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

probability amplitudes for each of the two possible paths for each particle, while in the latter case it will be determined by adding the squares of the probability amplitudes for each path separately taken by each electron. Again, because the square of a sum of numbers is different from the sum of squares of these numbers, the former probability can differ dramatically from the latter. And as we have seen, if the particles are electrons, the results are indeed different if we don’t measure the particle between beginning and end points compared to what happens if we do.
    Quantum mechanics works, whether or not it makes sense.
    I T IS PRECISELY this seemingly nonsensical aspect of quantum mechanics that Richard Feynman focused on. As he later put it, the classical picture is wrong if the statement that the position of the particle midway on its trip from a to c actually takes some specific value, b , is wrong. Quantum mechanics instead allows for all possible paths , with all values of b to be chosen at the same time.
    The question that Feynman then asked is, Can quantum mechanics be framed in terms of the paths associated with probability amplitudes rather than the probability amplitudes themselves? It turned out that he was not the first one who had asked this question, though he was the first to derive the answer.

CHAPTER 5
    Endings and Beginnings
    Instead of putting the thing into the mind, or psychology, I put it into a number.
    —R ICHARD F EYNMAN
    A s he was struggling to come up with a way to formulate quantum mechanics to accommodate his strange theory with Wheeler, Richard Feynman attended what he later called a “beer party” at the Nassau tavern in Princeton. There he met the European physicist Herbert Jehle, who was visiting at the time and asked Richard what he was working on. Feynman said he was trying to come up with a way to develop quantum mechanics around an action principle. Jehle told him about a paper by one of the fathers of quantum mechanics, the remarkable physicist Paul Dirac, that might just hold the key. As he remembered it, Dirac had proposed how one might use the quantity from which the action is calculated (which, recall from chapter 1, is the Lagrangian—equal to the difference in kinetic and potential energies of a system of particles) in the context of quantum mechanics.
    The very next day they went to the Princeton library to go through Dirac’s 1933 paper, suitably titled “The Lagrangian in Quantum Mechanics.” In that paper Dirac brilliantly and presciently suggested that “there are reasons for believing that the Lagrangian [approach] is more fundamental” than other approaches because (a) it is related to the action principle, and (b) (of vital importance for Feynman’s later work, but a fact he wasn’t thinking about at the time) the Lagrangian can more easily incorporate the results of Einstein’s special relativity. But while Dirac certainly had the key ideas in his mind, in his paper he merely developed a formalism that demonstrated useful correspondences and suggested a vague analogy between the action principle in classical mechanics and the more standard formulations of the evolution in time of a quantum mechanical wave function of a particle.
    Feynman, being Feynman, decided right then and there to take some simple examples and see if the analogy could be made exact. At the time he was just doing what he thought a good physicist should do—namely, work out a detailed example to check what Dirac meant by what he said. But Jehle, who was watching this graduate student carry out his calculations in real time, faster than Jehle could follow, in that small room in the Princeton library, knew better. As he put it, “You Americans are always trying to find out how something can be used. That’s a good way to discover things.”
    He realized that Feynman had carried Dirac’s work one stage further, and in the process had indeed made an important discovery. He had