the Schwinger functions for a periodic box of length l/p and temperature zero.

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THERMAL STATES OF VECTOR MESONS

2. THE REAL SCALAR FIELD

By E+[82(lR)] we shall mean the symmetric Fock-Cook Hilbert space built up from 9rz(R) [40]. Consider the Weyl system over 9,.“(R) viz., pairs of unitary operators on E+[Z2(R)], f - {U,(f), V,(f)} satisfying the integrated form of the CCR:

UF(f>vdg> = VFWU&“) expMf,g)l.

(2.1)

These unitary groups { U,( f), V,(f):f~ 6pl”(R)} are weakly continuous t - U,(q), t ‘- V,(tf) in t at zero, for t E R, so that Stone’s theorem applies and we write

udf> = ew(2-1~2Mf) + a,*(f)l>, v&3 = exp(i2-1’2Mf)- +*(f)l.

(2.2a) (2.2b)

The uF* satisfy the CCR on E+[82(lR)] in the form bF(f)~ %*k)l

c (f, gY Y

(2.3)

other commutators vanishing. The FC-vacuum vector is defined to be Sz, = 1 @ 0 @ 0 @ *’ so that we have the familiar result .

,

@b, WAf, d QF> = exp(--t

llfl12 - t II g 112>,

(2.4)

where W, is given by WAL g) = ud.f)

VAg) expK@)(f, dl.

(2.5)

Let h = (-(d2/dx2) + m2)lL2 - TVbe the one-particle Hamiltonian on P(R), where p < m is the chemical potential. The inclusion of TVis sensible as there is a conserved number operator NF = @,>,@ on FC-space. Obviously, the chemical potential will destroy any Lorentz covariance, but that is a concomitant of finding thermal equilibrium states. Our notation for Fourier transform on 5?2(lR2)is

F(Y) =

(271-l sM F(x) exp(--ix

. y) dx;

(2.6)

we have introduced the Minkowski space scalar product (x0, 3) . ( y”, yl) = 9~0 - x’J+. In this regard it is useful to use the abbreviations6 = (e(k), k) E l%A7 (R2, .) for Minkowski space. Here k E R is the momentum and E the energy function e(k) = (k2 + m2)1/2 - p

(k E W;

(2.7)

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thus (hF)“( p) = l ( p’)i< p) forp E l&fin the domain of h, i.e., such that SWl ( p)” x I.f(~>l”d~ -=c~0. It is by means of h that we proceed to the equal time description of the real scalar “relativistic” field. To everyfe YT(Iw) we consider the unitary operators U(f) = U&ly) (2.8)

V(f) = v~(h+l~“f)

on E+[Zr2(R)]. We consider E+[OEP2(lR)]as the natural complexification of E+[=.Y$(R)], so that the vector J2r E E+[9,.2(R)]. Using Eq. (2.4) it follows that

(2.9) with W defined similarly to W, of Eq. (2.5):

W(f g) = UFW1?f)v&1/2g)expWN.L&I. Equation (2.8) furnishes a representation

(2.10)

of the CCR, since

WI W> = Vd V.0 exp W-1%W=dl = v(g) W> ew[i(f,

(2.11)

dl.

The canonical fields are defined to be the selfadjoint generators of U and Y

u(f) = wWoWl~ v(f) = exphLf)l. These are the “time zero” fields and are related to the annihilation operators as follows. ido

= MW-1/2

To(f) = ~,Kvw”

fl, fl - ~F*Kw)1’2fl. fl + u~*KW-~‘~

(2.12) and creation

(2.13)

The “covariant” field is then defined by means of the canonical fields. We write & for the subset of ,U;(M) whose elements satisfy the “dispersion” condition n’(p”

= i4P1h

P’) = J(Pl).

(2.14)

Then $o(f) = dfi9 r,(f) = v(aoF) defines a densely defined essentially selfadjoint operator-valued distribution: the relativistic real scalar field of mass m, spin 0. Defining the telegraphy wave operator to be 0, = -(a,

+ W

+ @d2,

(2.15)

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this field satisfies the inhomogeneous telegraph equation v([O,, - m2]F) = 0 for all FE 9#~!). Note that if F is such that E(i) = 0, then y(F) $2, = 0. Turning to an algebraic formulation [41], the equal time local rings are defined as follows. By JV we shall mean the set of all open relatively compact sets in R; by (IL C R we shall mean the orthocomplementary set defined by (1 @ LP- = R. Then the product lemma applied to (1 gives [40] E+[92(lR)] = E+[Y2(J)] @ E+[LZ”“(n’-)]. The W*-algebra generated by {U,(f), V,(f): f E 9&l)} is B(E+[92(~)]) @ 1, but that is not the algebra of interest. We defined &‘(A) to be the W*-algebra generated by { U(f, V(f): f E 9&l)}. Whilst the former algebra is a Type I, factor with Type I, commutant, Dell’Antonio has shown that &@I) is a hyperfinite Type III factor isomorphic to its cornmutant &‘(A)‘, and &‘(A) = &(/ll). As the quasilocal algebra J&’ for the scalar field we shall take the C*-inductive limit of the &(A), where &(A) is identified with a W*-subalgebra of &‘(A’) for all (I’ C II by means of .&(A) @ II(/l’\/l) acting on E+[dP2(d)] @ E+[ZY2@l’\~)]; let us write and d = 2% {d(A):

A E A-}.

(2.16b)

The time translations for the free field will be generated by the antilocal oneparticle Hamiltonian h. In order to consider local time translations we shall assume that some selfadjoint boundary condition has been chosen for each (1 EJV and write hn for the corresponding Hamiltonian. The inhomogeneous telegraph equation (0, - m2)F = 0 has the same characteristics as the wave equation and so the same domains of influence and dependence [42]. By (1, we mean the union of the domains of dependence of each x E (I, for (1 E JV. The time translations generated by h, is then a W*-isomorphism am: &‘(A) - sZ(L’Q, with rl, E JV, and follows from (2.17) for fo gT(II). As opposed to the nonrelativistic expGW.f~

case [l 11, we have1

%VL>.

This enables us to define r(t) E Aut(x2) for all t E R by

1 As the vector space under consideration is real, exp(irh,$ is often called orthogonal unitary on 9&f). If J is the natural complex structure corresponding to multiplication it is more accurate to write cos(thi) + J sin@hn).

rather than by (-l)1/a,

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for f, g E 5@,.(R). Moreover, all these automorphisms

dfwm

811= WWL gM-4,

where U(t) = exp(itH) with FC-space Hamiltonian which is the closure of [H@‘“‘p

are unitarily implemented:

(k, ,... ) k,) = [i:

(2.18b)

given by the selfadjoint operator

E(kj)] wykl

)...) k,);

(2.19)

j=l

here !.P E Yr(!R”), !Pra) = 0 for almost all n > 0, and the r.h.s. is in LP(IP). We note that 7(t) a,*(g) = a&exp( --ith)f) (2.20) It will prove convenient to introduce the formal “fields” transforms” of the aF , aF* by writing

&8(k), the “Fourier

the conjugation rule is that [l&+‘(k)]* = b:‘(k). Combining yields 7(t) bF’(k) = exp[ &its(k)] bF’(k).

Eqs. (2.20) and (2.21)

In terms of the b&k)

(2.20’)

we can write

r(t)(F) = i-l JR [e--itc(k)fl(L) bk’(k) + e+ik(k)~(-R) bk’(k)][2e(k)]-“”

dk

(2.22)

for the time evolution of the field 9. Setting t = 0 in this equation gives the expansion of q~in terms of the b,+(k), a result we shall need. Since (f, g) = Sf(k)z(-k) for realfand g, it follows from Eqs. (2.3) and (2.21) that the CCR for the Q+(k) are [l&-‘(k) , l@(q)]

= 6(k - q) 9

(2.23)

others vanishing. Turning to the states 8 on the quasilocal algebra, if w E Q we call f, g w[W(f, g)] its generating functional (f, g E ~#k)) [43]. Equation (2.9) defines the generating functional for the FC-state wF : L(J;

d = eM--8

This functional extends continuously

II h-Tll

-2 II h’/“g 112)-

(2.9’)

tof, g E %(R), which was noted in Eq. (2.9).

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Amongst the other states on ~2, the KMS states are distinguished as pertaining to thermal equilibrium. Recall [8-121 that a state w E Q is said to [/3, T(R)] - KMS if for any observables A, B E d there exist two functions F, G:C + @ which are analytic in the strips Im (z) E (-8, 0) and (0, /3), respectively, are continuous on their respective boundaries, and satisfy the conditions that the limits sp’(R) - ,lipioF(z> = w(T(t) A * B),

(2.24a)

Y’(R) - z$~m~O G(z) = w(B * 7(t) A),

(2.24b)

exist and the Fourier transforms of these limits are related by the formula fl(d = G@) expW

(s E i-8).

