Ivancevic_Applied-Diff-Geom

1088 Applied Differential Geometry: A Modern IntroductionAction is defined on LSF totalT A ≡ 〈Action|Intention〉 total : INT ENT ION t0 ⇛ ACT ION t1 , (6.132)given by adaptive generalization of the Feynman’s path integral [Feynmanand Hibbs (1965); Feynman (1972); Feynman (1998)]. The transition map(6.132) calculates the overall probability amplitude along a multitude ofwildly fluctuating paths, fields and geometries, performing the microscopictransition from the micro–state INT ENT ION t0 occurring at initial micro–time instant t 0 to the micro–state ACT ION t1 at some later micro–timeinstant t 1 , such that all micro–time instants fit inside the global transitioninterval t 0 , t 1 , ..., t s ∈ [t ini , t fin ]. It is symbolically written as∫〈Action|Intention〉 total := Σ D[wΦ] e iS[Φ] , (6.133)where the Lebesgue integration is performed over all continuous Φ i con =paths + field + geometries, while summation is performed over all discreteprocesses and regional topologies Φ j dis). The symbolic differential D[wΦ]in the general path integral (6.133), represents an adaptive path measure,defined as a weighted productN∏D[wΦ] = lim w s dΦ i s, (i = 1, ..., n = con + dis), (6.134)N−→∞s=1which is in practice satisfied with a large N corresponding to infinitesimaltemporal division of the four motivational phases (*). Technically, thepath integral (6.133) calculates the amplitude for the transition functorT A : Intention ⇛ Action.In the exponent of the path integral (6.133) we have the action S[Φ]and the imaginary unit i = √ −1 (i can be converted into the real number-1 using the so–called Wick rotation, see next subsection).In this way, we get a range of micro–objects in the local LSF at theshort time–level: ensembles of rapidly fluctuating, noisy and crossing paths,force–fields, local geometries with obstacles and topologies with holes. However,by averaging process, both in time and along ensembles of paths, fieldsand geometries, we recover the corresponding global MD & CD variables.Infinite–Dimensional Neural Network. The adaptive path integral(6.133) incorporates the local learning process according to the standardformula: New V alue = Old V alue + Innovation. The general weightsw s = w s (t) in (6.134) are updated by the MONIT OR feedback during the