A Modified Y-M Action with Three Families of Fermionic Solitons and Perturbative Confinement
aa r X i v : . [ h e p - t h ] A p r A MODIFIED Y-M ACTION WITH THREE FAMILIES OFFERMIONIC SOLITONS AND PERTURBATIVECONFINEMENT
C. N. RAGIADAKOSPedagogical InstituteMesogion 396, Agia Paraskevi, TK 15341, Greeceemail: [email protected], [email protected]
ABSTRACT
The dynamics of a four dimensional generally covariant modifiedSU(N) Yang-Mills action, which depends on the complex structureof spacetime and not its metric, is studied. A general solution of thecomplex structure integrability conditions is found in the contextof the G , Grassmannian manifold, which admits a global SL(4,C)symmetry group. A convenient definition of the physical energy andmomentum permits the study of the vacuum and soliton sectors.The model has a set of conformally SU(2,2) invariant vacua and aset of Poincar´e invariant vacua. An algebraic integrability conditionof the complex structure classifies the solitonic surfaces into threeclasses (families). The first class (spacetimes with two principal nulldirections) contains the Kerr-Newman complex structure, which hasfermionic (electron-like) properties. That is the correct fermionic gy-romagnetic ratio (g=2) and it satisfies the correct electron equationsof motion. The conjugate complex structure determines the antisoli-ton, which has the same mass and opposite charge. The fermionicsolitons are differentiated from the complex structure bosonic modesby the periodicity condition on compactified spacetime. The non-periodicity of the found solitonic complex structures is proved. Themodification of the Yang-Mills action has an essential consequence tothe classical potential. It generates a linear static potential insteadof the Coulomb-like r potential of the ordinary Yang-Mills action.This linear potential implies that for every pure geometric solitonthere are N solitonic gauge field excitations, which are perturba-tively confined. The present model advocates a solitonic unificationscheme without supersymmetry and/or superstrings.1 ontents
1. INTRODUCTION2. ACTION OF THE MODEL
3. THE G , GRASSMANNIAN MANIFOLDS SU (2 ,
2) classical domain3.2 Complex structures in G , context3.3 Induced metrics on spacetimes
4. VACUA AND EXCITATION MODES
5. “LEPTONIC” SOLITONS st family5.3 Solitonic features of the massive structures5.4 Hopf invariants of complex structures5.5 Massless complex structures of the 1 st family5.6 The 2 nd and 3 rd family solitons may be unstable
6. “HADRONIC” SOLITONS AND CONFINEMENT INTRODUCTION
Standard Model appears to be amazingly successful in all its experimental test-ings. These successes can be found in many recent books in Quantum FieldTheory. But it is generally believed that it is not a complete theory because itcontains too many independent parameters. On the other hand many apparentphenomena have not yet been proven or successfully described. Quark confine-ment, the three generations of leptons, the corresponding three generations ofquarks and the apparent correspondence between leptons and quarks are somecharacteristic physical phenomena, which have not yet been understood in thecontext of Quantum Field Theory. These characteristic features are provedto occur in the present slightly modified generally covariant Yang-Mills model,which has fermionic solitons without fermionic fields in its action.General Relativity is actually a well established macroscopic theory. It isbased on the Einstein equation E µν ≡ R µν − R g µν = 8 πk T µν (1.1)where T µν is the energy momentum tensor of the matter fields. T µν is an exter-nal non-geometric quantity, which is formally imposed by hand. The classicalmathematical problem of this theory is to compute the metric tensor g µν , whichsatisfies the Einstein equation. The external character of T µν has always beenconsidered as a drawback of the theory and many efforts have been undertakento derive it from geometry.The two successful mainstreams (Quantum Field Theory and General Rela-tivity) have been developed independently. Each branch has tried to incorporatethe other one without apparent success. The straightforward “covariantization”of the Standard Model action with the Einstein gravitational term is not renor-malizable. That is, it is not a self consistent Quantum Field Theory. Thesefailures led researchers to look for non-conventional Lagrangian models. Themainstream of research turned first into Supergravity without success and afterinto Superstrings, without apparent experimental tests up to now. The firsttests on supersymmetry are expected to be provided by the Large Hadron Col-lider (LHC) experiments. If supersymmetry is not found then research has toturn to more conventional models like the present solitonic one.General Relativity researchers tried to generate particles in the context ofGeometrodynamics, where matter is considered as a manifestation of geome-try. The fundamental idea and expectation is to derive all particles from puregeometric quantities. The Einstein equation (1.1) is seen as the definition ofthe energy-momentum tensor of these particles. The Einstein-Infeld-Hoffmantheory[3] of motion in General Relativity may be considered the origin of thegeometrodynamic ideas. In this context the particle appears as a singularity ofthe Einstein tensor E µν and the equation of motion is derived from the self-consistency identity E µν ; ν ≡ g = 2. These two results strongly suggested the identification of theelectron with the charged Kerr manifold but the appropriate Quantum FieldTheoretic model was missing. The value of the present model is that it mayplay this role or it may show the way how to find such a theory, which couldincorporate these extraordinary results into Quantum Field Theory. This unifi-cation procedure does not need supersymmetry, because the fermionic particlesappear as solitons of the model. That is the proposed particle unification schemeis of solitonic and not supersymmetric origin. In a solitonic unification schemesStandard Model is simply an effective action like the phonon actions in solidsand fluids[28]. If the next few years the LHC experiments do not find supersym-metric particles, we have to turn to solitonic unification schemes as suggestedby the present model.The model started[21] as a simple exercise to find a four dimensional gen-erally covariant action which would depend on the complex structure of thespacetime and not on its metric. Recall that this property characterizes thetwo dimensional string action. The purpose of this search was to find a renor-malizable generally covariant action without higher order derivatives. Metricindependence assures renormalizability[26], because the regularization proce-dure cannot generate non-renormalizable geometric terms. Only topologicalanomalies may appear. Calculations[26] of the first order one-loop diagramsin a convenient gauge condition show that they are finite. The action[22] ofthe model is reviewed in section 2, where the properties of the Lorentzian com-plex structure[7] are reviewed[8]. The new result of this section is the generalsolution of the complex structure integrability conditions using structure coor-dinates. A large part of the present paper is devoted to review the mathematical4ackground because it is no used in current particle physics.In section 3, the formalism of the Grassmannian manifolds and the classicaldomains is applied to reveal the invariance of the complex structures under thefour dimensional global SL (4 , C ) which is analogous to the SL (2 , C ) symmetryof the string action. This mathematical background is necessary for the reader tounderstand the vacua and soliton sectors of the model and how global SL (4 , C )breaks down to the conformal SU (2 ,
2) and the physical Poincar´e symmetries,which are studied in section 4. The natural emergence of the Poincar´e group isthe most interesting result of the present model. It permits to find massive andmassless stationary axisymmetric solitons and to classify the complex structuresusing the Hopf invariant. This rich physical content of the model is revealedin section 4. The complex structures with solitonic properties are classified[25]into three classes relative to the number of sheets of the complex structure. Thefirst two-valued class is extensively studied. They are solitons because theircomplex structures cannot be compactified. The antisolitons are simply thecomplex conjugate complex structures of the solitons. Solitons and antisolitonshave the same mass but opposite charges. The general forms of these massiveand massless stationary axisymmetric solitons are computed. An analogouscalculation indicates that the other two classes of solitons (with three and foursheets) do not contain stable massive solitons.The model contains only a Yang-Mills field and the ordinary (null) tetradwhich determines the Lorentzian complex structure. The symmetries of themodel do not permit the existence of fermionic fields. In section 5 we show thatthe modification of the Yang-Mills action, which makes it independent of themetric tensor, has a characteristic physical consequence. The static potentialof a source is no longer r but it is linear, which could confine the “colored”sources[25]. From the two dimensional solitonic models we know that the soli-tons may be excited by the field modes. Analogous excitations are expected inthe present case too. That is, the solitonic complex structures of the model maybe excited by the gauge field modes. Then these excited solitons are pertur-batively confined because of the linear gauge field potential. Only “colorless”states may exist free, which is a characteristic property of the hadrons. Noticethat this confinement mechanism implies a strict correspondence between “lep-tonic” pure geometric solitons with