Proc. Natl. Acad. Sci. USAVol. 77, No. 12, pp. 6943-6947, December 1980Applied Mathematical Sciences

Quantization of wave equations and hermitian structures in partialdifferential varieties

(quantum fields/scattering theory/physical vacuum/hyperbolic equations/infinite-dimensional varieties)

S. M. PANEITZ AND I. E. SEGALDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; and Institut fur Theoretische Physik, TechnischeUniversitat Claustal, 3392 Claustal-Zellerfeld, Federal Republic of Germany

Contributed by I. E. Segal, August 4, 1980

ABSTRACT Sufficiently close to 0, the solution variety ofa nonlinear relativistic wave equation-e.g., of the form 30 +m20 + gO P = 0-admits a canonical Lorentz-invariant her-mitian structure, uniquely determined by the consideration thatthe action of the differential scattering transformation in eachtangent space be unitary. Similar results apply to linear time-dependent equations or to equations in a curved asymptoticallyflat space-time. A close relation of the Riemannian structureto the determination of vacuum expectation values is developedand illustrated by an explicit determination of a perturbative2-point function for the case of interaction arising from curva-ture. The theory underlying these developments is in part ageneralization of that of M. G. Krein and collaborators con-cerning stability of differential equations in Hilbert space andin part a precise relation between the unitarization of givensymplectic linear actions and their full probabilistic quantiza-tion. The unique causal structure in the infinite symplecticgroup is instrumental in these developments.

Quantization of a differential equation, as normally understood,involves two qualitatively distinct aspects. The first, which maybe called "formal," is the construction of a "quantized" solutionof the differential equation. By quantized is meant operator-valued rather than numerically valued; by solution is meant inthe distribution-theoretic or other suitably generalized sense.Typically there are many such solutions, even with the impo-sition of canonical commutation relations, which prescribenumerical values for the commutators of field values of pointsof space-like separation, together with strong regularity andgroup-invariance restrictions. This plethora of solutions arisesfrom the failure of the conclusion of the Stone-von Neumanntheorem in systems of infinitely many degrees of freedom or,stated in another way, from the existence of many mutuallyinequivalent representations of the canonical commutationrelations that are entirely smooth and invariant.

In certain cases, this ambiguity in the quantization may beappropriately suppressed by using field values that are affiliatedwith a C*-algebra, which may be regarded as nascent operators,rather than values that are operators in a given Hilbert space.But this procedure does not in itself resolve a second aspect ofthe quantization problem, which is essential from a physicallyempirical point of view. This aspect, which may be called"probabilistic," is the determination of the expectation valuesof the solution and general functions thereof at arbitrary pointsof space-time. Unlike the case of classical wave equations-i.e.,those whose putative solution is numerically valued-there is

typically a unique such determination in the case of quantizedequations that is specified in a natural manner and plays afundamental role. The concept in question is known as the"vacuum state expectation value functional," or, for short, the"physical vacuum.

In the case of linear equations, the formal problem has beenrelatively well developed and understood. There is no essentialdifficulty in dealing with time-dependent equations, or equa-tions on a Lorentzian manifold, as a result of investigationsduring recent decades. The problem is simplified by the cir-cumstance that it is basically local, as a consequence of the finitepropagation velocity of wave equations. However, despite thisrelatively satisfactory state of the formal problem, in the caseof a time-dependent equation, or one on a curved space-timemanifold, the probablistic problem has remained, basicallywhere it was some decades ago. Considerable formal study ofthe issue, particularly in the case of the Klein-Gordon equationon a nonstatic Lorentzian manifold, has not led to a uniquedetermination, or mathematical specification, of the vacuum.The problem is inherently difficult, being not only nonlocal butalso admitting a solution only with suitable constraints on theequation. For example, in the case of the Klein-Gordon equa-tion o/ + k b = 0 in Minkowski space, the formal problem issoluble for all real values of k, but the probabilistic problem hasa solution only when k > 0 a vacuum state being otherwisenonexistent. In the case of the static, space-dependent equationo 4 + V(x)+ = 0, the vacuum applicable in the vicinity of apoint x, when it exists, depends on the values of V(y) at all pointsy of space.The present note develops, for the class of equations indi-

