Generalized free fields and models of local field theory

7/29/2019 Generalized free fields and models of local field theory

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AXK.~LS OF PHYSICS: 16: 158-176 (1961)

Generalized Free Fields and Models of Local Field Theory*0. \;li. GREESBERGt$

Phy sics Department and Laboratory for Nuclear Scien ce, dlasrach ~~seftsInstitute oj Technology, Cambridge, Massachuseffs

C;eneralized free fields are a generalization of the free field in which the com-mutator is a c-number hut does not satisfy a homogeneous Klein-Gordonequation . Some properties of generalized free fields are examined includ ing theappropriate generators of the inhomo geneo us Lorentz group, the number opera-tor, spectrum, locality, and asymptotic fields. These generalized free fieldsare used to construct a family of local field theory models which is not unitarilyeyuivalent to any (generalized) free field.

I. INTRC)I)UCTIOiXLocal quant,um field theory is thr main theoretical tool in the study of ele-

mentary particle physics. However, up to now, there is no example, which isknown to he consistent, of a local quant,um field theory in which wattrring orreact,ions occur. Seit,her is there a theorem which states the existence of suchtheories. liurt~hcrmorr, there are few examples of any kind which ratisfy thereyuirement,s of local field theory. In this paper WC introduce generalized frrrfields, &nd use t,hese fields to construct a set of models which satisfy the rcquirc-ments of local field theory, hut, in which no scattering or reactions occur.

k?rst we drfinc these generalized free fields and rsnmillr romc of their proper-tics, including their generators of translat8ions and Lorcntz transformations,number operator, spectnlm, locality, and asymptotic fields. WC then uw thcscgeneraliztd free fields to con&xct a set of field throry models which satisfy therequirements of local field theory and are not (unitarily) cquivnlcnt to auy(generalized) free field. We t)ake as the requirements of local fie ld theory thefollowing four properties: I---relativistic transformation properties, II-unique,uormalizable, invariant vacuum state \ko , and no nrgativr energy states OIstates of space-like momenta, III-local commutation relations, Iv-complcte-ncss of the set of states obtjained by applying polynomials in the smeared field

* Th is work is supported in part through U. S. Atomic Energy Comm ission (ontract.At(301-1)2098, hy funds provided by t,he U. S. iltom ic Energy Com mission , the Office ofNaval Research, and the Air Force Office of Scien tific Research.

t National Science Foundation Post-doctoral Fellow.1: Now at Phy sics Ikpartment, University of Maryland, College Pz,rk, Maryland.

158


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(~ESER.~I,IZICI) FREE FIELIS 159

operat*ors to the vacuum state \EO Our models are characterized by an arbitrarymeasure and an arbikary symmetric function of two variables. Sonc of theremodels lead to scattering or reactions; they all have N-matrix equal to onr; hon-ever the vacuum expectation values and Greens functions of there models differfrom t)hose of the free field theory. One reason that S = 1 is, roughly speaking,that, the vacuum expectation values for these models, considered in momentumspace, agreewit,h the vacuum expectation values of the free fie ld in the neighhor-hood of t,hc stable particle mass shell p* = We, even though the vacuum csperta-tion values of these models differ from those of the free fie ld away from this massshell. Xone of these models have canonical commutat,ion relations.

In Section 2 we state properties I through IT, define the generalized free fields,and study their properties. Section 3 discusses the set of models of loral fieldt,heory constructed using generalized free fields. The last section sunnnarize::t,hc work of this paper.

II. REQUIREMENTS OF LOCAL FIELI) THEOR Y; I)EFINITION ASDPROP ERTIES OF (:ESERALI%E ;I) FREE FIEI,l)S

A. REQUIHEMENTA OF LOCAL FIELD THEORY'Property I requires that the fie ld A (s ) transform mider a Gtary representa-

tion of tbe inhomogeneous Lorentz group, I( a, A), where a is a space-time trans-laGon, and h t L+r (the orthochronous group of Lorentz tra,nsformations ofdet,erminant# one), as a scalar field

C(a, A),4(x)U(a, A- = A(Ax + a). (1)Property II requires that

CT(u, ii)*0 = \Il, (galor

I%0 = 0, and dd\ko = 0, (2b)where 90 is the unique, normalizahle, invariant vacuum st,at,e of the theory, andI and JP are the generators of spare-&me translations (i.e., the momentum-energy operat)ors) and generat)ors of homogeneous Lorentz transformations (i.e.,the moment, of energy and angular momentum operators) of the theory respec-tively. The spectrum of P0 must be bounded from below by zero. Property IIIrequires that the commutator of the field operators at two different points vanishfor space-like separation of the points,

