Representations of the poincaré group by higher-order field equations and unified field models of matter

Physica 114A (1982) 184-196 North-Holland

REPRESENTATIONS HIGHER-ORDER

Publishing Co.

OF THE POINCARk

FIELD

EQUATIONS

MODELS

GROUP

AND UNIFIED

BY FIELD

OF MATTER

H. STUMPF Unioersity of Tiibingen, Institute for Theoretical Physics, Auf der Morgenstelle 14, D-74 Tiibingen, West Getmany

Higher-order field equations produce representations of the Poincart group which regularize non-linear quantum field interactions but which in general are not unitary ones. These representations can be included in a new method of calculation for quantum field theory called functional quantum theory which permits quantum fields with indefinite metric to be treated. The method is outlined and afterwards applied to a unified lepton-quark model formulated by higher-order field equations. It will be explained in which way the hierarchy of forces can be derived in this model. The necessity of eliminating the unobservable new representations of the PoincarC group in asymptotic observable expressions leads to confinement. So by these representations regularization of non-renormalizable quantum field theories and confinement are closely connected.

1. Introduction

Due to the energy-mass equivalence of Einstein, all physical forces and particles must have a common origin. Hence the problem of high-energy physics is to find a unified dynamical law which governs the formation and reactions of matter and which leads to the derivation of the hierarchy of forces and particles. In the past decade the main effort in high-energy physics was devoted to the development of unified gauge quantum field theories. Such theories incorporate a priori the hierarchy of forces into one dynamical law, i.e. in this case into a basic Lagrangian functional. However, the price to be paid for the unification of forces is an increasing number of basic fermions and gauge bosons. In addition, the proliferation of “basic” particles is continued by the increase of fermion generations which have to be taken into account, cf. ref. 1. Hence it is obvious that new ideas and models must be used in order to master the unification problem. The way to reduce an increasing number of basic particles is the idea of the fusion of such particles by even more elementary particles. On the level of relativistic elementary particles the first step was the fusion of integer spin particles, i.e. bosons from elementary spin-half fermions. This was inaugurated by the group-theoretical spin-fusion analysis of de Broglie2) with respect to the fusion of the photon from two neutrinos. In this case bosons must not occur in the fundamental dynamical law, i.e. this law must describe 03784371/82/0000-0000/$02.75

@ 1982 North-Holland

REPRESENTATIONS

OF THE POINCARfi

GROUP

185

the interactions and formation of basic fermions only. However, the formulation of fermion interactions was prevented by the non-renormalizability of the corresponding non-linear fermion field equations. In the 1950’s Heisenberg3) proposed that leptons and nucleons be considered as the fundamental particles and all other elementary and non-elementary particles are obtained by fusion of these fundamental particles. To circumvent the nonrenormalizability of his spinor field equation, Heisenberg assumed a regularization using dipole ghosts. He identified the dipole ghosts with leptons, but the meaning of this identification remained unclear. The general development of quantum field theory was not influenced by Heisenberg’s approach. Nevertheless, the search for basic particles also became an important topic in this development, although boson fusion from fermions was not seriously taken into account. Gell-Mann4) and Zweig’) proposed a composite structure for the nucleons due to their origin by fusion from quarks. In this scheme the basic particles are leptons and quarks to which, in a gauge theory, gauge bosons, etc. have to be added. Schwinger6) proposed a composite structure for nucleons, in particular the hadrons owing to their structure as a fusion of magnetic monopoles. In recent years there has been an increasing number of attempts to reach a still more elementary level of composition at which already leptons and quarks are assumed to be built up from a few basic particles to which in most cases gauge bosons, etc., are added. These new models which are in most cases only classification schemes were developed by many authors’-“). These models use only invisible basic fermions and bosons. Insisting on regarding bosons as a fusion of fermions, we have to formulate the new models by means of non-linear spinor field equations, and these equations can be or have to be regularized by dipole ghosts. In contrast with the original Heisenberg approach these new models provide strong support for the dipole ghost regularization. Since the basic fermions of the new models have to be partly or completely unobservable, they must be confined, i.e. also theoretically be made unobservable. They share this confinement with dipole ghosts which must be confined in order to maintain a probabilistic interpretation of the theory. It is therefore reasonable to identify the invisible basic fermions with dipole ghosts. For a consistent formulation of a corresponding spinor field theory it turns out that higher-order field equations are an appropriate starting point. 2. Poincark group representations

by higher-order

field equations

The dipole ghosts play the role of a compensating field which prevents the rise of singularities from an original field. The regularizing properties of

