An Explanation by a Fermionic Substructure of Leptons and Quarks

Why Quarks are Different from Leptons – An Explanation by a Fermionic Substructure of Leptons and Quarks

H. Stumpf Institute of Theoretical Physics, University Tuebingen, Germany Reprint requests to Prof. H. S.; E-mail: [email protected]

Z. Naturforsch. 59a, 750 – 764 (2004); received August 14, 2004

To explain the difference between leptons and quarks, it is assumed that electroweak gauge bosons, leptons and quarks are composites of elementary fermionic constituents denoted by partons (not to be identified with quarks) or subfermions, respectively. The dynamical law of these constituents is assumed to be given by a relativistically invariant nonlinear spinor field theory with local interaction, canonical quantization, selfregularization and probability interpretation. According to the general requirements of field operator algebraic theory, this model is formulated in algebraic Schroedinger representation referred to generating functionals in functional state spaces. The derivation of the corresponding effective dynamics for the composite particles is studied by the construction of a map between the spinor field state functionals and the state functionals of the effective theory for gauge bosons, leptons and quarks. A closer examination of this map shows that it is then and then only selfconsistent if certain boundary conditions are satisfied. The latter enforce in the case of electroweak symmetry breaking the difference between lepton and quark states. This difference can be analytically expressed as conditions to be imposed on the wave functions of these composite particles and leads ultimately to the introduction and interpretation of color for quarks, i.e., the characteristic of their strong interaction.

Key words: Substructure of Quarks and Leptons; Effective Dynamics; Difference of Leptons and Quarks.

1. Introduction At present the general tendency prevails to embed the standard model into theories with grand unifica- In the standard model of elementary particle physics tion, supersymmetry, superstrings, etc., see [1]. But to leptons and quarks are assumed to constitute the solve the above problem we prefer the compositeness fermionic part of its basic particle set, i.e. in this model hypothesis as a possible approach of model building. leptons and quarks are considered to be the ultimate While the former model building leads to an en- fermionic constituents of matter without substructure. largement of symmetries and (or) dimensions, etc., the In spite of the enormous success of this model with latter approach should lead to a simplification of the respect to the experimental verification, it raises a lot of basic particle set and of the interactions. Thus from an theoretically and experimentally unsolved questions, economic point of view the compositeness hypothesis see [1], Sect. 3.10, [2], Sect. 5.7, [3], Sect. 12, and is very attractive. Nevertheless, before starting a the- these have led to further alternative model building oretical discussion one should have a look on experi- since several decades, see [1]. ments. Does there exist any experimental hint in favour One of these unsolved problems is the existence of of this hypothesis? lepton and quark families, or more simply: Why quarks In high energy physics deep inelastic scattering pro- are different from leptons and why do they exist in gen- cesses of photons by photons and similarly of quarks erations? by photons seem to provide signals of composite- If one tries to answer these special questions, one ness of the photons themselves, [4 – 6], as well as is confronted with the basic problem by which kind of the quarks, [7 – 9]. In addition an unexpected con- of model the standard model should be replaced, be- firmation of compositeness comes from the study of cause within the standard model itself one cannot find superfluids with an enormous variety of experimen- any reason for the difference of leptons and quarks and tal facts and analogies to high energy physics and their replication. even cosmology, [10]. We consider these facts as a

0932–0784 / 04 / 1100–0750 $ 06.00 c 2004 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com H. Stumpf · Why Quarks are Different from Leptons 751 justification of theoretical research in compositeness models cited in [15], because the mathematical for- models. mulation and evaluation is different from previous at- In the past decades such a model was developed tempts in this field. which we use as a suitable candidate to treat quanti- In previous attempts, see [15], the difference be- tatively the substructure of elementary particles. This tween leptons and quarks was generally explained by model is based on a relativistically invariant nonlin- attributing to quarks special algebraic and group the- ear spinor field theory with local interaction, canoni- oretical degrees of freedom, whereas in the case un- cal quantization, selfregularization and probability in- der consideration we start with a globally invariant terpretation. It can be considered as the quantum field SU(2) ⊗ U(1) spinor theory, and the difference be- theoretic generalization of de Broglie’s theory of fu- tween leptons and quarks is generated by the dynamics sion, [11], and as a mathematical realization and phys- of the model itself. Being composite particles, the ef- ical modification of Heisenberg’s approach, [12], and fective dynamics of leptons, quarks and gauge bosons is expounded in [13, 14]. will be studied. In this case the consistence of the map- Its basic particle set consists of partons (not to be ping from the basic parton theory to the corresponding identified with quarks) or subfermions, respectively, effective theory imposes subsidiary conditions which which are assumed to be the elements of the substruc- enforce the difference between lepton and quark states ture of the elementary particles of the standard model. and hence offers an explanation of the difference of In this model the formation of additional new partonic leptons and quarks themselves. bound states is not excluded, where the latter can act as Thus, before discussing the latter problem we first additional “elementary particles” in the corresponding develop the formal proofs for the derivation of the ef- effective theory replacing the standard model. fective dynamics of these composite particles and, sub- In analyzing the problem why quarks are different sequently, based on the results of these investigations, from leptons, we refer to this model. In particular, we can treat the actual problem in the last section. in its simplest version the electroweak gauge bosons are assumed to be two-parton bound states, while the 2. Algebraic Representation of the Spinor Field fermion families are assumed to arise from three parton bound states. The latter assumptions are made in The algebraic representation is the basic formula- several compositeness models, [15]. Among other au- tion of the spinor field model and the starting point for thors, the three fermion substructure of leptons and its evaluation. In order to avoid lengthy deductions we quarks was proposed by Harari [16] and Shupe [17]. give only some basic formulas of this formalism and But apart from this assumption our model has no- refer for details to [13, 14, 18]. The corresponding La- thing in common with the Harari-Shupe model or other grangian density reads, see [14], Eq. (2.52)

3 −1 ¯ µ L(x):= λi ψAαi(x)(iγ ∂µ − mi)αβδABψBβi(x) i=1 (1) 1 2 3 − h h ¯ ( ) ( ) ¯ ( ) ( ) 2g δABδCDvαβvγδ ψAαi x ψBβj x ψCγk x ψDδl x h=1 i,j,k,l=1 with v1 := 1 and v2 := iγ5. The field operators are To obtain a uniform transformation property with re- assumed to be Dirac spinors with index α =1, 2, 3, 4 spect to Lorentz transformations, the adjoint spinors and additional isospin with index A =1, 2 as well as are replaced by formally charge conjugated spinors, auxiliary fields with index i =1, 2, 3 for nonperturba- which are defined by tive regularization. The algebra of the field operators is c ¯ ψ (x)=CαβψAβi(x), (3) defined by the anticommutators Aαi and the index Λ is introduced by [ + (r ) (r )] = (r−r) ψAαi ,t ψBβj ,t + λiδij δABδαβδ (2) ( ); =1 ( )= ψAαi x Λ ψAΛαi x c ( ); =2 . (4) resulting from canonical quantization. All other anti- ψAαi x Λ commutators vanish. Then the set of indices is defined by Z := (A, Λ, α, i). 752 H. Stumpf · Why Quarks are Different from Leptons

