# Isospinor Preon Field M Odel I H. Stumpf Institut Für Theore

Gravitation as a Composite Particle Effect in a Unified Spinor- Isospinor Preon Field Model I H. Stumpf Institut für Theoretische Physik der Universität Tübingen Z. Naturforsch. 43a, 345-359 (1988); received February 5, 1988 The model is defined by a selfregularizing nonlinear preon field equation, and all observable (elementary and non-elementary) particles are assumed to be bound (quantum) states of fermionic preon fields. Electroweak gauge bosons, leptons, quarks, gluons as preon composites and their effective dynamics etc. were studied in preceding papers. In this paper gravitons are introduced as four-preon composites and their effective interactions are discussed. This discussion is performed by the application of functional quantum theory to the model under consideration and subsequent evaluation of a weak mapping procedure, both introduced in preceding papers. In the low energy limit it is demonstrated that the effective graviton dynamics lead to the complete homogeneous Einstein equations in tetrad formulation. Key words: PACS 04.50 Unified field theories and other theories of gravitation, PACS 04.60 Quantum theory of gravitation, PACS 11.10 Field theory, PACS 12.10 Unified field theories and models, PACS 12.35 Composite models of particles

Introduction General discussions of the gauge concept for gravity are contained in [11-16]. Theories with The quantum theory of gravitation is one of the Lagrangians analogous to those of conventional most challenging problems of modern physics. gauge fields were first introduced by Weyl [3] and Due to the nonrenormalizability of the Hilbert- further developed along various lines, see for Einstein Lagrangian the suspicion grows stronger instance [17-32]. that the original Einstein equations are not the true In spite of the enormous effort of applying microscopic description of gravitational pheno modern group theory to derive modified gravita mena. Thus in the last decades numerous ap tional theories, all the above-mentioned ap proaches were made to replace the Hilbert-Einstein proaches can be considered as being conservative Lagrangian by a Lagrangian of a modified insofar as they do not doubt the role of the gravita renormalizable field theory and to explain the tional field as an elementary quantity. It is just this classical theory of gravitation as a long wave or assumption which is abandoned in more radical mean value approximation etc. Among the most approaches to explain gravity and quantum prominent recent approaches are supergravity and gravity. For instance, Bars and MacDowell [33] superstrings which provide a modified theory of reproduced classical relativity as an approximation gravitation by means of gauging supersymmetric of a quantized field theory of interacting Rarita- groups etc. [1, 2]. More conventional approaches Schwinger spinors and gauge potentials. Amati consider gravity simply as an effect of a local non- and Veneziano [34] proposed a rather complicated abelian gauge group, such as the conformal group, nonpolynomial nonlinear spinor field equation for the Poincare group, the de Sitter group, etc. These Dirac-Spinors and approximately derived the Hil approaches originate from the work of Weyl [3] bert-Einstein action etc. Sen [35] used only spinors and its extension by Yang and Mills [4] and led to and spinor connections to describe classical rela numerous versions of nonrenormalizable or tivity. D'Adda [36] derived the spinorfield Lagran renormalizable gravitational theories. With respect gian of [34] by a symmetry breaking mechanism, to the Poincare group, see for instance [5-10]. and Sailer [37] tried to give a deeper foundation of relativity theory by using Weyl-spinors with a Reprint requests to Prof. H. Stumpf, Institut für Theoreti sche Physik der Universität Tübingen, Auf der Morgen Hermitean spinor metric, etc. Furthermore, maps stelle 14, D-7400 Tübingen. between classical Einstein-equations and classical