L4(2), A(!/)1 = 0, (.r - y)' < 0. 1.3)(Note that we use the metric for which s = so2 - x2. ) Propert,y IV requires that

1 For an extensive disc uss ion of the requirements of loca l field theory, see Ref. 1


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160 GHEENBERGany state \k can be approximated in the mran by a state of t,he form

5 / dr, . . . d4.#(x1, . . . rj)A(sl) . . . A(r,)qoj= U

for N finite. Finally, we introduce t,he notions of canonical fields and asymp-tot,ic fields (1-s). Property V- the canonical property-requires that (a) thereare operators A (x, t), T(X, t j which satisfy the equal time commutation relations

M(x, t), dY, Gl = iS(x - Y),M(x, t), A(Y, t)l = Idx, t), T(Y, Gl = 0,

and (b) this pair of operators at a given time suffices t#odefine the whole theoryin the sense of Property IV, where A(x, t) and a( X, tj replace A(s) and theintegrat#ions run over d3x. Property VI-the asymptotic condit,ion-requires thatout(a) the asymptotic fields Ai (2) of stable mass WLdefined by

0tAi (.r) = limr-*CC C-Uc-3(where B $ C = B(aC/&r ) - (aB/&r ) C> exist in t#he sense of weak operatorconvergent; of the smeared operators (that is integrate both sides of Eq. (4)wit,h f(x)dx, where .f is an arbitrary testring functiou in the Schwartz space s),outsatisfy the free Klein-Gordon equation of mass 1~2, 0 + o?)A = 0, trans-form as scalar fields under ((a, A), have free field commutaGon relations,

out out[Ain (x), A (yj] = iA,z(.c - y) ,and have a posit#ive requency part, which annihilates the vacuum,

2 We define positive and negative frequency and spacelike parts of a field A (2) byA()(L) = 11/(27rPl s flJkej(k)e-~-d(l/) CP?/,

wherej = zt OP S, and e,(k) = ti(z!zk) or e(-P),

respectively. For a vector k.

and for a numbere(k) = 1, x.2 2 0, ko > 0,0, otherwise


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GEh-ERALIZEU FREE FIELDS 161Out

A in +(.r)\k = 0;(1)) the asymptotic fields are complete in the sense that Propert,y IV holds withwith iloUt or il in (s) replacing i4(.r).B. DEFKITIOK OF GEXEHAL IZED IHEE FIELDS

We define generalized free fields to be field operators3 which satisfy t,hecommutation relation4

[4(x), 4(y)] = iA(x - y) = i lrn dap(u)A,z(.r - y), (5)where

A,z(s) = --[i/(&r)] [ d%e-~(k)6(k* - a)is the free field commutator function for mass a, and ~(a) is a positive measure5of not too fast increase. We cal l p the Lehmann weight (5) of the field 4. Inorder that the generator of time kanslations, PO, have a positive def inite spec-trum (Propert,y II), it, is necessary that

f#J(+yr)\k = c#F(a)\ko = 0, (6)where Pa is the vacuum stat,e. From Eq. (ci j it follows that [+(x), 4(y)] = 0.This fact together with Eq. (6) implies that al l matrix elements (x, @q) vanish,where x and Q are any st,ates in the Hilbert space belonging t,o 4. Thus +((x) =0.

As a concret,e example of a generalized free field to which we will refer later,we use the case A where there is a single disc&e mass II) in addition t,o a con-tinuum, and

p(a2) = s(u - 777) + fJ(u2);fJ(u2) = 0, a < (2m); du) > 0, CL22 (h)*.

For t#his example A we define a separation of the field 4(s) into its discrete andcont~inuous mass parts, &(.r) and +1(a), respectively:

3 For simplicity we assume that a ll the fields we discu ss are neutral, scalar fields.4 Wellner (4) has used similar commutation relations in a model of decaying elementary

particles. Licht and Toll have shown that a complete, local field 4(s) w hich has a translationinvariant commu tator is a generalized free field.

5 In the present paper we assume that the space of state vectors is a Hilhert space, i.e.,a space with a positive definite metric. If this assumption is not made. then p need not he apositive measure.