186

H. STUMPF

compensating fields were first recognized by Mie”) who formulated a divergence-free classical electrodynamics for charged point particles. Bopp 14) showed that this approach leads to higher-order field equations for the vector potential. Independently Podolski”) introduced the same equations. Pais and Uhlenbeck16) started an investigation of higher-order field equations on the quantum level. Applying the usual canonical quantization procedure to such equations they obtained an indefinite metric state space. Numerous attempts were made to work with such indefinite metric field theories which exhibit, in analogy with the classical case, self-regularizing properties on the quantum level. A review of the older literature is given by Nagy”) and Nakanishi”) and various other recent papers on this topic have been published’s25). In addition, higher-order field equations are indirectly used if the regularization procedure of Pauli and Villars26) is applied. A review of preceding literature with respect to this method is given by Pais*‘). RaiskiB), Gupta*p and other authors regularized quantum electrodynamics using compensating fields and also Heisenberg’s approach to the regularization of the non-linear spinor field can be considered as a special case of a Pauli-Villars regularization, although it was motivated by a more basic idea. Recently, Slavnow applied it to quantum chromodynamics, i.e. a non-Abelian gauge theory. For a thorough understanding of the properties of higher-order field equations and corresponding theories a group-theoretical analysis has to be performed as any kind of relativistic form-invariant field equation produces its own characteristic representation and vice versa. For free-spinor fields, Niederer and 0’Raifeartaigh3’) have shown that the field equations are projections which guarantee the unitarity and the irreducibility for the representations of the PoincarC group and lead to an invariant positive-definite scalar product. An analysis of higher-order field equations and the corresponding new representations would be of great interest for the present theory but has not yet been performed. It is clear that these representations and the corresponding fields will lack the three properties mentioned above. For asymptotic observable particles to be discussed later one must of course recover the standard representations of the Poincare group. Here we restrict our discussion to linear higher-order field equations and consider the effects of interaction in the next section. As a basic field must be a spin-i field we consider only the equations corresponding to such fields. Wildermuth3*) first discussed a linear multi-mass equation for Dirac spinors. If two masses are equal this leads to a dipoleghost equation. Froissart33) investigated a dipole-ghost equation for massive scalar fields. Diir?) treated a massless third-order spinor-field. Ferge3? analysed a monopole-dipole ghost massive scalar field and performed a perturbation calculation for a non-linear self-interaction of this field. Stumpf

REPRESENTATIONS

OF THE POINCARh

and Scheerer%) and Pouradjam3’) treated where the free wave equation reads (-iJ

+ m)(-i$

GROUP

the corresponding

+ ~)~+(x) = 0.

187

spinorial

case

(1)

The solutions are given by 4(x) = C,,(x) + (Y&,(X)- /.L-‘cx(A- x”&)Ao(x),

(2)

where A,(x), B,,(x) and C,(x) are the solutions of corresponding homogeneous equations and (Y: = (p - m)-‘. Clearly 1+5(x)belongs to a non-unitary representation of the Poincare group as it is unbounded. If quantization is performed according to the usual canonical scheme one obtains3’) [I/J(X), 6(x’>]+ = -icf2S(x

- x’,m)+ia[a-p$]S(x-x!,p)=F(x-x’), (3)

where S(x - x’, CL)and S(x - x’, m) are the anticommutators of the free fields. Direct evaluation of (3) shows that we obtain a regularized anticommutator for the $-field for which F(0) = 0. Furthermore the regularization of the G-field anticommutator due to the higher-order field equation (1) produces an indefinite state space. According to ref. 36 we have