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Yang-Mills field equations, nonlinear c-models sions about a composite graviton. Later on, etc. were discussed, for instance, by Sanchez [38] Kurdgelaidze and Funke [55] and Danilyuk [56] and Percacci [39]. made first quantitative calculations of composite The drawback of these unconventional ap spin-2-mesons and some elementary graphs in the proaches apparently is their ad hoc character. A framework of a nonlinear spinor theory, while more systematic approach is the concept of fusion Dürr [57] and Sailer [58] made some speculations of elementary fermions. In this concept all observ about the generation of a composite graviton by able elementary particles are assumed to be bound symmetry breaking effects. From these papers no states of the elementary fermions and the existence stringent conclusions can be drawn, whether the and the processes of observable particles are composite graviton is meaningful or not and which governed by the formation and reactions of these dynamical laws govern its reactions. bound states. For relativistic particles this hypo Recently there has been a renewed interest in the thesis was inaugurated by Jordan [40] who inferred group theoretical analysis of graviton states of the the compositeness of the photon by a statistical Bargmann-Wigner equations (de Broglie! equa argument. Subsequently, de Broglie [41] put forth tions) by Nous [59], Rodriguez and Lorente [60] the proposal that the photon is composed of a and Doughty and Collins [61]. As these equations neutrino and an antineutrino. This was further de describe only free pointlike gravitons the question veloped by Jordan [42], Kronig [43] and other arises whether these "free particle" gravitons can authors. Later on, de Broglie [44] extended his be used as building blocks for the construction of proposal to a general theory of fusion of particles the full gravitational Einstein equations or not. of spin 1/2, and along these lines Proca [45] and This question was answered affirmatively. Starting Kemmer [46] derived wave equations for com with free pointlike gravitons in Minkowski space it posite mesons. In this context it was assumed that was shown by Gupta [62], Kraichnan [63], Arno gravity is generated by a spin-2-field or particle, witt [64], Weinberg [65], Wyss [66], Ogievetsky resp., the graviton, and Tonnelat [47], Petiau [48] and Polubarinov [67], Deser [68], Groenewold and the Broglie [44] already discussed a composite [69], Boulware and Deser [70], Papini and Valluri graviton, i.e. a graviton composed of four spin- [71], von der Bij, van Dam and Ng [72], that 1/2-fermions. A more formal approach to higher interactions of such gravitons necessarily lead to spin wave equations was performed by Dirac [49], the Einstein equations. Furthermore, it was Fierz [50], Pauli and Fierz [51], Bargmann and demonstrated by Mittelstaedt [73] that even cos- Wigner [52] and other authors. All these equations mological solutions can be obtained from this only describe local one-particle states, and in par formulation of Einstein's theory in Minkowski ticular the graviton was "free". With the further space. This means: the assumation of the existence development of relativistic quantum field theory it of gravitons does not contradict general relativity. became obvious that fusion need not be strictly Therefore, the attempt to derive Einstein's equa local but can be a nonlocal phenomenon and that tions from a theory of composite gravitons is interaction has to be taken into account. However, neither selfcontradictory. In the following we the theory of the composite graviton did not demonstrate this for a spinor-isospinor preon field participate in this general progress of quantum model, where gravitons arise from four-preon field theory, perhaps because this problem was too states. complicated or the research concentrated on other The use of composite gravitons is of consider fields. The only attempts to include a composite able advantage compared with point-like gravi graviton into current research were made by Bopp tons. While according to the above-mentioned [53] who generalized de Broglie's ansatz to many- authors the pointlike gravitons strictly lead to Ein particle equations with relativistic potentials and stein's equations and thus a nonrenormalizable Heisenberg [54] who assumed any elementary par theory, the internal structure of composite gravi ticle to be a bound state of elementary spinor-iso- tons leads to formfactors in their mutual inter spinor fields in his nonlinear spinor field ap actions which prevent an ultraviolet catastrophe. proach. But neither these two authors nor their co In this way Einstein's equations are a low-energy workers were able to derive quantitative conclu limit which just corresponds to the ideas men 347 H. Stumpf • Gravitation as a Composite Particle Effect tioned at the beginning and the composite gravi- [(-iy Mdfl + m1)(-iy ede+m2)]aßVß(x) tons correspond to the most economic way of a modification of the laws of classical relativity. = 9 KßyS Vß(x) Vö(x) > 0-1) Within the context of a unified preon field where the indices a, ß , ... are superindices describ model [74-79] the dynamical laws which govern ing spin and isospin. Due to the mass terms in (1.1) the reactions and interactions of composite par the corresponding spinor field has to be a Dirac- ticles are effective field equations which are spinor-isospinor. derived from the preon field equations by means of In contrast to the nonrenormalizability of first a weak mapping procedure in functional space [75, order derivative nonlinear spinor field equations 76, 80]. The weak mapping procedure for generat and the difficulties connected with this property, ing functional is a mathematically well-defined the model (1.1) exhibits self-regularization, relativ- method which is the quantum field theoretical istic invariance and locality for common-canonical analogon to the resonating group method in quantization. Due to the self-regularization the nuclear physics [81 -86] and other nonrelativistic model is renormalizable, but we need not make use multiparticle branches. Both in nuclear physics of this property on account of our nonperturbative and in quantum field theory the critical point are calculation techniques. In particular, it was de the wavefunctions of the composite particles which monstrated by direct nonperturbative calculations are used as elements of the map. With respect to [74, 79] that relevant matrix elements of local com such wavefunctions two-preon-states correspond posite operators as occurring in (1.1) are finite and ing to electroweak gauge bosons and scalar bosons need no special renormalization apart from nor- were quantitatively calculated in the framework of malordering. the above mentioned preon model [74, 79]. The The regularization of the spinor field dynamics four preon states have at present not yet been by (1.1) leads to indefinite metric in the cor quantitatively investigated. Thus we propose the responding state space. It was, however, shown internal structure of these states guided by the [74, 79] that for observable states the probability results of [79] and of investigations of the interpretation can be maintained. For the further Bargmann-Wigner equations [59-61]. The papers evaluation equation (1.1) has to be decomposed of Lord [87], Wilson [88], Dehnen and Ghaboussi into an equivalent set of first order derivative [89] which at first sight are only losely connected to equations. It was proved by the author [96] and that problem provided hints for controlling and Grosser [97] that the set of nonlinear equations justifying our assumptions. #■=1,2 Finally, in particular in connection with general relativity, some authors, for instance Gürsey [90], (-iy " d ^ + m r)aß(Pßr(x) Braunss [91], Datta [92], Hehl and Datta [93], = Kßyö (Pßs (*) Vyt (*) Qöu (*) (1 -2) Ulmer [94], Borneas [95], have derived spinor field stu equations from torsion or more general principles etc. We do not exclude the possibility of deriving is connected with (1.1) by a biunique map where spinor field equations from a more basic principle. this map is defined by the compatible relations But at present we follow the de Broglie-Bopp- Heisenberg program to derive the entire high- Va(x) = (Pa\ 00 +