Of course /3 = (kT)-’ is interpreted as the inverse temperature. [/3, T(R)] - KMS it is often possible in practice to use the formula c+(t)A

. B) = w(B . T(f + zy3)A).

(2.244

When w is (2.25)

We shall do this, even using such formal objects as the b,+(k) introduced in Eq. (2.21). These manipulations will lead unambiguously to distributions and welldefined generating functionals, and we shall take this as their justification. First, though, we must consider the question of the uniqueness of the “solutions” to KMS conditions equations (2.24a-2.24c). For the nonrelativistic ideal Bose gas [6, 11, 44-471, the KMS state is unique for chemical potential p < 0. For p = 0 there is a phase transition associated with a spontaneous breakdown of gauge symmetry. The Gibbs state, defined as a thermodynamic limit of local grand canonical states, is a gauge invariant KMS state but not the only one, for now the set of KMS states with ,u = 0 is a nontrivial Choquet simplex [48]. The Gibbs state is a direct integral of gauge invariant canonical states, weighted with Kac’s density function; and each (/I, p)-canonical state is a direct integral of extremal KMS states I,&~@associated with pure gauge angles 0 < 6 < 2~. Formally this seems to happen for the relativistic model as well, but one may check that the limit p = m is too singular to be associated with a finite mean density [35]. Moreover, for p < m it is possible to achieve densities as great as desired, whereas in the nonrelativistic case densities above a critical value p,(p) can occur only if a condensate is present [49]. This means that on physical grounds we consider only p < m, and will find a unique KMS state w(~,~,~)by calculating its generating functional L(B,U,p). For brevity we shall often write v for the thermodynamic parameters
= 0.

(2.26)

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DANIEL

DUBIN

As p < m, m - TV> 0 and m - p < E(k) < co, so that 1 vanishes; then it follows that %(G(f)>

exp@e(k)) never (2.27a)

= 0.

Similarly, the choice A = blp+‘(k), B = II leads to 4%*u->)

(2.27b)

= 0.

Setting t = 0 in Eq. (2.22) then gives (2.27~) We summarize results (2.27a-2.27c) by saying that the one-point functions vanish. The two-point functions can be found from Eq. (2.25) also. The choice A = by’(k), B = b:-‘(q) yields an equation analogous to (2.26), viz. L-1 + exp@4k))l

dbk’(k)

b?(q))

(2.28)

= W - 4).

In order to arrive at this form we have used the CCR, Eq. (2.23): w,(&)(q)

Z&s+‘(k))= WV(&)(k) b;-)(q)) + 6(k - q).

(2.23’)

It is clear that we may define an operator pL+’ on LFr2(R) through the formula

[d+)fl” (k) = [exp(/W)) - ll-l.fW.

(2.29a)

It is useful to introduce the corresponding momentum space density function p:‘(k)

= [exp@r(k)) - 11-l,

so that pt+’ is multiplication by pj+) in k-space. This density operator solves the KMS equation for the case at hand, as Eqs. (2.28) and (2.29) yield

m&*(f) 69) = j&k,

d+‘(k)g”(k)dk (2.30)

= (f, d+)&d. Clearly, the two-point function with aF and a,* in reverse order follows from this and the CCR, so that %(~Fk)

e*(f))

= u (Ps” + 1) id.

(2.31)

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To find the two-point function for the field y we may use the explicit form Eq. (22) with t = 0 to reduce it to the above cases. Then w,(cp(F) y(G)) = - jR&i)[2p!+)(k)

+ l] e(-i)[2+)]-’

dk.

(2.32)

Recall that F, G E 5 C Y,(M) (cf. Eq. (2.14)) and A = (c(k), k). It is possible to cast these results into two-dimensional form. Let us take it that F and G E 5 are related to f and g E Y7(R), respectively, by Eq. (2.14). Note that Eq. (2.32) corresponds to the KD-1 function of field theory. Heuristically, field theory is the limit /3 -+ cc and in this limit pt+’ -+ 0; we see that the expression (2.30) will then vanish and (2.31) will coincide with (2.32) up to a factor [2e(k)]-l. With this in mind we introduce the distributions in sP,(mln) formally written as i-‘E@‘(x) = (27i--l jM e-i”zp”!“(~‘> &PO) 6[(p”)’ i-TEt~-j(x) = (21~)~~jM eIDz[I + pii)(

I”]

B(pO) a[(~‘)~ -

dp, c(p’)“] dp;

(2.33)

(2.34a)

we also write [Eg Y = B(O) ” + 5(-) ” 7

(2.34b)

as this combination occurs quite frequently. It is not hard to see that lB,O- 0 in the 9” topology as /3 + cc, recovering the field theory result. Let us write cc = exp[ -/3(m - p)] for the characteristic fugacity. A very crude estimate, sufficient for our purposes here, is sup I fiz+‘(k)l < ] 1 - 5;’ 1-l. We also write c--l = w-l[w/c] with w = (k2 + m2)lj2 and bound c-r by w-l . (1 - p/m)-l. This gives a comparison with the usual relativistic distribution i-lDk,- for a scalar particle of mass m [30-321. Then for FE 9’( Ml), I i@“(F)/

< I D!,?(G)1

. (1 -

(p/m))-’

11 -

5;’ 1-l.

(2.35a)

This inequality isolates the temperature variation in 5, , where G is defined by e(w, k) = P(E, k) so that I? is p-dependent. Since I 1 - <;’ 1-l -+ 0 with fl --f co, B”‘(F) --f 0 in the same limit. Y The distribution BP has an important physical interpretation: it is the density distribution of F-particles. More precisely, the density of T-particles in the state w, is p = 47%%90)

(2.36a) = 595/102/1-6

IR

,5?‘(k) dk/c.

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DANIEL

DLJBIN

To see that the density is well-defined for p < m but not at p = m we again replace dk/c by dk/w with corresponding factor (1 - ,u/m)-’ in an inequality. We also expand p”(+)(k) so as to obtain a series of modified Bessel functions of the third kind, K, [SO]. This will require the substitution k = m sinh (y.as for four dimensions [30]. Then, writing 5 = eQufor the fugacity,

Using the asymptotic formula K,(Npm) - (vr/2j?mN)1~z e-B”N(l + o(iV-l)) this series converges by the ratio test: aN+,/ a N -+[,
I B?(F)1< (1 - (p/m))-’ Dk’(G)[l + I 1 - 5;’ I-l1

(2.35b)

and I B,(F)1 d (1 - (p/m))-’

&‘(G)[l

+ 2 I 1 - 5F1 I-‘].

(2.35~)

It is customary to use the notation [cf. Eqs. (2.21)-(2.23)] i-$+‘(F)

= ap[(2h)-“2 f]

(2.35a) =

sR

P(A) &‘(k)[2s(k)]-“”

dk;

and similarly for c#+) j-$‘+‘(G)

The two-point

=

IR

G(-l)

&+‘(k)[2c(k)]-“”

dk.

(2.35b)

function Eq. (2.31) can be rewritten in the form

d&)(~) F,‘+‘(G)) = - j-,,

F(x) i-%!-‘(x

- y) G(y) dx dy,

(2.36a)

- y) G(y) dx dy.

(2.36b)

and from Eq. (2.30) we find

4d+)(F) d-‘(G))= - jM,, F(x)

i-%~‘(x

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Finally, Eq. (2.32) becomes, using v(F) = #-J(F) + q(+)(F),

(2.36~) ~xu F(x) i-'b(x - u) G(Y)dx &,

=-I

as F and G are real-valued. We shall also write w ”(p)“‘(X) c&‘(y)) = i-WO)(x - y) 3 ”

(2.37a)

wv(cp(x)

(2.37b)

9’(+)(y)) = i-‘5j-‘(x

- y).

It remains to determine the generating functional L, of the state w, . In terms of v we can write w(F, G) = ,imfF)eim(a~cfei/2(f,g).

(2.10’)

Then UF, G) = exp (--i jM,, F(x) G(Y) aob(x - v) dx dv + d2(f, 8)).

