Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1109quantum mechanical. We scale them by the coupling constant e, by settingN to eN, φ to eφ etc.. To the order o(1) in e 2 , we haveZ = ∑ exp(− 1 ∫e 2 S(p) cl) DF ′ exp(−S q (p) ),pwhere S q(p) is the quadratic part of the action in the quantum fields anddepends on the gauge orbit of the classical configuration A o , M o , which welabel by p. Explicitly [Zhang et. al. (1995)],S (p)q =∫X{[−∆φ + M o M o φ − iNN]λ − [∇ k ψ k + i 2 (NMo − M o N)]η + 2iφ¯µµ+ (iD A oN − γ.ψM o )µ − ¯µ (iD A oN − γ.ψM o )[ (− χ kl ∇ k ψ l − ∇ l ψ k) + i (+ ¯M o Γ kl N +2¯NΓ kl M o)]+ 1 4 |f + + i 2 ( ¯mΓM o + ¯M o Γm)| 2 + 1 }2 |iD A om + γ.aM o | 2 ,with f + the self–dual part of f = da. The classical part of the action isgiven by S (p)cl= S 0 | A=A o ,M=M o.The integration measure DF ′ has exactlythe same form as DF but with A replaced by a, and M by m, ¯M by ¯mrespectively. Needless to say, the summation over p runs through all gaugeclasses of classical configurations.Let us now examine further features of our quantum field theory. Agauge class of classical configurations may give a non–zero contributionto the partition function in the limit e 2 → 0 only if S (p)clvanishes, andthis happens if and only if A o and M o satisfy (6.167). Therefore, the SWmonopole equations are recovered from the quantum field theory.The equations of motion of the fields ψ and N in the semi–classical approximationcan be easily derived from the quadratic action S q(p) , solutionsof which are the zero modes of the quantum fields ψ and N. The equationsof motion read(dψ) + kl + i 2( ¯M o Γ kl N + ¯NΓ kl M o) = 0, D A oN + γ.ψM 0 = 0,∇ k ψ k + i (NM − MN) = 0. (6.175)2Note that they are exactly the same equations which we have alreadydiscussed in (6.168). The first two equations are the linearization of themonopole equations, while the last is a ‘gauge fixing condition’ for ψ. The