F+(x- x’) :=

Functional

quantum

theory

Due to the lack of suitable explicit state representations beyond the free-particle states in relativistic quantum field theory, an attempt was made in the past decades to obtain general and numerical information about quantum observables without the use of explicit state representations, i.e. to obtain representation-free results. However, such a programme cannot be performed consistently due to the fact that a non-trivial (non-linear) quantum

H. STUMPF

188

field produces its own metrical state space which is determined by the field dynamics and cannot be postulated at will. Discrepancies arise in unforeseen ways. For instance, some of the Bethe-Salpeter solutions exhibit negative norms’*), with the consequence that the derivation of both the equation itself and the norm expression become inconsistent. Oehme and Zimmermann3’) showed that due to the indefiniteness of certain expectation values, gauge theories are only consistent for a number of flavours NF with 10 c Nr < 16. Such problems arise a fortiori for theories which presuppose indefinite metric and bound states. The latter theories have been treated using mainly models in the Hamiltonian, non-relativistic formulation, cf. Nagy17). Hence a relativistic quantum theory containing explicit state representations beyond the free-particle representations has to be developed in order to treat these problems adequately. A model or method which satisfies these conditions of explicit relativistic invariance and explicit state representations, has been proposed by Stumpf’p and further developed?. For the evaluation of this model we assume a non-linear spinor field equation with the spinor field operator 4(x) to be given. Then a proper relativistic state representation can be achieved by introducing the generalized Fock states {1x1,. . . , x,), : = N$(x,) . . . $(x,)10), 1 G n < m}

(9

as the set of base states for the system under consideration where IO) is the true vacuum state of the system and N means the formal normal ordering of the corresponding time-ordered product by means of the two-point function of the spinor field. Since the ground state may be a complicated functional of the field, the set (5) is not in general orthonormalized. In this case (according to the theory of linear vector spaces) a dual set {Ix,, . . . , x,)“, 14 n Cm} can be introduced which is defined by orthonormality conditions n(j)m = 8:. Then any state Ia) is required to admit a representation with respect to the set (3, i.e.

la) =

$I$ f unh

or equivalently

la>=

. . . , xh)lx,, . . . , x,), d4x,. . . d4xn

with respect to the dual set

$,h j-44x1,. . . , x,la)lx,,. . . , ~2” d4x,.. . d4x,

and the scalar product of two states la) and lb) is given by

(a(b)= $I&j- unh, . . . , x,laYMx,, . . . , &lb) d4x,. . . d4x,.

(7)

REPRESENTATIONS

OF THE POINCARI?

GROUP

189

By multiplication of (7) from the left with &xi, . . . , xi1 and use of the orthonormality relations we obtain 4n(x1,. . *, xnld := (OlNrCl(x,) . . . W,)b).

(9)

Normal ordered base states of the set (5) for equal times x’: = * * * = xf = t, were first introduced by Nishijima4’), who also considered the expressions (9). But, using a Green function formalism, Nishijima4*) tried to avoid any explicit state representation. Subsequently, Coester and Haag43) attempted to evaluate a scalar product for non-relativistic systems with respect to the state representation (6) by means of a functional integral. However, due to the very restricted class of integrable functionals, this method cannot be applied beyond perturbation theory for non-trivial systems and an application to non-trivial relativistic systems is not possible due to the lack of positivedefiniteness, etc., of the corresponding quantities. The state calculation for field systems has long been developing. As regards relativistic field systems calculation methods have only been given in the literature for the quantities (9). The first step was the derivation of an equation for &(x1x21a) by Salpeter and Bethe4) which was more thoroughly derived by Gell-Mann and Low”). Within the framework of the interaction representation, Freesee) was able to derive equations for the whole set (9) of +-functions. Without reference to the interaction representation, Matthews and Salam4’) derived the same system. An improved version of this proof was given by Stumpf4). For an advanced discussion functional quantum theory must be used. Functional quantum theory is defined to be a map of the ordinary linear state space X:= {la)} o f a q uantized system into a functional linear state space @ which is defined by the set of state functionals {Ig(j, a))}. These state functionals are given by

IfHi, a)) :=

glA I

ddxl, . . . , xnla)lDn(xl,.

