Ivancevic_Applied-Diff-Geom

1086 Applied Differential Geometry: A Modern IntroductionSecond, we formulate the least action principle as a minimal variationδ of the action S[Φ]δS[Φ] = 0, (6.129)which, using techniques from the calculus of variations gives, in the form ofthe so–called Euler–Lagrangian equations, a shortest (loco)motion path, anextreme force–field, and a life–space geometry of minimal curvature (andwithout holes). In this way, we effectively derive a unique globally smoothtransition functorT A : INT ENT ION tini ⇛ ACT ION tfin , (6.130)performed at a macroscopic (global) time–level from some initial time t inito the final time t fin .In this way, we get macro–objects in the global LSF: a single pathdescribed Newtonian–like equation of motion, a single force–field describedby Maxwellian–like field equations, and a single obstacle–free Riemanniangeometry (with global topology without holes).For example, recall that in the period 1945–1949 J. Wheeler and R.Feynman developed their action-at-a-distance electrodynamics [Wheelerand Feynman (1949)], in complete experimental agreement with the classicalMaxwell’s electromagnetic theory, but at the same time avoiding thecomplications of divergent self–interaction of the Maxwell’s theory as wellas eliminating its infinite number of field degrees of freedom. In Wheeler–Feynman view, “Matter consists of electrically charged particles,” so theyfound a form for the action directly involving the motions of the chargesonly, which upon variation would give the Newtonian–like equations of motionof these charges. Here is the expression for this action in the flatspace–time, which is in the core of quantum electrodynamics:S[x; t i , t j ] = 1 ∫2 m i (ẋ i µ) 2 dt i + 1 ∫ ∫2 e ie j δ(Iij) 2 ẋ i µ(t i )ẋ j µ(t j ) dt i dt jwith (6.131)I 2 ij = [ x i µ(t i ) − x j µ(t j ) ] [ x i µ(t i ) − x j µ(t j ) ] ,where x i µ = x i µ(t i ) is the four–vector position of the ith particle as a functionof the proper time t i , while ẋ i µ(t i ) = dx i µ/dt i is the velocity four–vector.The first term in the action (6.131) is the ordinary mechanical action inEuclidean space, while the second term defines the electrical interaction of