cated, the characterization of the vacuum proposed in refs. 1-3,as the state-putatively unique-that is invariant under thescattering operator. Curiously, its new point of departureoriginates in cosmology-more specifically, in the idea that thenotion of causality that is central in ref. 4 underlies the stabilitytheory of Hill's equation by virtue of the unique invariant causalstructure in the symplectic group (5). From Hill's equation tothe first variation of a nonlinear wave equation is a naturalgeneralization in the light of the important extension of classicalstability theory by Krein and his school (6). Relevant parts ofthe Krein theory are extended in ref. 5 and related to cau-sality.

As a consequence of this work, a unique vacuum is obtainedfor limited time-dependent perturbations of a general class ofevolutionary equations. A similar method applies to equationsin a curved space-time whose metric is appropriately per-turbed-e.g., as in the case of the Klein-Gordon equation rel-ative to an asymptotically Minkowskian metric. As an illus-tration, some details of the quantization of a purely time-de-pendent perturbation are given.

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6944 Applied Mathematical Sciences: Paneitz and Segal

However, our ultimate concern is the treatment of nonlinearequations, the case of linear time-dependent equations beinga preliminary stage. A direct, but necessarily nonlocal, con-struction is given for a canonical relativistic hermitian structurein the stable solution manifold, which is inclusive of all suffi-ciently small solutions, for suitable given nonlinear waveequations such as the prototypical equations 03) + m2q + gop=O (M2 >0 g > 0, p odd and >1). This greatly extends resultsin this direction in ref. 7. Indeed, the symplectic quantizationprocedure in field theory (8-11) has lacked a mathematicallyprecise version of the physically appropriate probability mea-sure on the solution manifold. The Riemannian part of thepresent hermitian structure provides a definite basis for thedetermination of this measure, via the corresponding diffusionprocess.

Since classical wave equations such as those cited have ageneral theory of global solutions and their temporal asymp-totics, valid in four as well as lower space-time dimensions, thepresent note represents a material step toward a realistic con-structive quantum field theory that is conceptually simple. Thealternative approach to this theory that involves the multipli-cation of quantized distributions (12, 13) has thus far beenrigorously established only in lower dimensions, and beyonddimension two at some cost to conceptual simplicity. Therelation between the approaches may be summarized by theirapproximate descriptions as extensions to the nonlinear case ofthe complex-wave and real-wave representations (14), re-spectively. There is a prospect of the eventual formal concur-rence of the results of the two approaches through the estab-lishment of the same local quantized field partial differentialequations of the type given in ref. 15 for the multiplication ofdistributions approach.

Although directed toward a theoretical physical problem,the establishment of canonical differential-geometric structuresin solution manifolds of nonlinear partial differential equationsis interesting mathematically, as an extension of similar de-velopments in algebraic geometry, well known from the workof Hodge and his successors. The present solution manifolds arenot conveniently realizable in the Banach genre, due to thepresence of unbounded operators (cf. ref. 16), singularities, and,in some cases, limitations on the existence of strict global solu-tions [although generalized solutions are obtainable by aquasi-algebraic-geometric method (17)]. Consequently theterm "partial differential variety" is used, tangent spaces beingdefined in terms of the first-order variational equation andother differential-geometric concepts similarly interpreted.Quantization of symplectic transformationsIn order to treat the S-operator, or, when nonlinear, its differ-ential, it is useful to develop a general theory of the quantizationof a given symplectic transformation. To summarize relevantaspects of this theory, the following technical preliminaries arerequired.By an H- (for Hilbert) symplectic space is meant the pair