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162where

GREESBERG

&J(X)4ke- ik(x--u)0,2( i?)cjb( y) d41J,and O,z(li) is one in a small neighborhood of /? = TI? and zero elsewhere. Frombhe definitions of +0 and +l , we find the commutation relations[ddt),~d~)l = iAmz(:r - Y), [~J~(x:), h(y)1 = i/mI+edaa($)Aa2(z - Y), (7a)and

bo(x.), &(!/)I = 0, Vb)together with

&lyzNfo = 0. (8)Equations (7) and (8) require that 40(z) be unitarily equivalent to a free fieldof mass m so that +o(z) satisfies t,he free Klein-Gordon equation,

(0 + m%o(5) = 0.If ~(a*) = 0, we recover the usual free field of mass 1)).

We conclude t#his subse&on by showing that Eqs. (5) and (6) determine thet,heory of the single field Q(X) completely, provided we assume that $J(x) is theonly field present in the theory. Our demonstration that, Eqs. (5) and (6) deter-mine the t,heory rests on the fact that tzhe knowledge of al l the vaccum expecta-t,ion values of the t)heory,

{F(Xl . . xn); I1 > O),where

pP)(.J.l ) . . xx) = c*o, 4421) . . . d.rvNo),determines the t,heory completely (1). With no loss of generality, we assumet,hat F{ki = 0. Using Eq. (5) and (6), the Hermiticity of +(x) and the defi-nit ion of +*(x), we computeF(x1 , x.2) = ($(x~)\Eo , $(xz)Po) = (PO, ++(.~,)4O+h)

= (qo, [$+(.I& c$(x2)]\ko) = iA+(.r, - x2) = i / m d&(i)A:t(.rl - x2),06 Th is assump tion involves no los s of generality beca use if Fll(s) = (UO , +(r)Po) # 0,

then in place of @J(Z) we can consid er d(s) = 4(x) - F(, sinc e by translation invarianceF(l) is indepen dent of s. Equa tions (5) and (6) stil l hold for C#J(CZ); but nowfit!)(s) = (qO , c$(z)Y~) = 0. If we wish we can i-ecover the origina l 6 by adding F( to 6later.


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GESEH.ILIZED FREE FIELDS 1G3

whereA:+'(X) = --[il(2~)"] 1 d'kC'k"@(1


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164 GREENBERGand7

(9d)where

K(y) = 4/2@7$1 I,, b2/&)1&~(~). (se>Note that the PF' and P:" are four-dimensional nonlocal functionals of 4(x)(even though the P,? are three-dimensional local functionals of 40(r) ) in con-trast to the three-dimensional local form of the usual free field translationoperators8

Next we consider Lorentz transformations, U(0, A) = exp [(i/2)M,,y~Y].Now Eq. (1) implies

Assuming that M,, is a bilinear functional of +(z), we find the expressionM,, = ML:' + Mb:',

where again M,, I is the free field functional of 40 ,Al;;' = - s LO d3X[Z,CP$yrCJ,) - X" 6:u'(x0, x)],

(lOa)

(lob)7 The double dots in Eq. (Yd) stand for normal ordering which is defined for generalized

free fields in the same way that it is for free fields ; that is all posit,ive frequency (annihila-tion) ope rators are to be moved to the right an d all negative frequency (creation) operatorsare to be moved to t,he left.

8 There i s another way to write the continu um ma ss contribution to the generalized freefield Hamiltonian (i.e., generator of time translations ) in which PA is expressed as anintegral over masses of free field Hamiltonians of a given mass. We write

u-here

is defined only for a* in the support of (r. Note that the operation which define s +.2(r) is anonlocal four-dimensional procedure so that PA .S not determined by 4(x) at a given t,imeeven though it is given in t,erms of the set ( +oz(s), a2 in the support of g } at a sing le value of~0. Expressions analogous to that for P, can be written for the other Pk and for the MOPand N(r).


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GENER-4LIZED FREE FIELDS 165and iu:; = - sd4.r[a,l :I) s,s&yx)1. (1Oc)?;ote that the point, (0, 0, 0,O) has been cho,sen as the center of rotation (Lorentztransformation) for the operators M,,, .

Equations (9) and (10) for P, and M,, define the representation of t,he in-homogeneous Lorentz group which is appropriate for generalized free fields; withthis representation Propertly I is satisfied. The number operator for generalizedfree fields is N = N 4 Nil, where

N = (i/2) 1 d3X [ml(x) i $b:+)(J)]is the usual free field functional of ~$0 and

wherey(1) - 12 C27r)2s da2 Ai% - y)c7(a2) d4.r d4y : 41 s) c#a,

A::(x) = [1/(27r)3] / d4k6(k2 - $)e-?