(2.38)

Thus w, is quasifree [lo, 49, 611. The structure analysis of these results differs from the nonrelativistic ideal Bose gase only in the particular nature of t$+) [6, 11. Writing w, - [ZV, rV, Q] for the GNS representation, the well-known result is

(2.39) (2.40)

% = E+[z”(Wl 0 E+WzGW, Q"=fiFoQF,

dWf)l = V,([l + PS+yf~ 0 ~F(-[PS+)11’2f), TP-(f)l = UFUl+ P!+Yf) 0 w~d+‘11’2f); writing a,(f)

(2.41a) (2.41 b)

for the generators 77,(~,[(2h)~/~f]) of n,(U) and r”(V),

Q”(f) = 41 + PSfY2f) 0 Q+ Q0 ~F(kd+‘11’2f), (2.42a) (2.42b) a,*(f) = a&l + fJS+)11’2f) 0 Q+ Q0 aF([P:+)l”2 f>. Let us note that w, is not “Fock-like”, 4.f)

Q”

=

QF

a,*(f)

Q”

=

a,*([1

0

as u,(f) Q, # 0. Indeed eY[P”‘1”“f) +

PW2f)

(2.43a)

Qnp 3 QF

0

Qp.

The IV*-algebra rV(cpB)”is a Type III factor for 5 # 0 or fl f co.

(2.43b)

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3. THE ORDERED EXPONENTIAL

FIELD

In constructing solutions for the vector meson model, we shall need the exponential field exp[igv(x)], x E Ml. Although this object is not well defined as it stands, Jaffe has shown that Wick ordering eliminates enough divergences so that : exp[igF(x)]: is an operator-valued distribution on the FC-space of the field cp [22, 371. Moreover, it is in the Borcher’s class of v [51], which is why the S-matrix for the vector meson model is trivial. The ordering here is constructed so as to be compatible with the FC-vacuum Q, . Indeed, w,(S~ : e@+‘(“):F(x) dx) = fM F(x) dx. However, it is the state w, and not wF which we are concerned with, so we shall consider the exponential field ordered with respect to w, “v-ordering” for brevity. This is actually necessary, as the Wick-ordered exponential is incompatible with o, as it requires an infinite field strength renormalization. This is easily seen:

W”(1M :eigm(“):F(x)

dx) = IM F(x) exp[2Sg21B-j(O)]

dx.

(3.1)

In Appendix A we show that El:-‘(0) is not defined. To construct the v-ordered fields we introduce the notation i;jy for the vector state on rVTT,(&)”defined by

h(Q) = (Qv 9 Qsz,)

Q E d~a2)“,

(3.2)

so that if A E LZZ, w,(A) = zZy0 V,(A). For typographical convenience we write { }” for v-ordering. Then as w, has the quasifree form indicated in Eq. (2.38), the v-ordered polynomials are defined recursively by the formulas [38]:

The {&)“>V are operator-valued distributions is defined by means of its power series:

on 9” . The v-ordered exponential

In view of the fact that a,(f) 52, # 0 (cf. Eq. (2.43a)), the decomposition 9 = F+ + #-) will not play the rBle for v-ordering as it does for Wick-ordering. For future reference we note that as G,({gl(@“},,) = 0 for N > 1, Eq. (3.4) implies that (3.%) B,(r’“‘(g)) = 1,

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or (3Sb) where we shall write (3.6) here and hereafter. We shall encounter the field P’)(x,) . . . ITI in what follows, and to deal with it we now define v-ordering for noncoincident arguments [38, 521. Suppose we write d,,, for the sum over ail partitions of (I,..., n> into disjoint subsets (iI ,..., i, : with 1 < r < n. Then the i, < i, < .I. < i,}, {j, ,..., j,-, : j, c.jz < ... <.L,), inductive definition of v-ordering is {&4h

(3.7a)

= ~Y[9JWl - 4dXlh

{PC%> d%?N”= ~“JTy[dXI) d-u2>1 - 4dxJ

d&N

{d-d -*. d&N” = %[dXl) *.- &?Jl - i ~7L,hJ”(dxiI~I’ dXi,)> P=l x MX?,) .*. d%JLl.

(3.7b)

(3.7c)

The connection with the previous v-ordering is given by the limit

It is clear that ij,{~(xJ ..* ~(x,J}~ vanishes. We can easily see this by induction. Assume it is true for n - 1. Then

from which the result follows immediately. Consider next the quantity %- = {dXlY%

~dxBYz$lY*-* ~9)hJY%l *

(3.9a)

We may expand this into a polynomial in the n,[&xi)] with coefficients which are distributions. By introducing extra arguments, e.g., xy),,.., x:i for xi and taking the limit x:i) ----fxi , xr’ - xi ,..., at the end of the calculation, it is clear that ~“WM

= 0.

(3.9b)

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With this in mind it is also clear from its multiple power series that (3.9c)

a result we shall need shortly. First, though, let us consider the existence of P’) as a field. The power series for P) yields II Wan

Q” II2 G s,,,

J’(x) F(Y) evk”W

- v>l dx dy,

(3.10)

which is finite. In fact all estimates involving these distributions can be reduced to those of the zero temperature case by using the estimate I iP(

< I exp[P(m - p)I -

1 1-l.

(3.11)

The only other complication is the factor p in E, but this is harmless as fi < m. Then Jaffe’s estimates [37] can be adapted to show that

fi I-$$&)sz,Eiq

(3.12)

i=l

is well defined; here the gi subscript indicates {exp igj&)}y is under consideration. Just as for Wick-ordering linear combinations of such vectors form a domain D C XV stable under ~~“(8’). Because w, is quasifree [lo, 611 the following basic product formula holds, just as for Wick-ordering [37, Appendix A],

Combining this with Eq. (3.9~) enables us to read off the N-point functions directly. The distribution form and smeared form are 4 (fi

C%ql)

and

respectively.

= fJ ew((-g”)

i-Wxi

- xi))

(3.14)

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This and Eq. (3.12) shows that P) is an operator-valued distribution on XV. Further, the estimates (3.10), (3.11) enable us to conclude that P) can be considered through a sharp time formalism following from the one for y. Recalling the space f, of test functions associated with the spectrum condition Eq. (2.14) we write

P(F) = #(f)

(F E$1,

(3.16)

or simply r’“)(f) when it is clear that t = 0 is meant. The duality condition &‘(A)’ = &‘(A”) implies that the y,,(f) with supp(f) CA belong to rr,[&(A)]“; even more, they give rise to W*-subalgebras P’)(A) of them. This follows from Jaffe’s work [37] cast into a sharp time form: “the bounded functions of y(“) can be got from the Stone characteristic matrices;” and the r(“)(f) with supp(f) CA commute with &‘(Al), which implies that the Stone matrices also do, so the Stone matrices are in &‘(/ll)’ = &(A”‘) = &(A) [41]. The C*-inductive limit of the &P(A) will be denoted P’), a subalgebra of VT”(&). In writing the correlation functions for w, in two-dimensional form, it is clear that a two-dimensional algebraic formulation is also possible [53]. Let /r be the closure of A in R, and let C(A) be the open double light cone with base /f. We may view &(A) as a subalgebra of B?(C(A)), where 9?(U) is the W*-algebra associated to the open region 0 C Ml. The R(U) are generated by the Weyl operators associated with the Hilbert space [,ul,(Ml) completed with respect to the inner product $ Re JRfl( -&?(A) l -l dk and its symmetric Fock-Cook space; the CCR utilize the symplectic form o(F, G) = QIm Jn F( -&e(i) E-I dk. For 9?(O) we use W(F, G) with F, G E Yr(0). The injection &(A) -+ W[C(A)] enables us to extend w, to 99, the C*-inductive limit of the B(U): it is the quasifree state whose two-point function has i-1B, as kernel. These considerations also hold for P), as evidenced by Eqs. (3.14), (3.15). Let us also note here that as G,(P)(F)) = JM F(t) de, 3, viewed as a state on B(“) is not quasifree, and so a priori for the two-dimensional variant too. The generating functional for the restricted state, also written as 8,) follows from Eq. (3.15) by using the power expansion for the exponential, B,(exp[iP(F)]) -= z. G”/n 9 j . . . jMn r;(&) . .. F(LJ fi expWW& - &N d5, ... d5, . (3.17) i
Qv= ddxl) ... dx,)l = n,(Q)

(3.18b)

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and consider Z- = Wd~Ylv

QA

(3.19a)

The power series expansion for P) will then give an affirmative answer to (3.18a) if it is true for (3.19a), II = 1, 2 ,... , The only reason we must actually consider this matter is to convince ourselves that the counterterms arising from v-ordering do not interfere with the KMS condition. To prove (3.19a) we consider the slightly more general correlation distribution 9” nm = Gvt{d vd -** d ~n>>vQA

(3.19b)

so that % = lim,,, snnz in the Sp’(tW) sense; the existence of the limit follows from the definition of v-ordering. Then using this definition, 9?%, can be expanded to give [cf. (3.7c)]

x dQdtj,-, + 4%An-,>*-. do, + iP9~01.

