Ivancevic_Applied-Diff-Geom

1052 Applied Differential Geometry: A Modern Introductiongenerating functional (6.85), one can follow a procedure of quantizationof gauge–invariant Lagrangian systems. In the case of the Lagrangian L Π(5.319), this procedure is rather trivial, since the space of momenta variablesp a (x) coincides with the translation subgroup of the gauge groupAut Ker σ 0 .Now let us suppose that the Lagrangian L N (5.316) and, consequently,the Lagrangian L Π (5.319) are invariant under some gauge group G X ofvertical automorphisms of the fibre bundle Y → X (and the induced automorphismsof Π → X) which acts freely on the space of sections ofY → X. Its infinitesimal generators are represented by vertical vector–fields u = u i (x µ , y j )∂ i on Y → X which induce the vector–fieldsu = u i ∂ i − ∂ j u i p α i ∂ j α + d α u i ∂ α i , d α = ∂ α + y i α∂ i , (6.86)on Π × J 1 (X, Y ). Let us also assume that G X is indexed by m parameterfunctions ξ r (x) such that u = u i (x α , y j , ξ r )∂ i , whereu i (x α , y j , ξ r ) = u i r(x α , y j )ξ r + u iµr (x α , y j )∂ µ ξ r (6.87)are linear first–order differential operators on the space of parameters ξ r (x).The vector–fields u(ξ r ) must satisfy the commutation relations[u(ξ q ), u(ξ ′p )] = u(c r pqξ ′p ξ q ),where c r pq are structure constants. The Lagrangian L Π (5.319) is invariantunder the above gauge transformations iff its Lie derivative L u L Π alongvector–fields (6.86) vanishes, i.e.,(u i ∂ i − ∂ j u i p α i ∂ j α + d α u i ∂ α i )L Π = 0. (6.88)Since the operator L u is linear in momenta p µ i , the condition (6.88) fallsinto the independent conditions(u k ∂ k − ∂ j u k p ν k∂ j ν + d ν u j ∂ ν j )(p α i F i α) = 0, (6.89)(u k ∂ k − ∂ j u k p ν k∂ν)(σ j ij0 λµ pα i p µ j ) = 0, (6.90)u i ∂ i c ′ = 0. (6.91)It follows that the Lagrangian L Π is gauge–invariant iff its three summandsare separately gauge–invariant.Note that, if the Lagrangian L Π on Π is gauge–invariant, the originalLagrangian L (5.312) is also invariant under the same gauge transforma-