Cl/):,

The Appendix gives momentunl space expressions for P, , M,, , and N.D. SPECTRUM ASD LOCALITY

Property II, that the theory has a vacuunl state and a positive energy spec-t,rum follows from t,hc construct,ion of P, , M,, which satisfy Eq. (2b) by virtueof Eq. (61, and the fact that any functional of generalized free field operatorshas a normal ordered expansion

The st)ate O\ko has contributions only from the crcaGon (negative frequency)parts of 4; thus the spectrum of PO is bounded below by zero. Property III,locality, holds because of the local property of A,z(.r - ~1) in the defin ition of 4,Eq. (5).E. POSITIVE I~EFIKITEXEM COXDITIOAX (1)

In t,his subsection we show that the positivity of the Lehmann weight, p is anecessary and sufficient condition that the states created by smeared poly-nomicals in the generalized free field + acting on the xTacuum state have posit#ive


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166 (:ItEENUEltGnorm, i.e., that t,his vector space be a Hilhert space. Necessity is immediatebecause it is already known (5, 1) that, the positivity of p is necessary for thepositivity of /j +(.f)90 11 = (\ko , +(.&(f)\ko) where 4(f) = jd4.r+(rjp(s), and ,fis a testing function in the Schwartz space S. We show sufficiency direct)ly bycomputing the norm of any vector and finding that) it, is positive. ,\ny vector \kcan be written \k = cz=O \Ir,, , where

9, = c crnimlr~ n,J-:2 n P(.fT, )90j rli,l.zi,= u ais a vect#or in the n-yuant,um subspace, (,n,,\ are the occupation numbers forquanta in t,he st,ates (iJ and CIN{ni,] is the probability amplitude that the statewith occupation numbers [ni,j is in t.he vector q,, . The wave functions f; arechosen to be a complete orthonormal set with the scalar product

Then the commutat)ion relations of 4(x), Eys (5) and (6), lead to the mani-festly positive norm

This completes the proof t,hat# a positive Lehmann weight in a necessary andsufficient, condition that the positive definiteness condit,ions are satisfied for thetheory of the fie ld 4; and also concludes our demonstration that Properties Ithrough IV are sat,isfied for a generalized free field 4.I?. hYM PTO TIC b. IELI )S

outIn this subsection we study the asymptotic fields +in associated with 4 fromthe point of \-iew of Property VI. We find that) thesc asympt,otic fields exist andhave t,hc properties which we collectively cal l Property VI (a). In particularr#loUt(.r) = p(s) = c#Yl(J),

out.and the S-matrix, which relates +in by$o(r) = S--$p(r)S

is t,he identity operator. Thus neither scattering nor reactions are described by ageneralized free field. The st#atements of this paragraph follow straightforwardlyby embedding the defin ition of the asymptotic fields, Eq. (4), in an arbitraryvacuum expectation value and performing the lim it 7 ---) f = on the distribu-tions t,hus obtained.

D he limit, in the sense of tlistrihution theory,


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GENERrZLIZED FREE FIELDS 167

However Property VI (b) is not satisfied; that is the asymptotic fields are nota complete set of fields for generalized free fields. Property VI (b) is equivalentto the existence of an expansion of 4(r) in terms of normal ordered products of@*l(r) = +0(x). If the expansion($(s) = g1 SC&, . . . &,f(x - gl, . . . .r - y ,,) :p(yl) . . . @(grt): (11)

exists, then t,he expansion coefficients can be computed in the following way:

(12a)= 4y, d4y,f(21(x -. gl, .r - ti)[iArnz(~l - zl)iA,z(u.~ - 2~) (12b)+ ;A,z(y, - zs)iA,n(y, - x,,]

and in a similar way the vacuum expectation value of the lath interated com-mutator leads to an equation for f. On the other hand, the expressions on t)heleft of Eq. (12) can be computed direct,ly from t#he commutation relation, Eq.(a), which yields the result(b#d~,,@(~~)l)o &z(.r - 21) (1Sa)

([. . . b(J), 4Ji(G)l, . . 4in(z,l)])0 = 0, n 2 2. (131~)Comparing Eqs. ( 12) and ( 13), we find t#hat

.fw = Y!/), (143).f(y1 ) j/2 ) . . . yn) = 0, n L 2, (14b)

sincef(kl, . . . I;,) has support kc = n?. (Throughout this paper a tilde meansFourier transform.) Thus the result, of our expansion is 4(x) = $+(x) whichis clearly wrong. We conclude that the asymptotic fields are not complete for ageneralized free field theory.lOm

limi+*oc(1) = ,,p

0, otherwiseallows a straightforward discuss ion of the asymptotic limit for generalized free fields.

