Nonlinear Spinor Theory Research Articles

Comment on the proliferation of gauge fields in a nonlinear spinor theory

It is noted that by merely rewriting the Lagrangian in a different form, D\"urr and Saller do not realize their aim of inducing an additional hypercharge degree of freedom. It is pointed out that such an increase in the dynamical degrees of freedom would indeed occur if the Weinberg-Salam model could be derived starting from a four-fermion interaction. Existing studies on nonlinear spinor theories indicate that it may be possible to induce such additional symmetries, perhaps only in an approximate sense.

Meronlike solutions of nonlinear spinor theories

Properties of meronlike solutions of nonlinear spinor theories are discussed — in particular, their relationship with static pointlike Minkowski solutions of the same theory.

Space-time symmetries and nonlinear field theory

The amplification of the space-time symmetry of nonlinear spinor theory culminates in a conformally covariant self-interacting spinor theory. Auxiliary equations are formulated which bring to light dynamical and topological features underlying the emergence of intrinsic nonlinear field patterns as a collaborative effect of higher symmetry. The existence of sets of classical solutions might have import for the corresponding quantum field theory.

On the nonlinear propagator

The two-point function of the nonlinear spinor theory is determined by the solution of coupled nonlinear second-order equations by a method of solution based on phase-space trajectories.

1Nexpansion and the theory of composite particles

Dynamical structures of nonlinear spinor and bosonic theories are studied in the framework of the 1/N expansion. It is shown that a wide class of four-fermion theories contains composite particles and that they can be cast into equivalent field theories with the so-called compositeness condition. With the formation of composite particles, the ultraviolet behavior of a large class of four-fermion theories is improved so that they become renormalizable and well-defined field theories beyond two (but less than four) dimensions in spite of their apparent nonrenormalizability in the conventional perturbation expansions.

Two-point function of nonlinear spinor theory

The influence of interaction on the nature of the quantum field is investigated for a self-interacting spinor theory. Nonlinear equations for the two-point function are formulated. The solutions of these equations demonstrate that fields in interaction exhibit novel features. In particular, the modified short-distance behavior indicates the absence of pathological features, implying self-regulation of nonlinear systems with concomitant noncanonical quantization.

Effects of the Higgs Scalar on Gravity

Let us consider a role of the Higgs scalar in the present elementary particle physics. The Higgs scalar has been used to generate particle masses through the spontaneous symmetry breakdown and couples to almost all fields. There is also a gravitational field which couples to all fields. These facts naturally lead us to the question Is there any role of the Higgs scalar played in gravity? or Is there any effect of the spontaneous symmetry breakdown on gravity? In this paper, we shall find how the Higgs scalar affects gravity. There have already been many papers in which a scalar field is incorporated with a Lagrangian for gravity. A heuristic method was used to construct a scalar­ tensor gravity by Brans and Dicken and others.) There have also appeared pa­ pers3l,4l in which a scalar field is combined with a scalar curvature so that a theory is conformally invariant. The latter theory has a severe defect that a scalar field becomes a ghost when an action for a gravitational field has a minimum. 4l There is another way in which one can determine an effective Lagrangian for any field. It is a nonlinear spinor theory of the Nambu-Jona-Lasinio type 5l which has recently revived and produced many interesting results one after another. 6l~sl In this paper, we shall incorporate the Higgs scalar with a gravitational Lagrangian, using a nonlinear spinor theory of the Nambu-Jona-Lasinio type. A gravitational field is taken as a weak field in our model. This is shown in § 2. Section 3 is devoted to discussion of main results of our model. Summary and discussion are given in § 4.