vanishing gauge field and the “hadronic”ones with non-vanishing gauge field. The ordinary (Euclidean) almost complex structure is a real tensor J νµ , normal-ized by the condition J ρµ J νρ = − δ νµ (2.1)It defines an (integrable) complex structure, if it satisfies the Nijenhuis integra-bility condition J σµ (cid:0) ∂ σ J νρ − ∂ ρ J νσ (cid:1) − J σρ (cid:0) ∂ σ J νµ − ∂ µ J νσ (cid:1) = 0 (2.2)5hen the manifold over which J νµ exists, becomes a complex manifold.A complex structure is compatible with the metric tensor g µν of the manifold,if the two tensors satisfy the relation J µρ J νσ g µν = g ρσ (2.3)at any point of the manifold. If the signature of spacetime is Lorentzian, thereis always a coordinate transformation such that the metric tensor takes the formof the Minkowski metric η µν at a given point. Then we see that the real tensor J νµ defines a Lorentz transformation at the given point. However there is noreal Lorentz transformation, which satisfies the normalization condition (2.1) ofthe complex structure. Notice that this incompatibility is a pure local propertyand it is not related to the global structure of spacetime.Hence the Lorentzian signature of spacetime is not compatible with a realtensor (complex structure) J νµ . The notion of the Lorentzian complex structurehas been generalized[7] to include complex tensors J νµ . I anticipate that theexistence of antisolitons in the present model is based on this particular propertyof the Lorentzian complex structure. This (modified) complex structure hasbeen extensively studied by Flaherty[8]. It can be shown that there is alwaysa null tetrad ( ℓ µ , n µ , m µ , m µ ) such that the metric tensor and the complexstructure tensor take the form g µν = ℓ µ n ν + n µ ℓ ν − m µ m ν − m µ m ν J νµ = i ( ℓ µ n ν − n µ ℓ ν − m µ m ν + m µ m ν ) (2.4)The integrability condition of this complex structure implies the Frobenius in-tegrability conditions of the pairs ( ℓ µ , m µ ) and ( n µ , m µ ). That is( ℓ µ m ν − ℓ ν m µ )( ∂ µ ℓ ν ) = 0 , ( ℓ µ m ν − ℓ ν m µ )( ∂ µ m ν ) = 0( n µ m ν − n ν m µ )( ∂ µ n ν ) = 0 , ( n µ m ν − n ν m µ )( ∂ µ m ν ) = 0 (2.5)Frobenius theorem states that there are four complex functions ( z α , z e α ), α = 0 , dz α = f α ℓ µ dx µ + h α m µ dx µ , dz e α = f e α n µ dx µ + h e α m µ dx µ (2.6)These four functions are the structure coordinates of the (integrable) complexstructure. Notice that in the present case of Lorentzian spacetimes the coor-dinates z e α are not complex conjugate of z α , because J νµ is no longer a realtensor.The reality conditions of the Newman-Penrose null tetrad ( ℓ µ , n µ , m µ , m µ )imply 6 z ∧ dz ∧ dz ∧ dz = 0 dz e ∧ dz e ∧ dz ∧ dz = 0 dz e ∧ dz e ∧ dz e ∧ dz e = 0 (2.7)These relations are directly proven after a substitution of (2.6). They are equiv-alent to the existence of two real functions Ω , Ω e and a complex one Ω, suchthat Ω ( z α , z α ) = 0 , Ω (cid:16) z e α , z α (cid:17) = 0 , Ω e (cid:16) z e α , z e α (cid:17) = 0 (2.8)Notice that these relations provide an algebraic solution to the problem of com-plex structures on a spacetime. They are much easier handled than the PDEs(2.5). In the next section they will be transcribed in the G , Grassmannianmanifold context providing a powerful mathematical machinery for the compu-tation of complex structures.The integrability conditions of the complex structure can be formulated inthe spinor formalism. They imply that both spinors o A and ι A of the dyadsatisfy the same PDE ξ A ξ B ∇ AA ′ ξ B = 0 (2.9)where ∇ AA ′ is the covariant derivative connected to the vierbein e µa . Thisrelation is equivalent to the existence of a complex vector field τ A ′ B such that ∇ A ′ ( A ′ ξ B ) = τ A ′ ( A ξ B ) (2.10)Using the relation[18] ∇ A ′ ( A ∇ A ′ B ξ C ) = Ψ ABCD ξ D (2.11)one can show that both o A and ι A satisfy the algebraic integrability conditionΨ ABCD ξ A ξ B ξ C ξ D = 0 (2.12)Namely, they are principal directions of the Weyl spinor Ψ ABCD . Therefore acurved spacetime may admit a limited number of complex structures, which aredirectly related to its principal null directions. If the Weyl curvature vanishes,there is no restriction on the proper spinor basis. In this case the manifold isconformally flat and the integrability conditions are completely solved via Kerr’stheorem[8]. 7 .1 Tetrad and structure coordinate forms of the action
The string action describes the dynamics of 2-dimensional surfaces in a multi-dimensional space. Its form I S = 12 Z d ξ √− γ γ αβ ∂ α X µ ∂ β X ν η µν (2.13)does not essentially depend on the metric γ αβ of the 2-dimensional surface. Itdepends on its structure coordinates ( z , z e ), because in these coordinates ittakes the metric independent form I S = Z d z ∂ X µ ∂ e X ν η µν (2.14)All the wonderful properties of the string model are essentially based on thischaracteristic feature of the string action.The plausible question[21] and exercise is “what 4-dimensional action withfirst order derivatives depends on the complex structure but it does not dependon the metric of the spacetime?”. The additional expectation is that such anaction may be formally renormalizable because the regularization procedure willnot generate geometric counterterms. The term “formally” is used because the4-dimensional action may have anomalies which could destroy renormalizabil-ity, as it happens in the string action. Recall that the string and superstringactions are self-consistent only in precise dimensions, where the cancellation ofthe anomaly occurs.A four dimensional action which satisfies the above criterion was found. Thenull tetrad form of this action[22] of the present model is I G = R d x √− g { ( ℓ µ m ρ F jµρ ) ( n ν m σ F jνσ ) + ( ℓ µ m ρ F jµρ ) ( n ν m σ F jνσ ) } F jµν = ∂ µ A jν − ∂ ν A jµ − γ f jik A iµ A kν (2.15)where A jµ is a gauge field and ( ℓ µ , n µ , m µ , m µ ) is an integrable null tetrad.The difference between the present action and the ordinary Yang-Mills actionbecomes more clear in the following form of the action. I G = − Z d x √− g (cid:0) g µν g ρσ − J µν J ρσ − J µν J ρσ (cid:1) F jµρ F jνσ (2.16)where g µν is a metric derived from the null tetrad and J νµ is the tensor of theintegrable complex structure.Like the 2-dimensional string action, the metric independence of the presentaction appears when we transcribe it in its structure coordinates form I G = R d z F j F j e e + comp. conj.F j ab = ∂ a A jb − ∂ a A jb − γ f jik A ia A kb (2.17)8his transcription is possible because the metric and the integrable null tetradtake simple forms in the structure coordinates system.In the case of the string action we do not need additional conditions becauseany orientable 2-dimensional surface admits a complex structure. But in thecase of 4-dimensional surfaces, the integrability of the complex structure hasto be imposed through precise conditions. These integrability conditions maybe imposed either on the tetrad (2.5) or on the structure coordinates (2.7), us-ing the ordinary procedure of Lagrange multipliers. These different possibilitieswill provide the various forms of the action which are equivalent, at least in theclassical level. Its different variations should be seen as different ways to writedown the integration measure over the complex structures of the 4-dimensionalLorentzian manifolds. The additional action term with the integrability condi-tions on the null tetrad is I C = − R d x { φ ( ℓ µ m ν − ℓ ν m µ )( ∂ µ ℓ ν )++ φ ( ℓ µ m ν − ℓ ν m µ )( ∂ µ m ν ) + φ e ( n µ m ν − n ν m µ )( ∂ µ n ν )++ φ e ( n µ m ν − n ν m µ )( ∂ µ m ν ) + c.conj. } (2.18)This Lagrange multiplier makes the complete action I = I G + I C self-consistentand the usual quantization techniques may be used[24].The local symmetries of the action are a) the well known local gauge trans-formations, b) the reparametrization symmetry as it is the case in any generallycovariant action and c) the following extended Weyl transformation of the tetrad ℓ ′ µ = χ ℓ µ , n ′ µ = χ n µ , m ′ µ = χm µ φ ′ = φ χ χχ , φ ′ = φ χ χχ φ ′ e = φ e χ χχ , φ ′ e = φ e χ χχ g ′ = g ( χ χ χχ ) (2.19)where χ , χ are real functions and χ is a complex one. In order to make a selfconsistent paper we will present here some examples ofcomplex structures, which can also be found in the works of Flaherty. Theconfigurations of these complex structures will be used in the next sections.The spinorial form of the integrability condition of the complex structureis conformally invariant. It is invariant under a spinor ξ A multiplication withan arbitrary function, therefore we do not loose generality assuming the form ξ A = [1 , λ ]. Then, in the Cartesian coordinates of a conformally flat spacetime9he spinorial integrability conditions become the Kerr differential equations λ A λ B ∇ A ′ A λ B = 0 ⇐⇒ ( ∂ ′ λ ) + λ ( ∂ ′ λ ) = 0 and ( ∂ ′ λ ) + λ ( ∂ ′ λ ) = 0 (2.20)where the Penrose spinorial notation is used with x A ′ A = x µ σ A ′ Aµ = (cid:18) x + x ( x + ix )( x − ix ) x − x (cid:19) x A ′ A = (cid:18) x − x − ( x − ix ) − ( x + ix ) x + x (cid:19) ∂ A ′ A = ∂∂x A ′ A = σ µA ′ A ∂ µ = (cid:18) ∂ + ∂ ∂ − i∂ ∂ + i∂ ∂ − ∂ (cid:19) (2.21)Kerr’s theorem states that a general solution of these equations is any func-tion λ ( x A ′ B ), which satisfies a relation of the form K ( λ, x ′ + x ′ λ, x ′ + x ′ λ ) = 0 (2.22)where K ( · , · , · ) is an arbitrary function.Notice that in a conformally flat spacetime, the two solutions λ and λ ,which determine the spinor dyad o A ∝ (1 , λ ) and ι A ∝ (1 , λ ), completelydecouple. A characteristic example of a Minkowski spacetime complex structureis given by the two solutions of the quadratic (Kerr) function( x − iy ) λ + 2( z − ia ) λ − ( x + iy ) = 0 (2.23)where the ordinary Cartesian coordinates x = x, x = y, x = z are used. ThisKerr function is time independent and determines a static complex