I The fact that the LSZ asymptotic fields fail to be complete in this simple case leads tothe question: Are these asymptotic fields complete for a local field t,heory with interaction?The assumption that the answer is yes seems tjo be part o f the orthodox position in localfield theory (2); however, this com pleten ess assum ption has not been exploited in manyapplications of local field theory.


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168 GREENBERGIII. CLAS S OF MODELS OF LOCAL FIELD THEOR Y

Now we use the generalized free fields discussed above to construct models oflocal fie ld theory. In particular we construct1 models which can be exhibited inclosed form and are exactly local. The problem of finding Heisenberg fields whichare local in successiveorders of some approximation scheme s also interesting;however we do not discuss hat problem here. We study a Heisenberg field A (x)which can be expandedI in a series of normal ordered products of a generalizedfree field 4(x),A(x) = 4(x) + g2 1 d4yi . . . d4y,,

(15).f(x - y1 , . . . x - yr,):c$(.rl) ... $(yn):

where f((vl , . . . yn) = f( Ayl , . . . Ay,) is a real Lorentz invariant functionwhich is symmetric in it)s arguments. We assert t)hat Property III, locality, is theonly one of the four requirements of field theory that remains to be imposed onthis expression for A(x). Property I, relativistic invariance, is satisfied withU(a, A) equal to the operator determined for the generalized free field @B(Z)nSubsection II (C). This fact follows fromU(a,A)A(x)U(a, A)-

= gl j d4y1 . . . d4y,rf(.r - y, , . . . .I: - yn) :+(Ayl + a) ... +(Ay, + a):

= 2 j d4X1 . . . dx,,f (hx + a - x1 , . . . hx i- a - z,,) :$(G)u=l. . . $(z,) : = rl(Ar + a),

I1 Note added in proof: Haag and Schroer (to be pub lished) have proved that a gener-alized free field whose Lehmann weight contains a continuous contribution is not com-plete in a time slice.

12 An analogous situation holds with respect to the canon ical property. Property V(a)is satisfied with 4(x, t) and 4(x, 1) as can onica l field s and a nave-function renormalizationconstant in the relation

[6(x, tJ,t#i(y, t,] = i s co da* p(aM(x - y).0However Property V(b), com pleten ess of this pair of operat,ors at a sing le time, d oes nothold unless the field is an ordinary mass VL free field. We do not exclude the possibility thatsome other pair of canon ical fields at a single time is complete. Single time completeness isone of the assu mptio ns of Haa gs theorem (6) ; therefore there is no straightforward exten-sion of Haa gs theorem to field s related unitarily at a given time to a generalized free field.I3 Haag (6) has considered the expansion of a Heisenberg field in terms of a mass 7~ freefield. Sudarshan and Bardakci (to be published) have used the Haag m ass 17~ ree field espan-sion to construct local models analogous to t)hose considered in this section.


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GENERALIZED FREE FIELDS 169

where we have used the change of integration variables .zi = ~yi + a, the Lorentzinvariance of the fCn, and-to include the term $(.r) in the sum-have intro-duced f(r) = 6(s). Property II, positive energy spectrum, follows from t)hediscussion in Section II (D). The positive definiteness conditions follow froma discussion similar to that of Section II(E). Thus only locality requires adet,ailed discussion.

To discuss localit,y we observe that, the commutator [i4 (x), A(y)] can beromput#ed explicitly in terms of normal ordered products of 4(x). The result hasthe form

.I d4X1 . . . d4z, g (.le + y .r + ys - y; - x1 ) . . . 2 - 2, > :dzd . . 44X,8) :,

whereg(z; x1 , . . . x,,) = y(hr; & , . . . k) is real, Lorentz invariant, andsymmet,ric in ( z1 , . . . 2,). The locality of A (2) is equivalent to the requirement

gyx; 21 ) . . . x,,) = 0, x2 < 0, for all xi , and all n. 2 0.Since the gCn are funrt,ionals of the f and t,he Lehmann weight p, t,he localityof 9 imposes restrictions on the jand p.We now rest.rict ourselves to the simplest nontrivial case of Eq. (15) in whichonly the linear and bilinear terms in 4 are retained, and find sufficient, conditionsfor localit,y in this case. We discuss the locality of[il(.r), L4(y)] = / d41i1d4k2

