Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 999believe there is. What they would tell you to do, was find the momentumvariables and replace them by (/i)(∂/∂x), but I couldn’t find a momentumvariable, as there wasn’t any.”“The character of quantum mechanics of the day was to write things inthe famous Hamiltonian way (in the form of Schrödinger equation), whichdescribed how the wave function changes from instant to instant, and interms of the Hamiltonian operator H. If the classical physics could bereduced to a Hamiltonian form, everything was all right. Now, least actiondoes not imply a Hamiltonian form if the action is a function of anythingmore than positions and velocities at the same moment. If the action is ofthe form of the integral of the Lagrangian L = L(ẋ, x), a function of thevelocities and positions at the same time t,∫S[x] = L(ẋ, x) dt, (6.9)then you can start with the Lagrangian L and then create a HamiltonianH and work out the quantum mechanics, more or less uniquely. But theaction A[x; t i , t j ] (6.8) involves the key variables, positions (and velocities),at two different times t i and t j and therefore, it was not obvious what todo to make the quantum–mechanical analogue...”So, Feynman was looking for the action integral in quantum mechanics.He says: “...I simply turned to Professor Jehle and said, ‘Listen, do youknow any way of doing quantum mechanics, starting with action – wherethe action integral comes into the quantum mechanics?” ‘No”, he said, ‘butDirac has a paper in which the Lagrangian, at least, comes into quantummechanics.” What Dirac said was the following: There is in quantum mechanicsa very important quantity which carries the wave function fromone time to another, besides the differential equation but equivalent to it,a kind of a kernel, which we might call K(x ′ , x), which carries the wavefunction ψ(x) known at time t, to the wave function ψ(x ′ ) at time t + ε,∫ψ(x ′ , t + ε) = K(x ′ , x) ψ(x, t) dx.Dirac points out that this function K was analogous to the quantity inclassical mechanics that you would calculate if you took the exponential of[iε multiplied by the Lagrangian L(ẋ, x)], imagining that these two positionsx, x ′ corresponded to t and t + ε. In other words,K(x ′ , x) is analogous to e iεL( x′ −xε ,x)/ .