Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 923which are the familiar generalized Lorentz gauge.second-order differential operator (6.95) readsThe correspondingM r s ξ s = g λµ ( 1 2 cr pq(∂ α a r µ + ∂ µ a r α)ξ q + c r pqa p µ∂ α ξ q + ∂ α ∂ µ ξ r ).Passing to the Euclidean space and repeating the above quantization procedure,we come to the generating functional∫ ∫√Z = N −1 exp{ (p αµr Fαµ r − a mnG g µν g αβ p µαm p νβn |g|− 1 8 aG rsg αν g αµ (∂ α a r ν + ∂ ν a r α)(∂ α a s µ + ∂ µ a s α)− g αµ c r ( 1 2 cr pq(∂ α a r µ + ∂ µ a r α)c q + c r pqa p µc q α + c r αµ)+ iJ r µ a r µ + iJµαp r µαr )ω} ∏ [dc][dc][dp(x)][da(x)].xIts integration with respect to momenta restarts the familiar generatingfunctional of gauge theory.5.11.7 Gauge Conservation LawsOn–shell, the strong equality (5.356) becomes the weak Noether conservationlawof the Noether currentd α [(u A p ξ p + u Aµp ∂ µ ξ p )π α A] ≈ 0 (5.374)J α = −(u A p ξ p + u Aµp ∂ µ ξ p )π α A. (5.375)Therefore, the equalities (5.357) – (5.359) on–shell lead to the familiarNoether identitiesd µ (u A p π µ A) ≈ 0, (5.376)d α (u Aµp π α A) + u A p π µ A≈ 0, (5.377)u Aαp π µ A + uAµ p π α A = 0 (5.378)for a gauge–invariant Lagrangian L. They are equivalent to the weak equality(5.374) due to the arbitrariness of the gauge parameters ξ p (q).The expressions (5.374) and (5.375) shows that both the Noether conservationlaw and the Noether current depend on gauge parameters. Theweak identities (5.376) – (5.378) play the role of the necessary and sufficient