consisting of a real linear space L together with a real bilinearantisymmetric form A on L, having the properties that L is(real-) isomorphic to a complex Hilbert space H in such a waythat A is carried into the imaginary part of the inner product.The group of all bounded invertible real-linear transformationson L that preserve the form A will be denoted as Sp(L,A); or,when L has a given complex Hilbert space structure, asSp(H).The concepts of "essentially unitarizable," etc., which are

next treated, are more complex than the corresponding morealgebraic concepts not involving the qualification "essentially;"unbounded operators in Hilbert space and dense subdomains

are involved in the former concepts but not in the latter.However, these features appear to be inevitable in a realisticrelativistic theory.The element S of Sp(L,A), where (L,A) is an H-symplectic

space, is said to be:(i) Essentially unitarizable if there exists a dense real-linear

submanifold D of L that is invariant under S; a pre-complex-Hilbert structure on D, relative to which AID is subordinateand S ID is pre-unitary; and such that if x. e D and xn -p x inL and xn y in the Hilbert completion H of D, then x e Dand x = y. The system composed of D, H, the injection map jof D into H, and the (unitary) H-closure of SID, is called an"essential unitarization" of (S,L,A). Another essential unitari-zation, involving D', H', and an injection j' of D' into H', isequivalent to the first if D = D' and there exists a unitary op-erator T from H to H' that leaves D fixed and is such that forall x e D, j'Tx = Tjx and STx = TSx. The system (or simplyS. when the context is clear) is said to be "uniquely essentiallyunitarizable" if any two essential unitarizations are equiva-lent.

(ii) Essentially quantizable if there exists a system (D; K,W,A, v) where D is a dense real-linear submanifold of L that isinvariant under S; (K,W) is a Weyl systemt over (DA) with theproperty that, if xn e D and x. -x, and W(xn) - V E U(K),then x e D and W(x) = V; I v = v; I-W(x) 2; = W(Sx) forarbitrary x e D; and such that the only closed subspace of Kcontaining v and invariant under all the W(x), x e D, andunder X, is all of K.The system (D; K, W, Z, v) is said to be essentially a quan-

tization for S; and S is said to be uniquely essentially quanti-zable in case any two essential such quantizations are unitarilyequivalent, in the sense that if (D'; K', W', 2% v') is any otherquantization, then D = D', and there exists a unitary operatorT from K to K' that intertwines W, 2, v with W', A', v'.

(iii) Essentially C*-quantizable if there exists a regular statetof the Weyl algebrat W over (D,AID), where D is a densereal-linear submanifold of L that is invariant under S, whichstate E is invariant under the C*-automorphism induced fromS; and, if in addition, D has the following property. If xn e Dand xn x, and ifW(xn) --*V e U(M),M = directsumof all(Hilbert) representation spaces ME (where ME is the space inwhich the canonical representation ofW that is associated withE acts), E ranging over the set of all regular states of W, andWdenotes the corresponding Weyl system with representationspace M, then x E D and W(x) = V.

S is said to be uniquely essentially C*-quantizable in case any

t A Weyl system over a symplectic space (LA) is a pair (K,W) in whichK is a complex Hilbert space and W is a map from L into U(K)(where, for any complex Hilbert space K, U(K) denotes the groupof all unitary operators on K, in the strong operator topology] satis-fying the relations W(z)W(z') = efA(zwZ')/2W(Z + z') and has theproperty that the restriction ofW to any finite-dimensional subspaceis continuous. When L has a given topology it is often required inaddition that W be continuous as a map from L into U(K), but thistype of consideration is here replaced by a closure condition appro-priate to submanifolds of a Hilbert space.

* The Weyl algebra A of a symplectic space (L,A) is a unique algebrainvariantly associated with (LA), so that there exists a canonicalhomomorphism (which is in fact an isomorphism) from Sp(L,A) intothe C*-automorphism group of A. Specifically, ifM is an arbitraryfinite-dimensional subspace of L such that A IM is nondegenerate(M is then called nondegenerate), let A(M) denote the W*-algebragenerated by the [W(z): z e M], where (K,W) is any Weyl systemover (LA). A is then the C*-closure of the union over nondegenerateMs of the A(M) and is independent of the defining Weyl system(K,W) (cf. refs. 1-3). A regular state E of A is one such that for everyM there is a relative-trace-class operator D in A(M) such that E(B)= tr(BD) for all B e A(M).