structure.The two solutions are λ , = − ( z − ia ) ± √ ∆ x − iy , ∆ = x + y + z − a − iaz (2.24)The corresponding spinor basis (dyad) is o A = [1 , − ( z − ia ) + √ ∆ x − iy ] , ι A = − x − iy √ ∆ [1 , − ( z − ia ) − √ ∆ x − iy ] (2.25)The corresponding null tetrad is L ∝ (cid:2) (1 + λ λ ) dt − ( λ + λ ) dx − i ( λ − λ ) dy − (1 − λ λ ) dz (cid:3) M ∝ (cid:2) (1 + λ λ ) dt − ( λ + λ ) dx − i ( λ − λ ) dy − (1 − λ λ ) dz (cid:3) N ∝ (cid:2) (1 + λ λ ) dt − ( λ + λ ) dx − i ( λ − λ ) dy − (1 − λ λ ) dz (cid:3) (2.26)10hich is the “flatprint” null tetrad of the Kerr-Newman manifold. In the caseof a = 0 it becomes the trivial “spherical” complex structure.In the case of conformally flat spacetimes the structure coordinates z α aretwo independent functions of ( λ , x ′ + x ′ λ , x ′ + x ′ λ ) and the struc-ture coordinates z e α are respectively two independent functions of ( λ , x ′ + x ′ λ , x ′ + x ′ λ ). It is convenient to use the following structure coordinates z = t − √ ∆ − ia , z = − ( z − ia )+ √ ∆ x − iy z e = t + z + x + y z + √ ∆ − ia , z e = − x − iyz + √ ∆ − ia (2.27)Notice that this complex structure cannot be defined over the whole Minkowskispacetime, because it is singular when o A ∝ ι A , which occurs at z = 0 , x + y = a (2.28)We will see below that these points do not belong to the Grassmannian manifold.Using the Lindquist coordinates ( t, r, θ, ϕ ) x = ( r cos ϕ + a sin ϕ ) sin θy = ( r sin ϕ − a cos ϕ ) sin θz = r cos θ (2.29)the structure coordinates take the form z = t − r + ia cos θ − ia , z = e iϕ tan θ z e = t + r − ia cos θ + ia , z e = − r + iar − ia e − iϕ tan θ (2.30)The Minkowski spacetime null tetrad takes the form L µ dx µ = dt − dr − a sin θ dϕN µ dx µ = r + a r + a cos θ ) [ dt + r +2 a cos θ − a r + a dr − a sin θ dϕ ] M µ dx µ = − √ r + ia cos θ ) [ − ia sin θ ( dt − dr ) + ( r + a cos θ ) dθ ++ i sin θ ( r + a ) dϕ ] (2.31)A simple way[23] , [25] to find a curved space complex structure is the Kerr-Schild ansatz ℓ µ = L µ , m µ = M µ , n µ = N µ + f ( x ) L µ (2.32)where the null tetrad ( L µ , N µ , M µ , M µ ) determines an integrable flat complexstructure. In the case of the static (2.31) complex structure, ( ℓ µ , n µ , m µ , m µ )is integrable for 11 = h ( r )2( r + a cos θ ) (2.33)where h ( r ) is an arbitrary function. Notice that for h ( r ) = − mr + e theKerr-Newman space-time is found. A set of structure coordinates of the curvedcomplex structure, which are smooth deformations of the Minkowski complexstructure, are z = t − r + ia cos θ − ia , z = e iϕ tan θ z e = t + r − ia cos θ + ia − f , z e = − r + iar − ia e iaf e − iϕ tan θ (2.34)where the two new functions are f ( r ) = Z hr + a + h dr , f ( r ) = Z h ( r + a + h )( r + a ) dr (2.35)In the present model these configurations are seen as solitons. The com-plex structure J ρµ describes a fermionic soliton with charge e and its complexconjugate J ρµ describes an antisoliton with charge − e .12 THE G , GRASSMANNIAN MANIFOLD
The present work is heavily based on projective spaces and the classical do-mains, therefore a short review of the projective Grassmannian manifolds andthe SU (2 ,
2) classical domain is needed. The projective space CP is the setof non vanishing 4-d complex vector Z m , m = 0 , , , X m ∼ Y m if there exists a non vanishing complex number c such that X m = cY m . Then the natural topology of C induces a well defined topologyin CP . The coordinates Z m are called homogeneous coordinates and the threecoordinates y I = [ Z Z , Z Z , Z Z ] are called projective coordinates in the Z = 0coordinate neighborhood. Every two elements X m and X m of CP determinea 2 × r A ′ B such that X mi = (cid:18) λ Ai − ir A ′ B λ Bi (cid:19) (3.1)where they are written in the chiral representation. Penrose[16] has observedthat a general solution of the Kerr theorem, which determines the geodetic andshear free congruences, take the simple form K ( Z m ) = 0, where K ( Z m ) is ahomogeneous function. In the case of a first degree polynomial K ( Z m ) = S m Z m with S m = [ S A , S B ′ ] we may define ω A ω A λ A = S m Z m = ( S A − iS B ′ r B ′ A ) λ A (3.2)After a straightforward calculation we find that ω A ≡ ρ A − iτ B ′ r B ′ A satisfy thedifferential equation ∂ A ′ ( B ω C ) = ∂ ω C ∂r A ′ B + ∂ ω B ∂r A ′ C = 0 (3.3)Penrose points out that the inverse is also true. The space of the solutions ofthis differential equation is CP . He called this differential equation “twistorequation” and the projective space CP twistor space. One can easily showthat if X mi of CP satisfy the relations X i † EX j = 0 , ∀ i, j (3.4)with E = (cid:18) II (cid:19) (3.5)the generally complex 2 × r A ′ B becomes Hermitian and it transformsas the Cartesian coordinates of the Minkowski spacetime under the Poincar´esubgroup of the projective transformations SL (4 , C ) of CP . Penrose tried toextract physical meaning from these relations. I will not continue on twistormode of thinking, because I want to avoid confusion of the present conventionalquantum field theoretic model with the Penrose twistor program. But manytimes in the present work we will use the twistor formalism and its spinornotation, because it is computationally very effective.13n the case of a second degree polynomial K ( Z m ) = S mn Z m Z n with S mn = (cid:18) S AB S B ′ A S A ′ B S A ′ B ′ (cid:19) where S AB = S BA , S A ′ B = S A ′ B (3.6)we define the spinor ω AB ω AB λ A λ B = S mn Z m Z n = ( S AB − iS A ′ A r A ′ B − iS B ′ B r B ′ A − S A ′ B ′ r A ′ A r B ′ B ) λ A λ B (3.7)which satisfies the relation ∂ A ′ ( B ω CD ) = 0 (3.8)In the case of a fourth degree polynomial K ( Z m ) = S mnpq Z m Z n Z p Z q wefind ω ABCD = S ABCD − iS A ′ ( ACD r A ′ B ) − S A ′ B ′ ( CD r A ′ A r B ′ B ) ++ iS A ′ B ′ C ′ ( D r A ′ A r B ′ B r C ′ C ) + iS A ′ B ′ C ′ D ′ r A ′ A r B ′ B r C ′ C r C ′ C (3.9)which also satisfies the twistor equation ∂ A ′ ( B ω CDEF ) = 0 (3.10)It is proved that a spinor λ A , which satisfies the fourth degree homogeneouspolynomial ω ABCD λ A λ B λ C λ D = 0 (3.11)determines a geodetic and shear free congruence in Minkowski spacetime. Tworoots of this polynomial define a complex structure. This relation is useful,because it will coincide with the algebraic integrability condition on the curvedspacetime in the weak gravity approximation. That is, we expect the Weylspinorial tensor Ψ ABCD to become proportional with ω ABCD in the weak gravitylimit and in an appropriate (Cartesian) coordinate system.Let us now turn to the definition of the Grassmannian projective manifold G , . Consider the set of the 4 × T = (cid:18) T T (cid:19) (3.12)with the equivalence relation T ∼ T ′ if there exists a 2 × S such that T ′ = T S (3.13)The coordinates z = T T − (3.14)completely determine the points of the set. The topology of the 4 × G , . The coordinates T are called homogeneous coordinates and the coordinates z are called projectivecoordinates. Under a general linear 4 × (cid:18) T ′ T ′ (cid:19) = (cid:18) A A A A (cid:19) (cid:18) T T (cid:19) (3.15)14he inhomogeneous coordinates transform as z ′ = ( A + A z ) ( A + A z ) − (3.16)which is called fractional transformation and it preserves the compact manifold G , , which is called Grassmannian manifold. The points of G , with positive definite 2 × (cid:0) T † T † (cid:1) (cid:18) I − I (cid:19) (cid:18) T T (cid:19) > ⇐⇒ I − z † z > SU (2 ,
2) classical domain[19], because it is bounded in the gen-eral z -space and it is invariant under the SU (2 ,
2) transformation (cid:18) T ′ T ′ (cid:19) = (cid:18) A A A A (cid:19) (cid:18) T T (cid:19) z ′ = ( A + A z ) ( A + A z ) − A † A − A † A = I , A † A − A † A = 0 , A † A − A † A = I (3.18)The characteristic (Shilov) boundary of this domain is the S × S [= U (2)]manifold with z † z = I . The ordinary parametrization of this boundary is U = e iτ (cid:18) cos ρ + i sin ρ cos θ i sin ρ sin θ e − iϕ i sin ρ sin θ e iϕ cos ρ − i sin ρ cos θ (cid:19) == r − t +2 it [1+2( t + r )+( t − r ) ] (cid:18) t − r − iz − i ( x − iy ) − i ( x + iy ) 1 + t − r + 2 iz (cid:19) (3.19)where τ ∈ ( − π, π ) , ϕ ∈ (0 , π ) , ρ ∈ (0 , π ) , θ ∈ (0 , π ).In the homogeneous coordinates H = (cid:18) H H (cid:19) = √ (cid:18) I − II I (cid:19) (cid:18) T T (cid:19) T = (cid:18) T T (cid:19) = √ (cid:18) I I − I I (cid:19) (cid:18) H H (cid:19) (3.20)we have (cid:18) II (cid:19) = 12 (cid:18) I − II I (cid:19) (cid:18) I − I (cid:19) (cid:18) I I − I I (cid:19) (3.21)and the positive definite condition takes the form (cid:0) H † H † (cid:1) (cid:18) II (cid:19) (cid:18) H H (cid:19) > ⇐⇒ − i ( r − r † ) = y > r A ′ B = x A ′ B + iy A ′ B are defined as r = iH H − , which implies H = − irH and r = i ( I + z )( I − z ) − = i ( I − z ) − ( I + z ) z = ( r − iI )( r + iI ) − = ( r + iI ) − ( r − iI ) (3.23)The fractional transformations which preserve the unbounded domain are (cid:18) H ′ H ′ (cid:19) = (cid:18) B B B B (cid:19) (cid:18) H H (cid:19) r ′ = ( B r + iB ) ( B − iB r ) − B † B + B † B = I , B † B + B † B = 0 , B † B + B † B = 0(3.24)The characteristic boundary in this ”upper plane” realization of the classicaldomain is the ”real axis” y A ′ A = 0 (3.25)Using the SU (2 ,
2) generators, the SL (4 , C ) infinitesimal transformationshave the form δY = i ǫ µ γ µ (1 + γ ) Y , δY = − i ǫ µν σ µν YδY = − i κ µ γ µ (1 − γ ) Y , δY = − ργ Y (3.26)The real parts of the infinitesimal variables ( ǫ µ , ǫ µν , κ µ , ρ ) provide the 15 SU (2 ,
