Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1087the charges, representing the Maxwell–like field (it is summed over eachpair of charges; the factor 1 2is to count each pair once, while the term i = jis omitted to avoid self–action; the interaction is a double integral over adelta function of the square of space–time interval I 2 between two pointson the paths; thus, interaction occurs only when this interval vanishes, thatis, along light cones [Wheeler and Feynman (1949)]).Now, from the point of view of Lewinian geometrical force–fields and(loco)motion paths, we can give the following life–space interpretation tothe Wheeler–Feynman action (6.131). The mechanical–like locomotionterm occurring at the single time t, needs a covariant generalization fromthe flat 4D Euclidean space to the nD smooth Riemannian manifold, so itbecomes (see e.g., [Ivancevic (2004)])S[x] = 1 2∫ tfint inig ij ẋ i ẋ j dt,where g ij is the Riemannian metric tensor that generates the total ‘kineticenergy’ of (loco)motions in the life space.The second term in (6.131) gives the sophisticated definition of Lewinianforce–fields that drive the psychological (loco)motions, if we interpret electricalcharges e i occurring at different times t i as motivational charges –needs.Local Micro–Level of LSF total . After having properly definedmacro–level MD & CD, with a unique transition map F (including a uniquemotion path, driving field and smooth geometry), we move down to the microscopicLSF–level of rapidly fluctuating MD & CD, where we cannot definea unique and smooth path–field–geometry. The most we can do at thislevel of fluctuating uncertainty, is to formulate an adaptive path integral andcalculate overall probability amplitudes for ensembles of local transitionsfrom one LSF–point to the neighboring one. This probabilistic transitionmicro–dynamics functor is defined by a multi–path (field and geometry,respectively) and multi–phase transition amplitude 〈Action|Intention〉 ofcorresponding to the globally–smooth transition map (6.130). This absolutesquare of this probability amplitude gives the transition probability ofoccurring the final state of Action given the initial state of Intention,P (Action|Intention) = |〈Action|Intention〉| 2 .The total transition amplitude from the state of Intention to the state of