It is supposed that there exists a system O′ (intrinsic system) in which the field equation for a spin ½ representation has the simple form γµ ∂Ψ′/∂ϰμ′=0. This system is related to the physical system (in which all measurements are performed) by an affine connection which is induced by a certain group of local transformations. The investigation given here deals with the group of local four-dimensional complex orthogonal transformations. Subjecting ψ' to such a transformation Ω one gets with ψ' (x′) = Ω (x) ψ (x) the following equation γλ ∂Ψ/∂xλ+γλ Ω-1 ∂Ω/∂xλ ψ=0. The interaction term splits up into a vector and a pseudovector part: γλ Ω-1 ∂Ω/∂xλ ≡ γλ Vλ+γλ γ5 Ρλ. The special cases of real local orthogonal (LORENTZ-) transformations (ξλμ= - ξμλ; ξkl real, ξ4l imaginary; ψ → χ) and special complex local orthogonal transformations (ηλμ=- ημλ; ηkl imaginary, η4l real; ψ → φ) are first separately considered. It is required that Vλ and Pλ are to be built up from the fundamental covariants of the field. In order that certain conservation laws hold at least approximately, the following assumptions are made:Im{Vk}=±k2 ɸ̅γkφ, Re {V4}=±k2γ4φ, Im {Pk} = ± l2 χ̅ γk γ5 χ, Re {P4}= ±l2χ̅γ4γ5χ together with the symmetry conditions for the transformation parameters, ξλ[μυ] ≡ 0, η〈λμ,υ〉 ≡ 0, which can be fulfilled by setting, for example, ξλμ,υ = π[λπμ,υ],ηλμ = ϑ[λ,μ]. The remaining parts of Vλ and Ρλ, which are determined by these relations, are of higher order and can be assumed to describe weaker interactions. Neglecting these terms one obtains the following set of equations:(a) γλ ∂χ/∂xλ±k2γλ(ɸ̅γλφ) χ±l2γλγ5(χ̅γλγ5χ) χ≈0(b) γλ ∂χ/∂xλ±k2γλ(ɸ̅γλφ) φ±l2γλγ5(χ̅γλγ5χ) φ≈0Since the pseudovector coupling possesses a greater symmetry, it is assumed that χ represents the baryon and φ the lepton states. Within the approximation, which holds with (a) and (b), it follows the conservation of χ̅γλχand ɸ̅γλφ resp. (conservation of electric charge) and χ̅γλγ5χ and ɸ̅ γ·λγ5φ resp. (conservation of baryonic and leptonic charge resp.). These conservation laws are exact only if the mentioned terms of higher order are neglected; this is equivalent to a strict “local” conservation as can be shown. As to the isospin it is proposed to replace one of its components by a bounded state, i. e. a mixture of χ- and φ-states which would lead in the case of the neutron for example to the components of the /?β-decay. Due to the relations ± k2 ɸ̅γλφ = ¼ηλρ,ρ +O(η2) and ηλu = ϑ[λμ], and in agreement with the reality conditions, it is possible to connect the parameters ϑλ with the electromagnetic field Aλ by setting ϑλ= 8 iAλ. Taking into consideration terms of higher order this would lead to a type of nonlinear electrodynamics.
The Godbillon-Vey invariant as topological vorticity compression and obstruction to steady flow in ideal fluids