(3.19d)

In the last step the field q(tk , vlc’) were moved successively from the leftmost position in 8,(a) to the rightmost where their arguments became (tl, + ifI, yi) by virtue of w, being KMS for elements of &. The form (3.19d) may now be recognized as &(QV{~(tn + i/3, JJ~‘)... v(tl + i& yl’)}J. The limit yj - y yields 3 =

~vtQY{d~ + 4% v’>lJ,

and therefore (3.18a) is verified. It is clear from this computation

(3.19e)

that were we to take

Eq. (3.18a) would still hold. In this sense, ~5,is unrestrictedly KMS: the v-ordering is compatible with the KMS condition. The GNS construction for w, on &’ has been written w, N [ZV, 7rV, Sz,]. For the extension to A?(~)”we write 6, and +, . It is our intention to use these same symbols for the restriction to W) and LW”.

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4. MISCELLANEOUS

FREE BOSON FIELDS

Our model solution requires the ordered exponential of a zero-mass scalar field. Whereas in the usual field theory for such an object there is an infrared problem [22, 541 leading to an indefinite metric Pontriagin space for the one-particle space [55], the chemical potential p < 0 serves to regularize the distributions which appear, reducing the singularities and preserving positivity. In what follows we shall consider the theory of the zero-mass field modified so as to appear to have a chemical potential p < 0. The distributions that result are perfectly well defined; the GAS representation 8, - [ZV , rr, , QJ obviously so; the density &!I, 5) associated with the state is given in terms of the inverse temperature and the fugacity by p@, 5) = (l//3) ln(1 - 5); and yet the introduction of p cannot be for the usual reason, since there is no number operator on FC-space even for local regions. However, we do not mean the chemical potential p appearing here to fix the “number of particles.” The field is an auxiliary entity to be used in defining the operator solutions to the vector meson model. The chemical potential p must therefore be viewed as being associated with the thermal state of the vector meson solution. It remains to decide how p shall appear. We choose to arrange matters so as to be compatible with the modification of the temporal evolution indicated by the replacement 0 -+ 0, . This will result in concomitant changes in the equations of motion of the vector meson model. Fields containing a chemical potential p in the above sense will be referred to as p-modified. For the p-modified massless field, then, we do not have to introduce p as a Lagrange multiplier. By analogy with the massive field we may sutract p from the one-particle energy and consider the p-modified energy I k 1 - p leading to the use of the measure &/(I k / - p): the question of the nonexistence of the number operator is thereby avoided. Recall that if no chemical potential is present, the two-point function must be regularized to avoid the logarithmic singularity at k = 0. One modifies the field by regularizing to (the notation is that of [54])
1@z-t2

+ a”)

lM e- i”‘wq~l

+ c

(4.1)

%P’) + (P’L1 Q-1 + c~2(p)l4% (4.2)

where ,$ = x - y, the distribution (r);’ = (d/dr)[o(r) log r] and p* = p” & p1 are light-cone coordinates.2 One uses this distribution to form the one-particle space: calling the momentum space distribution J@&p), for F, G E YT(M), set [F, G] = 2 The constant C in Eq. (4.1) is an index relating to the regularization. It is possibleto choose c = 0.

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p) PC(p) dp. The space so constructed [55] is the direct sum-corresbnn-pm sponding to the two light-cone lines -of two Pontriagin spaces n1 0 n1 . Recalling [56,57] that a Pontriagin space U1 can be decomposed so that [ , ] is negative definite on a one-dimensional subspace, and is a separable pre-Hilbert space on the [ , ]-complement, hence the indefinite metric. Consider instead the situation when t.~< 0. Proceeding formally, the corresponding two-point function is

(4.3) where $ = (e( pl), p’) and E(pl) = 1p1 ] - p. Defining the modified exponential integral hi(x) = Ci(x) + i si(x) (4.4) =-m dt/t (cos t + i sin t), sz (y,(x)

~p,(y))~ = (1/2~)[e-i”~‘~i(--~+)

+ e+iuc’&i(-p(-)],

(4.5)

where 5 = x - y, [’ = 5” f 5’ just as forp* above. The fact that E(pl) 2 1 p 1 > 0 implies that e-ldpl is a positive measure and is well behaved enough to be used as is without modification. This means that we can work with the same FC-space as for the massive field, Section 2, but replacing the one-particle Hamiltonian used there by one appropriate to this field: m-

w = 4w m, 44 = I k I - I-G

(4.6)

wherefis in the domain of this operator: Ju e(p)” 1f(p)/” dp - co. The Hamiltonian h may be used here precisely as in the massive case and all the corresponding equations, e.g., (4.8)-(4.14) are identical in form; in particular, Eq. 2.8 with h as in (4.6) above defines a representation of the CCR. The first change occurs for the equation of motion: it is now the homogeneous telegraph equation 1421 RKI,F) = 0, (4.7) where 0, = -(a, + i~)~ + (Q2 was defined in Eq. (2.15). As this change does not affect the characteristics, the construction of the algebras leads to the same forms, &(A), & as in Eq. 2.16 or B(8), W defined at the end of Section 3, as the massive case. In fact, even the KMS state has the same form as in Sections 2 and 3, but now e(k) = I k I - p as in Eq. (4.6).

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Proceeding as in Section 2 we define v = [/3, TV,m = 0, s = 0] for the physical parameters. Then [cf. Eqs. (2.27)-(2.34)] the one-point functions vanish, and the density function in question is p”‘+‘(k) = [exp@(k))]-l ”

(4.8)

= [exp@(l k I : PL))- 1I-5 and this determines the density operator Bi”. With this density and this energy, the form of the two-point function is as for m # 0, and the distributions gV# modified accordingly. The prescribed mean density for this state is given by (2.36) Recalling the method used to estimate p for the massive case, we find in the same way that

= gl EWP I CLI), where Ei is the indicated exponential integral [50]. Using the asymptotic expansion Ei(x) - e-*(1/x + 0(x-2)), the ratio test uhr+JuN + 5 = e@ < 1 shows that the series converges for TV< 0, with fugacity 5 = e--Q1 < 1. We see from this that any attempt to consider TV= 0 in two dimensions comes to nought, as was the case for the limit p = m in the massive case. We continue further as in the massive case and define the v-ordered exponential F’) corresponding to our p-modified massless field qM . The estimate [cf. Eq. (3.1 l)]

I id%

d {exp(--PA - 11-l = [l-l - 11-l

(4.10)

enables us to conclude that F(“) is an operator-valued distribution on J$ and construct a stable domain for it. Note that w, - [&$ , rrV, Qn,] is given by Eqs. (2.39)-(2.43) but with the pertinent pt+’ and E. Similarly F(“) is defined as in Section 3. Let us note that i-l& leads to a closable form, and its exponentiation for the F(“) correlation functions does not lead to branch point singularities as for p =o. It is also necessary for us to consider a mass zero scalar field C, with reversed commutation relations [22]. If we proceed in the sharp time formalism we modify

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the FC-space by equipping the direct sum of n-particle spaces with the sesquilinear form

(0 Q(n)@Q(n)) = .@~*@(o) + c (-I)%JR.cv(kl

... k,)* P)(kl

... k,) dk, ... dk, . (4.11)

n>1

We also define the corresponding annihilation bdf>

@P (k ,..., k,) = +-l/2

operator with a minus sign.

i f(-kj) @n-l)&... ,$ -*- k,),

(4.12)

j=l

and leave a=*(f) as before. Thus a,*(f) is not the adjoint of a,(f) as before, but is defined by its familiar action on FC-vectors. The field C, is then found by zTT,(F)

= a,[(2h)-1/2f]

+ a,*[(2h)-91,

(4.13)

where FE $3 satisfies the spectrum condition Eq. (2.14) and h is the zero mass one-particle Hamiltonian given in Eq. (4.6). The overall result of this is to affect the positivity (which vP did not do) and put minus signs in front of certain of the correlation functions in the solution to the model. Because of the difficulty with positivity we have not been able to complete the structures in a topology so that a natural quasilocal C*-algebra emerges. If we content ourselves with incomplete *-algebras, we can proceed unambiguously as follows. Because the one-particle space Z2(Iw) is a Hilbert space and the energy 1k I - p, p < 0 is positive on it, the FC-vectors of the form E(f) = x,“,, [ - l]“f@‘“(n !)- lJ2 are linearly independent and total in C,“,, [-l]nLP([Wn) symm ; note that the [-I]” acts so that (E(f), E(g)) = exp[-(f, g)]. With this formalism we can define W,(f) by W,(f)E( g) = exp[*(f,f)]E( g + f) with vacuum generating functional (E(O), W,(j)E(O)) = exp[$(f, f)]. There has been a 2/2 scale change here, andf can be complex-valued; but the point to note is the positive sign in the argument of the exponential. Given the W,(f) we can form W(f) by setting W(f) = W,([2h]-112Jl + i[h/2]‘l”fi) with f =fi + ifi . This replaces Eqs. (2.1)-(2.19). If we now reintroduce the spectrum condition Eq. (2.14) and the space $3, we are led to the field C, . Restricting the test functionsf, resp. F, to La(A) and Y(U) with A C Iw, resp. 01C Ml, the W(f), resp. W(F), (abuse of notation!) generate polynomial algebras which we denote by JzZ(A)~ and W(0)‘, the embellishment indicating incompleteness. There is an involution assumed here, following by linearity from W(f)* = W(--f*). For [w and Ml we can either use the disjoint unions to form &“’ = Un &‘(A)‘, W” = Ue L~Z’(O)~ for suitable local regions. The “states” w FCand w, are linear functionals on these algebras, continuous in the sense that their action on generators leads to distributions. We do not demand a GNS construction: the Wightman reconstruction

