Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1001each point and add them together. We are to take the limit as ε → 0.Therefore, the connection between the wave function of one instant andthe wave function of another instant a finite time later could be get by aninfinite number of integrals (because ε goes to zero), of exponential whereS is the action expression (6.9). At last, I had succeeded in representingquantum mechanics directly in terms of the action S[x].”Fully satisfied, Feynman comments: “This led later on to the idea of thetransition amplitude for a path: that for each possible way that the particlecan go from one point to another in space–time, there’s an amplitude. Thatamplitude is e to the power of [i/ times the action S[x] for the path], i.e.,e iS[x]/ . Amplitudes from various paths superpose by addition. This thenis another, a third way, of describing quantum mechanics, which looks quitedifferent from that of Schrödinger or Heisenberg, but which is equivalent tothem.”“...Now immediately after making a few checks on this thing, what wewanted to do, was to substitute the action A[x; t i , t j ] (6.8) for the otherS[x] (6.9). The first trouble was that I could not get the thing to workwith the relativistic case of spin one–half. However, although I could dealwith the matter only non–relativistically, I could deal with the light orthe photon interactions perfectly well by just putting the interaction termsof (6.8) into any action, replacing the mass terms by the non–relativisticLdt = 1 2 Mẋ2 dt,A[x; t i , t j ] = 1 ∑∫m i (ẋ i2µ) 2 dt i + 1 2i∑i,j(i≠j)e i e j∫ ∫δ(I 2 ij) ẋ i µ(t i )ẋ j µ(t j ) dt i dt j .When the action has a delay, as it now had, and involved more than onetime, I had to lose the idea of a wave function. That is, I could no longerdescribe the program as: given the amplitude for all positions at a certaintime to calculate the amplitude at another time. However, that didn’tcause very much trouble. It just meant developing a new idea. Insteadof wave functions we could talk about this: that if a source of a certainkind emits a particle, and a detector is there to receive it, we can give theamplitude that the source will emit and the detector receive, e iA[x;ti,tj]/ .We do this without specifying the exact instant that the source emits orthe exact instant that any detector receives, without trying to specify thestate of anything at any particular time in between, but by just finding theamplitude for the complete experiment. And, then we could discuss howthat amplitude would change if you had a scattering sample in between, as