Proc. Natl. Acad. Sci. USA 77 (1980)

Applied Mathematical Sciences: Paneitz and Segal

two such essential C*-quantizations involve the same sub-manifold D and same state E.THEOREM 1. Let H be a complex Hilbert space. The fol-

lowing conditions on a linear symplectic transformation S eSp(H) are all equivalent: (i) S is essentially unitarizable; (ii)S is essentially quantizable; (iii) S is essentially C*-quantizable.Moreover, any one of these holds uniquely if and only if theothers do; and this is the case of an essentially unitarizableoperator, S, if and only if its spectrum is disjoint § from thatof S5-.

In particular, all of the foregoing holds in the special casewhen D = H throughout-i.e., the qualification "essentially"is removed. The proof is in fact an extension of that for this case(18) and is based in part on work of Weinless (19).

Stability theory of evolutionary equationsIn order to apply Theorem I to wave equations, it is necessaryto develop criteria for S-operators to be essentially unitarizable.This can be done by suitably combining causality considerationsin Lie groups (5) with extensions of the stability theory of theKrein school (6), resulting in generalizations of the Kreintheory.The key observation connecting causality and stability theory

is that, for any sympletic space (L,A), Sp(L,A) may be givenan invariant causal structure [which is essentially unique (5)]by defining the future cone C in its Lie algebra sp(L,A), con-sisting of bounded infinitesimal symplectics-i.e., generatorsof one-parameter uniformly continuous subgroups of Sp(L,A )--as [a e sp(L,A): A(ax,x) > 0 for all x e L]. A wide class ofdifferential equations to which stability theory applies definetime-like arcs in Sp(L,A)-i.e., have tangent vectors in C. Theelements of exp C , where Co denotes the interior of C, areuniquely unitarizable. However, the elements of exp C are ingeneral not unitarizable at all or even essentially unitarizable.The time-like arcs defined by wave equations typically havetangent vectors in the boundary of C, rather than in CO, com-plicating significantly the application of the unitarizationconcept. The possibility of making this application rests on thefact that if a E sp(L,A) and A(ax,x) >0 for all x p 0 in L, thena is essentially skew-hermitizable and uniquely so [in sensesanalogous to those defined for finite elements of Sp(L,A)]. Ina general scattering theory context, a basic results (5) em-bodying these ideas is Theorem 3.THEOREM 2. Let H be a separable complex Hilbert space.

Let t - a(t) be a measurable mapfrom R' to sp(H) such thata(t) E Cfor all t, and such that f -,, I1a(t)II dt < 2. Then thereexists a unique continuous solution S(t) of the equation S(t)= I + S co, a(s)S(s)ds having values in Sp(H); limt-A S(t) exists;this limit is uniquely essentially unitarizable, and has spectrumconfined to the set [eil: 0 < 0 < r - el, for some e > 0, where0 is not in the point spectrum, if there is no nonzero vector inH that is annihilated by all a(t).The proof uses in part generalizations of methods in ref. 6,

treating the equation essentially as if periodic but with (-oo,m)as the interval over which the Floquet matrix is defined. Asnoted, for applications it is necessary to permit the operators

7a(t) to have values in the closed cone C and not assume thevalues to lie in CO; but operators on the boundary of C need notbe conjugate to skew-hermitian operators even in an extendedsense. The additional hypothesis that no nonzero vector beannihilated by all a(t) is necessary to ensure essential unitari-zability, and it automatically implies unicity. More specifically,

Cayley transform considerations, whose applicability rests inpart on the fact that the map X -- (1 + X)(1 - X)-' is bicau-sality-preserving from sp(H) into Sp(H), determine the indi-cated spectral range, which by Theorem 1 implies unicity.The constant 2 that intervenes in the theorem is best possible.