2) charges and their imaginary parts the remaining 15 charges of the SL (4 , C ) transformations.Considering the explicit forms of the homogeneous coordinates we have H = X X X X X X X X = λ λ λ λ − i ( r ′ λ + r ′ λ ) − i ( r ′ λ + r ′ λ ) − i ( r ′ λ + r ′ λ ) − i ( r ′ λ + r ′ λ ) (3.27)where everything has been arranged such that the spinor transformations implythe corresponding spacetime transformations and vice-versa.If we restrict the above z ij → r A ′ B transformations at the Shilov boundarywe find the form t = sin τ cos τ − cos ρ x + iy = sin ρ cos τ − cos ρ sin θ e iϕ z = sin ρ cos τ − cos ρ cos θ (3.28)16dditional formulas are r = sin ρ cos τ − cos ρ = − sin ρ τ + ρ sin τ − ρ p t + r ) + ( t − r ) = τ − cos ρ (3.29)where now r = p x + y + z denotes the ordinary radial component and takespositive values. Notice that the Cartesian coordinates ( x, y, z ) are the projectivecoordinates from the center of S . We essentially need two such tangent planesto cover the whole sphere (but equator). The two hemispheres are covered bypermitting the radial variable r to take negative values too.We also see that t − r = − cot τ − ρ t + r = − cot τ + ρ (3.30)Through the above transformation Minkowski spacetime is conformally equiv-alent to the half of S × S . It is easy to prove that the following two points of S × S correspond to the same point ( t, x, y, z ) of the Minkowski space.( τ , ρ, θ, ϕ ) = ⇒ ( t, x, y, z )( τ + π , π − ρ , π − θ , ϕ + π ) = ⇒ ( t, x, y, z ) (3.31)In the τ , ρ axes the r = ∞ boundaries are τ − ρ = 0 and τ + ρ = 0. These arethe two Penrose boundaries J ± of Minkowski spacetime. The two triangles atboth sides of the ρ axis have r > r < J + and J − . This is naturally done throughthe identification of compactified Minkowski spacetime with the characteristicboundary of the type IV SO (2 ,
4) invariant classical domain, which is the partof CP determined by the relations[19] t ⊺ Ht = 0 , t † Ht > , Im t t > t is a 6-dimensional complex column (the homogeneous coordinates of CP ) and H = diag [1 , , − , − , − , − SU (2 ,
2) and O (2 , X mi ) of G , arerelated to the homogeneous coordinates of CP with the following relations[18] t = i √ ( R − R ) , t = R + R t = R − R , t = i √ ( R − R ) t = √ ( R + R ) , t = − i √ ( R + R ) (3.33)17here R mn = X m X n − X n X m . Recall that the homomorphisms betweenthe three conformal groups are SU (2 , → === > O ↑ + (2 , → === > C ↑ + (1 ,
3) (3.34) G , context In the simple case of conformally flat spacetimes, the integrability conditionof the complex structure can be solved by Kerr’s theorem[7]. Using the G , homogeneous coordinates X mi this general solution takes the form[18] X mi E mn X nj = 0 K i ( X mi ) = 0 (3.35)where the first line relations fix the surface to the Shilov boundary or a part ofit, and the second line relations are the two Kerr homogeneous function. Thestructure coordinates z α are then two independent functions of X m X and z e α aretwo independent functions of X m X .This particular solution indicates the form of a general solution for the in-tegrability condition of the complex structure on a generally curved spacetime.In this case the G , homogeneous coordinates X mi have to satisfy relations ofthe form Ω ij ( X mi , X nj ) = 0 K i ( X mi ) = 0 (3.36)where all the functions are homogeneous relative to X n and X n independently.That is, they are defined in CP × CP . The rank-2 condition on the matrix X mi defines the solutions as SL (2 , C ) fiber bundles on 4-dimensional surfacesof G , . The structure coordinates ( z α , z e α ) are determined exactly like in thesimple case of conformally flat spacetimes given above.In General Relativity the asymptotic flatness condition is imposed using themetric. In the present case of complex structures this condition is imposedthrough the assumption that there are independent homogeneous transforma-tions of X n and X n such that X m E mn X n = 0 X m E mn X n = 0 X m E mn X n = 0 (3.37)That is the two functions Ω ( X m , X n ) and Ω ( X m , X n ) take the flat spaceforms. The first two annihilations will be used below to restrict the forms ofstationary axisymmetric complex structures.The central problem of the present work is to find solutions X mi ( x ) whichsatisfy relations of the form (3.36). The topological classes of these solutionswill determine the soliton sectors of the model. The algebraic nature of the18wo homogeneous Kerr functions K i ( X mi ) is a powerful mathematical propertywhich will be used below. But from the physical point of view it is somehowobscure because it hides physical intuition. Therefore one way to replace them isthe parametrization (3.1) of X mi where λ Ai are functions of r A ′ A which satisfythe Kerr differential equations λ A λ B ∂∂r A ′ A λ B = 0 (3.38)which was our intuitive procedure for the discovery of the form (3.36) of thegeneral solution.Another physically very intuitive form, which replaces the Kerr functions, isthe following trajectory parametrization of X mi X mi = (cid:18) λ Ai − iξ iA ′ B ( τ i ) λ Bi (cid:19) (3.39)where ξ iA ′ B ( τ i ) are two complex trajectories in the Grassmannian manifold G , .A combination of this parametrization with the Grassmannian one (3.1) impliesthe two conditions det[ r A ′ B − ξ iA ′ B ( τ i )] = 0 for the two linear equations [ r A ′ B − ξ iA ′ B ( τ i )] λ Bi = 0 to admit non-vanishing solutions. Notice that this condition(restricted to the Shilov boundary) is identical to the relation η µν ( x µ − ξ µ ( τ ))( x ν − ξ ν ( τ )) = 0 (3.40)which was first used by Newman and coworkers[14] to determine twisted geodeticand shear free null congruences in Minkowski spacetime.It is to prove that the quadratic Kerr polynomial Z Z − Z Z + 2 aZ Z = 0 (3.41)is implied by the trajectory ξ µ ( τ ) = ( τ , , , ia ).In the case of one trajectory, we will have one Kerr function. In this caseand for a trajectory normalized by ξ ( τ ) = τ , the asymptotic flatness conditionimplies i ( τ − τ ) − i ( ξ − ξ ) λ + λ λλ + ( ξ − ξ ) λ − λ λλ − i ( ξ − ξ ) 1 + λλ λλ = 0 (3.42)which fixes the imaginary parts of the two complex parameters τ and τ .Using the G , formalism, the (2.23) complex structure of Minkowski space-time is implied by the quadratic homogeneous polynomial (3.41). Notice thatthe points of spacetime (2.28) with det( λ Ai ) = 0 do not belong to the Grass-mannian projective manifold, because the corresponding 4 × G , , thesepoints do not belong to the spacetime!Using the Kerr-Schild ansatz, we have derived a general curved complexstructure (2.33). Assuming that the Kerr function of the curved complex19tructure is the same with that of the corresponding flatprint complex struc-ture we can find the 4-dimensional surface of G , . Its explicit form for theKerr-Newman complex structure ( h ( r ) = − mr + q ) in Lindquist coordinates( t, r, θ, ϕ ) is x = t + f cos f +cos θ sin f x + ix = ( r + f ) cos f + a sin θ sin f cos f +cos θ sin f sin θe iϕ e − if x = r cos θ + f cos θ cos f +cos θ sin f ++ sin f sin θ cos f +cos θ sin f [( r sin f − a cos f ) cos θ + f sin f ] y = 0 y + iy = ( r + f ) sin f − a cos f cos f +cos θ sin f cos θ sin θe iϕ e − if y = cos f sin θ cos f +cos θ sin f [ a cos f − ( r + f ) sin f ] (3.43)where the two functions entering the configuration are f = a √ a + q − m arctan r − m ) √ a + q − m r − mr ++2 m − a − q f = − m ln r − mr + a + q m + m − q a f (3.44)Notice that this surface is outside the classical domain because y = 0.As a second example we will present below the “natural” complex structureof the U (2) surface. The 1-forms of the left invariant generators of the group U (2) are defined by the relation U † dU = ie aL σ a = i ( e L σ − e iL σ i ). In the (3.19)parametrization the 1-forms are e L = dτe L = − sin θ cos ϕdρ + (sin ρ sin ϕ − sin ρ cos ρ cos θ cos ϕ ) dθ ++(sin ρ cos θ sin θ cos ϕ + sin ρ cos ρ sin θ sin ϕ ) dϕe L = − sin θ sin ϕdρ + ( − sin ρ cos ρ cos θ sin ϕ + sin ρ cos ϕ ) dθ ++(sin ρ cos θ sin θ sin ϕ − sin ρ cos ρ sin θ cos ϕ ) dϕe L = − cos θdρ + sin ρ cos ρ sin θdθ − sin ρ sin θdϕ (3.45)20n Cartesian coordinates the generators take the form [ C = t + r )+( t − r ) ] e L = C [(1 + r + t ) dt − txdx − tydy − tzdz ] e L = C [ − xtdt + (1 + t + x − y − z ) dx ++2( xy + z ) dy + 2( xz − y ) dz ] e L = C [ − ytdt + 2( xy − z ) dx + (1 + t − x + y − z ) dy ++2( yz + x ) dz ] e L = C [ − ztdt + 2( xz + y ) dx + 2( yz − x ) dy ++(1 + t − x − y + z ) dz ] (3.46)The 1-forms satisfy the following differential relations de L = 0 , de iL = ǫ ijk e jL ∧ e kL (3.47)which imply the relations (cid:0) e i e j ∂e k (cid:1) = e iµ e jν (cid:0) ∂ µ e kν − ∂ ν e kµ (cid:1) = 2 ǫ ijk (3.48)The “natural” complex structure on S × S is defined by the following tetrad L µ = e µL − e µL N µ = e µL − e µL M µ = e µL + ie µL (3.49)This complex structure is generated from the following degenerate quadraticpolynomial, which is the product of two linear polynomials( Z + Z )( Z + Z ) = 0 (3.50)The surface is the boundary of the classical domain y A ′ A = 0 with thefollowing homogeneous coordinates of G , H = x − iyt − z + ix + iyt + z + i − i [ t − z − ( x − iy ) λ ] − iλ [ t − z − ( x − iy ) λ ] − i [ − ( x + iy ) + ( t + z ) λ ] − iλ [ − ( x + iy ) + ( t + z ) λ ] (3.51)These structure coordinates are valid over the whole Shilov boundary spacebecause λ = λ everywhere, while the corresponding surface of CP , definedby the Kerr polynomial (3.50) is singular.21 .3 Induced metrics on spacetimes The characteristic property of the present model is that it depends on the inte-grable complex structure J νµ and not on a metric g µν . The above approach tothe solution of the complex structure equations through the G , mathematicalmachinery permit us to look at spacetimes from a different point of view. Themetric should be seen as a product and not as a “primitive” dynamical variable.In fact we may define more than one metric on the 4-dimensional surface of G , . The complex structure determines its eigenvectors ( ℓ µ , n µ , m µ , m µ ) up tofour independent Weyl factors. These vectors permit us to define a symmetrictensor g µν through formula (2.4). This tensor may be used as a metric of thespacetime. The tetrad is apparently null relative to this metric, which is com-patible with the complex structure J νµ . In fact the most general metric wemay use is g µν = Ω (cid:2) ( ℓ µ n ν + n µ ℓ ν ) − ω ( m µ m ν + m µ m ν ) (cid:3) (3.52)where Ω( x ) and ω ( x ) are two arbitrary real functions. I want to point out thatthis metric arbitrariness saves the present model from the scalar solitons whichcaused the most serious difficulties to the Misner-Wheeler geometrodynamicmodel. Notice that the spherically symmetric spacetimes (e.g. Schwarzschild)are compatible with the Minkowski metric. Therefore they do not differ fromthe “vacuum” surface.The above metric is very useful because it is directly related to the complexstructure. In fact it determines the complex structure through the algebraic con-dition (2.12), and in the “vacuum” surface (Minkowski spacetime) it may takea form which respects the remaining Poincar´e symmetry. But it is not the onlymetric we may define. The rank-2 matrices X mi ( ξ ) permit us to induce the wellknown SU (4) and SU (2 ,