. . , x,,)) d4x1. . . d4x,

(10)

with

{IDnh . . . , x,))

:=

-$j(x,)...ihAld& 16 It < 4,

(11)

where j(x) is a generating source operator of the functional space and I&,) the functional ground state. For a complete description of a state la) (6) is also needed. We therefore define the dual functional states

IG(j, a)> =

$I$1 a,(~.

. . , x,la)lD&,

. . . , x,N d4x,. . . d4xn.

(12)

190

H. STUMPF

Then due to the orthonormality of the functional base set (11) the scalar product (a(b) of two states is invariant against the map, i.e. (alb) = W,

a)lW, b)).

(13)

State functionals (11) which are the relativistic generalization of Fock states were introduced by Stumpf48*4p. This extension from the non-relativistic Fock states to the relativistic functional states is, however, by no means trivial as the states (11) must show the proper relativistic transformation properties48*4p. This problem was solved for Bose functional spaces by Rieckers5@) and for Fermi functional spaces by Stumpf and Scheerer”). An approach which is not based on the Grassmann algebra was performed by Garbaczewski52) but he did not take into account the problem of transformation properties of state functionals. If the transformation properties are secured, then the representation U(A, d) of a Poincark transformation (A, d) in the state space %‘, is mapped into a corresponding representation V(A, d) in 6. In particular, from this it follows that the generator eigenvalue equations for a state la) which define the good quantum numbers of la) with respect to the Poincare group are mapped into corresponding equations in functional space for Ig(j, CX))‘~)

WY:(i,a)>= pIi%, a)); VItYti,a))= m’lW,aI), (14) G3ltW, a)) = s31S(La)); 8pWIB(i,

a) = s(s + OIW,

a)>.

If furthermore I,!+) is assumed to be a spinor field operator which satisfies the non-linear field equation (19), then for a superspinor representation ($, $)e q the Freese-Matthews-Salam equations for the &components can be mapped into functional spaces4) and give D(x)W)

+ Vd(xMxMx)llW,

aI> = 0

(1%

with d(x):=

1 F( x, x’)j(x’) d4x’ + a(x),

(16)

where F(x, x’) is the time-ordered two-point function of the field. In addition to this equation the I& a)) state functional must satisfy the eigenvalue equation m$Y(j, a)) = .9@(j, @INi, a)),

(17)

where 2)X2&a) can explicitly be derived from (15)*8*55). If it is assumed that the state la) belongs to a monopole, then it can be concluded? that the dual state functional @(j, a)1 must be a left-hand solution of eq. (17) Wj,

a)Jm’ = (Wj, a)l.@(j, a).

(18)

REPRESENTATIONS

OF THE POINCARk

GROUP

191

If la) belongs to a multipole, then @(j, a)1 must be calculated by means of the metrica fundamental tensor @(j) which is the map of g,, in functional space’7). Eqs. (17) and (18) are non-trivial eigenvalue equations for the mass operator of (14). By combination of (15) and (14) similar equations can be derived for the other eigenvalue conditions if all eigenvalues belong to a monopole. In any case both kinds of state functional can, at least approximately, be calculated and from this via the scalar product (13) the observables of interest.

4. The hierarchy of forces in a lepton-quark

model

We apply functional quantum theory to the treatment of a unified leptonquark model where the bosons are formed by fusion from fermions. This model was developed by Stumpf”) and is based on a higher-order field equation with a non-linear interaction term. The input to the model are the various lepton-quark generations which one does not expect the model itself to explain. The model offers a self-consistent explanation of the hierarchy of forces which are generated by local and non-local bosons arising from fermion fusion. While the local bosons are thought to be responsible for gravitational and electro-weak interactions, the non-local bosons should lead to the strong interactions. The model is not meant to give a completely realistic description of nature. Rather we intend to study its dynamics which is characterized by bound state reactions and the confinement problem. To explain the derivation of the hierarchy of forces it is sufficient to consider a simplified version where only one lepton and one quark particle are assumed to exist. The corresponding equation then reads54 (- i,8 + P)*(- id + m)+(x) = VJl(x)$(x)Jl(x).

