Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1175Thus, we can conclude that the action is∫S = −γ√ ∣∣∣∣ ∣ ∫dσ 0 d n σ i det ∂xµ ∂x µ ∣∣∣= −γ∂σ a ∂σ b√dx 0 dV 1 − −→ v 2 ⊥. (6.291)6.6.4 String DynamicsAs it was mentioned above, the evolution operator U ω can be constructedfor strings and branes in the same way as above [Goto (1971)]. For p =1 action (6.284) is the action of free bosonic string with the Lagrangiandensity (see, e.g., [Kaku (1988)])L = −γ √ √−g = −γ (ẋx ′ ) 2 − ẋ 2 x ′2 . (6.292)We assume that ẋ is time–like and x ′ is space–like, so that σ 0 can beregarded as a time parameter. In this case the momentump µ ≡ ∂L∂ẋ µ = −γ (ẋx′ )x ′ µ − x ′2ẋ µ√(ẋx′ ) 2 − ẋ 2 x ′2 (6.293)will also be time–like. One easily gets two constraintsp µ x ′ µ = 0, (6.294)p 2 + γ 2 x ′ 2 = 0. (6.295)The second one is obtained by squaring the (6.293) and hence some informationis lost. As in the case of a pointlike particle (6.295) yields p 0 = ±E pwithE p =√−→p 2 − γ2 x ′2 .The sign of p 0 follows from the definition of the momentum. We havep 0 + E p (x, −→ p )sgn(y 0 ) = 0,wherey µ = (ẋx ′ )x ′ µ − x ′ 2ẋµ .The vector y is obviously time–like (indeed, y 2 = x ′2 g > 0 and y µ ẋ µ =−g > 0, so that sgnẋ 0 = sgny 0 ). We get the ‘constraint’ analogous to(6.287):p 0 + E p (x, −→ p )sgn(ẋ 0 ) = 0. (6.296)