2) invariant metrics of G , down to the 4-dimensionalsurface. In the bounded (Dirac representation) coordinate neighborhood thesurface is z = X X − X X X X − X X , z = X X − X X X X − X X z = X X − X X X X − X X , z = X X − X X X X − X X (3.53)In this coordinate neighborhood the SU (4) and SU (2 ,
2) invariant metricsof G , are ds ± = ∂∂z ij ∂∂z kl ln[det( I ± z † z )] dz ij dz kl (3.54)where the “+” and “-” denote the SU (4) and SU (2 ,
2) invariant metrics re-spectively. These metrics may easily be transcribed in the unbounded (chiral)coordinate neighborhood. After the direct substitution of z ij ( ξ ), the inducedEuclidean metrics on the 4-dimensional surface may be found. These metricsdo not seem to be directly related to the complex structure of the surface.22 VACUA AND EXCITATION MODES
A physically interesting geometrodynamic model must generate the electromag-netic field and the intermediate vector bosons of the Standard Model from thefundamental equations of the model itself, without introducing anything byhand. In the Misner-Wheeler model, the electromagnetic potential is directlyand exactly derived from the Rainich conditions. In fact this derivation was theessential reason behind the assumption of the Rainich conditions, as the funda-mental equations of the Misner-Wheeler model[10]. In Quantum Field Theorythe vacuum excitation modes are the periodic configurations which diagonal-ize energy and momentum. In the present context periodicity is understood incompactified Minkowski spacetime M . Therefore we first have to define energyand momentum, and after to look for the model vacua with vanishing energyand momentum, the excitation modes and the solitons with finite energy.
It is well known that in any generally covariant model the translation genera-tors are first class constraints, which must vanish. Therefore energy, momentumand angular momentum cannot be defined using Noether’s theorem. The successof the Einstein equations strongly suggests that energy-momentum has to bedefined through the Einstein tensor E µν . The direct relation of the Einstein ten-sor with the classical energy-momentum and angular momentum is also stronglyimplied by the derivation of the equations of motion in the harmonic coordi-nate system, imposed by the condition ∂ µ ( √− gg µν ) = 0, using the contractedBianchi identities ∇ µ E µν = 0. But E µν depends on the metric and it is notdirectly related to the Poincar´e generators of the present model. Therefore forthe definition of the energy in the present model we proceed as follows[25].We first consider the coordinate system imposed by the relation ∂ µ (cid:0) √− gE µν (cid:1) = 0 (4.1)We next consider the conserved quantity E ( g µν ) = Z t √− gE µ dS µ (4.2)where the time variable t is chosen such that E ( g µν ) ≥
0. This quantity dependson the metric g µν and it does not characterize the complex structure, thereforeit cannot be the energy definition of the configuration. We think that the energyof a complex structure is properly defined by the following minimumE[ J νµ ] = min g µν ∈ [ J νµ ] E ( g µν ) (4.3)where the minimum is taken over all the class [ J νµ ] of metrics (3.52).Apparently this conserved quantity depends only on the moduli parametersof the complex structure. In a vacuum sector it vanishes, E[ J νµ ] = 0. From23he 2-dimensional solitonic models[5], we know that the minima of the energycharacterize the solitons. Assuming that E[ J νµ ] is a smooth function of themoduli parameters, we can always expand it around a minimum.E[ J νµ ] ≃ E + X q ε q a q a q (4.4)where E and ε q are positive parameters. These variables and a q are moduliparameters of the complex structure. E is defined to be the energy of thesoliton characterized by the minimum and ε q are the energies of the excitationmodes. In the special metric where the minimum (4.3) occurs, we can definethe 4-momentum and the angular momentumP ν = R t √− gE µν dS µ S µν = R t √− gE ρσ x τ Σ µνστ dS µ (4.5)These quantities are conserved in the precise coordinate systems, which sat-isfy (4.1). But this is not enough to identify them with the Poincar´e groupgenerators! Recall that the Poincar´e transformations are well defined in thepresent model. They form a subalgebra of sl (4 , C ) and a part of the infinite al-gebra of the complex structure preserving transformations. The relation of thesePoincar´e group generators with the present conserved quantities is implied bythe transformation of the Einstein tensor under the Poincar´e transformations.Recall that in the unbounded coordinate neighborhood of G , the Poincar´etransformations do not mix the Hermitian x A ′ A and the anti-Hermitian iy A ′ A parts of the projective coordinates r A ′ A . Therefore we must first fix the coordi-nates to be the Cartesian coordinate system defined as the real part of the r A ′ A projective coordinates of the Grassmannian manifold G , . But in this coor-dinate system energy-momentum is not exactly conserved. It is approximatelyconserved in the “weak gravity” limit. Therefore we will consider the modeswhich diagonalize this “weak gravity” limit of energy-momentum. These modesbelong to irreducible representations of the Poincar´e transformations, properlydefined on the Cartesian coordinate system as δx µ = ω µ ν x ν + ε µ . Then theQuantum Theory relation i [ εQ ε , E µν ] = δ ε E µν = E µρ ∂ ρ ε ν + E ρν ∂ ρ ε µ − ε ρ ∂ ρ E µν (4.6)implies that P ν approximately behaves as a vector and S µν as an antisymmetrictensor. P µ and S z commute with the corresponding Poincar´e generators. Hencethe approximative relation (4.4) and the preceding Poincar´e group transforma-tions imply the formsP µ ≃ k µ + P i,s R d k k µ a + i ( → k , s ) a i ( → k , s )S z ≃ s z + P i,s R d k s a + i ( → k , s ) a i ( → k , s ) (4.7)24here the summation is over the momentum, the spin and the irreducible rep-resentations i of the Poincar´e group. k µ is the 4-momentum and s z is thez-component of the spin of the soliton. k µ is the 4-momentum and s is the z-component of the spin of the excitation modes a i ( → k , s ). In the quantized theorythe variables a + i ( → k , s ) and a i ( → k , s ) become the creation and the annihilation op-erators of the approximative modes, which diagonalize the 4-momentum. FromQuantum Field Theory we know that the second parts of (4.7) are formally gen-erated by the ordinary energy momentum tensors of free quantum fields. Theyshould be bosonic because they represent excitation modes. This procedurepermit us to write down the Einstein tensor as the energy-momentum tensorof the excitation modes. Notice that this effective energy-momentum tensorhas to contain interactions, because the field excitation modes diagonalize theapproximative Einstein tensor. We will refer to this effective Lagrangian againbelow in relation to the computation of the soliton form factors.The tetrad vectors ( ℓ µ , n µ , m µ , m µ ) are the two real and one complex vec-tor fields, which appear in the action of the present model. The number ofthe Poincar´e representations of the excitation modes may be found looking forthe independent variables of the tetrad ( ℓ µ , n µ , m µ , m µ ) which determines thecomplex structure. It has (4 × × A direct consequence of the present definition of energy and momentum is thatall the complex structures which are compatible with the Minkowski metrichave zero energy. They determine vacuum configurations. In the context ofthe Grassmannian manifold formulation of complex structures we see that onlythe Shilov (characteristic) boundary of the classical domain or its subsurfacesmay be vacua. All the complex structures on the closed S × S are vacuumconfigurations, which are SU (2 ,
2) symmetric, because the surface is invariant.These vacua will be called conformal vacua, because they break global SL (4 , C )symmetry group down to SU (2 ,
2) symmetry.The complex structures on the open ”real axis” of the Shilov boundary breakthe conformal SU (2 ,
2) symmetry down to the [
P oincar ´ e ] × [ dilatation ] group.Recall that the “real axis” subsurface is characterized by a point of the closedShilov boundary, which fixes the Cayley transformation (3.23). It is the pointof the characteristic boundary in the bounded realization, which is sent to “in-finity” in the unbounded realization of the classical domain. A general theorem,valid for all classical domains, states that the automorphic analytic transfor-mations, which preserve a point of the characteristic boundary in the bounded25ealization, become linear transformations in the unbounded realization of theclassical domain[19]. In the present case of the SU (2 ,
2) classical domain theselinear transformations form the [