(19)

Introducing superspinors the map of (19) and of its adjoint equation into functional space is given by (15) if we choose the kinematical operator to be D(x) := (-iG’8,

+ p)2(-iGP8p + m),

(20)

where the {G} are the superspinor Dirac matrices. The functional calculation method which is applied for the solution of the corresponding functional equation may be characterized to be an interaction representation with inclusion of bound states. An attempt to formulate such a method was undertaken by Dtirr and Wager@‘) who analysed graphically the lowest equation of the &functions in order to extract the bound states. A complete analytical solution of this problem in any order of the +-functions

192

H. STUMPF

by means of functional quantum theory was given by Stumpf’p. We can only describe the principle of this calculation and refer to ref. 59 for details. In a first step certain local interaction terms are removed from the functional equation by the application of exact partial resolvents. This procedure corresponds to the introduction of forces between the basic fermions which are mediated by local bosons. The exchange of local bosons is a universal coupling to both leptons and quarks. The local bosons which are allowed to occur follow as a direct consequence of a group-theoretical analysis of their local interaction vertex. Due to the spin fusion of Dirac spinors in combination with the electro-weak isospin the bosons of the gravitational and electro-weak interactions can arise. Turning now to the strong interactions we must make a distinction between leptons and quarks. The linear equation (1) can easily be shown to be consistent with the following decomposition and projections: rcI(x) = -cJ(x)

+ q(x),

P(p)+

: = (-id

+ p)*+(x) = a-‘l(x),

P(m)+

:= (-id

+ m)+(x)

(21)

= -&q(x),

if we identify the dipole ghosts with quarks. Similar relations for linear higher-order field equations were first used by Wildermuth3z) and by Marx6’). We assume that this decomposition holds also for the non-linear equation (19), as we work in the interaction representation. This leads to a unique map of (19) into the system (-i$

f m)l(x) = - CYVE-al(x) + q(x)]*[-

(-i,J

+ r_L)*q(x)= OL-’VI-

d(x)

+ q(x)]+

d(x) J(x)

+ q(x)], + q(x)].

(22)

B&unique maps are also possible, but we will not discuss them here. Eqs. (21) and (22) can be equally well formulated for superspinors and applied to the state functionals where corresponding lepton and quark sources have to be introduced. If now in a second step certain non-local interaction terms are removed from the functional equation by the application of exact corresponding partial resolvents these non-local boson resolvents depend explicitly on the representation, i.e. we obtain lepton-lepton, lepton-quark and quark-quark boson resolvents and therefore corresponding bosons. Application of these resolvents then leads to equations where the local as well as the non-local bosons are incorporated into the interaction representation. If we identify the non-local bosons as the transmitter of the strong interactions, the leptonlepton and the lepton-quark bosons must be very heavy in order to reproduce experimental results. This assumption is also made in the grand unified gauge

REPRESENTATIONS

OF THE POINCARI?

GROUP

193

theories which had started with the work of Pati and Salam6’). But numerical calculations are not yet available so far. The universality of the local boson coupling can immediately be concluded from these projected equations, as the corresponding coupling constants are invariant with respect to these projections. In addition, functional quantum theory provides a clear prescription for the formation of scalar products of scattering functionals, i.e. the S-matrix construction, which avoids the inconsistencies of the attempts to solve the scattering problem for bound states via Green functions. In a first step the L.S.Z.-reduction technique was derived for scalar products of scattering functionals of point particles. Furthermore, a calculation method of state functionals for bound state formation and reactions beyond the interaction representation of ref. 59 was developed63).