Quarks and flavours? A dissident interpretation of higher symmetries

In the framework of a model theory it is demonstrated that the spontaneous breakdown of fundamental symmetries in local quantum field theories leads to a dynamical rearrangement of the degrees of freedom of the basic-field operators in such a way that a larger set of effective local Bose and Fermi field operators is generated, adapted for the phenomenological description of particles, which differ in their transformation property under the broken symmetries from the basic fields. In a nonlinear spinor theory of the Nambu type with a fundamentalSU 2 isospin symmetry—as suggested by the gauge-type interactions—the effective fields can be classified according to anSU 2 ⊗SU 2 group with aU 1 ⊗SU 2 invariance subgroup. The phenomenological approximate higher symmetries are a direct consequence of the spontaneous breakdown of the fundamental symmetries. Natural dynamical interpretations offer themselves for many phenomenological features usually attributed to the quark model, like badly brokenSU 4 andSU 3 symmetries with well-conserved quantities «strangeness» and «charm», four different local constituent Fermi fields, eightfold way, low-lying boson and baryon octets, 2-constituent character of bosons, 3-constituent character of fermions and others. The constituent fields require no parastatistics, colour, confinement; they carry integral charges, and may have direct physical manifestations. Investigations of the asymmetry effects suggest a simple explanation of why «strangeness» and «charm» in lowest order are only violated in charged currents, and point a way for a dynamical determination of the Cabibbo angle.

Gravitational interaction in a model of nonlinear spinor theory with derivatives of the fundamental fields

To eliminate the difficulties that arise when particles of high spin (J ≥ 2) are treated in nonlinear spinor field theory, it is proposed to include terms with derivatives of the fundamental fields in the self-interaction Lagrangian of the subparticles. It is shown that with this generalization the four-fermion model of nonlinear spinor theory describes the gravitational interaction of subparticles in the approximation of single-graviton exchange. A relation between the gravitational constant and the parameters of the model is found.

On the geometric and algebraic foundation of the spinor formalism and the application to relativistic field equations

The Hilbert calculus of segments plays an important role in the axiomatic foundation of the Euclidean geometry, as the relationship to some fundamental agebraic structures can be made more apparent. An extension of the Hilbert calculus to the field of the quaternionsU2 or biquaternionsU4 leads to some new aspects on the spinor formalism. By that, a geometrical interpretation of the Dirac equation is obtained. Including the torsion of the Minkowski space (Cartan geometry), the affine connection of the spinor spaceU4 also can be interpreted with the help of a generalized Hilbert calculus. These considerations lead to a simple geometrical access to the nonlinear spinor theory, proposed by Ivanenko, Heisenberg, Durr, etc.

Some counterterms in the nonlinear spinor theory of heisenberg

Standard methods of renormalizable theories are (unlegitimately) applied to a four-fermion interaction in four dimensions. The nonlinear spinor theory of Heisenberg is taken as an example to work with. This is to show that then, except for inducing new types of counterterms, it is even the structure of the original Lagrangian which is badly destroyed. The results of some one-loop calculations are quoted and a discussion is presented.

New approach to collective phenomena in superconductivity models

A new approach to collective phenomena in superconductivity models is presented. In this approach nonlinear spinor models are converted into equivalent theories involving fermions and collective bosonic states. For a wide class of nonlinear spinor theories, interactions among fermions and collective states are of the renormalizable kind, and hence these models themselves turn out to be renormalizable. The equivalence of various four-fermion theories to known renormalizable models is pointed out. The original Nambu-Jona-Lasinio model, for instance, is shown to be equivalent to the linear $\ensuremath{\sigma}$ model. Detailed discussions are given on the origin of massless fields and the local gauge invariance. Implications of Pauli-G\ursey-type symmetry in spinor theories are also expounded.

On the Calculation of Nonlinear Spinor Field Functionals. III

The limitations of the Green functions method concerning nonlinear spinor theory are discussed. To remove these difficulties, functional quantum theory of the nonlinear spinor field was introduced in preceding papers. To obtain numerical values for the global observables in this theory, the functional eigenstates have to be calculated. In this paper especially for functional scattering states appropriate equations are derived. A general integration method of the spinor field functional equation is introduced, leading to an equation for the irreducible part of the state functional. Generating functionals are defined, allowing a separation of selfenergy and interactionenergy terms in the equation of the irreducible part. By spectral decomposition of the scattering functionals the boundary conditions are examined, which lead in connection with selfenergy and interactionenergy terms to the construction of channel equations for the irreducible part of elastic scattering functionals. The method is extented to inelastic processes, it can be tested in the case of nonrelativistic scattering theory. The procedure for the three particle case is discussed in some details

Mass Renormalization and Chiral Symmetry Breaking in Nonlinear Spinor Theory

Abstract The chiral invariance of conventional nonlinear spinor theory is assumed to be broken. This results in a finite mass renormalization of the nucleon mass. The self-consistency of the assumption is demonstrated. The meson mass eigenvalue equation is solved in lowest approximation. The spectrum which is no longer parity degenerate is in reasonable agreement with experiment.