with a = (a,A) = (spinor, isospinor index) etc., one-time description can be performed, so that and eventually the following equation results

vaß = Vaß <5AB > Ä = 1,2, Po\%> = l j h KIil2dl2\%> M o .2 - 5 (1-5) vaß'--Oaß, Vaß \-iy aß. hh This kind of coupling is needed to obtain non- + i h w \hhi

D%Z2: = i}'axa2^AiA2^ixi2^AiA2 > If gravitons are assumed to be composite preon states then the following requirements have to be mzxz2'-~ mi\öaxa2öA^Alöixi2öAxÄ2, (1.10) fulfilled: ^z, z2z3z4 '-= 9 hax ai <^AJA2 SAiA2 ^ c )«2a3 i) In the low-energy limit the graviton selfcou- pling has to coincide with classical relativity theory; The quantum states of the model (1.1) or (1.8) ii) in the low-energy limit the graviton-matter respectively are described by state functionals coupling has to reproduce the classical cou tf]> with respect to the states {|tf>} where plings of the corresponding fields. j = j z (x) are sources with corresponding Z-indices. In the high energy limit deviations from the For concrete calculations it is necessary to intro classical couplings are admitted and lead to form- duce normal transforms by \Z) = Z0 [j] | and factors in the interactions, see [75]. the energy representation of the spinor field in According to our model all kind of observable terms of state functionals. Both procedures were matter is build up of preons. For a first exploration discussed in detail in [78] and yield a functional of gravity as a composite particle effect we concen equation for In this equation the limit to a trate on the coupling of gravitons with these 349 349 H. Stumpf • Gravitation as a Composite Particle Effect