THERMAL

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using these distribution kernels suffices, provided we do not take null-equivalence class for purposes of completion, as this demands positivity. Similarly, the v-ordered exponentials are well defined apart from positivity and lead to exponentials of distributions as in Eq. (3.15). The difference between these and the p-modified massless field with standard commutation relations is the difference in sign multiplying the distributions; this difference in sign is the same as in the usual “field theory” state for the model. Free quantum fields transforming variously under the two-dimensional Lorentz group can be described in terms of (fractional) derivatives of scalar fields, local or antilocal as they are the two-dimensional analogs of integral or half-integral spin fields in four dimensions [58]. When the chemical potential is built into the fields there is no longer PoincarC covariance and the covariant theory, as such, does not apply. There does seem to be analogous decompositions available, modeled from the covariant results, which reduce to the expected limits when p = 0. Consider a p-modified real massive vector field, U = (UO, V). We choose the metric go0 = 1, gll = -1 (which has been implicit in the work so far, so that, e.g., a0 = 3, and 31 zzz-a, ; U0 = U, and U1 = -VI). Each component U” of U must satisfy the inhomogeneous telegraph equation satisfied by the p-modified massive scalar field uqo, - mZ]F) = 0, (4.14) for FE Y!(M); this is in keeping with the spirit of the usual generalization from scalar to vector field. Now in the usual situation (CL= 0), the nonpositivity of the “energy” is eliminated by some sort of subsidiary condition, usually taken to be a,U” = 0, or lP(a,F) = 0 in smeared form. The natural p-modification would seem to be the replacement of a, by a, + ip as in the p-D’Alembertian operator 0, . To this end we define the derivatives 2, = a, + ip = a0 + ip = 8, a,

--

so that, e.g., 0, = -a,% condition

=

a,

=

-a1

(summation

=

(4.15)

-21,

convention).

qJJ~ = 0

We impose the subsidiary (4.16)

on the vector field. Along with this, the differential operator polynomial, characteristic of the two-point functions is now go’ + rn-‘%~. It may be verified that if y is the p-modified scalar field of Section 2, then we may write ua = m-l&1 2

(4.17)

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where 8 = a0 by definition. The subsidiary condition follows from this expression directly: --a,U~ = m-l[iYob81q,+ a,ap] = m-l[aoal~ - a1a%p’l. Clearly the equation of motion follows from (4.17) as well, and the two-point -functions will also be of the correct form, since m-2%+18r+1 = guT + m-Wa* when acting on a solution of the inhomogeneous telegraph equation (for then (8)” - (Z”)” = m2 is an identity). The validity of Eq. (4.17) reduces all quantities we shall need involving U to quantities involving go’, which we either have calculated already, or very soon can. For example,

= [go’ + m-‘%W] i-YBv([),

(4.18)

where t = x - y and derivatives are with respect to 5. Another example arises from the existence of the tensor (E,, = -E, ; eol = + 1) in two dimensions. For the field q,,8U" occurs in the model solution; but Eq. (4.17) allows us to write

coT3Uu= &(1/m) 8~1 - ZOb[(l/m) 8Ov] (4.19)

= my.

Thus the FC-space E+[~~(R)] and the construction of the CCR used in Section 2 serves to define U simply by appending Eq. (4.17), defining U in terms of q~ Let us also note here that the field C, defined above [cf. Eqs. (4.1 l)-(4.13)] appears in the model solution acted upon by E,,% at one point; we emphasize that it is the modified derivative % which must be used in our p-modified model.

5.

THE

FREE

MASSLESS

SPINOR

FIELD

We come next to consider the p-modified zero mass spinor field. In two dimensions, even though there is no spin, the Poincart group has (reducible) representations which satisfy the Dirac equation [5.8]. With p-modification, the replacement &, -+ &, is formally made and existence is assured. In what follows, our notation shall be

(5.1)

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a = y”Ao + ylAl , the Hermitian conjugate of a Hilbert space operator A is A*; and if relevant A = A*y”. We shall be using the one-particle Hamiltonian h associated with the energy function c(k) = / k 1 - p, p < 0 as we are considering mass zero. The thermal states are not Fock-like, and so there are more correlation functions to keep track of. For this reason we shall use the full index notation [30]. Our spinor field 4 is an operator-valued distribution on the antisymmetric K-space associated with a twofold copy of the CAR [32]. We shall use the indices Y, s = 1,2 for the fixed purpose of distinguishing amongst them. The symbols (&), on functions and fields, will invariably refer to f-frequencies (SE actually). We shall also use E” = k” + p, IE1= k1 arising from the derivatives 80 ; thus 2 = yak) + y% . Working in the momentem representation, let W**(k) be smooth positive energy solution of the Dirac equations (yoEo + ylE1) U/“r*+(k,) = 0, (y”lEo - ylfE1) FV-(-k,)

= 0

(k” = E),

(5.2)

by with adjoint rules [W**(k,)]* = W*‘*“(k,), or upon post multiplication 70, V~**(kJ = [Wan]* 7”. Amongst the many relations obeyed by the W(k,), W(k,), which are supposed to form an orthonormal spinor basis for a one-particle space, the most important for us is

(5.3) 1

(a,b=

W,‘*-(k,) W;*+(kl) = koJ24kJ

1,2).

r=1.2

We also have the two CAR representations (a,#) r = 1, 2; in k-space we have

bT-(W, as+(d+ = %Ak, - a),

(5.4a)

W(W,

(5.4b)

d+(qdl+

conjugated according to [a,*(kl)]* valued (on Fe-space) distribution @(x)

= LV,

= a:‘(k,).

The spinor field is the operator-

W,‘**(k,) dk, ,

(5.5a)

I, efiR- rz2 a,**(k,) FV;*s(kl) dk, .

(5.5b)

= (2~)-l’~ s, eitiL- v;2 c&k,)

z,&,(x)= (27r-“2

- qd,

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The CAR obeyed by the fields # are determined by the distributions the chemical potential and m = 0) irlD;&)

(cl < 0 is

= (277-l JM e-W(kO) 6[(kO)Z - E(k1)2] dk, (5.6)

= (277-l JR e-ik’c(dk1/2e(k1)), and D,,,([)

= lLD,($([) - D:;A(-Q

(t E Ml). Direct calculation gives

{p(x),

$(‘)(v>>+ = cia i-lD:::(x

M”‘(X),

$‘-‘WI+

{K4

= <@ i&(Y

- y), - 4,

$(Y>>+ = (ii9 i-1b,o(x

(5.7)

- Y).

The reason for writing down these well-known formulas here is for comparison with the thermal distributions. The algebras involved here follow by weak closure of polynomials either in the a7 or the 4, $ with functions of suitably restricted class and support. For functions in g(A), A C R, the former operators lead to the equal time ring &(A), and thence by C*-inductive limit to the quasilocal C*-algebra &. The latter choice leads to the space-time ring W(0) for ICI(F), $(F) with FE Y(0), 0 C M and thence to 9’. The differential equation of evolution is iJ,,G = 0, so there is no difficulty with time translations. This leads us directly to the computation of the thermal states w, viewed, by an abuse of notation, as a state either on d or 6% context distinguishing. Introducing the one-particle Fermi-Dirac density operator pi-’ by (&)f)” Y

(k) = j+‘(k) f(k) Y

(k E RI,

p?(k) = [l + exp@&W,

(5.8a) (58b)

we may easily find that (5.9a)

w,(afB*(k)) = 0;

4dW

a,-(d)

= WXk

- 4) &‘W;

(5.9b)

4a,‘(k)

4%))

= LW

- d P”?)(k);

(5.9c)

w&-(q)

a?(k))

= 8,&k

- q)[l - it!-‘(k)];

(5.9d)

4a,-(k)

a%))

= 0;

and so on. Recalling our discussion concerning the density associated with w, for the scalar boson field (any mass) we see the (5.9b, c) indicates a nonzero

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particle density for the two sorts of number operators a:+(k) a,-(k) = n&), a,+(k) a:-(k) = n,*(k) in a formal notation [30]. We now introduce the distributions [F,# corresponding to the B,-# for bosons: i-‘@“(f)

= (277)-l j-M /%(k”)

6[(k”)’ -

= (277)-l JR e-i”p”;-‘(kl) i-‘F:-‘(0

= (2z-)-l s, ,@[l = i-l[D;y;(()

E’] &‘(kl)

dk1/2c;

- $‘(kl)]

dk1/2c,

- l@‘(f).

dk,

(5.10) (5.11) (5.12)

We are now in a position to compute the two-point functions. For example,

x 4a,+(k,) d+)(q,N dk, 4, iS,,e“~‘“-“~~-‘(k,)(dk,/2E)

= (27r)-ll,

= (i-l~~~)(i-lF~)(-R).