. exp [.r + y-i(kI + li,) . 2 - i(k, - k,) . y 1X(lcl), B(k,)]

in terms of the commutjator of the momentum space fields, and rall t,his momen-tum space commutatjor loral if the corresponding x:-spacae commutator islocal. Note that, it, is the dependence of [A( k,), d(k,)] on p = li , - kc, which iprelevant for locality. The moment~um space field is given by

J(k) = l+(k)+ [ dpl(; + 23,; 23) :,(;+,>,(; -9): (16)where a numerical fact or has been absorbed in f. The commutator [A( kl), A( ,+?)Ihas t,hree terms cont,aining normal ordered products of zero, one and t,wo + oper-ators. The term wit,h no 4 operators is necessarily local, since it is the vacuumexpectation value of the commutator (6). The terms with one or two + operatorsare

2[t(lil)p(X:j)J( -kl , kl +li?) - t(li?)p(X:>').T( -kXr , k1 + X2)] r$(lil + kz) (17a)


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170 GREENBERG

respectively.To discuss the locality of these terms we use the theorem t,hat a necessary and

sufficient condition for a function 6(q) to have a Fourier transform which van-ishes for x2 < 0 is that, i(q) be representable in the form (7)

I;(q) = s s4U ds c(qO - .u)d[(y - uy -s]\k(u, s).[If, in addition, /i(q) vanishes in a given region bounded by t,wo spacelike surfacesc1 and g2 , then \k(u, s) vanishes outside a region which depends on (or and (~2We do not have t,o insert explicitly the analog of t,his support condition for ourcommutator [A( aI), B( k2)] ; the support condition for matrix elements of[A(h), &WI gs uaranteed since we st,art# from an operat,or expression forA(lz) .] We now exhibit the most general local expressions with one or two 4operators using t.he above t#heorem: for one + operator,

and for two 4 operators,

p (Mb). -6( kl + Ii2h*kpp 4 -2-p> ( ) :.

If we can find expressions for 1, and for II and \k2 so t,hat Eqs. ( 17a, b) havethe same form as Eys. (18a, b), respect#ively, then our held A (x) will he localThe ansatzes


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C;ESER.~LIZED FREE FIELW 171

change both expressions with one + operator to

and change both expressions with two + operators to

Therefore

is a suflicient condition for t,he locality of A (x). In r-space our result, is that, anyfield of the form

A(x) = 4(e) + 1 r((.c - yd?,(.x - yz)?) :+(y&(yd : d4yld4ya (19,is loc+al.

For the case where 4(z) is chosen to he a free field, 4(r), of mass nz, the localfield .?jk) reduces toX(/i) = &(k)d(k - WL) + y(??Z, VI) /d4dk(;+,)

.q;+q - wl$&(; -+[(g -q - $1 :,C:c do not know if there is a mno;iical fie ld a( x, t) m-hic*h, together with

4 (x, t 1, satisfies Iroperty 1. However, W can auswcr the cluestim: Is A(x, t)the cnnoni~nl field T-(X, t )? A%\swe might espivt for a theory which differs sotnur*h fro:11 a c~onvc~~tionnl uonderivati rc mipling theory, the nuswer is m. \\rshow t hi+ hy finding the co!iditioiis that

[;l(x, t), A(y, f,J = ciS(x - yl.


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172 GREENBERGThese conditions, which follow from a straightforward calculation, are

s10p(s)y(s, s) ds = 0, all s (20a)0

and

s*

p(s)y(s, s)y(s, s) ds = 0, all s, s. (20b)0If we set s = a in Eq. (20b), we find

sm

p(s)[y(s, s)] = 0, all s.0Then the reality and symmetry of y, and the positivity of p require

y (s, s) = 0, all s, sn,so that our field A(z) reduces to the generalized free fie ld 4(r).