The operator formulation of the theory by means bra with corresponding duals ∂I := ∂Z (r) which sat- of (1) and (2) cannot be evaluated without further isfy the anticommutation relations preparation. In order to obtain definite results from this theory a state space is needed in which the dynamical [jI , ∂I ]=δZZ δ(r − r ), (8) equations can be formulated. This is achieved by the use of the algebraic Schroedinger representation. For while all other anticommutators vanish. ∂ |0 =0 a detailed discussion we refer to [13, 14, 18]. Here we With I f the basis vectors for the generat- present only the results and definitions. ing functional states can be defined. The latter are not To ensure transparency of the formalism we use the allowed to be confused with creation and annihilation symbolic notation operators of particles in physical state spaces, because generating functionals are formal tools for a compact | (ψI1 ...ψIn ):=ψZ1 (r1,t) ...ψZn (rn,t) (5) algebraic representation of the states a : To each state |a in the physical state space we associate a functional with Ik := (Zk, rk,t). Then in the algebraic Schroe- state |A(j; a) in the corresponding functional space. dinger representation a state |a is characterized by the The map is biunique, and the symmetries of the orig- set of matrix elements inal theory are conserved. Nevertheless this map does not induce any equivalence of the kind indicated above. ( ):=0|A( )| =1 ∞ τn a ψI1 ...ψIn a ,n ... , (6) For details see [13, 14]. In order to find a dynamical equation for the func- A where means antisymmetrization in I1 ...In. tional states, we apply to the operator products (5) the By means of this definition the calculation of an Heisenberg formula eigenstate |a is transferred to the calculation of the set of matrix elements (6) which characterize this state. ∂ i A(ψI ...ψI )=[A(ψI ...ψI ),H]−, For a compact formulation of this method, generating ∂t 1 n 1 n (9) functionals are introduced, and the set (6) is replaced n =1...∞, by the functional state where H is the Hamiltonian of the system under con- ∞ n i sideration. |A(j; a) := τn(I1 ...In|a)jI ...jI |0f , n! 1 n If |0 as well as |a are assumed to be eigenstates of n=1 I1...In (7) H, then between both states the matrix elements of (9) can be formed, and subsequent evaluation leads to the := (r) where jI jZ are the generators of a CAR-alge- functional equation, see [13, 14]

a E0 |A(j; a) =[KI1I2 jI1 ∂I2 − WI1 I2I3I4 jI1 (∂I4 ∂I3 ∂I2 + AI4J1 AI3J2 jJ1 jJ2 ∂I2 )]|A(j; a) (10) a with E0 = Ea − E0 and summation convention for I, etc. The symbols which are used in (10) are deﬁned by the following relations: := 0 ( k∂ − ) KI1I2 iDI1I D k m II2 (11) with µ := µ (r − r ) := (r − r ) DI1I2 iγα1α2 δA1A2 δΛ1Λ2 δi1i2 δ 1 2 ,mI1I2 mi1 δα1α2 δA1A2 δΛ1Λ2 δi1i2 δ 1 2 , (12) and := − 0 WI1I2I3I4 iDI1I VII2I3I4 (13) with 2 := h ( h ) (r − r ) (r − r ) (r − r ) VI1I2I3I4 gλi1 Bi2i3i4 vα1α2 δA1A2 δΛ1Λ2 v C α3 α4 δA3A4 δΛ31δΛ42δ 1 2 δ 1 3 δ 1 4 , h=1

Bi2i3i4 := 1,i2,i3,i4 =1, 2, 3 (14) H. Stumpf · Why Quarks are Different from Leptons 753 and the anticommutator matrices := ( 0) 1 (r − r ) AI1I2 λi1 Cγ α1α2 δA1A2 σΛ1Λ2 δi1i2 δ 1 2 . (15)

So far this functional equation holds for any algebraic representation. A special representation can be selected by specifying the corresponding vacuum. This can be achieved by the introduction of normal ordered functionals. Using the summation convention for I they are deﬁned by

∞ 1 in |F(j; a) := exp[ jI FI I jI ]|A(j; a) =: ϕn(I1 ...In|a)jI ...jI |0f , 2 1 1 2 2 ! 1 n (16) n=1 n where the two-point function

FI1I2 := 0|A{ψZ1 (r1,t)ψZ2 (r2,t)}|0 (17) contains an information about the groundstate and thus ﬁxes the representation. The normal ordered functional equation then reads

a E0 |F(j; a) = HF (j, ∂)|F(j; a) (18) with

HF (j, ∂):=jI1 KI1I2 ∂I2 + WI1I2I3I4 [jI1 ∂I4 ∂I3 ∂I2 − 3FI4K jI1 jK ∂I3 ∂I2 +(3FI4K1 FI3 K2 1 1 (19) + AI K AI K )jI jK jK ∂I − (FI K FI K + AI K AI K )FI K jI jK jK jK ]. 4 4 1 3 2 1 1 2 2 4 1 3 2 4 4 1 3 2 2 3 1 1 2 3

Equation (19) is the algebraic Schroedinger repre- Then one will ask whether in consequence of this sentation of the spinorfield Lagrangian (1), written in procedure relativistic covariance is completely lost. In functional space with a fixed algebraic state space. perturbation theory this is not the case, because the With respect to the physical interpretation of this for- equivalence of the Hamilton formalism and the covari- malism two comments have to be added: ant formulation can be shown. A similar result can be i) The algebraic Schroedinger representation is the obtained for the composite particle theory in algebraic formulation of the Hamilton formalism for quantum Schroedinger representation: The corresponding effec- fields independently of perturbation theory. Neverthe- tive theories for composite particle reactions can be co- less, for the justification of this procedure the findings variantly formulated too. of perturbation theory are indispensible. ii) In general, in the literature spinor field models, From the perturbation theory one learns that during like the NJL-model in nuclear physics, are considered the time interval where mutual interactions between as effective, low energy theories. If spinor field models the particles take place, these particles cannot be kept are to play a more fundamental role, they need a special on their mass shell. This means that in order to get preparation to suppress divergencies. In the case under nontrivial interactions one is not allowed to enforce the consideration this preparation is expressed by the in- particles on their mass shell all the time. troduction of auxiliary fields with constants λi in (1), The formal treatment of perturbation theory starts (2) and (14), (15), which are designed to generate a with the Schroedinger picture, i. e., the Hamilton nonperturbative intrinsic regularization of the theory. formalism which explicitly avoids the on shell fix- This regularization is closely connected with the ing of particle masses. Hence, if this treatment probability interpretation of the theory. As the λ i are of the perturbation theory is extended to the case indefinite, an indefinite state space of the auxiliary of composite particle interactions, the use of the fields results. Hence these auxiliary fields are unob- Hamilton operator leads to the formalism introduced servable, and a special definition of a corresponding above. physical state space is needed. This definition is iden- 754 H. Stumpf · Why Quarks are Different from Leptons tical with the intrinsic regularization prescription and the ordinary Schroedinger equation in Fock space, and leads in turn to probability conservation in physical the corresponding equation is defined by the functional state space. Furthermore it can be shown that the latter equation (18), (19) in functional space. property can be transferred to the corresponding effec- A transformation of the latter functional equation tive theories themselves. For details we refer [18]. to describe fermion pairs leads to a corresponding bosonic functional equation in algebraic Schroedinger 3. Bosonization in Functional Space representation. For this case, exact mapping theorems were derived in [21, 22] and [23]. In [23] the functional With respect to an application in section 5, we first mapping formalism was formulated in close analogy to treat the simple case of collective modes where bound ordinary bosonization in Fock space. states are represented by fermion pairs. In nuclear In the algebraic Schroedinger representation the physics this treatment has a long history, [19, 20], and central formula which defines the map into the boson in solid state physics it is applied in numerous ver- representation was originally given by the expansion, sions. Following [19] we shortly give the central for- see [13, 14], mulas of the conventional method of approach, which afterwards will be modified to treat the problem under ϕn(I1 ...I2n|a)= consideration. 1 {I1I2 I2n−1I2n} (24) Let $(k1 ...kn)C ...C , (2n)! k1 kn + + + + k1...kn | f = |0f n aβ1 aβ2 ...aβ2n−1 aβ2n (20) where the wave functions C II are single time func- be an even numbered fermion state in fermionic Fock k + tions which result from fully covariant wave func- space with creation operators aβ and the correspond- tions ϕk(x1,x2) by formation of the symmetric limit ing Fock vacuum |0f , see [19] Eqs. (2.5), (2.6), and t1 → t, t2 → t, and by separation into parts which de- let scribe the wave functions of collective variables. So the II + + wave functions C reflect in their structure their rel- |nB = b ...b |0B (21) k {β1β2 β2n−1β2n} ativistic origin by containing the relativistic deforma- be the corresponding bosonic Fock space with creation tions, but no genuine energy eigenvalue equation can + be derived for them. In addition the state space of the operators bαα and the corresponding Fock vacuum |0B auxiliary fields is indefinite, see Section 2. , where the brackets in (21) mean antisymmetriza- ∗ tion ≡A, see [19], Eqs. (2.1), (2.3). Then the operator From these facts it follows that in general Ck is not the dual of Ck. In contrast to the Fock space mapping ∞ 1 + + methods in nuclear physics, it is thus necessary to in- U := bα α ...bα α k n! 1 2 2n−1 2n (22) troduce dual states RII , where both types of states are n=0 α1...α2n assumed to be antisymmetric functions |0Bf0|aα1 aα2 ...aα2n−1 aα2n II II k k Ck = −Ck ; RII = −RII , (25) transforms the fermion state (20) into the boson state (21), see [19], Eq. (2.4), i. e. and where the relations