(5.13a) (5.13b)

In the four dimensional case, the operators a,*-(k) a,+(k), where I = 1, 2, correspond to the number of antiparticles, m,(k). From (5.9~) we see that the density of antiparticles in the state w, is associated with the density function 1 - p”:-‘(k) as in IF:-‘; IFso’on the other hand is associated with the density of particles. However, the Wick normal ordered form [30] for the total number of particles and antiparticles, :nr(k) + m,(k): = n,(k) + m,*(k) (this statement is only formal, as we really mean the distributions they lead to), so that the total number density (per unit energy) of momentum k is 2p”l-‘(k) from (5.9b, c), and not 1, as would result from forgetting the ordering. The average “charge,” on the other hand, is

dQ> = e J‘, dk c w&(k)

- n,*(k))

e1.2

=

0,

as it should. We write [cf. Eq. (2.36a)] ,@, p) = 2[4rri+“‘(O)] zz

595/dI-7

2 lR p;-‘(k) dk/c

(5.14)

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The other two-point wy(&‘(x)

DUBIN

functions of interest are computed in the same way $2’(y))

= (27~)~’ JR &/+-ff)[l

- p”S-‘(kl)](dkl/2~)

(5.15b)

= t&,wF:-‘(+~)); 4p

&%>>

(5.15a)

(5.16)

= 0;

4~dx) $b> = &bF-‘(n - c%n1.

(5.17)

This last result is to be contrasted with the result for the FC-state (5.18)

QJFC(W) $dYN = 7aDl:ko, to which it reduces in the limit Y’(M)-limit Using the CAR (5.7~)

(/3 -+ a).

it follows that

%(?Jb(4hz(YN= ii&P(-t3

+ &4.,(-n + Pm.

(5.19)

Note that (5.18) and (5.19) are thus compatible with the CAR. We must say a few words about the free field current defined from the ,u.-modified spinor field 4 [22]. It is sufficient to use Wick ordering (with respect to the FC-state) in order that (5.20) j?)(x) = :f/$qh: (x) be a well-defined operator-valued distribution: the p-modification this. It does change the differential identities satisfied byj(O) to

a_qo) = E,,~y’“‘” = quj$)= 0. The operator commutation

does not affect (5.21)

relations on FC-space will be similarly affected, e.g.,

(f)* [j?)(x))j’“‘(y)] A = (iTr)-la02hDLL,o

(5.22)

In our usual notation, we write w, N [SV, VT~,QJ for the GNS association. The GAS triple so obtained is explicitly constructible by reference to the ideal Fermi gas [6,41,59]. Let F1 , &. be two copies of the relativistic FC-space on which a free spinor field may be realized, but with ,u-modified energy function E = I k1 I - CL.Let x1, xZ be the corresponding spinor fields, and set YV = S1 @ FZ, a = QF.1 0 G.2 * Recalling the definition of the density operator p:-‘, let

97

THERMAL STATES OF VECTOR MESONS

us set pi-‘(f) = pi-‘(F) by abuse of notation, when f and F are related be the spectral condition (2.14). The representation Z-, is given by T”[tpi’(F)]

= xi”([l

- p;-)]l’2 F) @ II - 4 @ n?([p!-‘I””

F),

(5.23a)

n,[@‘)(F)]

= #‘([l

- P!-)]~‘~ F) @ II - 21@ ~~)([py(-‘]~‘~F).

(5.23b)

We observe that this reproduces the proper results; to see this it is sufficient to verify the two-part functions, in view of the quasifree nature of w, (meant in the sense of fermions). Clearly all one-point functions vanish. Next,

(Q, , TM”‘(~) $‘-‘@)I JAI= w&+)([l

- P;-)]~‘~ F) #([l

+ ~~,~[#(pj-‘~‘~F)

- p;-)ll”

G)]

xp’(p:-“‘“G)]

= w,(yb(+)(F) 4’-‘(G)),

and so on.

6. THE VECTOR MESON MODEL In formal terms, the vector meson model [22-261 is that of a massless spinor field # intereacting with a massive vector field A, . The equations of motion are icPa,# = -ga:A,#:

w

= --yOyo)

(0 - mo2)A, = gj”.

(6.la) (6.lb)

The field A, must be constrained by some divergence condition: Let us assume the “Lorentz gauge” %“A, = 0 (6.2) unless otherwise stated. The current jU is a regularization of $+&

“m, l > = Mx + 4 Y”$w + $(x - 6)Y”$w, then the two regularizations [22, 60, 61, 251

If we write 6.3)

which have been considered in this context are

and

due to Schwinger [60] and Johnson [61], repectively. The vectors cS and 6t are

98

DANIEL

DUBIN

spacelike and timelike, and K is a factor related to a field strength renormalization for $. It turns out that Et may be substituted for E, in the definition ofjsU. The fields I/, A, are supposed to be definable at sharp times and to satisfy equal time (anti-) commutation relations

[--Ayr,

Kh4a (f, x1), hdt, .a+ = Lm(xl - v’), xl), (PA1 - @AO)(t, yl)] = i-%3(x1 - yl),

(6.5a) (6.5b)

with renormalization factor Z, , possibly zero. The gauge condition implies that in the commutation relations A0 should be replaced by A0 = -m,Z(gjO

+ a,[aOA - &IO]).

(6.5~)

All this is well known, (see [22, 251, for example) and pertains to the case /3 = cc and no chemical potential. In accordance with our approach to the thermal states of this model, we must recast these equations in p-modified form. Without further examination of the solution of the model in the form above, we cannot know just how # and A, depend upon p, and the possible effect on the regularizing limit for the current [Eq. (6.4)] and the ordering in (6.la): indeed, we may well suspect that the ordering will depend upon both p and p as in the Vordering of Section 3 et seq. What we can say, however, is that the replacements 8, -+ 80 and 0 + 0, are to be made, and that the a-matrices remain unchanged. Let us assume that the equal time commutation relations (6.5a, b) are chosen so that the right-hand side is that of the corresponding p-modified free field: since the p-modification of the energy k” = E does not enter here, the commutation relations stand unchanged. Equations (6.5b, c) change by the use of p-modified derivatives, however. Thus [--Al@, x), [aok! - SAO](t, y’)] = i-16(x1 - y’), - A0 = -m&go + a,[aoA’ - ~‘AO]).

(6.6a) (6.6b)

To proceed further requires knowledge of the particular operator solution to the model. We shall use the details in [25], supplemented by material from [22]. One starts with certain auxiliary free fields v(l),..., @“) acting on FC-spaces F1 ,..., gN . In the case of the Schwinger current and Lorentz gauge, for example, one may take: v(l) = U”, a vector field of mass m, = (mo2 + g2/n)1/2; I$~) = v, a scalar field of mass zero; v(3) = C, a scalar field of mass zero with reversed commutation relations; v(4) = yfN) = #PO),a spinor field of mass zero; and all fields are p-modified, with the same p < 0. Next one constructs the tensor product Hilbert space F = s1 @ ... 0 FN and the corresponding tensor product of FC-representations. In this regard we shall always use the following conventions. Any field v(j) is

THERMAL

STATES

OF

VECTOR

MESONS

99

identified with its image on 9, as is any element of the algebra .JzP. Moreover, any otherwise undefined operators under this identification are taken to be the unit operators, e.g., v(l) @ v(2) is identified with y(l) @ y(2) @ II, @ ... @ Ii ,,, , etc. As mentioned in Section 4, where a field such as C = I$~) appears, the space 3ZY3 and the algebra &@) are incomplete. The tensor products relating to d(3) will then be meant in the algebraic sense. From the point of view of this work, the essential point is that the distributions relating to #j) @ q~(L)factorize, and this is unaffected by the positivity difficulty. The auxiliary fields y(j) are used to construct the field I/Jand A, on 9 in the case when no chemical potential is present, but this will not do for us. As ordered exponentials must appear, we shall use v-ordering. This results in the construction of I/Jand A, on the tensor product of the GNS Hilbert paces Xtj’ of the auxiliary fields Z” = & ,y.

