Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 9195.11.5 Lagrangian Gauge TheoryClassical gauge theory of unbroken symmetries on a principal G−bundleP → Q deals with two types of fields. These are gauge potentials identifiedto global sections of the connection bundle C → Q (5.45) and matter fieldsrepresented by global sections of a P −associated vector bundle Y (5.330),called a matter bundle. Therefore, the total configuration space of classicalgauge theory is the product of jet bundlesJ 1 (X, Y ) tot = J 1 (X, Y ) × J 1 (Q, C). (5.354)Let us study a gauge–invariant Lagrangian on this configuration space.A total gauge vector–field on the product C × Y readsξ Y C = (−∂ α ξ r + c r pqξ p a q α)∂ α r + ξ p I i p∂ i = (u Aλp ∂ α ξ p + u A p ξ p )∂ A , (5.355)where we use the collective index A, and put the notationu Aλp ∂ A = −δ r p∂ α r , u A p ∂ A = c r qpa q α∂ α r + I i p∂ i .A Lagrangian L on the configuration space (5.354) is said to be gauge–invariant if its Lie derivative L J 1 ξ Y CL along any gauge vector–field ξ (5.351)vanishes. Then the first variational formula (5.109) leads to the strongequality0 = (u A p ξ p + u Aµp ∂ µ ξ p )δ A L + d α [(u A p ξ p + u Aµp ∂ µ ξ p )π α A], (5.356)where δ A L are the variational derivatives (5.233) of L and the total derivativereadsd α = ∂ α + a p λµ ∂µ p + y i α∂ i .Due to the arbitrariness of gauge parameters ξ p , this equality falls into thesystem of strong equalitiesu A p δ A L + d µ (u A p π µ A) = 0, (5.357)u Aµp δ A L + d α (u Aµp π α A) + u A p π µ A= 0, (5.358)u Aλp π µ A + uAµ p π α A = 0. (5.359)Substituting (5.358) and (5.359) in (5.357), we get the well–known constraintsu A p δ A L − d µ (u Aµp δ A L) = 0for the variational derivatives of a gauge–invariant Lagrangian L.