|nB = NU|nf , (23) I1I2 k = k Ck RI1I2 δk , (26) I I where N is an appropriate normalization factor, and 1 2 where the exponential in [19], (2.4), is represented by 1 I1 I2 k I1 I2 I1 I2 Ck RI I = (δI δI − δI δI ) (27) its power series expansion. 1 2 2 1 2 2 1 If the formulas (21) – (23) are systematically eval- k uated, then Schroedinger equations in fermionic Fock have to be satisfied by definition, provided the wave space can be transformed into Schroedinger equations functions {Ck} are a complete set of antisymmetric in bosonic Fock space. functions in (I1,I2)-space. In the case under consideration the algebraic In order to show the equivalence of both sides of Schroedinger representation of the spinor field replaces (24), we study the inversion of (24) by multiplying H. Stumpf · Why Quarks are Different from Leptons 755 (24) with the corresponding duals and summing over which leaves the boson state (33) invariant: I1 ...I2n. Using here and in the following the summa- P|F( | )=|F( | ) tion convention for k, I, etc., this gives b a b a . (36) Therefore, by the definition (16) of the fermion func- k1 kn R ...R ϕ(I1 ...I2n|a)=˜$(k1 ...kn|a) I1I2 I2n−1I2n tional state, by the definition of the mapping rela- (28) tion (32) and the definition of the boson functional state (33) a bijective map between (16) and (33) is es- with tablished.

k1...kn It remains the task to transform the eigenvalue equa- $˜(k1 ...kn|a)=Sk ...k $(k1 ...kn|a), (29) 1 n tion (18) for the fermion state (16) into a corresponding where eigenvalue equation for the boson state (33). The first step must be to replace the mapping defi- 1 k1...kn k1 kn nition in configuration space (32) by a corresponding Sk ...k = RI I ...RI I 1 n (2n)! 1 2 2n−1 2n mapping definition in functional space. I1...I2n By generalizing relation (22) the following defini- {I1I2 I2n−1I2n } · C ...Ck . tion was introduced by Kerschner [23]: k1 n ∞ f 1 (30) T (b, ∂ ):= bk ...bk |0B ! 1 n n=1 n (37) f f The latter tensors have the projector properties · k1 kn 0|∂ ∂ RI1I2 ...RI2n−1I2n f I1 ... I2n , k ...k j ...j k ...k S 1 n S 1 n = S 1 n . (31) j1...jn l1...ln l1...ln which leads in analogy to (23) to the mapping defini- From this it has to be concluded that a bijective map- tion f ping for the functions appearing in (24) can only be |F(b|a) = T (b, ∂ )|F(j|a), (38) achieved if (24) is replaced by, see [21, 23]: 1 and owing to the general definition of duals (26), (27) ϕn(I1 ...I2n|a)= $˜(k1 ...kn|a) the inverse relation of (38) reads (2n)! (32) b |F(j|a) = S(j, ∂ )|F(b|a) (39) · {I1I2 I2n−1I2n } Ck1 ...Ckn . with Then according to (31) formula (28) is indeed the in- ∞ verse of (32). b 1 S(j, ∂ ):= jI ...jI |0f (2 )! 1 2n In the next step we define the generating functional n=1 n (40) state for bosons by I1 I2 I2n−1I2n b b · B0|∂ ∂ Ck1 ...Ckn kn ... k1 . ∞ 1 |F(b|a) = $˜(k1 ...kn|a)bk ...bk |0B In contrast to nuclear physics the inverse operator of T ! 1 n (33) n=1 n is not its Hermitean conjugate, but (40). For these operators the following relation can be de- with the functional creation operators bk and the func- rived, [23]: |0 ∂b tional Fock vacuum B, where with the duals k of f b bk the commutation relations T (b, ∂ )S(j, ∂ )=P (41) b [bk, ∂k ]− = δkk (34) and S( ∂b)T ( ∂f )=1( ∂f ) are postulated, while all other commutators vanish. j, b, j, . (42) Furthermore, for later use we define the projector Using these relations, equation (18) can be mapped ∞ into equation 1 b b P = |0 l1...ln 0|∂ ∂ bl1 ...bln BSl ...l B l ... l , (35) a b ! 1 n n 1 |F( | ) = H ( ∂ )|F( | ) n=1 n E0 b a B b, b a (43) 756 H. Stumpf · Why Quarks are Different from Leptons for the boson functional states (33) with and

b f f b f f f b f HB ( ∂ )=T ( ∂ )HF ( ∂ )S( ∂ ) KI b, b, j, j, . (44) T (b, ∂ )∂I ∂K = Ck ∂kT (b, ∂ ) (46) To evaluate (44), the commutator between T and H F has to be derived. This can be achieved by the combi- and repeated application. Similar commutator relations nation of two special commutation relations, [23]: were evaluated and applied in nuclear physics, [19], Eqs. (2,7), (2,8), etc. f k f f T (b, ∂ )jI =2RIKbkT (b, ∂ )∂K (45) For the case under consideration one obtains

a kk b k ll kll b b kk ll E0 |F(b|a) := {K bk∂k + Q (bk + Γkk blbl ∂k )+W1 bk∂l ∂l + W2 (bk + Γkk blbl ∂k )∂k