elementary constituents. Because if their coupling investigation it was demonstrated that the cluster satisfies ii), one can expect that due to the com- state transform Jfbf the functional energy operator positeness all other graviton-matter couplings will J f leads to a hierarchy of interactions which is satisfy ii), too. We thus confine ourselves to generated by the different magnitudes of the cou graviton-preon systems. pling constants of the individual terms in This We assume the graviton to be composed of four property in combination with high preon masses preons in agreement with the idea of spin fusion, allows the application of the leading term approxi see [44, 47, 48, 59-61]. In contrast to the mation. This approximation was thoroughly dis solutions of the corresponding de Broglie-Barg- cussed in [75]. It is performed in two steps: mann-Wigner ( = BBW) equations our graviton states are, however, assumed to be nonlocal fusion states. The properties and the physical meaning of i) The complete sets of cluster states, i.e. of such composite preon states can be studied by the scattering states as well as of bound states are investigation of the effective interactions. In [75, reduced to the subsets of bound states; 76] it was demonstrated that the appropriate ii) the complete set of transformed functional method for such an investigation is the weak energy operator terms is reduced to the subset mapping procedure. This procedure is defined by a of highest magnitude terms. transformation of the set of functional preon source operators {jj} into a set of functional cluster While the first step is justified by energetic source operators \XR). For the case of a graviton- estimates, the second step is justified by interaction preon system the corresponding cluster source estimates. Furthermore, it is convenient to assume operators for the bosons read that only the physically relevant bound states occur nhhhh . . . . and that possibly exotic bound states can be bK-= I Ca: Ji^Ji2JijJi4 > (2.1) excluded by more detailed bound state investiga hhhh tions. These assumptions were already partly where the set of coefficient functions {CK, K verified by direct calculation [79] and are a = 1...} is defined to be a complete set of four- peculiar feature of the model under consideration. preon cluster states, i.e. this set contains bound With respect to the weak mapping of the state clusters (gravitons) as well as scattering state graviton-preon system we observe that the term clusters of four preons, etc. The fermion cluster hierarchy of is the same as that for the boson- operators are trivial fj = jj due to the restriction to fermion system. For a first exploration of the preons. graviton-preon system we assume for brevity that The transformation of the preon source opera the leading term approximation works also for this tors {j/} into boson and fermion source operators system in analogy to the boson-fermion system. {bK,fj} induces a transformation of the set of The cluster transform J f of J f allows the functional states {| g>} as well as of the correspond following decomposition ing functional equation (1.11). This transforma tion is characterized by the invariance conditions Jl — Jlp f — b, (2.2) h S f Sb and f S , S + Jfßp f, — , b ,----- (2.4) S f Sb J f J, bh ,-----, 3 /, f — 0 (2.3) Sj Sb Sf where is the pure preon part, the pure where | §> and J^are the cluster transforms of graviton part, while JfBF contains the interactions |g> and J f of (1.11). between both kinds of constituents. By direct cal The weak mapping of a boson-fermion system culation we obtain 3fF= Jfand in the leading term with two-preon scalar boson states and three-preon approximation for the pure graviton part JfB = fermion states was discussed in [75]. By a detailed J fj+ J ff i with 350 H. Stumpf ■ Gravitation as a Composite Particle Effect

I R7xi2hi,KhI[c V ^ b t This is achieved by means of an appropriate V2W1 dbn approximation of the state functional | Since, apart from formfactors etc., (2.9) is to describe a - 6 E conventional field theory of pointlike particles, we hhhhh can apply a convenient approximation scheme LxL2L^nm which is provided by conventional quantum field theory for this case. Such a scheme is defined by (2.5) the application of connected functional states. For öb„ gravitons the connected functional Zc [b] is defined by J f B : = 24 E Ä hhhUhk WhL\L2h hLxL2L^nmn' (2.10) W 3W 6

