Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1053tions. Indeed, one gets at once from the condition (6.89) thatu(F i µ) = ∂ j u i F j µ, (6.92)i.e., the quantity F is transformed as the dual of momenta p. Then thecondition (6.90) shows that the quantity σ 0 p is transformed by the samelaw as F. It follows that the term aFF in the Lagrangian L (5.312) istransformed exactly as a(σ 0 p)(σ 0 p) = σ 0 pp, i.e., is gauge–invariant. Thenthis Lagrangian is gauge–invariant due to the equality (6.91).Since S i α = y i α − F i α, one can derive from the formula (6.92) the transformationlaw of S,u(S i µ) = d µ u i − ∂ j u i F j µ = d µ u i − ∂ j u i (y j µ − S j µ) = ∂ µ u i + ∂ j u i S j µ. (6.93)This expression shows that the gauge group G X acts freely on the spaceof sections S(x) of the fibre bundle Ker ̂L → Y in the splitting (5.309).Let the number m of parameters of the gauge group G X do not exceed thefibre dimension of Ker ̂L → Y . Then some combinations b r µi Si µ of Sµ i canbe used as the gauge conditionb rµ i Si µ(x) − α r (x) = 0,similar to the generalized Lorentz gauge in Yang–Mills gauge theory.Now we turn to path–integral quantization of a Lagrangian systemwith the gauge–invariant Lagrangian L Π (5.319). In accordance with thewell–known quantization procedure, let us modify the generating functional(6.85) as follows [Bashkirov and Sardanashvily (2004)]Z = N −1 ∫∫exp{ (L Π − 1 2 σ 1 ijαµp α i p µ j+ 1 2 Tr(ln σ) − 1 2 h rsα r α s + iJ i y i + iJµp i µ i )ω} (6.94)∆ ∏ × r δ(b rµ i Si µ(x) − α r (x))[dα(x)][dp(x)][dy(x)]x∫ ∫= N ′−1 exp{ (L Π − 1 2 σ 1 ijαµp α i p µ j + 1 Tr(ln σ)2− 1 2 h rsb rµ i bsα j SµS i α j + iJ i y i + iJµp i µ i )ω}∆ ∏ [dp(x)][dy(x)],x∫ ∫where exp{ (− 1 2 h rsα r α s )ω} ∏ [dα(x)]x