(47) kk l l l k1k2 l1l1 l2l2 + ( + ∂ ) ∂ + ( + ∂ )( + ∂ )}|F( | ) W3 bk Γkk bl bl k bk l W4 bk1 Γ bl1 bl k bk2 Γ bl2 bl k b a , k1k1 1 1 k2k2 2 2 where the various tensor coefﬁcients are deﬁned by the following relations

kk := 2 k I2K k =2 k K KI1I2 RI1K Ck ,Q KI1I FII2 RI1I2 , kll := 2 k I4 K I2I3 kk := −6 k I2I3 W1 UI1I2I3I4 RI1K Cl Cl ,W2 UI1I2I3I4 FI4 I RI1I Ck ,

1 kk l k k I2K W := 4UI I I I (3FI I FI I + AI I AI I )R R C , (48) 3 1 2 3 4 4 3 4 4 3 I1I I K l 1 k1k2 k1 k2 W := −4UI I I I (FI I FI I + AI I AI I )FI I R R , 4 1 2 3 4 4 3 4 4 3 2 I2I I I ll I1I2 l I3I4 l := −2 Γkk Ck RI2I3 Ck RI4I1 .

The effective boson functional (47) can be transformed into those equations in [21, 22], which were derived earlier without using the formalism introduced above.

4. Mappings with Two-fermion and Three-fermion dynamical law for the composite bosons and fermions States has to be derived. Concerning the literature of such more compli- Phenomenological theories describe the interactions cated mappings, in physical Fock space the two- of bosons and fermions. If a partonic substructure of fermion-one-fermion problem was discussed in detail these particles is assumed, this leads to the necessity by Meyer, [24]. The same problem, referred to the to derive effective dynamical laws for the simultane- algebraic Schroedinger respresentation in functional ous occurrence of composite bosons and composite space, was treated by Kerschner, [23]. In addition Ker- fermions. In this respect, at present, the most promi- schner, [23], developed a derivation of an effective the- nent example is given by the quark structure of mesons ory for the two-fermion-three-fermion system, which and nucleons in nuclear physics. In the simplest case is of interest with respect to our problem. Unfortu- this leads to the problem of a simultaneous treatment of nately, in the latter case the corresponding proof has to two-fermion (quark) and three-fermion (quark) states. be rejected, because it does not properly take into ac- A similar situation arises in the parton model of ele- count the boundary conditions and the effect of dressed mentary particles, where gauge bosons are assumed to particle states on the mapping. Thus a new line of ap- be composed of two partons, while leptons and quarks proach is necessary to solve this problem. are assumed to have a three parton structure. In this section we first treat the formal derivation In both cases the elementary quark or parton dy- of the mapping procedure, while in the following sec- namics, respectively, can be described by appropriately tions the physical implications of this proceeding are regularized NJL-models like that defined by the La- to be studied, and corresponding modifications are to grangian (1), and thus from these models an effective be introduced. H. Stumpf · Why Quarks are Different from Leptons 757 The elements of the map are the two-fermion and and the completeness relations the three-fermion states. The former are defined by the q 1 boson states (25), (26) and (27), while for the latter I1 I2I3 I1I2I3 Cq RI I I = δ{I I I } . (50) 1 2 3 3! 1 2 3 as states we postulate the general orthonormality relations q q between wave functions Cq and their duals R The map of the parton state (16) onto a corresponding q I1I2I3 q functional state for composite bosons and fermions is Cq RI I I = δq , (49) 1 2 3 achieved by a generalization of the transformation (32) to the case under consideration. The boundary conditions will be incorporated afterwards.

For brevity we replace the parton indices Il simply by l , i. e., RI1I2 is replaced by R(1,2), apply the summation convention for repeated numbers, and deﬁne the transformation by ∞ ∞ {(1,2) (2m−1,2m) n( 1 n| )= n,2m+3r ˜( 1 m 1 r) ϕ I ...I a δ $ k ...k ,q ...q Ck1 ...Ckm m=0 r=0 (51) · (2m−1,2m+2,2m+3) (2m+3r−2,2m+3r−1,2m+3r)} Cq1 ...Cqr with

k1...km,q1...qr $˜(k1 ...km,q1 ...qr)=Sk ...k ,q ...q $(k1 ...km ,q1 ...qr ) (52) 1 m 1 r and 1 k1...km,q1...qr k1 km q1 qr Sk ...k ,q ...q := R(1,2) ...R(2m−1,2m)R(2m+1,2m+2,2m+3) ...R(2m+3r−2,2m+3r−1,2m+3r) 1 m 1 r ! n (53) {(1,2) (2m−1,2m) (2m+1,2m+2,2m+3) (2m+3r−2,2m+3r−1,2m+3r)} · Ck ...C Cq ...Cq . 1 1 r In analogy to (31), the matrix (53) has the projector property. Using the the functional formulation, in this case the state (16) is mapped into its image functional state

∞ ∞ ∞ 1 |F(b, f; a) := δn,2m+3r $˜(k1 ...km,q1 ...qr|a)bk ...bk fq ...fq |0BF ( !)( !) 1 m 1 r (54) n=2 m=0 r=0 m r by application of the operator

∞ ∞ ∞ 1 f k1 km T (b, f, ∂ ):= δn,2m+3r bk ...bk fq ...fq |0BF R ...R (m!)(r!) 1 m 1 r (1,2) (2m−1,2m) n=2 m=0 r=0 (55) q1 qr f f · R(2m+1,2m+2,2m+3) ...R(2m+3r−2,2m+3r−1,2m+3r)f 0|∂n ...∂1 , where the boson operators satisfy the commutation re- It should be noted and it is remarkable that the value lations (37), and the fermion operators satisfy the anti- n =1is not included in equation (54) as well as commutation relations in (55), because it is impossible to find within the com- F q posite particle set a composite particle combination for [fq, ∂ ]+ = δ , (56) q q n =1. As in the general case all n-numbers are re- while all other anticommutators vanish, and the defini- quired for a nondegenerate map, the ommision of the tion |0BF := |0B ⊗|0F for the functional vacuum n =1term has to be justified in order to obtain a state is used. selfconsistent theory. After having completely derived 758 H. Stumpf · Why Quarks are Different from Leptons the effective dynamics of two- and three-fermion states closely connected with the requirement that the three we will treat this problem in section 6 in detail. Here, parton states describing leptons and quarks have to be we only indicate that the omission of the n =1term is genuine three particle states and are not allowed to have projections into the one particle (parton) sector. If one defines ∞ ∞ ∞ b f 1 (1,2) (2m−1,2m) S(j, ∂ , ∂ ):= δn,2m+3r j1 ...jn|0f C ...C n! k1 km n=2 m=0 r=0 (57) · (2m+1,2m+2,2m+3) (2m+3r−2,2m+3r−1,2m+3r) 0|∂f ∂f ∂b ∂b Cq1 ...Cqr BF qr ... q1 km ... k1 and ∞ ∞ ∞ 1 P := |0 k1...km,q1...qr 0|∂f ∂f ∂b ∂b bk1 ...bkm fq1 ...fqr BF Sk ...k ,q ...q BF q ... q k ... k , (58) ( !)( !) 1 m 1 r r 1 m 1 n=0 m,m=0 r,r=0 n m analogous relations to equations (36), (38), (39), (41) – Obviously QF has to be interpreted as a generalized (44) hold for these operators, while after rather lengthy quantization term, where the anticommutators A II , rearrangements equations (45) and (46) are replaced by which govern and define the abstract algebra, are mod-