A stronger conclusion, with 2 replaced by 4, is obtained if theHilbert-Schmidt norm, which is symplectically invariant, isused; this generalizes a beautiful result of Krein.COROLLARY 2. 1. If - tr (fr-O a(t) dt)2 < 4, where a(t) is as-

sumed to be Hilbert-Schmidt, then the conclusion of Theorem2 remains valid.

In the case of wave equation applications, a(t) is of Hil-bert-Schmidt class only in two space-time dimensions. How-ever, the theorem itself applies and yields the following criterionfor similar equations in a Hilbert space.COROLLARY 2.2. Let B be a given self-adjoint operator on

the real separable space H such that B > eIforsome e > 0. LetC = B'/2, let H denote the real Hilbert spaces ID(C)] 1[D(C-')], and let H denote the complex Hilbert space that isidentical to H as a real space, and in addition is such that forZ = UI C U2 and z' = vi (G V2,

Im (zz') = (CvI,C-'u2) -(CUC-IV2),iz = -B-u2 @ Bul.

Let t - V(t) be a measurable map from RI to the non-negative bounded operators on H such that fabs 11C-'V(t)C-' IIdt < 2. Then the scattering operator for the differentialequation

un(t) + B2u(t) + V(t)u(t) = 0

exists for arbitrary data u(to) @ u'(to) e H; is in Sp(H); and isuniquely essentially unitarizable if and only if there exists nonon-zero solution w(t) of the differential equation w"(t) +B2w(t) = 0 such that V(t)w(t) = 0 for all t.

This follows by writing the given second-order equation asa first-order equation, applying the method of variation ofconstants (interaction representation), and referring to Theorem2. Applying Corollary 2.2 to a wave equation of a type inclusiveof the tangential equation for scalar relativistic equationsyieldsCOROLLARY 2.3. Let V(x,t) be a bounded non-negative

continuous function on Rn X R', let m > 0, set B = (m2I- A),where A is the Laplacian on Rn, and suppose thatfro IIC-MV.t)C-' II dt <2, where for anyfunction F on Rn,MF denotes the operation of multiplication by F.Then the scattering operator exists and is uniquely essen-

tially unitarizable, for the differential equation

304 + m20 + V(x,t)o = 0

in the complex Hilbert space given in Corollary 2.2, providedthere exists xo e Rn and to e R' such that V(xo,t) # Ofor al-most all t > to.

This follows directly from Corollary 2.2 together with thevanishing of any solution 4l' in the space H that vanishes in aforward light cone (20). This hypothesis is implied by hyper-bolicity and the vanishing in a neighborhood of almost all of fxojX (to,@O) of t6.Quantization in curved space-timeIn the foregoing there has been a fixed linear symplectic spacein which the data lie at all times. However, the procedure isadaptable to cases in which the Cauchy data space varies, at

I The notation [D(T)l, where T is an operator in a Hilbert space Hdenotes the completion of the domain D(T) in the inner product(x,Y)T = (Tx,Ty).

§ The spectrum of an essentially unitarizable symplectic may be de-fined as that of any essential unitarization of it. Disjointness of spectrais in the Hilbert space sense.

Proc. Natl. Acad. Sci. USA 77 (1980) 6945

6946 Applied Mathematical Sciences: Paneitz and Segal

least as regards the expression for the symplectic structure, withtime or space. An interesting and representative instance of thisis quantization on a curved space-time.