xWihu A ____ 1 where Zc [b] is a power series in the graviton source operators b. The usual quantum field theoretic (2.6) approximation techniques [98] are based on an a priori given Lagrangian of the field under con whereas the graviton-preon interaction is given by sideration, a precondition which obviously is not fulfilled in our case, as the only a priori given Lagrangian is that of the preons and not that of y^RF — 6 y r L\L2hUDk. iy h , , hLL\L2L^ gravitons. Thus we are forced to apply an approxi I2I3kn mation method of statistical mechanics to Zc [b] where the lowest approximation reads A fk (2.7) Sfh Sbn Zc[b] = E z„b„ (2.11) where #™/2/3/4 are the dual graviton states, while ^/2/3/4/5/6 and /? /2/3 are the first order polariza with {zn} as the one-particle graviton transition tion cloud states of gravitons and preons, resp. matrix elements, i.e. wave functions. By means of Furthermore, the definition this approximation in statistical mechanics " _ tii/n ) classical hydrodynamic equations can be derived WIL^LjLT, - ( W /L1Z.2Z.3ias(Z.l£2£3) (2.8) and we expect in our case to obtain the classical is used, i.e., the vertex is antisymmetrized in the equations of gravity. last three superindices. In this paper we treat only The operators (2.5) (2.6) can formally be written the effective graviton selfcoupling which follows from evaluation of In the next paper the graviton-preon coupling (2.7) will be discussed. ^ B = 4 kB+ 4 pB = E [Kmn + P mn]bmdn (2.12) Due to the above mapping-procedure the func and tional energy equation (1.11) is mapped into the equation and if we confine our E [Vmnn bmdndn] . (2.13) selves to the graviton dynamics, the corresponding equation reads Substitution of (2.10) - (2.13) into (2.9) then gives the equation (2.9) This functional equation describes a nontrivial nonlinear quantum field theory which is rather e lz,b'\ E [Kmn + Pmn)bmzn (2.14) difficult to evaluate. For a first exploration it is, however, sufficient to derive gravity from (2.9) as + E Vmnn' bmZ„Z„-- E ----bmZm M 0 > = 0 a mean value or as a classical approximation, resp. mnn' m dt 351 H. Stumpf • Gravitation as a Composite Particle Effect

This equation is solved if the equation where U is the auxiliary field operator, S the symmetrical spinor basis and T the antisymmetri- ^m j S f >mn\Zn~^~ Y, Vmnn'ZnZn' cal super-spinor-isospinor basis. These quantities [ mn mnn' are discussed in detail in [76, 79]. The polarization amplitude of the center of mass-motion is not - l — zm\ = 0 (2.15) included in (3.1). In [76] it was rather shifted into m at the vector boson source operators themselves. In the following we will do the same for the graviton is satisfied for arbitrary bm, i.e., we obtain a set of case. nonlinear equations for the determination of z„. In By means of (3.1) a general bound state of two the following we will evaluate these equations. vector bosons can be written in the following form 3. Low Energy Graviton States '2 r3 '2 h '1 '2 '3 '4 i\ •2 For the computation of the effective graviton Cr = j c(k,k')«c dynamics by means of the weak mapping pro «1 «2 a3«4 «1 a2 cedure the explicit form of the graviton wave func u *2 *3 w Ui *2 tions is needed. As long as these wave functions are not derived from a direct calculation, they have to be postulated. Since the calculation of four preon y dkdk' bound states is rather complicated, in a first exploration we postulate them. For a physically (3.2) meaningful guess of the graviton wave functions where the bracket means antisymmetrization in the assumption of being composed of four preons /i 72/3/4. In order to have a definite center of mass is too general. It must be supplied by a more momentum the wave function c{k,k') must have detailed information. Such an information is the special form provided by considering the pointlike composite graviton states which are solutions of the BBW- c(k,k') = Ö(K -K ')c(w ) (3.3) equations [44, 47, 48, 59-61], and by proposals contained in [87 - 89] which support this ap with k = \ K'+w, k' = \ K '-w and K the true proach. For brevity we cannot draw the con center of mass momentum. Then (3.2) goes over clusions in detail. Summarizing the results we are into forced to assume that graviton states are the direct Cn = +'2+'3+'4)1/4^ <,<2 (ri _ ri) product of two vector boson states. With this con clusion we can relate graviton states to detailed • cp '3'4 (r3 - r A)(p(rx + r2- r i - r 4) quantitative calculations. Vector boson states were ■SaS a'T 'T '' las (3.4) extensively treated in [76, 79] as two-preon bound state solutions in the low energy approximation. In where (p is the Fourier transform of c(h>). this approximation the dependence of the internal If (3.4) is to describe a graviton state, in the wave function on the center of mass momentum is pointlike approximation this function must reduce neglected, and taking over such vector boson wave to the direct product of two pointlike vector boson functions for the construction of graviton wave states. We are going to show this by evaluation of functions we work within the same approximation the diagonal part of which corresponds to the scheme. common BBW-equation. According to [76, 79] the wave functions of the For the evaluation of Jf% the diagonal part (2.5) vector boson states are given by of JfBj the dual states R m are needed. These states can be obtained from (3.1) and (3.2) by replacing tp 1 r r' by its dual counterpart <73, i.e. the left-hand i i' C a _ fjii'e ik(r+r') 1/2 solution of the ^eigenvalue equation, see [74, 79]. a a' The evaluation of itself by substitution of C„ \x x' '1 •