k q ified by the influence of the vacuum, i. e., the influ- T =2 T ∂ +3 T ∂ ∂ jK bkRIK I fqRIIK I I , (59) ence of the special representation which is under con-

k b sideration. In addition, if this representation is mapped T ∂ ∂ = ∂ T K K CKK k (60) into the two-fermion-three-fermion sector, further de- and formations of the original operator quantization rules are to be expected. T ∂ ∂ ∂ = q ∂f T I3 I2 I1 CI1I2I3 q . (61) Because we are primarily interested in the effective dynamical law of the two-fermion-three-fermion sec- For brevity all details of these calculations were sup- tor, we postpone the investigation of these modified pressed. quantization terms and concentrate on the transforma- Thus, by means of these relations equation (18) can tion of the terms in (63) which decribe the system dy- be mapped into the two-fermion-three-fermion sector. namics (under the influence of the vacuum!). Before doing this in detail, we rewrite the energy In particular one recognizes that under the influence operator (19) in the form of the vacuum the system develops two-body forces H ( ∂):=Hd ( ∂)+Q ( ∂) which are given by the last term in (63). F j, F j, F j, (62) d Transformation of HF leads to the equation with a b d E0 |F(b, f|a) = (Kkl + Mkl)bk∂l HF (j, ∂)=jI1 KI1I2 ∂I2 f b b (63) +( + ) ∂ + kl1l2 ∂ ∂ Kqp Mqp fq p W1 bk l2 l1 + WI1I2I3I4 [jI1 ∂I4 ∂I3 ∂I2 − 3FI4K jI1 jK ∂I3 ∂I2 ]

qlp b f k1k2l1l2 b b + W fq∂ ∂ + W bk bk ∂ ∂ and 2 l p 3 1 2 l1 l2 (65) q1q2p1p2 f f + q q ∂ ∂ QF = WI I I I W4 f 1 f 2 p1 p2 1 2 3 4 1 qpkl b f · (3FI K FI K + AI K AI K )jI jK jK ∂I + W5 bk∂l fq∂p |F(b, f|a) 4 1 3 2 4 4 1 3 2 1 1 2 2 1 (64) − ( + ) FI4K1 FI3K2 AI4K1 AI3K2 with 4 k l · FI2K3 jI1 jK1 jK2 jK3 . := 2 Kkl RII1 KI1I2 CI2I , H. Stumpf · Why Quarks are Different from Leptons 759 := −6 k l Mkl WI1I2I3I4 FI4K RKI1 CI2I3 , over the formulas and results needed for our discus- q p sion from [14]. In particular, the dressed boson states Kqp := 3R KI I C , II I1 1 2 I2II are of interest with respect to the modiﬁcation of the

Mqp := −9WI1I2I3I4 FI4K effective dynamics in (65), i. e., we concentrate on the q p q p treatment of these states in the following. · (R C − R C ), IKI1 I2I3I II K I2I3I As an intermediate step we deviate from the method kl1l2 := 2 2 k l1 l2 of Sect. 2 and 3 and introduce an explicit func- W1 WI1I2I3I4 RII1 CI I CI I , 4 2 3 tional representation of the dressed boson states by the qlp q q l W := 3WI I I I R C C , 2 1 2 3 4 II I1 I4 II I2I3 (66) ansatz, see [14], Eq. (5.50) k1k2l1l2 := 12 k1 k2 l1 l2 ∞ W3 WI1I2I3I4 FI4K RII RIK CII CI I , 1 2 3 = I1...I2j+2n q1q2p1p2 q1 q2 b2j,k C2j,k jI1 ....jI2j+2n , := −27 (67) W4 WI1I2I3I4 FI4K RII I1 RK1K2K n=0 p1 q2 j =1, 2, ...∞. · CIK K CI I I , 1 2 2 3 qpkl := −24 q p k The index j describes the number of fermion pairs W5 WI1I2I3I4 FI4K RIIK CI I I RII 2 3 1 which lead to irreducible boson states, while n char- l q p k l acterizes the contribution of the polarization part re- · CII − RIII CI I I RKK CIK . 1 2 3 sulting from the excitation of fermion pairs, etc., in For the evaluation of the coefficient functions (66) the the vacuum. Furthermore, by the definition of dressed wave functions of the corresponding particles are re- boson states any term in the expansion (67) must be- quired. With respect to these wave functions and their long to the same quantum numbers, as otherwise a calculation, detailed investigations were performed in unique description and characterization of dressed bo- several papers. The boson wave functions can be ex- son states would be impossible. actly calculated, see [13, 14]. The fermion functions The inversion of (67) reads, see [14], Eq. (5.52), are discussed in Section 6. In Sects. 5 and 6 the sum- r mation convention will again be applied. 2j,k I I = 2j,k j 1 ...j 2r RI1...I2r b , (68) k j=1 5. Constraints by Dressed Particle States where the R-functions in (68) are the duals of the C- Formally, the mapping calculations of Sect. 3 are ex- functions in (67), see [14]. For fixed k and j, the du- act, provided the necessary preconditions are observed. als R2j,k are projectors, which from the set of possible But even if all these conditions are satisfied, physically, boson generators b2h,l project out the generators rep- the map is inadequate. This inadequacy stems from the resenting the quantum numbers k, j and no other ones. fact that by its construction the mapping formalism of Hence the duals R2j,k must be elements of the same Sect. 4 does not incorporate the effects of dressed par- irreducible representation to which b2j,k belongs. ticle states. For r =3one obtains from (68) The existence of dressed particle states induces additional correlations which must be incorporated into = 2,k + 4,k jI1 ...jI6 RI ...I b2,k RI ...I b4,k the dynamics in order to narrow down the set of pos- 1 6 1 6 k (69) sible solutions of Equation (65) to those solutions 6,k + RI ...I b6,k . which are physically meaningfull. Therefore the for- 1 6 malism of Sect. 4 has to be supplemented by consider- ing the influence of such dressed states on the effective To resolve this formula, we represent the fermion states dynamics. of Sect. 3 in the explicit form Two decades ago a proposal was made, [25], to := I1I2I3 solve this problem. Based on the development of the f3,l C3,l jI1 jI2 jI3 (70) formalism of dressed particle states, [26], [13], [14], with the inversion this proposal can now be elaborated in more detail, and 3,l the result can be incorporated in (65). Concerning the I I I = 3,l j 1 j 2 j 3 RI1I2I3 f . (71) dressed particle formalism, we refer to [14] and take l 760 H. Stumpf · Why Quarks are Different from Leptons Then, multiplying (69) by the antisymmetrized product In so doing, we consider the boson operators in (65) C3,l ⊗C3,l of two fermion states, for the left hand side as spectators and transform the F-space into an F- of (69) the following relation space and a boson space with generating sources ˜b. After having performed this map, we identify the ˜b- 1 (−1)P 3,l 3,l generators with the original (spectator) b-generators CI I I CI I I jI1 ...jI6 6! λ1 λ2 λ3 λ4 λ5 λ6 λ1...λ6 (72) to achieve selfconsistency with the constraint (74). 3,l 3,l This step is justified as long as phenomenologically = C jI jI jI C jI jI jI = f3,lf3,l I1 I2I3 1 2 3 I4I5I6 4 5 6 the electroweak bosons are only characterized by their quantum numbers. In the effectivce theory it is then results, and thus (69) is transformed into ˜ impossible to discriminate b2,k from b2,k. 1 I I I I I I P λ1 λ2 λ3 λ4 λ5 λ6 The exact treatment of this (2,1)-map runs along the f3,lf3,l = (−1) C C 6! 3,l 3,l same lines as that of Sect. 2 and was performed by Ker- λ1...λ6 k 2,k 4,k schner in [23]. To avoid lengthy deductions we refer · 2,k + 4,k (73) RI1...I6 b RI1...I6 b directly to the corresponding formulas of [23]. 6,k To apply the (2,1)-map to the effective energy oper- + R b6,k . I1...I6 ator of (65) we rearrange it into the form