Formal quantization for the Klein-Gordon equation waseffected by Lichnerowicz (21) using the general theory of Lerayas a basis. However, the question of the vacuum has remainedunresolved, except in the case in which there exists a one-pa-rameter group of temporal isometries; in this case the posi-tive-energy concept (1, 3) suffices to determine the vacuum,when it exists. In general, however, there is no such group, andour present concern is precisely with a perturbation of a sym-metrical space-time that eliminates its symmetries. Indeed, thephysical understanding of general relativity is fairly completeonly in the case of local perturbations of an asymptotically flatspace-time, which is treated in Theorem 3.THEOREM 3. The Klein-Gordon equation for a perturbation

of the Minkowskl metric propagates Cauchy data symplecti-cally relative to the form

A(0,V) = (0* dA - V * dO) di

where * is with respect to the perturbed metric, which is takento have theform g = go + h, h being assumed to be twice dif-ferentiable and to vanish, together with its first two deriva-tives with respect to time; t, at -t = I 00; and assumed also notto alter the hyperbolicity of the metric.The scattering operator for the perturbed relative to the

(flat) Minkowski metric exists and is symplectic with respectto the given form, in the same Hilbert space as in Corollary2.3. It is uniquely essentially unitarizable for all perturbedmetrics of the form (1 + h)(dx4 - dx2 - dx2 - dx2) providedh satisfies certain inequalities, which in the case when h ispurely time-dependent take the form

co

h> O 2m2h 2 h", 2/m 2 h(t)dt.

The proof depends on the formulation of the Klein-Gordonequation in curved space-time as a Hilbert space valued dif-ferential equation of second order, with the same (Lorentz-invariant) real Hilbert space as earlier. This results in anequation similar to but more generic than that involved inCorollary 2.3. The propagation of the symplectic structure fromearly times can be explicitly determined from Stokes' theorem.Finally, Theorem 2 is applied.The inequalities on h are satisfied, for example, by h(t) = b(I

+ ct2)-n, b > 0, c > 0, n an integer > 1, provided c < 2m2(2n2+ n)-' and b is small enough that fCla h(t)dt < 2/m. Thecomplex structure of the unitarization, and thereby the 2-pointfunction of the quantization, may be determined to first orderin such h, in terms of half-integral Bessel functions. When n =1, the complex structure, as an operator on [D(C)] + [D(C-1)],takes the form

_ 0-B-1M-1/2 M= m2 + (m2 + 2B2)e-2Bc-1/2= IJIM1/20 m2 - (M2 + 2B2)e-2Bc-1/2

This is a bounded perturbation of the Lorentz-invariant (flatMinkowski space) complex structure. The same method applieswhen h is both space and time dependent, apart from theevaluation of the results in closed analytic form.

In particular, there is well-defined particle production; suchproduction was proposed in earlier intuitive treatments of theeffects of curvature.Nonlinear wave equationsThe existence of the classical, nonlinear, S-operator is known(22-24) for equations of the form

o 4 + m240 + gq5P = 0 (m > 0, g> 0, p odd, n . 3)

in n + 1 space-time dimensions, when the asymptotic value Xi,1of X in the infinite past is sufficiently small in a certain topology.By the "stable solution variety" of the equation will be meantthe totality of solutions whose Cauchy data are in the domainsof all powers of the space-Laplacian at all times and in all Lo-rentz frames; and the tangent spaces to which (as defined bytheir first-order variational equations, of the form treated inCorollary 2.3) are stable in the sense of having a uniquely es-sentially unitarizable S-operator. The Lorentz-invariance ofthis variety follows in certain cases from ref. 25 and probablyis valid in general.THEOREM 4. The stable solution manifold includes all

sufficiently small smooth solutions and has a canonical her-mitian structure extending its given symplectic structure.The proof depends on the earlier treatment of linear wave

equations, together with nonlinear scattering theory for waveequations and considerations to verify the nonvanishing hy-pothesis of Corollary 2.3. The latter are based on an extension(26) of the earlier cited unique continuation from the forwardcone for solutions of the Klein-Gordon equation, in conjunctionwith decay estimates for the Klein-Gordon equation (27) andthe present nonlinear one. For a smooth solution X of theKlein-Gordon equation, there exists a Lorentz frame and vectorx e Rn for which tn/2 q5(x,t) is asymptotically bounded andnonvanishing a.e. as t -- 00; for a sufficiently small solution ofof the nonlinear equation, there exists a solution X of theKlein-Gordon equation such that tn/2(q6(x,t) - (x,t)) -O 0 ast .00

Similar results are applicable to wave functions that are pe-riodic in time, such as suitable solutions of nonlinear waveequations in chronometric space-time, the advance of chro-nometric time from 0 to 27r corresponding to that of Minkowskitime from -co to c.

