Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 995Fig. 6.2 Analysis of all possible routes from the source A to the detector B is simplifiedto include only double straight lines (in a plane).∫where the sum–integral Σ is taken over all possible routes x = x(t) from theinitial point A = A(t ini ) to the final point B = B(t fin ), and S = S[x] is theclassical action for a particle to travel from A to B along a given extremalpath x. In this way, Feynman took seriously Dirac’s conjecture interpretingthe exponential of the classical action functional (De iS ), resembling acomplex number (re iθ ), as an elementary amplitude. By integrating thiselementary amplitude, De iS , over the infinitude of all possible histories, weget the total system’s transition amplitude. 77 For the quantum physics associated with a classical (Newtonian) particle the actionS is given by the integral along the given route from a to b of the difference T − V whereT is the classical kinetic energy and V is the classical potential energy of the particle.The beauty of Feynman’s approach to quantum physics is that it shows the relationshipbetween the classical and the quantum in a particularly transparent manner. Classicalmotion corresponds to those regions where all nearby routes contribute constructively tothe summation. This classical path occurs when the variation of the action is null. Toask for those paths where the variation of the action is zero is a problem in the calculusof variations, and it leads directly to Newton’s equations of motion (derived using theEuler–Lagrangian equations). Thus with the appropriate choice of action, classical andquantum points of view are unified.Also, a discretization of the Schrodinger equationi dψdt = − 2 d 2 ψ2m dx 2 + V ψ,leads to a sum–over–histories that has a discrete path integral as its solution. Therefore,the transition amplitude is equivalent to the wave ψ. The particle travelling on thex−axis is executing a one–step random walk, see Figure 6.3.