If we are not interested in the coupling of the higher b f Heff (b, f, ∂ , ∂ ):= order boson states b4,k and b6,k to the fermionic cur- b f b f (75) rent, we can suppress the corresponding terms in (73) H1(b, f, ∂ , ∂ )+H2(b, f, ∂ , ∂ ) and obtain from (73) with I1 I2I3 I4I5I6 2,k = f3,lf3,l C3,l C3,l RI1I2I3I4I5I6 b2,k ˜ f q1q2p1p2 f f k H1 := Kqpfq∂ + W fq fq ∂ ∂ (74) p 4 1 2 p1 p2 ˜k b b b = Rll b2,k, b b kl1l2 H2 := (Kkl + Mkl)bk∂l + W1 bk∂l ∂l (76) k 1 2 b b + k1k2l1l2 ∂ ∂ 2,k W3 bk1 bk2 l1 l2 , if the antisymmetry of R in I1,...,I6 is observed. 3,l 3,l In (74) C ⊗ C acts as projector which then and and then only gives a nonvanishing expression if the ˜ f f qlp b qpkl b quantum numbers attributed to this projector coincide Kqp := (Kqp + Mqp + W2 ∂l + W5 bk∂l ). (77) with those of b2,k. For electroweak bosons b2,k and 3,l 3,l fermion functions C , C , this is the case if the Denoting the corresponding mapping operators by T˜ fermions are assumed to be leptons or quarks, and if in and S˜, from [23], Eq. (3.305) it follows the corresponding projectors these functions are cho- f qr b f sen in an appropriate group theoretical combination. T˜ ∂ S˜ =[2˜k1 ˜ ˜ ∂˜ + ∂ ]P˜ fq p RprCk bk1 k fq p . (78) Hence from these fermion state combinations, a non- 2 2 trivial contribution to the polarization cloud of the elec- Furthermore, from [23], Eq. (3.303) one obtains troweak bosons is expected. Having derived relation (74), we consider it as ˜ f f ˜ ˜k ˜ ˜ll ˜ ˜ ˜b T fq fq ∂ ∂ S = 2R (bk + Γ blbl ∂ ) a constraint which relates (composite) fermions to 1 2 p1 p2 q1q2 kk k f bosons, and we return to the exact mapping theorems: − 2( ˜k r1 − ˜k r1 )˜ ∂ Rq1r2 δq2 Rq2r2 δq1 bkfr1 r2 (79) We disregard the functional operator relation (74) ˜k f f ˜ { } + ∂ ∂ P and consider the set Rll as the starting point for a fq1 fq2 p1 p2 . bosonization in the functional F-space of the effective theory (65). Having derived these relations, one can transform (65) { ˜k } ˜ If the set Rll is assumed to be complete, one can into the b-f space by applying the rules of Sect. 2 and 3. ˜k construct the corresponding set of dual states {Cll } This gives with and can thus apply the mapping procedure to Equa- tion (65). T|F˜ (b, f|a) = |F(˜b, b, f|a) (80) H. Stumpf · Why Quarks are Different from Leptons 761 for equation (65) the expression formula (51). For this case several explicit evaluations were performed which allowed to identify bosons and a ˜ ˜ b f ˜˜ E0 |F(b, b, f|a)=THeff (b, f, ∂ , ∂ )ST|F(b, f|a) fermions with the elementary particles of the standard ˜b b f model. For a review see [14]. =H˜(˜b, b, f, ∂ , ∂ , ∂ )|F(˜b, b, f|a), (81) 6. Boundary Conditions for Fermion States P˜ because (36) holds likewise for the projector . From (66) it follows that for the application and ˜ ≡ Introducing the constraint b b means identifying evaluation of the mapping theorem the explicit knowl- ˜b the ˜b generators and their duals ∂ with the spectator edge of the wave functions of the various particles is b generators b and their duals ∂ . required. In this section we concentrate on the role of Under these constraints one obtains from (81) fermion states within this mapping formalism. In particular we will demonstrate why the omission of the ˜ f b b f H = Hf (f,∂ )+Hb(b, ∂ )+Hbf (b, f, ∂ , ∂ ) (82) n =1term in (54) and (55) can be justiﬁed and leads to a discrimination of leptons and quarks. with Starting with the Lagrangian density (1), by

f f f q1q2p1p2 f f means of the ﬁeld theoretic formalism generalized Hf := (K + M )fq∂ + W fq fq ∂ ∂ , qp qp p 4 1 2 p1 p2 de Broglie-Bargmann-Wigner (GBBW)-equations for (83) three-parton states can be derived, the solutions of which are assumed to represent lepton and quark states. H :=[( b + b )+( f + f )2 ˜k ˜pr] ∂b b Kkl Mkl Kqp Mqp RqrCl bk l The Lagrangian (1) and the corresponding GBBW- (2) ⊗ (1) kl1l2 ql1p ˜k ˜pr b b equations are invariant under the global SU U +[W +2W RqrC ]bk∂l ∂l (84) 1 2 l2 2 1 superspin-isospin symmetry group, and the solutions qpk l pr b b +[ k1k2l1l2 + 1 1 2 ˜k2 ˜ ] ∂ ∂ of the GBBW-equations are to be representations of the W3 W5 Rqr Cl bk1 bk2 l l 2 1 2 corresponding permutation group, i.e., for three-parton and states under the group S3. qlp b f q1q2p1p2 ˜k f f With respect to these groups as well as the Lorentz Hbf :=W2 ∂l fq∂p +W4 2Rq q bk∂p ∂p 1 2 1 2 group, a group theoretical analysis was performed qpkl b f q1q2p1p2 + W5 bk∂l fq∂p +W4 (85) in [27], the result of which leads to the table