We thank Prof. Daoxing Xia of the Research Institute of Mathe-matics (Fudan Universitz, Shanghai, People's Republic of China) forhis valuable collaboration on the section Quantization in curvedspace-time, during his visit to the Massachusetts Institute of Tech-nology in June 1980. We thank Prof. Stuart Nelson of the Departmentof Mathematics (Iowa State University, Ames, IA) for a fruitful dis-cussion of ref. 23. The present research was supported in part by theNational Science Foundation.

1. Segal, I. E. (1959) Mat.-fys. Medd. Danske Vid. Selsk 31 (12),39.

2. Segal, I. E. (1961) Can. J. Math. 13, 1-18.3. Segal, I. E. (1962) ll. J. Math. 6,500-523.4. Segal, I. E. (1976) Mathematical Cosmology and Extragalactic

Astronomy (Academic, New York).5. Paneitz, S. M. (1980) Dissertation (Massachusetts Institute of

Technology, Cambridge, MA).6. Daleckii, J. L. & Krein, M. G. (1974) Translations of Mathe-

matical Mongraphs (American Mathematical Society, Provi-dence, RI), Vol. 43.

7. Segal, I. E. (1974) Symplectic Structures and the QuantizationProblem for Wave Equations, Symposia Mathematics (Aca-demic, London), Vol. 14, pp. 93-117.

8. Segal, I. E. (1960) J. Math. Phys. 1, 468-488.9. Segal, I. E. (1965) J. Math. Phys. 5, 269-282.

10. Balaban, T. & R4czka, R. (1975) J. Math. Phys. 16,1475-1481.11. Streater, R. F. (1966) Comm. Math. Phys. 2,354-374.12. Segal, I. E. (1967) Proc. Natl. Acad. Sci. USA 57, 1178-

1183.13. Segal, I. E. (1969) Bull. Am. Math. Soc. 75, 1383-1389; 1390-

1395.14. Segal, I. E. (1978) The Complex-Wave Representation of the

Free Boson Field, Adv. Math. Supp. Studies (Academic, NewYork), Vol. 3, pp. 321-343.

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15. Segal, I. E. (1971) Invent. Math. 14,211-241.16. Segal, I. E. (1965) J. Math. Pures Appl. 44, (9), 71-132.17. Segal, I. E. J. Integral Eq., in press.18. Segal, I. E., Quantization ofSymplectic Transformations, Ad-

vances in Mathematics (Academic, New York), in press.19. Weinless, M. (1969) J. Funct. Anal. 4,350-379.20. Goodman, R. W. (1964) Proc. Am. Math. Soc. 15,653-60.21. Lichnerowicz, A. (1961) Propagateurs et Commutateurs

en Relativiti CGn&ale, Inst. Hautes Etudes Sci. Publ. Math. No.10,56.

Proc. Nati. Acad. Sci. USA 77 (1980) 6947

22. Segal, I. E. (1966) Dispersion for Non-Linear RelativisticEquations, Proc. Conf. Math. Th. El. Particles (M.I.T. Press,Cambridge, MA), pp. 79-108.

23. Segal, I. E. (1968) Ann. Sci. Ecole Norm. Sup. 1, (4), 459-497.24. Strauss, W. A. (1974) in Scattering Theory in Mathematical

Physics, (Reidel, Dordrecht, The Netherlands), pp. 53-78.25. Morawetz, C. S. & Strauss, W. A. (1973) Comm. Pure Appl. Math.

26,47-54.26. Morawetz, C. S. (1963) Comm. Pure Appl. Math. 16,353-362.27. Nelson, S. (1971) Proc. Am. Math. Soc. 27, 110-116.