P oincar ´ e ] × [ dilatation ] group. This argumentdemonstrates that the ”real axis” vacuum surface breaks global SL (4 , C ) downto the [ P oincar ´ e ] × [ dilatation ] group. One may understand the above theoremlooking at the following general form of the Cayley transformation which trans-forms the upper half-plane realization of the SU (2 ,
2) classical domain onto itsbounded realization z = U (cid:0) M rM † − N † (cid:1) (cid:0) M rM † − N (cid:1) − (4.8)where det M = 0 , i ( N † − N ) is negative definite and U is the point of theShilov boundary, which is sent to infinity. This clear cut emergence of thePoincar´e group, through a symmetry breaking mechanism, makes the presentmodel physically very interesting. Recall that the asymptotic flatness conditiongenerates the BMS group, which does not appear in Particle Physics.The SU (2) × U (1) transformation z ′ = U z changes the characteristic point U of the Cayley transformation (4.8) to U U , while it does not affect thePoincar´e transformation. That is, it changes the Minkowski spacetime, while itdoes not change the explicit form of the Poincar´e transformation. This impliesthat the SU (2) × U (1) transformation commutes with the Poincar´e transforma-tion in the following sense: [First make a U preserving Poincar´e transformationand after an “internal” U transformation] =[First make an “internal” U trans-formation and after a U U preserving Poincar´e transformation]. I want to pointout that these two subgroups of SU (2 ,
2) do not commute in the ordinary sense.I think that this clear cut emergence of the Poincar´e group and the “internal” SU (2) × U (1) group may have some physical relevance.The dilatation symmetry is broken by the parameter a of the static Kerrpolynomial which will be described in the next section. But this proof will bepresented here because of the importance of the Poincar´e group in physics.In the next section we will see that the first soliton family is generated by aquadratic polynomial A mn Z m Z n = 0 with A mn = (cid:18) ω AB p B ′ A p A ′ B (cid:19) (4.9)where p AA ′ = ǫ AB p A ′ B . This form is determined assuming invariance underthe Poincar´e transformations (cid:18) B B B (cid:19) , with B † B = I , det( B ) = 1 and B † B + B † B = 0. If we try to impose dilatation symmetry, which has theform (cid:18) e − ρ e ρ (cid:19) , we find ω AB = 0 . Apparently this makes the complex struc-ture trivial. Hence ω AB , which generates the spin of the static soliton, breaksthe dilatation symmetry leaving the Poincar´e group as the largest symmetry.26 ”LEPTONIC” SOLITONS Standard Model provides a description of weak and electromagnetic interactionsthrough the classification of the left-handed and right-handed field componentsin the representations of the U (2) group. The electron, muon and heavy lepton(tau) doublets are trivially repeated without any apparent reason. This is thewell known family puzzle. No theoretical explanation exists of this dummyrepetition of only three representations of the U (2) group. The appearance ofthe same representations for the quarks obscures the situation. The extensionof the unitary group has not yet provided any experimentally acceptable model.The present solitonic model provides a new way to look for a solution to thisproblem. The dynamical variables of the present model are the gauge field A jµ ( x ) and theintegrable tetrad ( ℓ µ , n µ , m µ , m µ ) up to the extended Weyl symmetry. Noticethat the field equations have the characteristic property to admit pure geometricsolutions with A jµ ( x ) = 0. These are the complex structures defined by theintegrable tetrad or equivalently a rank 2 matrix X mi ( x ). The integrable tetradpermits us to define the symmetric tensor (3.52), relative to which the tetrad( ℓ µ , n µ , m µ , m µ ) is null. This symmetric tensor may be used as a metric, whichcontains a large information from the complex structure. It has been pointedout the essential difference between the Euclidean complex structures and thepresent Lorentzian ones. In the case of Euclidean complex structures the metricis somehow independent of the complex structure. But the Lorentzian complexstructure is essentially algebraically fixed by the metric g µν because the spinordyad o A and ι A , which determines the integrable tetrad, satisfy the algebraicequation (2.12). They are principal directions of the Weyl spinor Ψ ABCD .The cornerstone of the soliton theory is the regularity of the solitonic config-urations. In the present case this is translated to the regularity of the complexmanifold. Therefore the Weyl spinor Ψ
ABCD must be regular and ( o A , ι A ) areroots of a homogeneous fourth degree polynomial with regular coefficients.Namely, the complex manifold is a covering space of R with a maximum offour sheets and a minimum of two sheets. That is the solitons of the presentmodel are algebraically classified into three classes (families) according to thenumber of principal null directions of the Weyl tensor as follows: • The fourth degree polynomial (2.12) is reduced to the square of a ho-mogeneous second degree polynomial (Φ AB ξ A ξ B ). These are the type Dspacetimes which admit a regular complex structure. We will call it ”typeD family” of the model. • The Weyl tensor has three principal null directions, which are geodeticand shear free. These are type II spacetimes in the Petrov classification.27
In the third class the Weyl tensor has four distinct principal null directions,which must also be geodetic and shear free. These are type I spacetimes.We should notice the amazing similarities of these three classes with thethree families of leptons and quarks indicating a completely different approachto the family problem. In conventional Quantum Field Theoretic models thesolution to this problem was searched in the context of large simple groups forGrand Unified Theories and supergroups for recent supersymmetric models. Inthe present model the proposed solution is topological, based on the Petrovclassification of the spacetimes, which is well known in General Relativity. Thepresent model shows for the first time that Quantum Field Theory and GeneralRelativity may be intimately related without Grand Unified Theories, Super-symmetry, Supergravity, Strings and Superstrings. The study of the stationaryaxisymmetric solitons of type D family in the present section will be in thecontext of this new point of view. st family The charged Kerr metric has been extensively studied. After a mass and chargemultipole expansion it was observed[1],[11] that this spacetime had the correctelectron gyromagnetic ratio g = 2. Notice that this extraordinary result cameout in the context of pure General Relativity without any reference to QuantumMechanics or any other particular assumption. This result triggered many at-tempts to generate particles in the context of pure General Relativity withoutapparent phenomenological success.The knowledge of the Poincar´e group permit us to look for stationary (static)axisymmetric solitonic complex structures, which will be interpreted as particlesof the model with precise mass and angular momentum. In the case of vanishinggauge field, we may use the general solutions (2.8) to find special solutions. Inthis case the convenient coordinates are z = u + iU , z = ζ , z e = v + iV , z e = W ζ (5.1)where u = t − r , v = t + r and t ∈ R , r ∈ R , ζ = e iϕ tan θ ∈ S are assumed tobe the four coordinates of the spacetime surface. Assuming the definitions z = i X X , z = X X , z e = i X X , z e = − X X (5.2)we look for solutions which are stable along s µ = (1 , , , δX mi = iǫ [P ] mn X ni (5.3)where P µ = − γ µ (1 + γ ). It implies δX i = 0 , δX i = 0 δX i = − iǫ X i , δX i = − iǫ X i (5.4)28he above definition of the structure coordinates implies δz = ǫ , δz = 0 δz e = ǫ , δz e = 0 (5.5)and consequently δu = ǫ , δU = 0 δv = ǫ , δV = 0 δζ = 0 , δW = 0 (5.6)This procedure gives stable (time independent) solutions. The little grouprelative to the vector s µ is the SO (3) subgroup of the Lorentz group. Thereforewe may look for solutions, which are “eigenstates” of the z-component of thespin. In this case the homogeneous coordinates satisfy the following transfor-mations δX mi = iǫ [Σ ] mn X ni (5.7)where Σ µν = σ µν = i ( γ µ γ ν − γ ν γ µ ). That is we have δX i = − i ǫ X i , δX i = i ǫ X i δX i = − i ǫ X i , δX i = i ǫ X i (5.8)The above definition of the structure coordinates implies δz = 0 , δz = iǫ z δz e = 0 , δz e = − iǫ z e (5.9)and consequently δu = 0 , δU = 0 δv = 0 , δV = 0 δζ = iǫ ζ , δW = 0 (5.10)A general solution of (2.8), which satisfies these symmetries, is given by therelations U = U [ z z ] , V = V [ z e z e ] W = W [ v − u − i ( V + U )] (5.11)A static complex structure is expected to be determined by a Kerr function K ( X m ) globally defined on CP . Chow’s theorem states that every complex29nalytic submanifold of CP n is an algebraic variety (determined by a polyno-mial). Hence the present complex structure will be determined by a quadraticpolynomial invariant under (5.3) and (5.7), which turns out to be Z Z − Z Z + 2 aZ Z = 0 (5.12)The asymptotic flatness condition (3.37) implies U = − a z z z z , V = 2 a z e z e z e z e (5.13)A quite general solution is found if W W = 1 ( V + U = 0). In this case wehave the solution U = − a sin θ , V = 2 a sin θ W = r − iar + ia e − if ( r ) (5.14)A simple investigation shows that this complex structure is (in different coordi-nates) the static solution (2.32) found in section II using the Kerr-Schild ansatz.One may easily compute the corresponding tetrad up to their arbitrary factors N , N and N . ℓ = N [ dt − dr − a sin θ dϕ ] n = N [ dt + ( r + a cos 2 θr + a − a sin θ dfdr ) dr − a sin θ dϕ ] m = N [ − ia sin θ ( dt − dr ) + ( r + a cos θ ) dθ + i ( r + a ) sin θdϕ ] (5.15)The corresponding projective coordinates are r ′ = i X X − X X X X − X X = z +( z e − ib ) z z e z z e r ′ = i X X − X X X X − X X = ( z − z e + ib ) z e z z e r ′ = i X X − X X