(6.7)

j=l

For ease of notation, when a field is taken as acting on its space .%‘tj’ we shall write 9:) for 7rsj)(@)): the previous convention concerning identifying y’sj’ with II 0 ... @ vy) 0 ... 0 1 still holds force. With this change we proceed as for the T = 0 field theory situation. We define N-l

N-l

1 a&) 1

+ y5 c 1

(6.8)

b&j I

on XV (compare [2.5, Eq. (1.2)]. We choose the coefficients aj, bk so that di, satisfies @, = 5 ,

(6.9)

7)” = {eigo}v I@’

(6.10)

for this will enable the choice

to satisfy the first equation of motion (6.la). In expression (6.8) is the 2 x 2 unit matrix, and normally a field will appear in only one of the sums. This does not solve the problem yet as the second equation of motion has not been considered. This, in turn, requires the choice of regularized current. With the ansatz chosen, it turns out that (js?” = (j’o’o) + (ghwTRX

(6.11a)

CjJ?”= ~j’“‘o>y + (d24K~TRx - (ALxx

(6.11b)

and

where (AL), and (ATR), are the longitudinal

and transverse components

of A,.

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For the T = 0 case, these data are used in constructing the functional solution for the generator of time-ordered products, and from these the Wightman functions [Z, 63-681. It is then a straightforward and essentially unique computation to determine the coefficients of @, . In addition to changes of current, j, or j, , there is a freedom of gauge, but we shall consider only the Lorentz gauge here. We shall take these results for the operators as the dynamical solution for the model, i.e., the same choices of auxiliary fields and coefficients, save only the changes of p-modification and v-ordering. The fact that $j)(-)Qsj’ # 0 does not cause any difficulties. With these solutions &and AVOwe find the thermal states for the model as follows. The &“-vector (6.12) determines the state w, = @f ~lj’ -its extension 8, as a vector state on ~2’: = [ 0: &‘:)I” actually-and the correlation functions by directly computing (Q” 9 MY =

W"(Xl

. . . $&n> $L(Vl> -.* ~“b%L) 4x-a ...

x,

, Yl

*.* Yn

; (al

3 m,...,

** * &%G) (%I

Q”) (6.13)

3 ZJ).

This construction is such that Gi, is KMS with respect to the time translation automorphism group of -Ce,following from #,(x0, x1) -+ &(t + x0, x1) and AVu(xo,x1) + A,“(t + x0, xl), and may legitimately be called the thermal state for parameters v. By keeping the chemical potential p < 0, no phase transition occurs and w, is probably unique, but the positivity problem prevents the proof of this via the theory of mudular Hilbert algebras. The actual computations follow those of the T = 0 case: one replaces the field theory commutator function A(-)(m; x) by B,(m; x) = uQ’(m; x) + lp(m;

x)

(see Eqs. (2.33, 2.34)) and the fermion anticommutator function $A(-)(O; x) by $p(O; x) = &F-+0; x) - lp’(0; -x)1,

SC-‘(0; x) = (6.14)

where the zero refers to the null mass of the auxiliary free spinor field S/I(O). Letting X = g2/mo2, 0 < h < 2 and rnX = ma2 * g2/2r, the fermion n-point is ~~(~~(XJ ... G4)

PP”(Xj - ypJ lFpjqypj - XR) = l$ (sgn P) rJ yxj _ xk) ~Pj,Pyypj _ YPK) x

n

1

w%

-

YPJ

i-W@;

XL -

yp2),

(6.15)

THERMAL

STATES

OF

VECTOR

101

MESONS

where P is an element of the permutation group in n-letters acting as indicated. This is precisely the same as [25, Eq. (4.3)] except for the above indicated replacements. The same is true of the [F’3~~, which depend on the choice of current. From [25, Eq. (4.4)] we have Fjs(xj - J,~) = exp(K?EK,(O; xj - yk) $ yt’yp)h(l

ffj(xj

f h/7rP1 i-l[B,(nls ; Xj - vk) - B,(O; Xj - vk)]), (6.16a) - yk) = exp(h(1 - h/2~)-l iPB,(O; xi - yk) + ##‘A(1

+ h/257-l i-l[B,(mJ+ ; xj - JQ”)- B,(O; xj - y,)]). (6.16b)

The y5 factors act on the indicated spinor indices, and [F is obtained from Pk by replacing $#’ by unity. This solution results from the choice of operator solution @ = y5(m;2c”‘2,U,

+ milC)

+ rn,lv,

A,” = (V’ + mS1d’78TC + milpy)v

,

(6.17) (6.18)

for the Schwinger current; the substitution m, + m,, , m, --+ m,- will give the solution for the Johnson current. Solution (6.18) enables us to write down the meson correlation functions. The one-point function LGi,(Avu)= 0. The two-point function is &,,(A,“(x) A,‘(y))

= c$)(U”(x)

U’(y)) + (l/ms)2 E~‘E~~~~+~~‘w,(~)(C(X) C(y))

+ WmJ2 &“%‘P(dx)

KY))

(6.19)

= (i/ms2) ~+l~+lBv(mS ; 4) - (i/ms2) ~+ltY+liB,(O; .$) + (i/m,2> IOWA

0,

similarly for the others. The operator solutions we have given do not satisfy the second equation of motion (6.1 b) as an operator identity on all of ZV. The difference operator on XV, R,” = (0,

- m,2) A,” - gj,”

= -(rn,a,

+ msa,,PC - gj$‘)y ,

(6.20a) (6.2Ob)

which is not identically zero. Let 8, be the polynomial algebra generated by #,,$,, and A,, so that B, C dV . Direct calculation shows that R,“(F) E 9”‘: note that we are proceeding as if R, were bounded, which it isn’t. Conceivably

102

DANIEL

DUBIN

one could supply the details either of considering algebras of unbounded operators, or of considering the spectral projections associated with R, . It seems very unlikely that any conclusions we come to by treating with R, directly would be modified if that were done. As R, is selfadjoint (formally, anyway), let us write R,“(F) = (S”)*S” (a = 0, 1). We also see directly that b,(R,“(F))

(6.21)

= 5,(S=‘*P) = 0.

Then S” E 9L , the left ideal associated with the GNS construction for o, . Using B,(ZP) = 0 for every T E 8, it follows that R,” E 9L : iYiQR,“*R,“) can be written as &,(TSh) with T = R,O*S h* . Consequently G,(TRva) = 0 for all T E LP:,. This means that the matrix elements of the second equation of motion (6.lb) restricted to the “physical” subset of XV given by 3” = 8,l-J”

(6.22)

will all vanish:

(6.23)

This is as strong a conclusion as is true for the T = 0 field theory situation. Let us compute the density of A-particles. To do this, suppose we consider the term %p, alone. Now the density associated to 9 in the w,-state is JR/%-‘(0;wkw40; WI, where the zero refers to the null mass. The density associated with m&180pl is determined by a quadratic expression in %q in which -@8, appears, the first acting on ekibx, the second on eFilEs;this leads to an extra is then factor (&)(-i&J = I k I2 in the integrand. The density contribution E)) p”b-‘(0; k) dk. The field C appears similarly, but with an SWNE” - k2Mmo2 additional factor (-1) as uF and aF* for C are “equal” to -uF and +F* , respectively, for IJJ.Using the decomposition U0 = 8°+1tp(m,), we can find the density contribution for this field as well. Adding them together gives H4

= J-,

c2(ms) - k2

ms24md

(-)

ri; h;k)dk

+

~~(0) - k2 ---1 ( mo2 c(O)

1 ms2 1

p”F’(O; k). (6.24)

Note that as TV-+ 0, [r2(0) - k2]/c(0) -+ lim,,,(S - p2/8), so we must take t..~< 0. For p < 0 the above density is finite and nonzero.

THERMAL STATES OF VECTOR MESONS

103

We could similarly compute the density p(#) of #-particles. Because the same chemical potential p would appear in both expressions, the densities would not really be independent of one another. This points the way to using two potentials, pL1and p2, the first in # the second in A, as expressed in terms of the auxiliary fields. To do so is possible, and all the necessary computations have already appeared in previous sections. There would now be twice as many auxiliary fields, one set for p1 and another for pz ; similarly we have Q1 and Qz . The difficulty is that # and A would no longer be related by the same equations of motion, for now $h = {eig@l}”a)?‘,

(6.25a)

but k = (s@J” .

