Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1095where S = S[g ij ] is the n−dimensional geodesic action on M,∫ √S[g ij ] = d n x g ij dx i dx j . (6.148)The corresponding Euler–Lagrangian equation gives the geodesic equationof the shortest path in the manifold M,ẍ i + Γ i jk ẋ j ẋ k = 0,where the symbol Γ i jkdenotes the so–called affine connection which is thesource of curvature, which is geometrical description for noise (see [Ingber(1997); Ingber (1998)]). The higher the local curvatures of the LSF–manifold M, the greater the noise in the life space. This noise is the sourceof our micro–level fluctuations. It can be internal or external; in both casesit curves our micro–LSF.Otherwise, if instead we choose an n−dimensional Hilbert–like action(see [Misner et al. (1973)]),∫ √S[g ij ] = d n x det |g ij |R, (6.149)where R is the scalar curvature (derived from Γ i jk), we get then−dimensional Einstein–like equation: G ij = 8πT ij , where G ij is theEinstein–like tensor representing geometry of the LSF manifold M (G ij isthe trace–reversed Ricci tensor R ij , which is itself the trace of the Riemanncurvature tensor of the manifold M), while T ij is the n−dimensional stress–energy–momentum tensor. This equation explicitly states that psycho–physics of the LSF is proportional to its geometry. T ij is important quantity,representing motivational energy, geometry–imposed stress and momentumof (loco)motion. As before, we have our ‘golden rule’: the greaterthe T ij −components, the higher the speed of cognitive processes and thelower the macroscopic fatigue.The choice between the geodesic action (6.148) and the Hilbert action(6.149) depends on our interpretation of time. If time is not included in theLSF manifold M (non–relativistic approach) then we choose the geodesicaction. If time is included in the LSF manifold M (making it a relativistic–like n−dimensional space–time) then the Hilbert action is preferred. Thefirst approach is more related to the information processing and the workingmemory. The later, space–time approach can be related to the long–termmemory: we usually recall events closely associated with the times of theirhappening.