Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1157• | ˆV 2 〉 X 12For the N = 2 case, we are to do the same argument as in the N = 1 case.The continuity conditions mean thatP (1)i (σ) + P (2)i (π − σ) = 1 π (p(1) i + p (2)i ) + 1π √ α ′∞ ∑n=1cos(nσ)(p (1)n,i + (−1)n p (2)n,i ),X (1)j (σ) − X (2)j (π − σ) = G jk (y (1)k− y (2)k) + 1 π θjk (σ − π 2 )(p(1) k+ p (2)k )+ 2 √ ∑∞ [α ′ G jk cos(nσ)(x (1)n,k − (−1)n x (2)n,k )n=1+ 12πα ′ θjk sin(nσ) 1 ]n (p(1) n,k + (−1)n p (2)n,k )should be zero for 0 ≤ σ ≤ π. It turns out again that the conditions forthe modes are identical with those in the Neumann case:p (1)i + p (2)i = 0, p (1)n,i + (−1)n p (2)n,i = 0,y (1)i − y (2)i = 0, x (1)n,i − (−1)n x (2)n,i = 0,for n ≥ 1. Thus we have the solution [Sugino (2000)][]| ˆV 2 〉 X 12 = |V 2 〉 X 12 = exp−G ij∞∑n=0(−1) n a (1)†n,i a(2)† n,j|0〉 12 . (6.257)