(6.25b)

Thus there seems very little doubt that modifying the thermal Wightman functions by introducing pI and pz will give well-defined distributions with independent densities; the more important structure of the equations of motion and the KMS condition would be lost thereby. For this reason we take it that a density constraint between p(A) and p(4) occurs if the equation of motion and the KMS condition for this model are those we have stated, i.e., using the potential p and derivatives 3. As we do not understand the origin of this constraint we cannot claim that it ought to occur: conceivably a different analysis could avoid it. The only constructive remark we can make is that p seems to regulate the density of particles associated with 0 as a primary object.

7. LIMIT TO THE THIRRING MODEL The Thirring model [22, 25, 26, 391 is that of a massless spinor field interacting with itself via the current-current interaction of the theory of weak interactions. For the T = 0 case, the operator solution and the Wightman functions for this model may be obtained from those of the vector meson model by taking the simultaneous limit m, -+ co, g -+ co and having g2/m2 = h remain fixed between 0 and 27r. The limit is meant in the Y’ distribution sense [25]. If we consider the limit m + co, g --f 00 and h fixed for the exponential T,(x) = exp(fi(

g2/m2) D(-)(m,

x)),

(7.1)

we find that in this limit T,(x) -+ 1 in Y’ and that (T, : m = 1,2,...} is weakly bounded. This suffices for the convergence of the Wightman functions. For the thermal states we have a similar estimate to make, but for f%(x) = expW(

g2/m2) Mm, xl).

(7.2)

104

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DUBIN

The estimate (2.35a) can be used here. Calling (1 - (p/m))-l[2 b, and noting that G = e--ituF, we have

[ 1 - .$;I 1-l + l] = (2.35d)

(g21m4 I Wm, 41 e (g2h2) bv I ~i?(x)l;

this enables the computation for T, to be used to show that {S, : m = 1,2,...} is weakly bounded and 9” - lim,,, S, = 1. All the considerations in [25, Sect. V] for the limit solution to the Thirring model remain valid with the replacement 3, + a, and the use of the p-modified fields. For the Schwinger current and Lorentz gauge, this entails replacing the vector field by zero, U0 + 0; and its exponential by unity, I’(“) -+ 1. The field AyU--f 0 and the fermion field is given by [25, Eqs. (5.11-5.12)]. This solution is valid only on the subspace of Xv indicated there, and this can be proven precisely as we dealt with the equation of motion (6.lb).

APPENDIX

The distribution

A

I&-’ can be written Et,!-’ = BL”’ + A, with A”(f)

= j)(k)

c-l dk.

We must show that B,(O) = j-- c-l dk

diverges. For then B:-)(O) = &,(O) + p, where p is the density. Writing Am < +I, A”(O) = -(d/dt)F(t

p =

= 0),

with F(t) = 1-T exp(- t cash x)(cosh x - $)-l dx. Now F satisfies the differential equation (dF/dt) + $F = -2&(t)

with [50] modified condition is

Bessel function

K. as inhomogeneous

term. The “initial”

THERMAL STATES OF VECTOR MESONS

where 01= cash-l(-fi)

105

is a characteristic real constant. Then F(l) = e-fit [E(O) - 2 jot efiSKo(s)ds],

so that -(d/dt)F(O) = [--2Fa csch 01+ 2&(O)]. But K,(O) = lim,,, K,,(t) = co does not exist. For the region 0 < $ < 1 there is a much simpler way of proceeding: for then cash x/[cosh x - 61 > 1 so that A,(O) 3 j’: dx = co.

ACKNOWLEDGMENTS I would like to thank G. Sewell for a number of profitable discussions. Part of this work was done while I was a guest at the Institut fur Theoretische Physik in Vienna. It is a pleasure to thank Professor W. Thirring and the Overseas Travel Fund Committee of the Open University for financial support. Amongst my many colleagues in Vienna who were generous with their time, I should like particularly to thank M. Breitenecker, H. Grosse, H. Narnhofer, A. Pflug, W. Thirring, and A. Wehrl.

REFERENCES 1. G. VELO AND A. S. WIGHTMAN Eds., “Constructive Quantum Field Theory” Lecture Notes in Physics, Vol. 25, Springer-Verlag, Berlin, 1973. 2. G. IVERSON, A. PERLMUTTER, AND S. MINTZ, “Fundamental Interactions in Physics and Astrophysics,” Plenum Press, New York-London, 1973. 3. B. SIMON, “The p(& Euclidean (Quantum) Field Theory,” Princeton University Press, Princeton, N.J., 1974. 4. D. RUELLE, “Statistical Mechanics, Rigorous Results,” Benjamin, New York, 1969. 5. G. G. EMCH, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory,” Wiley-Interscience, New York, 1972. 6. D. A. DUBIN, “Solvable Models in Algebraic Statistical Mechanics,” Clarendon Press, Oxford, 1974. 7. R. N. SEN ANLI C. WEIL Eds., “Statistical Mechanics and Field Theory,” Halsted Press (Wiley), Israel Universities Press, 1972. 8. R. HAAG, N. HUGENHOLTZ, AND M. WINNINK, Comm. Math. Phys. 5 (1967). 9. D. KASTLER, J. C. T. POOL, AND E. THUE POULSEN, Comm. Math. Phys. 12 (1969). 10. F. ROCCA, M. SIRUGUE, AND D. TESTARD, Comm. Math. Phys. 13 (1969), 19 (1970). 11. D. A. DUBIN AND G. L. SEWELL, J. Math. Phys. 11 (1970). 12. M. TAKESAKI, “Tomita’s Theory of Modular Hilbert Algebras and its Applications,” Lecture Notes in Mathematics, Vol. 128, Springer-Verlag. Berlin 1970. 13. E. B. DAVM, Comm. Math. Phys. 27 (1972); 33 (1973); 39 (1974). 14. K. HEPP, Results and problems in irreversible statistical mechanics of open systems in

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15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

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J. T. LEWIS AND J. V. PIJLE, Comm. Math. Phys. 36 (1974). 47. J. T. CANNON, Comm. Math. Phys. 29 (1973). New York, 1966. 48. R. L. PHELPS, “Lectures on Choquet’s Theorem,” Van Nostrand-Reinhold, 49. R. P. MOYA, Thesis, Queen Mary College, London, 1974. 50. F. MAGNLJS AND F. OBERHETTINGER, “Functions of Mathematical Physics,” Chelsea, New York, 1949. 51. H. J. BORCHERS, Nuovo Cimento 15 (1960). 52. A. S. WIGHTMAN AND L. GARDING, Ark. Fys. 28 (1964). 53. H. ARAKI, J. Mathematical Phys. 4 (1963); 5 (1964); Progr. Theoret. Phys. 32 (1964); 32 (1964). 54, J. TARSKI, J. Math. Phys. 5 (1964); 7 (1966); “Lectures on The Infrared Problem in Quantum Field Theory,” Imperial College Lecture preprint ICTP/64/42, 1964. 55. D. A. DUBIN AND J. TARSKI, J. Mathematical Phys. 7 (1966). 56. L. S. PONTRIAGIN, Izv. Akad. Nauk SSSR Ser. Mat. 8 (1944). 57. I. S. IOHVIDOV AND M. G. KREIN, English transl., Am. Math. Sot. Transl. Ser. 2, 13 and 34. 58. D. A. DUBIN, Nuovo Cimento Ser. X, 673 (1970). 59. H. ARAKI AND W. Wvss, Helv. Phys. Acta 37 (1964). 60. J. SCHWINGER, Phys. Rev. 82 (1951); 128 (1962); Phys. Rev. Left. 3 (1959). 61. K. JOHNSON, Nuovo Cimento 20 (1961). 62. J. MANUCEAU, F. ROCCA, AND D. TESTARD, Comm. Math. Phys. 12 (1969). 63. I. BIALYNICKI-BIRULA, J. SNIATYCKI, AND S. TATUR, BUN. Acad. Polon. Sci. 11 (1963). 64. C. SOMMERFIELD, Ann. Physics 26 (1963). 65. W. THIRRING AND J. Wsss, Ann. Physics 27 (1964); J. WESS, Acta Phys. Austriaca, Suppl. II (1966). 66. G. WRAITH, Thesis, Cambridge Univ., 1964. 67. J. SCHWINGER “Theoretical Physics” (A. Salam, Ed.), IAEA, Vienna, 1963. 68, H. M. FRIED, “Functional Methods and Models in Quantum Field Theory,” MIT Press, Cambridge, Mass., 1972.

46.