r2'rl'r4~*r extensively discussed in [75, 76]. So here we will = e*{q>i (0)ilm S Z aS £ T ,^ T ' ^ „ t (3.8) not repeat explicit calculations, but give only their results. i.e., in this expression antisymmetrization rests Due to (3.4) we observe that in full notation we only on the T' (x) T' -product. The only combina can write bm = an(^ dn = dat a-t' and define tion for any T ' (x) T' -product that survives this antisymmetrization is ' ata2aja4(r) 1-^*1*2 ^*3*4'as = ^3*3^4*4}as V t,t'. (3.9)

w Thus by this limes condition the superspin-isospin (3.5) dependence is uniquely determined. Furthermore, '"' (r) it will be demonstrated that in all relevant inter a co actions only t = t' is admitted and that the coupling E ct^Sa3a^sym(o,o')dot,o't'- ac constant is independent of t. In accordance with (3.9) we are thus allowed to choose an arbitrary By strictly observing antisymmetry properties etc. product T' (x) T' as a representative of (3.9). In the and using definitions (3.5) we obtain for (2.5) following we use T' = T = ( ^ * and omit = I \ b t" ( r ) { [(akdk- M ß - g ßa, ) a, it' the t, t' indexing. V- * Due to the special structure of (3.6), Eq. (3.7) ^a2a2 ^Ojaj ^0404] symt^ a2a3a4) can be interpreted as the functional energy repre + 9 i(ßy5)a,ai y«2«2^ym sentation of a local field theory with local field operators V a^a2a3a4(x) • If we represent |§ / b y the + 9<5a,ai<5a2i [iß?5) a3ai V l^ h y m Q ^ )} connected functional defined by (2.10) (2.11) and da{aiaia\(r)dr . (3.6) substitute this in (3.7) we obtain the linear part of (2.15) which reads The corresponding energy eigenvalue equation for [/' r " a^ - M - g] a{ Vala2a3a4 (*) free gravitons then reads = - g y at a{ y a2a2 (X), E | § / = ^ | S / . (3.7) (3.10) [i y ^ d ^ -M - g]aia. y/a^ a 3a4 (*) As can be easily seen, in (3.7) the superspin-isospin = - g y «I o.\ y a2a2 (X), dependence drops out, i.e., our effective equations for the determination of the free graviton states and equations with exchange of 1 2->3 4. In partic give no hint how to choose these states with respect ular for linear fields the solutions of the functional to the set {T'(x) T''}. This dependence can, how equation and of the corresponding classical equa ever, be deduced from a consideration of the tions coincide exactly as is the case with (3.7) and BBW-states. According to Rodriguez and Lorente (3.10). [60] for massless gravitons with the special mo Equations (3.10) are generalized BBW-equa- mentum p = (E, 0,0, E) only the local amplitudes tions. The advantage of having derived generalized with a.\ = a 2 = or3 = a4 = a give a non-vanishing and not the original BBW-equations will soon contribution to the graviton states. If this condi become obvious. The expansion of the multispinor tion is transferred to the microscopic theory it reads means that for ax = a2 = a3 = a4 = a the limes Va]a2a3a4 = ^ O a , ^ ( / O a 3a4 Va.ö /*,-»/•, /= 1,2,3,4 of (3.4) for all combinations of auxiliary fields must exist. In particular for the + (yac ) axai( z a'b'c ) a^ v a,a.b. original field strengths of (1.1) we have (p\ = + (ZabC)a a (ya C)a a lf/ab,a' ii' -a'b' and for these fields the limes of (3.4) reads + (ZabC)axai ( I a 0 C)0304 y,abia.„.. (3.11) 353 H. Stumpf • Gravitation as a Composite Particle Effect