Now we discuss the scattering matrix for these model? from the point ofview of the LSZ formalism (2, 3, 1) and show that S = 1, i.e., no scattering orreactions occur. Since the generalized free fie ld does not produce scattering orreactions, we can conclude that S = 1 if we can show that only the first term onthe right-hand side of Eq. (16) or Eq. (19) contributes to the asymptotic fields.Now the term in A(z) which contains :~(YJ~)c$(Y~): contributes only to matrixelements (9, :+(yl)+(yl) : x) in which t(he numbers of 4 quanta in \k and x differby zero or two. Thus t,o determine the comribution to the asymptotic limi t fromthis term it suffices tjo consider the asymptotic lim it appl ied to ,4 (x) in the matrixelementsM2 = ($*(pl)\ko ,A(.r)&*(pz)\ko) and IIf1 = ($*(P~)c$*(P~)\~o, -4(x)*0).We now compute the asymptotic fields in these two matrix elements in momen-tum space using the momentum space equivalent of Eq. (4))

outA (12) = limt+*%- e(k)G(k - m) / dqX(q, k) (k + ~~)e-~~*~-~~,

as the defin ition of tfhe in and out fields. We find

(21). lim s dq,y(p;,(p: + CJ)~ - (pl + k))(k + ~)~-ino+k2+m2r~.t+*Z?

14 It can he shown quite generally that any model which is constructed in terms of a finitedegree normal ordered expansion in (generalized) free fields e ither has S = 1, or has a non-unitary S-matrix.


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GENERALIZED FREE FIELDS 173According to the Riemann-Lebesque lemma this limi t vanishes unlessApt, (PI + q12 - (PI + W21 h as a singularity at which q + (k + v?)~ = 0.However, a function y depending on the variables indicated cannot have such asingularity, and thus the limits in Eq. (21) vanish. A similar argument showsthat the asymptotic limit appl ied to t.he matrix element M2 also vanishes. Weconclude that

out outA (32) = fp (z) = c#q(2-)and X = 1. The heuristic reason for this result is simple. To get AoUt # A(k)we must have principal value singularit,ies in matrix elements of A(k) at k2 = w?.Equation (16) for A(k) contains the c-number functionf( (k/2) + p, (k/2) - p)which a priori might depend on the three scalar products ((k/2) + p),((k/2) - p), and k2. Our condition for locality is that there be no dependenceon this third variable, li. Thus locality conspires to eliminate from our controljust that variable which is relevant for the difference between AoUt and A.

In this paragraph we show that the field A (2) which we have constructed is inhhe Borchers class (8) of both its generalized free fie ld 4(z) and t#he mass mpart, &(.r), of this generalized free field. To do this, we compute the commutator[A(k,), ~$(k?)] and show that it is locIal. The calculat,ion is

where we have introduccd the variables p = ,& + k, and (1 = k1 - k2 , and t,hesecond term is in the local Jest-Lehmann-Dyson form (7). Thus [A (z), 4(y)] =0, (X - g) < 0 and il is in the Borchers class of 4. ,4 similar comput8at,ion yieldsthe same result for the commutator [A(lz,), $o(l~s)6(ksZ - UZ)].

We have studied a simple generalization of the free field, in which t)hc com-mutator is a c-number but can have t)he space-timc dependence of the vacuumexpectat)ion value of a commutator of arbitrary Heisenberg fields, rather thansatisfying a homogeneous KleinGordon equat8ion. For the example A in whichthc generalized free field is taken to have a discrete mass part 40 at mass 1~1, nd acontinuous mass part, +1 starting at some higher mass, the generators of trxnsla-tions and Lorentz transformations al l have a term bilinear in 4 which has anonlocal four dimensional kernel. The asymptotic fields exist and are given by


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174 CREENBERC r

4in (x) = 40(x); however, these asymptotic fields are not, complete. Similarly4 and 4 at a given t#ime satisfy t,he canonical c*ommut8ation relations, hut, are notcomplete.

Csing these generalized free fields we constructed a set of models of fie ld theoryin which the Heisenherg field sat,isfies the requircment,s of rclat,ivist,ic covariancc,spectrum, localit#y, and part of the asymptotic condit,ion. The asymptotic fieldsfor these models are not, complete. Xo equation of mot,ion or time slice conditionis known for t,hese models. These models have S = 1, and are in the Borchersclass of their generalized free field.