Scalarproducts of Spinor Field Functionals and State Representations in Nonlinear Spinor Field Quantum Theory

Abstract In preceding papers it was shown that in nonlinear spinor theory cross-sections of elementary particle scattering processes can be calculated only if the state representations and their scalar products are explicitly known. To obtain these quantities, functional quantum theory of the non-linear spinor field was introduced-In this paper it is demonstrated that the introduction of functional scalar products in functional quantum theory is equivalent to impose restrictions to the spinor field operator itself concerning its groundstate behaviour. Performing this, explicit state representations of spinor field states as well as corresponding scalar products can be derived, leading thus to functional quantum theories of the spinor field in dependence on the groundstate model. It follows from these considerations that a spinor field quantum theory is in principle in-complete, as long as no additional assumptions on the groundstate are made, which cannot be derived from the general dynamics of the field.

On a gauge and conformal invariant nonlinear spinor theory

A nonlinear spinor theory for a 2-component Weyl spinor field with anomalous dimension 1/2 is formulated which is invariant under (local) gauge transformations and the full 15-parameter conformal group. In this formulation a first attempt is made to incorporate all recent developments in this context: 1) establishment of gauge invariance without additional vector fields by virtue of bilinear forms of a subcanonical spinor field of dimension 1/2, 2) construction of a canonical framework for the subcanonical spinor field by introducing subsidiary fields and by a corresponding extension of the unphysical state space, 3) verification of conformal invariance in the enlarged framework. The present theory appears, in a certain sense, as a canonical imbedding of a nonlinear spinor theory of the Heisenberg type. In the present formulation the spinor theory can be approached by conventional approximation methods. The lowest Bethe-Salpeter approximation for boson states yields nondivergent results for masses and coupling constants but requires a mass scale which is introduced by the assumption of a dilatation asymmetric vacuum. The theory can be readily generalized to include higher internal symmetries and in this form may be appropriate for a fundamental theory of elementary particles.

Numerical Solution of the Spinor-Spinor Bethe-Salpeter Equation in Functional Nonlinear Spinor Theory

Abstract The mass eigenvalue equation for mesons in nonlinear spinor theory is derived by functional methods. In second order it leads to a spinorial Bethe-Salpeter equation. This is solved by a variational method with high precision for arbitrary angular momentum. The results for scalar mesons show a shift of the first order results, obtained earlier. The agreement with experiment is improved thereby. An excited state corresponding to the η' is found. A calculation of a Regge trajectory is included,too.

Nonlinear spinor theory and Weinberg's model

It is shown that the formalism for eliminating vacuum degeneracy can be used to reduce the Heisenberg-Ivanenko nonlinear spinor Lagrangian to a Weinberg-type model Lagrangian of the weak and electromagnetic interactions.

On the Calculation of Scalar Mesons in Functional Nonlinear Spinor Theory

Abstract The formulation of nonlinear spinor theory in functional space is used for the calculation of scalar meson masses. The second order equation used, requires an explicit angular momentum reduction. For illustration, this method is also applied to the first order equation. In second order, we get an integral equation of the Bethe-Salpeter type which is solved in Fredholm approximation.

Gravitation and nonlinear spinor theory

In the framework of a single model of nonlinear spinor theory of matter, an attempt is made to describe the gravitational interaction. It is shown that the scattering amplitude of two primary fermions contains a contribution corresponding to one-graviton exchange. A relation between the gravitational constant and the mass of a subparticle is obtained.