Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1165noncommutative structure. First, we consider the phase factor of the 3–string vertex which multiplies in front of |V 3 〉 (see (6.269)). Making use ofthe continuity conditions∞∑∞∑P 0,i = −2 Y 0n P n,i , ¯P0,i = −2 Ȳ 0n ¯Pn,i , (6.271)n=1it can be rewritten as [Sugino (2000)]n=1−θij4 √ 3α P ¯P ′ 0,i 0,j = θij4 √ ∑∞ (P 0,i Ȳ 0n ¯Pn,j + P n,i Y 0n ¯P0,j )3α ′= − θij24α ′n=1∑ ∞n=1X 0n [(−p (2)0,i − p(3) 0,i + 2p(1) 0,i )p(1) n,j+ (−p (3)0,i − p(1) 0,i + 2p(2) 0,i )p(2) n,j + (−p(1) 0,i − p(2) 0,i + 2p(3) 0,i )p(3)3∑ ∞∑= − θij8α ′r=1 n=1X 0n p (r)0,i p(r) n,j ,where we used the property of the matrix Y : Y 0n = −Ȳ0n = √ 32 X 0n forn ≥ 1 and the momentum conservation on |V 3 〉: p (1)0,i + p(2) 0,i + p(3) 0,i = 0. Wemanage to represent the phase factor of the Moyal type as a form factorizedinto the product of the unitary operators(θ ij ∑U r = exp8α ′n=1,3,5,···X 0n p (r)0,i p(r) n,j)n,j ]. (6.272)Note that the unitary operator acts to a single string field. So the Moyaltype noncommutativity can be absorbed by the unitary rotation of thestring field123〈 ˆV 3 ||ψ〉 1 |ψ〉 2 |ψ〉 3 = 123 〈V 3 |U 1 U 2 U 3 |ψ〉 1 |ψ〉 2 |ψ〉 3 = 123 〈V 3 ||˜ψ〉 1 |˜ψ〉 2 |˜ψ〉 3 ,(6.273)with |˜ψ〉 r = U r |ψ〉 r . It should be remarked that this manipulation hasbeen suceeded owing to the factorized expression of the phase factor, whichoriginates from the continuity conditions relating the zero-modes to thenonzero-modes (6.271). It is a characteristic feature of string field theorythat can not be found in any local field theories.Next let us see the kinetic term. In doing so, it is judicious to write thekinetic term as follows:12〈 ˆV 2 ||ψ〉 1 (Q|ψ〉 2 ) = 123 〈 ˆV 3 ||ψ〉 1 (Q L |I〉 2 |ψ〉 3 + |ψ〉 2 Q L |I〉 3 ), (6.274)