Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 959left ideal of Cl 1,3 on which this algebra acts on the left. We have therepresentation γ : M ⊗ V → V of elements of the Minkowski spaceM ⊂ Cl 1,3 by Dirac’s matrices γ on V .Let us consider a bundle of complex Clifford algebras Cl 1,3 over Xwhose structure group is the Clifford group of invertible elements of Cl 1,3 .Its subbundles are both a spinor bundle S M −→ X and the bundle Y M−→ X of Minkowski spaces of generating elements of Cl 1,3 . To describeDirac fermion fields on a world manifold X, one must require Y M to beisomorphic to the cotangent bundle T ∗ X of X. It takes place if there existsa reduced L subbundle L h X such thatThen, the spinor bundleY M = (L h X × M)/L.S M = S h = (P h × V )/L s (5.475)is associated with the L s −lift P h of L h X. In this case, there exists therepresentationγ h : T ∗ X ⊗S h = (P h ×(M ⊗V ))/L s −→ (P h ×γ(M ×V ))/L s = S h (5.476)of cotangent vectors to a world manifold X by Dirac’s γ−matrices on elementsof the spinor bundle S h . As a shorthand, one can writêdx α = γ h (dx α ) = h α a (x)γ a .Given the representation (5.476), we shall say that sections of the spinorbundle S h describe Dirac fermion fields in the presence of the gravitationalfield h. Let a principal connection on S h be given byA h = dx α ⊗ (∂ α + 1 2 Aab αI abAB ψ B ∂ A ).Given the corresponding covariant differential D and the representation γ h(5.476), one can construct the Dirac operator on the spinor bundle S h , asD h = γ h ◦ D : J 1 S h → T ∗ X ⊗ V S h → V S h , (5.477)ẏ A ◦ D h = h α a γ aA B(y B α − 1 2 Aab αI abAB y B ).Different gravitational fields h and h ′ define nonequivalent representationsγ h and γ h ′. It follows that a Dirac fermion field must be regarded onlyin a pair with a certain gravitational field. There is the 1–1 correspondencebetween these pairs and sections of the composite spinor bundle (5.469).