Covariant objects and invariant equations on fiber bundles

Abstract

Let P(M4,G) be a principal fiber bundle over the Minkowskian space–time M4 with the structural group G. The group G is supposed to be a compact and semisimple Lie group. Let A be a connection form on P(M4,G) and F=DA its curvature form. Let gG be the Cartan–Killing metric on G, and gM4 the Minkowskian metric on M4. Let us define dπ : TP→TM4, the differential of the canonical projection from P onto M4. Then we can define a scalar product for any two vectors from P(M4,G): gP(X,Y)=gG(A(X),A(Y))+gM4Q (dπ(X),dπ(Y)). In this metric the horizontal and vertical subspaces of the connection A are orthogonal to each other. Next, we construct the Clifford algebra corresponding to the metric gP. The metric gP can be always diagonalized locally to give diag((3+N)+,1−), where N is the dimension of G. The lowest faithful representation of this algebra, which we call C(3+N,1) is of the dimension K=2[(N+5)/2]. This K-dimensional vector space is called the space of spinors over P(M4,G). We study the decomposition of these spinors into multiplets of Lorentz spinors. We also define the generalized Dirac equation for such a spinor, construct an explicit representation in the case of G=SU(2), and give the formulae for the mass splitting. Finally, the invariant interaction with vector fields over P(M4,G) and scalar multiplets is discussed, together with the physical implications of the coupled equations.