Ivancevic_Applied-Diff-Geom

1006 Applied Differential Geometry: A Modern IntroductionWhen the potential V (x) ≠ 0 the propagator associated with (6.11) isformally given by∫∫ ∫W (x ′′ , t ′′ ; x ′ , t ′ ) = N D[x]e −(1/2ν) ẋ 2 dt− V (x) dt ,an expression which is well defined if V (x) ≥ c, -∞ < c < ∞. A mathematicallyimproved expression makes use of the Wiener measure and reads∫ ∫W (x ′′ , t ′′ ; x ′ , t ′ ) = e − V (x(t)) dt dµ ν W (x).This is an elegant relation in that it represents a solution to the differentialequation (6.11) in the form of an integral over Brownian motion pathssuitably weighted by the potential V . Incidentally, since the propagatoris evidently a strictly positive function, it follows that the solution of thedifferential equation (6.11) is nonnegative for all time t provided it is nonnegativefor any particular time value.6.2.5 Itô FormulaItô [Ito (1960)] proposed another version of a continuous–time regularizationthat resolved some of the troublesome issues. In essence, the proposal ofItô takes the form given by∫lim N ν D[x] exp{(i/) ∫ [ 1ν→∞ 2 mẋ2 − V (x)] dt} exp{−(1/2ν) ∫ [ẍ 2 + ẋ 2 ] dt}.Note well the alternative form of the auxiliary factor introduced as a regulator.The additional term ẍ 2 , the square of the second derivative ofx, acts to smooth out the paths sufficiently well so that in the case of(21) both x(t) and ẋ(t) are continuous functions, leaving ẍ(t) as the termwhich does not exist. However, since only x and ẋ appear in the restof the integrand, the indicated path integral can be well defined; thisis already a positive contribution all by itself (see e.g., [Klauder (1997);Klauder (2000)]).6.3 Standard Path–Integral Quantization6.3.1 Canonical versus Path–Integral QuantizationRecall that in the usual, canonical formulation of quantum mechanics, thesystem’s phase–space coordinates, q, and momenta, p, are replaced by the