X X − X X = ( z − z e + ib ) z z z e r ′ = i X X − X X X X − X X = z e +( z + ib ) z z e z z e (5.16)If these projective coordinates become a Hermitian matrix x A ′ A , then the com-plex structure is compatible with the Minkowski metric. Otherwise, it is acurved spacetime complex structure. The form (5.14) has been chosen suchthat for f ( r ) = 0 the complex structure becomes compatible with the Minkowskimetric.The soliton form factor f ( r ) is expected to be fixed by Quantum Theory,but we have not yet found the precise procedure. We think that any attempt30o give some physical relevance of the present model may come through theidentification of the effective energy-momentum tensor of the excitation modeswith the bosonic part of the Standard Model energy-momentum tensor. In thiscase the form factors f ( r ) of the solitons may be fixed, assuming the conditionthat the solitons are particle-like sources of the excitation modes. Then themassive static soliton (2.32) with spin S z = ma = h , should be identified withthe electron. Then it will have (in natural gravitational units c = G = 1) mass m = 6 . × − cm , a = 1 . × − cm and charge q = 1 . × − cm . Thecomplex conjugate complex structure would be the positron. In ordinary Lorentzian Quantum Field Theory the vacuum is determined as thestable state with the lowest energy. Solitons are stable states with finite energiesrelative to the vacuum. Their configurations are not smoothly deformable tovacuum configurations. We have already revealed the existence of two sets ofvacua. The conformally invariant vacua, which are complex structures definedon the closed Shilov boundary U (2) and the Poincar´e vacua which are complexstructures defined on the open “real axis” of the unbounded neighborhood. Inorder to reveal the solitons of the model, we have to use the periodicity criteria.Recall that the φ -model admits two vacua with φ = ± µ √ λ . It is well knownthat the vacuum configurations are periodic, while the soliton configurationsare not periodic. This characteristic difference will be used in the presentmodel. The kink configuration and its excitations satisfy the boundary con-ditions φ kink ( ±∞ , t ) = ± µ √ λ and the antikink configuration the opposite ones.The corresponding energy-momentum charges are related to the gap of the limitvalues of the field φ ( x ) at ±∞ .In the present model the excitation modes and the solitons are 4-dimensionalsurfaces of G , which admit integrable tangent vectors in pairs ( ℓ, m ) and( n, m ). Their essential difference will be on the periodicity of the complexstructures they admit. The vacuum surfaces will admit periodic complex struc-tures, while the solitonic complex structures are not periodic on the correspond-ing surfaces. Therefore we have to specify the precise compactification of theMinkowski spacetime.Minkowski spacetime is the Poincar´e vacuum of the model and it has alreadybeen identified with a precise open surface of G , . It is the “real axis” in theunbounded realization of the classical domain. In the bounded realization ofthe classical domain, it is an open part of the characteristic (Shilov) boundary.It is precisely limited by the “diagonals” τ + ρ = π , ( − π ≤ τ − ρ ≤ π ),which is J + , and τ − ρ = π , ( − π ≤ τ + ρ ≤ π ), which is J − . There is anessential difference between Minkowski spacetime and the other asymptoticallyflat spacetimes. Every null geodesic which originates at some point A − of I − will pass through the same point A + of I + . This association permits us toidentify A − with A + compactifying Minkowski spacetime[18]. Notice that allthe complex structures, which are compatible with the Minkowski metric, are31ell defined on compactified Minkowski spacetime M , because they smoothlycross J = J + = J − . The topology of the whole spacetime M turns out to beM ∼ S × S .In order to avoid any misunderstandings, I want to emphasize that there is anessential difference between the present analysis and the corresponding Penroseone. In the present model we deal with complex structures while Penrose dealswith Weyl (conformally) equivalent metrics. The present equivalence relation islarger than the Penrose one. A typical example is the Schwarzschild spacetime.It is not Weyl (conformally) equivalent with Minkowski spacetime and it can-not be metrically compactified, because the first derivatives of the metric do notsmoothly cross J + = J − . But the complex structure of Schwarzschild spacetimeis compatible with Minkowski spacetime because it is trivial. Therefore it cansmoothly cross J and the Schwarzschild spacetime is a trivial vacuum config-uration. In the following example of a static solitonic surface we will considerthe Kerr-Schild spacetime but the proof can be extended to any stationary ax-isymmetric spacetime with f ( − r ) = f ( r ). It will be shown that the integrabletetrad cannot be smoothly extended across J . Therefore J + and J − cannot beidentified and this complex structure belongs to a soliton sector. The proof ofthis failure goes as follows:In order to make things explicit the Kerr-Newman integrable null tetrad willbe used as an example. Around I + the coordinates ( u, w = r , θ, ϕ ) are used,where the integrable tetrad takes the form ℓ = du − a sin θ dϕn = − mw + e w + a w w (1+ a w cos θ ) [ w du − a w cos θ )1 − mw + e w + a w dw − aw sin θ dϕ ] m = √ w (1+ iaw cos θ ) [ iaw sin θ du − (1 + a w cos θ ) dθ −− i sin θ (1 + aw ) dϕ ] (5.17)The physical space is for w > I + up toa factor, which does not affect the congruences, and it can be regularly extendedto w <
0. Around I − the coordinates ( v, w ′ , θ ′ , ϕ ′ ) are used with dv = du + r + a ) r − mr + e + a drdw ′ = − dw , dθ ′ = dθdϕ ′ = dϕ + ar − mr + e + a dr (5.18)and the integrable tetrad takes the form32 = w ′ [ w ′ dv − a w ′ cos θ )1+2 mw ′ + e w ′ + a w ′ dw ′ − aw ′ sin θ ′ dϕ ′ ] n = mw ′ + e w ′ + a w ′ a w ′ cos θ ′ ) [ dv − a sin θ ′ dϕ ′ ] m = − √ w ′ (1 − iaw ′ cos θ ′ ) [ iaw ′ sin θ dv − (1 + a w ′ cos θ ′ ) dθ ′ −− i sin θ ′ (1 + aw ′ ) dϕ ′ ] (5.19)The physical space is for w < I − up toa factor, which does not affect the congruences, and it can be regularly extendedto w >
0. If the mass term vanishes the two regions I + and I − can be identifiedand the ℓ µ and n µ congruences are interchanged, when I + ( ≡ I − ) is crossed.When m = 0 these two regions cannot be identified and the complex structurecannot be extended across I + and I − . We will now consider a classification of the complex structures defined on the S × S surface (the Shilov boundary) of G , . They are determined by twolinearly independent functions λ Ai ( ξ ) in S . That is for any complex structurewe have two functions S × S → S (5.20)It is known that the homotopy group π ( S ) is trivial but π ( S ) = Z . TheHopf invariant is determined using the sphere volume 2-form ω = i π dλ ∧ dλ (1 + λλ ) (5.21)which is closed. This implies that in S there is an exact 1-form ω such that ω = dω . Then the Hopf invariant of λ ( x ) is H ( λ ) = Z λ ∗ ( ω ) ∧ ω (5.22)In the simple case of a linear polynomial bZ + Z = 0, we have λ ( x ) = t − z + ibx − iy , t = 0 (5.23)The exact 1-form is ω = ydx − xdy − bdz π ( x + y + z + b ) (5.24)and H ( λ ) = − b | b | (5.25)33here we have integrated over the two Minkowski charts, which cover S bysimply permitting r ∈ ( −∞ , + ∞ ). Notice that the present spinor λ A ( ξ ) is asolution of the linear Kerr polynomial in the unbounded realization and its Hopfinvariant is its helicity. In the bounded realization the function which definesthe mapping S → S is different and its Hopf invariant will be different. Asimple transformation shows that in the present case the corresponding mappinghas zero Hopf invariant.In the case of the solutions of the quadratic Kerr polynomial λ ± ( x ) = − z + ia ± p x + y + z − a − iazx − iy (5.26)the Hopf invariant can be computed using its relation to the linking coefficientof two curves in S determined by the inverse images λ − ( λ ) = { x i ( ρ ) } and λ − ( λ ) = { x i ( ρ ) } . Two general curves are determined using the Lindquistcoordinates ( ρ, θ, ϕ ) x i = (sin θ cos ϕ , sin θ sin ϕ , cos θ ) ρ + a (sin ϕ , − cos ϕ ,
0) (5.27)for two different values of θ, ϕ and the variable ρ ∈ ( −∞ , + ∞ ) in order to coverthe whole sphere. Then we know that H ( λ ) = 2 14 π Z ε ijk ( x i − x i ) dx j dx k |−→ x − −→ x | (5.28)The two curves can be smoothly deformed to the values θ = 0 and θ = π , ϕ = 0. Then the integral becomes H ( λ ± ) = a π Z Z dρ dρ ( ρ + ρ + a ) = ± a | a | (5.29)The curved complex structures which are smooth deformations of confor-mally flat spacetimes will have the same Hopf invariants. This is apparentlythe case of all the complex structures derived using the Kerr-Schild ansatz. Thecurved complex structures have Hopf invariant a | a | at J + and Hopf invariant − a | a | at J − . st family We will now consider configurations X mi which are covariant along a null vector s µ = (1 , , , X mi satisfy the relations δX mi = i ǫ P + P ] mn X ni (5.30)which imply δX i = 0 , δX i = 0 δX i = 0 , δX i = − iǫX i (5.31)34n this case the most general quadratic polynomial, which is invariant under theabove transformations is ( bZ + Z ) Z = 0 (5.32)Notice that this polynomial determines a singular surface in CP , which maybe considered as the limit of the corresponding massive Kerr polynomial A mn Z m Z n = 0 A mn = s − E + p s E + p E + p − E + p (5.33)with energy E and momentum p in the z-direction in the case of vanishing mass.The two solutions are X = 0 and X = − bX . In this case we cannotuse the (5.2) definitions of the structure coordinates. Instead we may use thefollowing structure coordinates z = i X X , z = − i X X , z e = i X X , z e = X X (5.34)Then they transform as follows δz = 0 , δz = 0 δz e = ǫ , δz e = 0 (5.35)and consequently δu = 0 , δU = 0 δv = ǫ , δV = 0 δζ = 0 , δW = 0 (5.36)This procedure gives stable solutions along the null vector s µ . In the presentcase the little group is the E (2)-like group with the third generator being thesame as the previously studied massive case with the SO (3) little group.The axial symmetry condition (5.7) gives the following infinitesimal trans-formations δX i = − i ǫ X i , δX i = i ǫ X i δX i = − i ǫ X i , δX i = i ǫ X i (5.37)The above new definition of the structure coordinates implies δz = 0 , δz = iǫ z δz e = 0 , δz e = − iǫ z e (5.38)35sing the same coordinates u, v, ζ a general complex structure solution is U = U [ u, z z ] , V = V [ z e z e ] , W = W [ u + iU ] (5.39)Notice that these relations are not interconnected and they define a generalsolution without additional conditions.In the case of the invariant Kerr quadratic polynomial (5.32) we have X =0 and X = − bX . Then the asymptotic flatness conditions (3.37) imply U = 0 , V = bz e z e , W = W [ u ] (5.40) nd and 3 rd family solitons may be unstable It has already been pointed out that the integrability condition Ψ
