Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1003lattice regularization procedure are given by∫K(x ′′ , t ′′ ; x ′ , t ′ ) = lim (m/2πiε) (N+1)/2 · · ·ε→0∫N∑· · · exp{(i/) [(m/2ε)(x l+1 − x l ) 2 − ε V (x l ) ]}l=0N∏dx l ,where x N+1 = x ′′ , x 0 = x ′ , and ε ≡ (t ′′ − t ′ )/(N + 1), N ∈ {1, 2, 3, . . . }. Inthis version, at least, we have an expression that has a reasonable chance ofbeing well defined, provided, that one interprets the conditionally convergentintegrals involved in an appropriate manner. One common and fullyacceptable interpretation adds a convergence factor to the exponent of thepreceding integral in the form −(ε 2 /2) ∑ Nl=1 x2 l, which is a term thatformally makes no contribution to the final result in the continuum limitsave for ensuring that the integrals involved are now rendered absolutelyconvergent.l=16.2.3 Hamiltonian Path IntegralIt is necessary to retrace history at this point to recall the introductionof the phase–space path integral by Feynman [Feynman (1951); Groscheand Steiner (1998)]. In Appendix B to this article, Feynman introduceda formal expression for the configuration or q−space propagator given by(see e.g., [Klauder (1997); Klauder (2000)])∫K(q ′′ , t ′′ ; q ′ , t ′ ) = M D[p] D[q] exp{(i/) ∫ [ p ˙q − H(p, q) ] dt}.In this equation one is instructed to integrate over all paths q(t), t ′ ≤ t ≤ t ′′ ,with q(t ′′ ) ≡ q ′′ and q(t ′ ) ≡ q ′ held fixed, as well as to integrate over allpaths p(t), t ′ ≤ t ≤ t ′′ , without restriction.It is widely appreciated that the phase–space path integral is more generallyapplicable than the original, Lagrangian, version of the path integral.For example, the original configuration space path integral is satisfactoryfor Lagrangians of the general formL(x) = 1 2 mẋ2 + A(x) ẋ − V (x) ,but it is unsuitable, for example, for the case of a relativistic particle withthe LagrangianL(x) = −m qrt1 − ẋ 2