Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1169sidered as continuous limits of ordered discrete sets of relativistic particlesfor which the tangential velocities were excluded from the action. The authorshave proved that p−branes might be considered as continuous limitsof discrete sets of relativistic particles and that the p−brane action∫S = −γ d p+1 σ √ (−1) p g,g = det ∂xµ∂σ a∂x µ∂σ b, (6.284)is a continuous limit of sum of properly modified relativistic particle actions.Here a, b = 0, 1, . . . , p and µ = 0, 1, . . . , n where p + 1 and n + 1 are thebrane world–sheet and the bulk space dimensions respectively; g is theinduced metric determinant on the world–sheet. Strings correspond to thep = 1 case in (6.284). The modification is such that particle motions alongthe brane hypersurface become un–physical. It yields p constraints; theremaining constraint (H = 0) is a consequence of arbitrariness of ‘time’ σ 0 .The authors have also found the linear constraints in un–physical momenta,which allowed them to derive the evolution operators for the objects underconsideration from the ‘first principles’.6.6.1 A Relativistic ParticleFollowing [Golovnev and Prokhorov (2005)], we start with the simplest caseof one relativistic particle which can be formally considered as a 0−brane.Recall that the motion of free relativistic particle is defined by the well–known action∫ √S = −m 1 − −→ v 2 dt,where −→ v = d−→ x (t)dt. The canonical momentum is−→ ∂L p ≡∂ −→ v = m−→ v√1 − −→ ≡ E −→ v 2 p v ,with L being the Lagrangian, and the Hamiltonian is√H = E p = m 2 + −→ p 2 .We can write down the action in the explicitly relativistic invariant form byparametrization of the world line: x µ = x µ (σ 0 ) (usually one uses τ insteadof σ 0 ) withx µ (σ 0 ) = (t(σ 0 ), −→ x (σ 0 )),(µ = 0, . . . , n).