˜ll b ˜k r1 ˜k r1 f f f ·[ ∂ +( − ) ∂ ]∂ ∂ 1 2 3 4 5 6 7 8 Γkk blbl k Rq1r2 δq2 Rq2r2 δq1 bkfr1 r2 p1 p2. χ χ χ χ χ χ χ χ t 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 For a complete identification of electroweak gauge t3 1/2 −1/2 1/2 −1/2 1/2 −1/2 1/2 −1/2 bosons, leptons and quarks it would be necessary to f 11−1 −1 1/3 1/3 −1/3 −1/3 evaluate their effective dynamics in detail, which hy- q 100−1 2/3 −1/3 1/3 −2/3 e¯ ννeud¯ d¯ u¯ pothetically is defined by the functional energy operator (82). This would exceed the scope of this paper. It is, however, possible to discriminate analytically lep- for the classification of leptons and quarks by their in- ton and quark states without explicit evaluation of (82), ternal superspin-isospin wave functions χ. as will be demonstrated in the following section. In this table the isospin quantum numbers t and t 3 Although in this paper the identification problem of are to be identified with the quantum numbers of the composite particles is not explicitly treated by evalu- unbroken electroweak SU(2) gauge group, and no dis- ating (82), it should be emphasized: for the supposi- tinction is made between lefthanded and righthanded tion that (82) is a meaningful description of the ef- particles. The latter difference comes about by parity fective dynamics of these bosons and fermions it ex- symmetry breaking and is not the topic of this inves- ists a strong evidence. If exchange forces are partly tigation. The fermion number f in this scheme is con- neglected an approximative expression for the effec- nected with the weak hypercharge by f =2y. Using tive dynamics can be formulated and derived by rep- this relation, the values of the table are in complete resenting boson and fermion states directly in the correspondence to the phenomenological values of the form (67) and (70) instead of using the exact mapping lefthanded particles, see for instance [28], Table 22.2. 762 H. Stumpf · Why Quarks are Different from Leptons Analogous values characterize the lepton and quark and states of the higher generations, where the occurence of 1 1 0 1 the latter states depends on the calculation of the mass = −1 = ≡ 3 ⊗ 1 F G FG 3 0 −1 3σ . (90) spectrum to be done in the last step. In a preliminary way this was shown in [29] by calculations with the According to Sect. 4, the two-fermion-three-fermion corresponding energy equation in the ultralocal limit. map does not apply to the n =1sector of the orig- Furthermore, by means of the Clebsch-Gordan coeffi- inal functional equation (18), (19). Hence this map is = 3 cients, isospin representations for t /2 can be de- then and then only compatible with this equation if the rived. After electroweak isospin symmetry breaking n =1sector vanishes identically. This can be consid- it is assumed that the corresponding masses are ex- ered as a boundary condition imposed on the map. tremely large and are thus at present unobservable, i. e., To simplify the notation we use the covariant bound- for getting a more realistic information about the par- ary condition for n =1, which reads ton structure of leptons and quarks, electroweak sym- µ ( ∂ − ) = metry breaking has to be taken into account. DII µ mII ϕI WII1I2I3 ϕI1I2I3 . (91) Nevertheless, even if this symmetry breaking is included in the calculation of eigenstates of GBBW- This condition is completely equivalent to the corre- equations, no explanation of the difference between sponding condition in the algebraic Schroedinger rep- lepton and quark states can be given within this for- resentation (18), (19) and thus can replace the latter. malism. If condition (91) is transformed into the G- conjugated representation, this yields However, it will be demonstrated that after electroweak symmetry breaking an explanation of the dif- µ (ΛA) (Dαα ∂µ − miδαα )δii δΛΛ δAA χα (x)i = ference between leptons and quarks can be given by observing the n =1term boundary condition for the h h i ( )α α gλ vαα1 δΛΛ2 δAA1 v C 2 3 cΛ2Λ3 cA2A3 mapping on to an effective theory. h h h To evaluate this condition it is necessary to change + ( ) (92) vαα2 δΛΛ2 δAA2 v C α3α1 cΛ3Λ1 cA3A1 from charge conjugated spinors to G-conjugated h h + ( ) spinors, as the group theoretical analysis in [27] is re- vαα3 δΛΛ3 δAA3 v C α1α2 cΛ1Λ2 CA1A2 ferred to these quantities. (Λ1A1)(Λ2A2)(Λ3A3) · χˆα α α (x, x, x). This transformation reads 1 2 3 One can interprete (91) or (92), respectively, in the fol- ( )= ˜ ( ) ψκαi x Gκκ ψκ αi x , (86) lowing way: Let ϕ(3) be a nontrivial three-parton state amplitude ˜ ( G) where ψ is defined by the set ψ, ψ and G is defined which yields after substitution in (92) a nonvanish- by ing right hand side. Then, if ϕ(3) describes a physical state, due to the field theoretic construction of (92) a 1 0 := wave function ϕ(1) has to exist with the same quantum G 0 − . (87) iσ2 numbers as ϕ(3). Hence ϕ(3) cannot be the most elementary description of this state. Therefore if leptons ϕI I I ϕˆ With (86) and (87) the three-parton states 1 2 3 or , and quarks are to be genuine three-parton states, their respectively, are transformed into wave functions must decouple from ϕ(1), i.e., substi-

κ1κ2κ3 tuted in (92) for these states the right hand side of (92) ϕˆ (x1,x2,x3)= α1α2α3 must vanish, which automatically includes the vanish- (88) (1) κ1κ2κ3 ing of ϕ . Gκ κ Gκ κ Gκ κ χˆα1α2α3 (x1,x2,x3), 1 1 2 2 3 3 Without worrying about the mechanism of the sym- and application to the superspin-isospin group genera- metry breaking one can accept its existence and ask tors leads to the transformation laws for the group theoretical consequences with regard to the construction of the set of states under consider- k 1 σ 0 1 ation. After the electroweak symmetry breaking has Gk = G−1GkG = ≡ 1 ⊗ σk (89) 2 0 σk 2 taken place only one algebraic constraint remains in H. Stumpf · Why Quarks are Different from Leptons 763

Θα1 α2 α3 β1 β2 β3 qf the superspinor-isospinor space, namely the eigenvalue 1 Θ 2111111 1e¯ condition of the charge operator, see [28], p. 302. I.e., 2 Θ 2211110 1ν¯ 3 in the state space only the charge operator has to be Θ 2112220−1 ν 4 diagonalized. This gives the condition Θ 221222−1 −1 e