In contrast to the solutions of the BBW-equations in the case of massless gravitons ij/a,a'b' and Wa'b'.a the multispinors (3.11) need not be fully vanish identically, [60], whereas in our case these symmetrized since these spinor amplitudes are quantities do not vanish! This difference is derived from the fully antisymmetric states (3.2) essential for obtaining a meaningful theory of and are not forced to be completely symmetric. By gravity. It is reasonable to interpret ij/ab,a' and substitution of (3.11) into (3.10) it can be shown Va.a'b' as spinorial affine connections and if these that free graviton states result only for i//aa< = 0. quantities vanish an essential geometrical quantity This assumption is compatible with the general is not available. It is this fact that caused earlier equations resulting from (3.10). For brevity we do approaches to gravity based on BBW-equations to not discuss this in length. For y/a>a> = 0 we obtain fail. An extensive geometrical interpretation of our from (3.10) with g = —M and rescaling iUaba'b1 by approach will be given in Section 5. (2 M )" 1

Vab.a'b'(x) = Q[a¥b],a'bl(x)+Q[alWab,b'\(x) , 4. Effective Nonlinear Graviton Equations

To evaluate the complete Eq. (2.15) we will use (3 12) the graviton states of Section 3. However, before doing any special calculation we have to discuss Va.a'ß (X) + V ^a'ix) = 0 . selfconsistency carefully. As the linear states are This system is solved by the ansatz completely fixed by the linear theory they cannot be compatible with the nonlinear theory. In the Wab,a'b'(.X) = da da>Xbb'-da db'Xa'b perturbation theoretic approach to gravity [65] this -dbda'Xb'a+dbdb'Xaa'. _ _ contradiction is ignored. In this approach we do (3.13) not use perturbation theory and thus solve this Va,a'b'(x) = da'Xab'-db'Xaa'+SaXa'b', problem explicitly. To reconcile the nonlinear Vab,a

