Topological Integrals of Motion in Gauge Theory

Abstract

The existing models combining strong, weak and electromagnetic interactions (grand unification models) are built using the same principles as the theory of electroweak interaction. We take the Lagrangian

$$L = {L_0} - \Gamma \bar \psi \psi \varphi - U\left( \varphi \right)$$

(12.1)

, where ψ = (ψ ¹,..., ψ ^m) is a multicomponent fermion field, φ = (φ ¹,..., φ ^m) a multicomponent scalar field, and L ₀ the free Lagrangian describing the interaction of these fields. Assume that (12.1) is invariant under an internal symmetry group G. This means that ψ and φ transform in a certain way under transformations in G (they take values in some representation space of G) and that the Lagrangian is a scalar with respect to this group; in particular, the polynomial U(φ) and the expression $\Gamma \bar \psi \psi \varphi = {G_{ijk}}\overline {{\psi ^i}} {\psi ^j}\varphi$ are G-invariant.