+ + = can be written in the same form with respect to the QI1I χII2I3 QI2I χI1II3 QI3I χI1I2I qχI1I2I3 , superspinor part. Furthermore, the symmetry breaking (93) has no effect on (92). By substitution of these states where the solutions of this condition can be con- into (92), the right hand side of this equation vanishes. structed by direct products of one particle states. Therefore in both cases the leptonic states decouple In the one-parton sector the eigenvalue condition from the one-parton sector, remain irreducible states reads and the boundary condition for the map is satisfied because this implies that ϕ1 vanishes too. ( ) = ( ) Qκ1κ χκ α x i qχκα x i (94) This situation is drastically changed for quark states. We consider the superspinor-isospinor representatives with of the set of quark states given by the table 1 1 1 3 3 Θα1 α2 α3 β1 β2 β3 qf Q = G3 + F ≡ δΛ1Λ2 σA A + σΛ Λ δA1A2 . 5 2 2 1 2 6 1 2 Θ 2111122/3 1/3 u 6 Θ 221112−1/3 1/3 d 7 ¯ (95) Θ 2111221/3 −1/3 d 8 Θ 221122−2/3 −1/3 u¯ The eigenvectors of the charge operator are derived l by means of the ansatz χκα(x)i = ζκχα(x, l)i with j l which is referred to the representation of Θ by one ζ = δκ,l, l =1,...,4. For the three-parton case κ particle superspinor-isospinor states (96). the eigenvectors are generated by the direct products l l l After symmetry breaking, the superspinor-isospinor ζ 1 ⊗ ζ 2 ⊗ ζ 3 with l1,l2,l3 =1,...,4. parts for quarks are expressed by such direct prod- For unbroken symmetry, in general such direct products, and substituting these states of the table into (92) ucts cannot be considered as appropriate eigensolu- one easily verifies that in this case the right hand tions, because the complete wave functions ϕ(3) or side of (92) does not vanish. Therefore after symmetry χ(3), respectively, have to be antisymmetric. But with breaking the quark states have lost their irreducibility. the loss of the full symmetry by symmetry breaking, There is only one way to restore this irreducibility: the permutation symmetry is lost, too, see [30], and The local amplitude χ(x, x, x) has to vanish. thus single direct products of one-parton functions can be considered as representatives of the various states. If In order to achieve this, a group theoretical argu- each of these direct products is an eigenvector of (93), ment is desirable. For instance it would be sufficient if then there exists a total of 64 states, which, however, the quark states had nonvanishing orbital angular mo- are highly degenerate in general. menta in their groundstates. Then the difference be- To illustrate this we give the representatives of tween leptons and quarks would be created if the lep- the set of degenerate states for the leptons, where tons had zero orbital momentum which is allowed only the eigenfunctions Θj of superspin-isospin are simply for the leptonic ground states, see above. given by direct products in the form The decision whether this group theoretical argument works, obviously depends on the group theo- j retical properties of the three-parton wave function ΘΛ A ,Λ A ,Λ A := δA ,αj δA ,αj δA ,αj 1 1 2 2 3 3 1 1 2 2 3 3 I1I2I3 (96) Cq of Section 4. These wave functions cannot be · δ j δ j δ j , Λ1,β1 Λ2,β2 Λ3,β3 chosen arbitrarily. On the contrary: they have to be calculated in accordance with the general field theoretic j j which with αi and βi as the superspin-isospin quan- formalism of the model under consideration. This is tum numbers of the single particle states leads to the guaranteed if the three-parton states are solutions of table above. the corresponding GBBW-equations, see above. Translated into the language of the original states, About the solutions of these equations exact state- see [27], after symmetry breaking the leptonic states ments can be made: by a slight generalization of the 764 H. Stumpf · Why Quarks are Different from Leptons results in [31], it can be shown that in the rest system The latter facts provide the basic for an approxi- 2 the angular momentum operators J and J3 commute mate construction of quark states, proposed and elab- with the operators defining the GBBW-equation, and orated in [34] and lead in consequence to a quantum by a slight modification of the results of [32], Sect. 5, mechanical explanation of color, [34]. This explana- withouut symmetry breaking the intrinsic partity oper- tion is substantiated by group theoretical considera- ator P commutes with the GBBW-operators, too. tions, see [35], Sect. 11.1, where it is demonstrated that The simultaneous existence of the associated quan- in quantum mechanics a representation space of the in- tum numbers leads to the conclusion that in this case variance group G with dimension n is simultaneously the orbital angular momentum quantum numbers must a representation space of SU (n).Forn =3in the case have definite values, too, see [33], Sect. 6.1 for a sim- O(3) this leads to a representation of SU (3). ple example. Owing to the relativistic invariance of the With respect to the interpretation of the effective dy- GBBW-equations and the angular momentum condi- namics this transmutation was already quantitatively tions these quantum numbers can be transferred to ar- studied in [34], see also [14], but a more detailed elab- bitrary k-vectors, so the complete set of three-parton oration should be performed, including the results of states can be classified in this way. recent progress. This will be done else where. Hence with respect to the three-parton solutions of GBBW-equations one can indeed verify the proposed Acknowledgement grooup theoretical argument, as in any case the solu- I would like to express my thanks to Prof. Alfred tions without symmetry breaking are basic for further Rieckers for a critical reading and discussion of the calculations. More details will be elaborated in [27]. manuscript. [1] R. N. Mohapatra, Unification and Supersymmetry, [16] H. Harari, Phys. Lett. 86 B, 83 (1979). Springer, New York 2003. [17] M. A. Shupe, Phys. Lett. 86 B, 87 (1979). [2] G. Boerner, The Early Universe, Springer, New York [18] H. Stumpf, Z. Naturforsch. 55a, 415 (2000). 2002. [19] D. Jansen, F. Doenau, S. Frauendorf, and R. V. Jolos, [3] B. Povh, K. Rith, C. Scholz, and F. Zetsche, Particles Nucl. Phys. A 172, 145 (1971). and Nuclei, Springer, New York 2002. [20] A. Klein and E. R. Marshalek, Rev. Mod. Phys. 63, 375 [4] H. Stumpf and T. Borne, Ann. Fond. L. de Broglie 26, (1991). 429 (2001). [21] R. Kerschner and H. Stumpf, Ann. Phys. (Germ.) 48, [5] M. Erdmann, The Partonic Structure of the Photon, 56 (1991). Springer, New York 1997. [22] W. Pfister and H. Stumpf, Z. Naturforsch. 46a, 389 [6] A. J. Finch, Photon 2000, Int. Conf. on the Structure and (1991). Interactions of Photons, AIP Conf. Proc. 571, 2001. [23] R. Kerschner, Effektive Dynamik zusammengesetzter [7] Z. Cao, L. K. Ding, Q. Q. Zhu, and Y.D. He, Phys. Rev. Quantenfelder, (Thesis) University of Tuebingen 1994. Lett. 72, 1794 (1994). [24] J. Meyer, J. Math. Phys. 32, 3445 (1991). [8] K. Akama and H. Terazawa, Phys. Lett. B 321, 145 [25] H. Stumpf, Z. Naturforsch. 40a, 294 (1985). (1994). [26] H. Stumpf, Z. Naturforsch. 42a, 213 (1987). [9] P. Grabmayr and A. Buchmann, Phys. Rev. Lett. 86, [27] P. Kramer and H. Stumpf, Group Theoretical Repre- 2237 (2001). sentation of the Partonic Substructure of Leptons and [10] G. E. Volovik, The Universe in a Helium Droplet, Univ. Quarks. To be published. Press, Oxford 2003. [28] O. Nachtmann, Elementarteilchenphysik, Phaenomene [11] L. de Broglie, Theorie Generale des Particules a Spin, und Konzepte, Vieweg, Wiesbaden 1986. Gauthier-Villars, Paris 1943. [29] W. Pfister, Nuovo Cim. A 108, 1427 (1995). [12] W. Heisenberg, Introduction to the Unified Field The- [30] H. Stumpf, Z. Naturforsch. 59a, 185 (2004). ory of Elementary Particles, Interscience Publ., London [31] H. Stumpf, Z. Naturforsch. 58a, 1 (2003). 1966. [32] H. Stumpf, Z. Naturforsch. 58a, 481 (2003). [13] H. Stumpf and T. Borne, Composite Particle Dynamics [33] F. Gross, Relativistic Quantum Mechanics and Field in Quantum Field Theory, Vieweg, Wiesbaden 1994. Theory, Wiley, New York 1999. [14] T. Borne, G. Lochak, and H. Stumpf, Nonperturbative [34] H. Stumpf, Z. Naturforsch. 41a, 1399 (1986). Field Theory and the Structure of Matter, Kluwer Acad. [35] M. Hamermesh, Group Theory, Addison-Wesley Publ. Publ., Amsterdam 2001. Comp., Dordrecht 1962. [15] L. Lyons, Progr. Particle Nuclear Phys.10, 227 (1983).