Free Quantum Man: Richard Feynman's Life in Science by Lawrence M. Krauss Page B

quantum mechanical effects tend to become insignificant, the formalism that Feynman developed would reduce to the classical principle of least action.
    How this happens is relatively straightforward. If we consider all possible paths between a and c , we can assign a probability amplitude “weight” to each path that is proportional to the total action for that path. In quantum mechanics many different paths—perhaps an infinite number, even crazy paths that start and stop and instantly change speeds and so on—can have nonzero probability amplitudes. Now the “weight factor” that is assigned to each path is expressed in terms of the total action associated with that particular path. The total action for any path in quantum mechanics must be some multiple of a very small unit of action called Planck’s constant, the fundamental “quantum” of action in the quantum theory, which we earlier saw also gives a lower bound on uncertainties in measuring positions and momenta.
    The quantum prescription of Feynman is then to add up all of the weights associated with the probability amplitudes for the separate paths, and the square of this quantity will determine the transition probability for moving from a to c after a time t .
    The fact that the weights can be positive or negative accounts not only for the weird quantum behavior, but also for the reason why classical systems behave differently than quantum systems. For if the system is large, so that its total action for each path is then huge compared to Planck’s constant, a small change in path can change the action, expressed in units of Planck’s constant, by a huge amount. As a result, for different nearby paths the weight function can then vary wildly from positive to negative. In general, when the effects of these different paths are added together, the many different positive contributions will tend to cancel the very many negative contributions.
    However, it turns out that the path of least action (the classically preferred path therefore) has the property that any small variation in the path produces almost no change in the action. Thus, paths near the path of least action will contribute the same weight to the sum, and will not cancel out. Hence, as the system becomes big, the contribution to the transition probability will be essentially completely dominated by paths very close to the classical trajectory, which will therefore effectively have a probability of order one, while all other paths will have a probability of order zero. The principle of least action will have been recovered.
    W HILE LYING IN bed a few days later, Feynman imagined how he could extend the analysis he made for paths over very short time intervals to ones that were arbitrarily large, again by extending Dirac’s thinking. As important as it was to be able to show that the classical limit was sensible, and that the mathematics could be reduced to the standard Schrödinger equation for simple quantum systems, what was most exciting for Feynman is that he now had a mechanism to explore the quantum mechanics of more complex physical systems, like the system of electrodynamics he devised with Wheeler, which they could not describe by traditional methods.
    While his motivation was to extend quantum mechanics to allow it to describe systems that couldn’t otherwise be described quantum mechanically, it is nevertheless true (as Feynman later emphasized) that for systems to which Dirac, Schrödinger, and Heisenberg’s more standard formulations could be applied, all the methods are completely equivalent. What is important, though, is that this new way of picturing physical processes gives a completely different “psychological” understanding of the quantum universe.
    The use of the word picturing here is significant, because Feynman’s method allows a beautiful pictorial way of thinking about quantum mechanics. Developing this new way took awhile, even for Feynman, who did not explicitly