Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1159for 0 ≤ σ ≤ π 2. In terms of the modes, the conditions for the sectors oft = 2 and 4 are identical with the Neumann case(1 − C)|Q 4,k )| ˆV 4 〉 X = (1 + C)|P 4,k )| ˆV 4 〉 X = 0,(1 + C)|Q 2,k )| ˆV 4 〉 X = (1 − C)|P 2,k )| ˆV 4 〉 X = 0,which can be seen from the point that the conditions (6.260) for the sectorsof t = 2 and 4 lead the same relations between the modes as those withoutthe terms containing θ jk . Here we adopted the vector notation for themodes⎡ ⎤⎡ ⎤⎢|Q t,k ) = ⎣Q t,0,kQ t,1,k.⎥⎢⎦ , |P t,k ) = ⎣P t,0,kP t,1,kand C is a matrix such that (C) nm = (−1) n δ nm (n, m ≥ 0). Thus there isneeded no correction containing θ ij for the sectors of t = 2 and 4, so it isnatural to assume the form of the phase factor in (6.258) as14∑2 θij(p (r)ir,s=1.⎥⎦ ,|Z rs |p (s)j ) = θ ij (P i |Z| ¯P j ) (6.261)with Z being anti–Hermitian.Next let us consider the conditions for the sectors of t = 1 and 3. Werewrite the mode expansions of Q j (σ) and ¯Q j (σ) as [Sugino (2000)]Q j (σ) = G jk ( √ 2α ′ Q 0,k + 2 √ α ′+ θ jk [ ∫ σ≡ θ jk ∫ σπ/2π/2∞∑n=1dσ ′ P i (σ ′ ) + 1π √ α ′¯Q j (σ) = G jk ( √ 2α ′ ¯Q 0,k + 2 √ α ′+ θ jk [ ∫ σπ/2cos(nσ)Q n,k )∑n=1,3,5,···]1n (−1)(n−1)/2 P n,kdσ ′ P i (σ ′ ) + ∆Q j (σ), (6.262)∞ ∑n=1dσ ′ ¯Pi (σ ′ ) + 1π √ α ′cos(nσ) ¯Q n,k )∑n=1,3,5,···]1n (−1)(n−1)/2 ¯Pn,k∫ σ≡ θ jk dσ ′ ¯Pi (σ ′ ) + ∆ ¯Q j (σ). (6.263)π/2