ABCD ξ A ξ B ξ C ξ D =0 classifies the complex structures into those with 2, 3, and 4 algebraic sheets.The complex structure with 2 sheets is the type D family and it has beenextensively studied in the previous subsections. The application of the sameprocedure to the type II and type I families will show that they may not havestable (static) configurations. That is we will look for an eigenconfigurationwhich will be invariant under time translation and z-rotation, and we will findthat they do not exist.The starting point is the reasonable assumption that for an asymptoticallyflat static spacetime there is a coordinate system such that the Weyl tensor hasto approach a Penrose twistor, that is Ψ ABCD ≃ f ω ABCD . This means that theKerr function can locally become equivalent to a quartic polynomial K ( Z ) = A mnpq Z m Z n Z p Z q = 0. The same result is found applying Chow’s theorem.The existence of an axially symmetric configuration implies the existence ofquartic polynomials such that δK ( Z ) = Cε K ( Z ) (5.41)This relation can be easily solved. The following five solutions are found:1. The first solution has C = 2 i and it contains only the components Z and Z . The polynomial K ( Z ) takes the form A ( Z ) + A ( Z ) ( Z ) + A ( Z ) ( Z ) ++ A ( Z )( Z ) + A ( Z ) = 0 (5.42)which defines a singular surface in CP .2. Another solution has C = − i and it is singular too because it containsonly the components Z and Z .3. The third solution has C = i and the polynomial K ( Z ) takes the form A ( Z ) ( Z ) + A ( Z ) ( Z ) + A ( Z ) ( Z )( Z )++ A ( Z ) ( Z )( Z ) + A ( Z )( Z )( Z ) ++ A ( Z )( Z ) ( Z ) + A ( Z )( Z ) + A ( Z ) ( Z ) = 0(5.43)36. The fourth solution has C = − i and K ( Z ) has a form analogous to theabove with Z ⇔ Z and Z ⇔ Z interchanged.5. The final solution has C = 0 and K ( Z ) takes the form A ( Z ) ( Z ) + A ( Z ) ( Z )( Z ) + A ( Z ) ( Z ) ++ A ( Z )( Z ) ( Z ) + A ( Z )( Z )( Z )( Z )++ A ( Z )( Z )( Z ) + A ( Z ) ( Z ) ++ A ( Z )( Z ) ( Z ) + A ( Z ) ( Z ) = 0 (5.44)The stability condition relative to time translation δK ( Z ) = Cε K ( Z ) (5.45)can now be applied on the above axially symmetric Kerr polynomials. I findthat the only regular (in CP ) surface comes from the fifth case. The invariantpolynomial is A ( Z Z − Z Z ) + B ( Z Z )( Z Z − Z Z ) + C ( Z Z ) = 0 (5.46)which may be written as the product of two quadratic polynomials which givethe type D complex structures. Hence we may conclude that the complex struc-tures (particles) with 3 and 4 sheets cannot be stable.37 ”HADRONIC” SOLITONS AND CONFINE-MENT Quark confinement is actually based on the SU (3) gauge group and the nonproven yet hypothesis that the non-Abelian gauge field interactions producea confining potential. The perturbative potential of the ordinary Yang-Millsaction is Coulomp-like r . The ordinary Yang-Mills action also generates thestrong P (CP) problem, because it admits instantons which permit tunnellingbetween the gauge vacua. The real vacuum of the model is a θ -vacuum whichgenerates a parity violation topological term in the action. The axion particlesolution of this problem is expected to be tested in a LHC experiment. Thepresent model trivially solves these problems because its modified Yang-Millsaction generates a linear static potential and it does not have instantons.The amazing similarity between the quark flavor parameters and the leptonsis also a puzzle. The quarks look like leptons with a “color”. The theoretical ef-forts to solve this quark-lepton correspondence in the context of Grand UnifiedTheories have affronted serious problems with the cosmological proton decaybounds. The present model provides a different way to approach this prob-lem. It seems to imply that in some approximation for each “leptonic” (puregeometric) soliton there should be a gauge field excited soliton, which must beperturbatively confined because of the linear static potential.The variables of the present model are the gauge field A jµ ( x ) and the inte-grable tetrad ( ℓ µ , n µ , m µ , m µ ). Notice that the field equations have the char-acteristic property to admit pure geometric solutions. These solutions may bereplaced back into the new covariant Yang-Mills equations and find the corre-sponding gauge field solutions. The simple solution A jµ ( x ) = 0 corresponds tothe pure geometric “leptonic” solitons without any gauge field interaction.In complete analogy to the 2-dimensional kinks, we may quantize aroundthe soliton complex structure[25]. Then the gauge field configurations have anasymptotically linear potential instead of the Coulombian ( r ). This is a clear in-dication that these excitation modes cannot exist free and they must be confinedinto “colorless” bound states which remind us the hadrons. These bound stateswill be hadronic-like solitons with non-vanishing gauge field strength which insome approximation look like bound states of the simple “leptonic” excitationsthrough a linear potential. That is in the present model picture the quarks couldbe gauge field excitations of the leptons and they are perturbatively confined.This very simple picture could explain the complete correspondence betweenleptons and quarks. Apparently in the present context, Standard Model shouldbe considered as an effective theory, like the phonon Lagrangians in solids andfluids[28]. In order to support the above picture of the soliton sectors, we willfirst compute the classical potential implied by the present action.In spherical coordinates ( t, r, θ, ϕ ) and in the trivial (vacuum) null tetrad ℓ µ = (1 , , , n µ = (1 , − , , m µ = r √ (cid:0) , , , i sin θ (cid:1) (6.1)38he dynamical variable of the gauge field is ( r sin θ m µ A jµ ). Assuming theconvenient gauge condition m ν ∂ ν ( r sin θ m µ A jµ ) + m ν ∂ ν ( r sin θ m µ A jµ ) = 0 (6.2)the field equation takes the form (cid:18) ∂ ∂t − ∂ ∂r (cid:19) ( r sin θ m µ A jµ ) = [ source ] (6.3)The dynamical variable apparently gives a linear classical (time-independent)potential. The other two variables ℓ µ A jµ and n µ A jµ of the gauge field decoupleand vanish.Exactly the same approach can be followed in the static soliton sector. Thedynamical variable is now A = (( r + ia cos θ ) sin θ m µ A jµ ) (6.4)and it satisfies the gauge condition( r − ia cos θ ) m ν ∂ ν A + ( r + ia cos θ ) m ν ∂ ν A = 0 (6.5)The corresponding linear part of the field equation is more complicated but inthe asymptotic limit coincides with (6.3). The other variables of the gauge fieldare r-independent and decouple.The emergence of the asymptotically linear classical potential implies thatthe gauge field modes cannot exist free. They must be confined. The gauge fieldexcitations of the pure geometric solitons will also be confined because of thelinear potential. An SU ( N ) gauge group implies that in some approximationthere should be N gauge field excitation modes. These states could look like thethree colored quarks. That is, N must equal three and the gauge group becomes SU (2). Notice that this mechanism implies the existing in nature correspon-dence between “leptons” and “quarks”. Namely, for each pure geometric solitonthere must be three “colored” structures which cannot exist free because of theirlinear interaction. The confining potential imposes that that the gauge field ex-citations cannot exist free. But the existence of “colorless” solitons with nonvanishing gauge field gauge field configurations has to be proved. These solitonsare expected to be described by complicated configurations of the tetrad andgauge field configurations, which satisfy the complicated field equations of thepresent action.We will now show that the Euclidean form of the present Yang-Mills actiondoes not admit finite action solutions which are called instantons and measurethe tunnelling between the gauge vacua. The proof is based on the fact that inthe Euclidean manifolds, the complex structure becomes the ordinary real onewith z e α = z α . Then the structure coordinate form (2.17) of the action becomes I G = 2 R d z F j F j F j = ∂ A j − ∂ A j − γ f jik A i A k (6.6)39hich is invariant under a complex gauge transformation A ′ jα = A jα + ∂ α Λ j + γf jik Λ i A kα , where Λ j are now N complex functions. Assuming the enlargedgauge condition A j = 0 the field equations become ∂ F j = 0 ∂ F j − γ f jik A i F k = 0 (6.7)We see that F j is an holomorphic function of z . On the other hand thefinite action solutions must satisfy the condition F j F j = ⇒ | z |→∞ F j must be bounded as a function of z . But we know that the constantis the only bounded holomorphic function. Hence the finite action solutions musthave F j = 0. Therefore the present model does not have instantons.40 eferenceseferences