5. Confinement

and unitarization

In order to provide a physical and probabilistic interpretation of the theory, the negative norm states, ghost states or multipole ghost states, etc., which are produced by the higher-order field equations must be removed from the physical state space. As these states are responsible for the regularization of the field, they ought not to be eliminated completely. The only way to reconcile these contradictory demands is to eliminate the unwanted states only asymptotically, i.e. to remove them from the S-matrix. As in this case the S-matrix provides a map of the positive definite state space of free ingoing particles into the unitarily equivalent positive definite state space of free outgoing particles, this procedure is called unitarization. If, furthermore, the ghost states, etc., are identified with quarks as in section 3 or with other invisible fermions, this procedure is equivalent to the confinement for these particles. Many proposals have been made concerning unitarization procedures, cf. Nagy”). Most of them lack any physical interpretation and are rough actions which destroy the analyticity properties of the unitarized S-matrix. The most gentle prescriptions seem to be the good ghost admixture introduced by Heisenberg3) for the Lee model and formulated by Stumpf and Scheerer@) for the relativistic case in functional quantum theory, the principal value description of Heisenberg3) and the shadow-state prescription of Sudarshan6’). For the case of dipole ghosts these three methods are equivalent. With respect to the latter method Sudarshan and coworkers investigated the consequences for the analyticity of the S-matrix and emphasized that no severe violation of the causality condition has to be expected, cf. refs. 66-72. Without reference to

194

H. STUMPF

other authors, Blaha73) also used this procedure. Later Newton”) and Stapp”) criticised these results so that a definitive decision on the causality-violating effects of this method cannot be given. Even if the arguments against it can be invalidated, presumably a complete affirmative answer will never be found as its application depends on the numerous bound states which occur in realistic quantum field theories and for which, a priori, no systematic classification can be given. Another important proposal for realizing quark confinement which can be transferred into functional quantum theory of a non-linear spinor field is concerned with the gluon propagators in gauge theories. It has been observed that gluon propagators belonging to higher-order field equations of the gluons lead, in certain approximations, to confining harmonic forces between quarks73376).These possibilities were further investigated by Blaha”), Bender, Mandula and Guralnik”), Mtiller-Kirsten’p, Narnhofer and Thirring**), Ragiadakosz3) and Mintchevz5). In addition, d’Emilio and Mintchev’@) indicated that gluons themselves should be confined due to the properties of their renormalized propagators. For a self-interacting spinor field the gluon propagator has to be replaced by the spinor field propagator itself, as this propagator is responsible for the mutual interactions of the ghost particles and their asymptotics. It is noteworthy that also in this direction new attempts have been made. The first attempts to calculate non-linear spinor field propagators were undertaken by Heisenberg, Kortel and Mitte?), Ascoli and Heisenberg*‘) and Mitter82). A new approach was made by Van der Merwea). Since all these methods are not sufficiently analysed with respect to their effects on causality, etc., it is reasonable to minimize the influence of ghost effects by physical assumptions. This is achieved if we postulate that any physical particle must occur as a bound state of the invisible particles and that the forces arise from exchange of physical particles only. Then all causality violating effects occur in the interior of physical particles but will never be set free, which has of course to be demonstrated. In any case, this leads to the distinction of an invisible subdynamics and the formation of an observable bound state dynamics85).

References 1) J. Ellis, Grand Unified Theories; Lectures at the 21st Scottish Universities Summer School in Physics, Preprint TH 2942-Centre EuropCene de Recherche Nucltaire, Geneva (1980). 2) L. de Broglie, Compt. Rend. Acad. Sci. (Paris),195 (1932) 862; 199 (1934) 813. 3) W. Heisenberg, Introduction to the Unified Field Theory of Elementary Particles (Interscience, London, 1966). 4) M. Gell-Mann, Phys. Lett. 8 (1964) 214.