<0 \(p^(Pi2(Pii \ . (4.3) Naturally, all these considerations need an improved justification by more detailed calcula 355 H. Stumpf • Gravitation as a Composite Particle Effect tions. For a first draft of a rather complicated explicit calculations, but give only the results. If theory it is, however, impossible to justify all in the properties of the polarization cloud wave func detail, because a precondition for carrying out tions (4.6) and of the original graviton wave func detailed calculations is a clear idea about the tions (4.9) are observed, in the leading term ap physical background that governs these calcula proximation only one integral of (4.7) is left. By tions. In this way we assume that the preliminary using the definitions (3.5) and Sa:= (s aC) and discussion of polarization cloud contributions integrating over the internal coordinates of the offers this possibility and we apply the results of wave functions {Cn} we obtain the following ex this discussion to the evaluation of (2.13). pression for this term Substitution of (4.5) into (2.13) and straightfor ward rearrangements by using symmetry and anti r2 ri r4 symmetry properties lead to the following expres m *B = k l\RJ \h1 ax '2a2 '3a3 Ua4 sion mn aa' \X\ *3 *4 # lB= 12 L\L-)L-> • [sa'y V - 5 V y V y 5]a3/?3 / r\ r2 r3 r4 (4.10) '1 '2 h >4 . — 4 Wr , r r C rV^lC C «1 «2 ßi a4 +^ 2W^ ry I5LiL2Li , , r , c^ bn W s C ^ 1^n 1* \*1 *2 *3 *4 b md n'd a a (4.7) where according to Sect. 3 for Cn the superspin- If the wave functions in (4.7) are correctly anti- isospin combination T ' = T ' = T has been used. symmetrized and the antisymmetry of the vertex is The factor k is a numerical constant whose value taken into account, each term of (4.7) formally follows from the integral. contains 3(4!)3 integrals. A number which would Formula (4.10) can still be simplified. According prevent any evaluation. Due to the formation of to Sect. 2 we first transform (4.10) to connected gravitons by two vector boson states, the graviton functionals by the substitution dn-+zn. Then we wave functions (3.4), however, possess such a high observe that za,a' ~ zab,a'b' and zab,a' drop out from structural symmetry that the number of indepen the beginning and that due to (3.13) wehavezfla< = 0. dent integrals is considerably reduced. Formally, So the only term that is left in (4.10) is za,a'~ we can write for Cn of (3.4) za,a'b'- Furthermore, (4.10) admits a symmetriza- tion and if we introduce the abbreviation Cn = eiKR{Zn ZUu( 12,34)}as(1,2>3)4), (4.8) X ^ X ( i) = öaia{. . .X arf.. (4.11) where R = + r2 + r^ -I- r4) 1/4 and Z and u follow by comparison with (3.4). If antisymmetrization is we can eventually write (4.10) in the abridged form explicitly performed and the symmetry properties of u and Z are taken into account, we obtain from (4.12) (4.8) E (2 a'b'a a)i Za,a'b>(ri) C-n / t)mZn Cn = {[Z12Zj4]sym«(12,34) /"= 1 + [Z14Zi3]w( 14,23) Owing to the compatibility condition initially dis + [Z3XZ±4]u(\3,\4)}eiKR. (4.9) cussed, the function za,a'b'(r ) can only have the With this special form of the wave functions each form term of (4.7) eventually contains only 27 indepen Za,a'b'(r) = 9 aXa'b'ir) (4.13) dent integrals. These integrals were individually estimated along the lines which were extensively with x antisymmetric. Defining then the spin discussed in [75, 76]. So here we will not repeat tensor 356 H. Stumpf • Gravitation as a Composite Particle Effect

Z(r): = k I [Zab'Xa-b\r)] (4.14) a'b' - I (M +9)ß(i) r z(r)dr = 0 , (4.19) /= l we obtain for (4.12) with (4.14) the expression where in particular the results of Sect. 2 were used.

* b= I ( R I §'aZ{rh i)a a(i) / bmZn /= l / (4.15) 5. Tetrad Equations of Gravity This expression can be combined with the kinetic energy term f)# of (2.12) to give the formula Equation (4.19) is the intended nonlinear gravi ton equation. It holds for arbitrary b(r) = b(r,0). Since in the energy-representation the center of /B+k . JT Tjpl B —_ (4.16) mass time t can be fixed at an arbitrary value we consider Eq. (4.19) for b(r, t) and z(r,t). Then (4.19) is equivalent to the system of equations Z \R m \ 1 [Va + V am U aii)\C n >bmZn mna \ \J = 1 J / (a k dp- M ß - gß)a^a{za{a2a3aA (*) (5.1) for connected functionals. The term (4.16) allows a rearrangement to the following form ~~~ Za^a2a3a4 (x) + g(ßy~)a^a^y a2a2Za{a2a3a4 (x) 91 ,k . ijpl _ f/B+ Ji B — (4.17) + k £ (Za'b\h ))z a,a.bix )a % a{zaia2a3a4(x) = 0 / - lZ(rhl) 4 lZ(rh,h) h = 1 Z cd (5.9) invariance condition. For infinitesimal local Lorentztransformations the most general form of if we put g = - M and remove the factors M and k Xab reads by rescaling. Furthermore, from symmetry con 3 siderations it follows Xab(x ) = E ek(x )Maa>rja'b k = 1 Zfjv,nd = 0 . (5.10) + E ek(x)eklmM'amb, (5.4) These equations are the tetrad version of Einstein's klm = 1 homogeneous vacuum equations, see Edgar [101]. 358 H. Stumpf • Gravitation as a Composite Particle Effect

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