We conclude by mentioning an idea, not yet developed, which is suggested bygeneralized free fields. Generalized free fields + can he considered to he the lowestorder case of models of fields whose commutat,or can be expanded in products ofHeisenberg fields cont,aining up to a given number of factors. For generalized freefieldstherelations[$(r), +(yj] = iA(r - y) and4+(r)% = +(x)@ = Ode-termine the field. Similarly, for the next more complicated case, the commutator

[A(r),A(y)l = iA(s - y) + [ d4zJ(x - y,$J - 2) A(x) (22)and the requirement,s

il+(r)k,, = .4(X)QS = 0 ( 23 )determine all the vacuum expectation values by recursion and t,hus again deter-mine the field. Using argumentIs similar to t,hose discussedabove, it, is straight-forward t)o choose f so t,hat the commutator of Eq. (22) satisfies translat,iouinvariance, 1,orentz invariance, and localit,y. One can also see that Eq. (23),toget,her wit,h translation invariance, insures a positrive energy spectrum. How-ever it is difficult t,o find conditions on p and j such that, (a) the commutator ofEq. (22) satisfies the Jacobi ident,ity, and (b) the vacuum expectation valuesgenerat,ed by recursion from Eqs. (22) and (2X satisfy the positive dtfinit)enrssconditions (1). Models satisfying Eqs. (22) and (23) have a considerably morecomplicat,cd structureI than those considered in Section 3 ; in particular t,hcydo not, have a finite degree normal ordered expansion in terms of a gcncraliseclfree field, and thlts may give rise t,o scaatteringand reactions.

APPlOY I)IS: MOMI~XTUM SPAC E P:SPR ESSIO iY;S FOR I, , iIf,,, ANI, 1VDefine t,hr Fourier transformed ( momentum space) ield $( I) by

I5 John son has remarked that the mmm ut:~tion relations of IlGi. (22) are those of thegenerators of a Lie group of ir1finit.e order.


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GENERALIZED FREE FIELDS 175

Corresponding to the separation @(.I+) = &I(X) + $I( x), write4(q) = io(qh%22 - )i) + 41(q).

The commutation relations in momentum space are[c&(p)6(p - d), &(q)6(qZ - n?)] = [l/(2a>lt(p)6(p2 - a4p + q),

[&(p)6(pJ - ))12), &(q)l = 0,k&(p), h(9)] = ~~/(2~)l~(P)~(P~~(P + 9).

The generators of translations are P, = Prco + P,?, wherepi) = K%r)/m [ d4pe(p) a -p)p, &(p)6(p2 - m) :

andIJ,:l) = [(2a)/2] 1 d4p14p)l&J2N :A( -P)P, 41(P) .

The generators of Lorentz transformations are M,, = AI,? + *lrb:, whereML;) = -y j d4pc(p) :$0(-p) aBo(p)p, dp - s(p2 - n?):

and(27r)iJ/f;; = -- s

aMp)2 P,----- -w

Finally, the number operator is N = N(O) + N(l), whereN = [(27r)/21/ d4p :&o( -pMo(pM(p - m):

andA+) = [(27r)3/21 [ dp[l/a(pN :d;1(-p)&(p):.

RECEIVED: June 23, 1961

We thank Professo rs P. G. Federbush, It. Haag, F. E. Low, and E. C. G. Sudarsha n forcritical discuss ions of this work. We also thank the National Science Foundation for a post-doctora l fellows hip during the tenure of which th is work was done, and Professor N. H.Frank for the hosp itality of the M.I.T. Phy sics I>epart,ment during the past two years.


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176 GREENBERGREFERENCES

1. A. S. WIGHTMAN, Phys. Rev. 101, 860 (1956); an d ProhlPmes mathe matique s de latheorie quan tique des cham ps, University of Paris lecture notes (1957).

2. H. LEHMANN, K. SYMANZIK AND W. ZIMMERMABN, Nuovo cimen to [lo], 1, 205 (1955). Werefer to this paper as LSZ.

PI. 0. W. GREENBERG, The sis, Princeton University, 1956 (unpublished).4. M. WELL NER, 1959 (unpub lished);A. L. LICHTAND J. S. TOLL, Nuovo cimento [lo], 21, 346 (1961).

5. H. LEHMANN, Nuovo cimen to [9], 11, 342 (1954).6. R. HAAG, Kgl. Danske Videnskab . Selskab, Mat.-fys. Medd. 24, No. 12 (1955);

I>. W. HALL AND A. S. WIGHTMAN, Kgl. Danske Videnska b. Selskab, Mat.-fys. Medd. 31,No. 5(1957) ;

0. W. GREENBERG, Phys. Rev. 116, 706 (1959);R. JOST, 1959 (unpublished).

7. F. J. DYSON, Phys. Rev. 110, 1460 (1958);R. JOST AND H. LEHMANN, Nvovo cimen to [lo], 6, 1598 (1957).

8. H. J. BORCHERS, Nuovo cimen to [lo], 16, 784 (1960).

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Generalized free fields and models of local field theory