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5) G. Zweig, Reports TH 401 and TH 412, Centre Europeene de Recherche Nucltaire, Geneva (1964). 6) J. Schwinger, Science 165 (1%9) 757. 7) H. Harari, Phys. Lett. 86B (1979) 83. 8) M. Shupe, Phys. Lett. 86B (1979) 87. 9) J. Taylor, Phys. Lett. 88B (1979) 291. 10) H. Terazawa, Progr. Theor. Phys. 64 (1980) 1388; Phys. Rev. D22 (1980) 184. 11) R. Casalbuoni and R. Gatto, Phys. Lett. 93B (1980) 47. 12) K. Akama, Progr. Theor. Phys. 64 (1980) 1494. 13) G. Mie, Ann. Physik 37 (1912) 511; 39 (1912) 1; 40 (1913) 1. 14) F. Bopp, Ann. Physik 38 (1940) 345. 15) B. Podolski, Phys. Rev. 62 (1942) 68. 16) A. Pais and G.E. Uhlenbeck, Phys. Rev. 79 (1950) 145. 17) K.L. Nagy, State Vector Spaces with Indefinite Metric in Quantum Field Theory (Noordhoff, Groningen, 1966). 18) N. Nakanishi, Progr. Theor. Phys. Suppl. 51 (1972) 1. 19) R. Palmer and Y. Takahashi, Can. J. Phys. 52 (1974) 1988. 20) E. Marx, Int. J. Theor. Phys. 10 (1974) 253. 21) I. Rabuffo and G. Vitiello, Nuovo Cimento 44A (1978) 401. 22) H. Narnhofer and W. Thirring, Phys. Lett. 76B (1978) 428. 23) C. Ragiadakos, Nuovo Cimento 49A (1979) 175. 24) A. Barut and P. Crawford, Phys. Lett. 82B (1979) 233. 25) M. Mintchev, J. Phys. Al3 (1980) 1841. 26) W. Pauli and F. Villars, Rev. Mod. Phys. 21(1949) 434. 27) A. Pais, The Development of the Theory of the Electron (Princeton Univ. Press, Princeton, 1948). 28) G. Raiski, Acta Phys. Polonica 9 (1948) 129. 29) S. Gupta, Proc. Phys. Sot. (London) A66 (1953) 129. 30) A. Slavnov, Teor. Mat. Fiz. 33 (1977) 210; Theor. Math. Phys. (USA) 33 (1978) 977. 31) H. Niederer and L. O’Raifeartaigh, Fortschr. Phys. 22 (1974) 111. 32) K. Wildermuth, Z. Naturforsch. 5s (1950) 373. 33) M. Froissart, Suppl. Nuovo Cimento 14 (1959) 197. 34) H.P. Dtirr, Nuovo Cimento 22A (1974) 386; 27A (1975) 305. 35) W. Ferge, Thesis, University of Ttibingen (1979). 36) H. Stumpf and K. Scheerer, Z. Naturforsch. 34a (1979) 284. 37) T. Pouradjam, Thesis, University of Ttibingen (1980). 38) R. Oehme and W. Zimmermann, Unitarity Relations and the Number of Flavours, Preprint University of Chicago (1981). 39) H. Stumpf, Ann. Physik 13 (1964) 294. 40) H. Stumpf, Funktionale Quantentheorie, in: Quanten und Felder, H.P. Dtlrr, ed. (Vieweg, Braunschweig, 1971) p. 189; Acta Phys. Austriaca Suppl. IX (1972) 195; New Representation Spaces of the Poincare Group and Functional Quantum Theory, in: Groups, Systems and Many-Body Physics, P. Kramer and M. Dal Cin, eds. (Vieweg, Braunschweig, 1980) p. 319. 41) K. Nishijima, Progr. Theor. Phys. 10 (1953) 549. 42) K. Nishijima, Progr. Theor. Phys. 12 (1954) 279; 13 (1955) 305; 17 (1956) 765. 43) F. Coester and R. Haag, Phys. Rev. 117 (1960) 1137. 44) E. Salpeter and H. Bethe, Phys. Rev. 84 (1951) 1232. 45) M. Gell-Mann and F. Low, Phys. Rev. 84 (1951) 350. 46) E. Freese, Z. Naturforsch. 8a (1953) 776. 47) P. Matthews and A. Salam, Proc. Roy. Sot. (London) A221 (1954) 128. 48) H. Stumpf, Z. Naturforsch. 25a (1970) 575. 49) H. Stumpf, Z. Naturforsch. 26a (1971) 623. 50) A. Rieckers, Z. Naturforsch. %a (1971) 631; 27a (1972) 7.

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H. STUMPF

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