## ‘Space-Time Approach to Non-Relativistic Quantum Mechanics,’ pp. 367-387 in: Reviews of Modern Physics, Vol. 20, no. 2, April, 1948.

Lancaster, PA & New York, NY: American Physical Society, 1948.

First edition, journal issue in original printed wrappers, and a virtually mint copy, of Feynman’s path-integral formulation of quantum mechanics, a “new formalism for quantum theory (presented in Feynman’s thesis) which … turned out to be the most powerful and elegant version of quantum mechanics; indeed, Feynman’s formalism includes the classical mechanics of Newton within the same mathematical framework. According to John Wheeler, Feynman’s PhD supervisor, Feynman’s presentation of these ideas in his thesis in 1942 marked the moment ‘when quantum theory became simpler than classical theory’ (but the ideas were not formally published in a journal until 1948 because of the delay caused by Feynman’s involvement with the Manhattan project). The essence of the path-integral approach is that, in order to calculate the probability that a particle will go from A to B, you have to take account of every possible path from A to B, not just … the trajectory described by Newtonian mechanics. Each path (each ‘world line’) has a certain probability [‘amplitude’], which is related to the ‘action’ associated with that path … The overall probability [amplitude] is calculated by adding up (integrating) the contributions from all the paths” (Gribbin, p. 272). “The Nobel Prize in Physics for 1965, shared [by Feynman] with Julian Schwinger and Sin-itiro Tomonaga, rewarded their independent path-breaking work on the renormalization theory of quantum electrodynamics (QED). Feynman based his own formulation of a consistent QED, free of meaningless infinities, upon the work in his doctoral thesis of 1942 at Princeton University … His new approach to quantum theory made use of the Principle of Least Action and led to methods for the very accurate calculation of quantum electromagnetic processes, as amply confirmed by experiment. These methods rely on the famous ‘Feynman diagrams,’ derived originally from the path integrals” (Brown, p. vii). “The essential parts of his thesis were published in 1948 in his paper ‘Space-time approach to non-relativistic quantum mechanics’. The Feynman path-integral approach to quantum mechanics was on a par with the formulations of M. Born, W. Heisenberg and P. Jordan, P. A. M. Dirac, and E. Schrödinger of 1925–26” (*Biographical Memoirs of Fellows of the Royal Society of London* 48 (2002), p. 107). “Mark Kac, the eminent Polish-American mathematician, wrote: ‘In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the ‘ordinary’ and the ‘magicians’. An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with magicians … the working of their minds is for all intents and purposes incomprehensible … Richard Feynman [was] a magician of the highest caliber’” (*ibid*., p. 99).

“While still an undergraduate at MIT, Feynman became aware of the so-called divergence problems of QED. That is, physical quantities which should have been calculable by the theory, such as the self-mass of the electron (the effect of the action of the electron’s own electromagnetic field on its mass), were predicted to have an absurd result: infinity. Feynman knew that a similar result was predicted classically; namely, the energy contained in the Coulomb field of a point charge is theoretically infinite. As he said in his Nobel Lecture, his ‘general plan was to first solve the classical problem, to get rid of the infinite self-energies and to hope that when I made a quantum theory of it, everything would just be fine.’ The idea which he embraced (‘fell deeply in love with’) was to replace the field itself by ‘delayed action-at-a-distance.’ In this view the electron would act only on *other charges, *not on itself, and the field would be only a useful invention for representing that delayed interaction” (*Selected Papers*, p. 33).

“From MIT, Feynman went to Princeton University, in New Jersey, to pursue his PhD, starting in 1939 … Feynman began his graduate studies and immediately impressed his young mentor, John Archibald Wheeler, who had arrived at Princeton himself, as an assistant professor, only the previous year … Bearing in mind that in the action-at-a-distance theories ‘we are faced with the circumstance that the equations of motion of the particles are expressed classically as a consequence of a principle of least action and cannot, it appears, be expressed in Hamiltonian form,’ Feynman concluded that we need ‘a formulation of quantum mechanics … which does not require the idea of a Hamiltonian or momentum operator for its expression. It has, as a central mathematical idea, the analog of the action integral of classical mechanics” (DSB).

For a classical mechanical system, the ‘Lagrangian’ is the difference between an object’s energy of motion (kinetic energy) and the energy it has by virtue of its location (potential energy). The ‘action’ associated with the object’s path is calculated by adding (integrating) the values of the Lagrangian from the beginning of the path to its end. The principle of least action in classical physics states that the path taken by any object between two points in any specified time interval is the one corresponding to the smallest value of the action.

“In the spring of 1941, Feynman was already actively looking for a new approach to quantum mechanics directly from the classical Lagrangian. One day, when he was struggling with this problem, he went to a beer party in the Nassau tavern in Princeton. There he met Professor Herbert Jehle, who had recently arrived from Europe. He sat down near Feynman and together they began to talk about various scientific problems. Feynman asked him if he knew any way of doing quantum mechanics starting with the action, or where the action integral came into quantum mechanics. Jehle said that he did not, but told him of a paper by [Paul] Dirac in which the Lagrangian, at least, came into quantum mechanics [‘The Lagrangian in quantum mechanics,’ *Physikalische Zeitschrift der Sowjetunion*, Vol. 3 (1933), pp. 64-72].

“Next day Jehle and Feynman went to the Princeton Library, and Jehle showed him Dirac’s paper, and they studied it together. The paper began with the words: ‘Quantum mechanics was built upon a foundation of analogy with the Hamiltonian theory of classical mechanics. This is because the classical notion of canonical coordinates and momenta was found to be one with a very simple quantum analogue … Now there is an alternative formulation for classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinates and velocities instead of coordinates and momenta. The two formulations are, of course, closely related, but there are reasons for believing that the Lagrangian one is more fundamental. In the first place the Lagrangian method allows one to collect together all the equations of motion and express them as the stationary property of a certain action function … Secondly, the Lagrangian method can easily be expressed relativistically, on account of the action function being a relativistic invariant; while the Hamiltonian method is essentially nonrelativistic in form, since it marks out a particular time variable … For these reasons it would seem desirable to take up the question of what corresponds in the quantum theory to the Lagrangian method in the classical theory.’ This was exactly what Feynman was looking for” (Mehra, pp. 134-5).

Feynman found in Dirac’s paper an infinitesimal time development operator involving the classical Lagrangian. Applications of this operator to the initial wave function generated the wave function at an infinitesimal later time, and the wave function was equivalent to finding the solution of the Schrödinger equation.

“Thus Feynman found the relation between the Lagrangian and quantum mechanics, which was an important result of his dissertation, but still for infinitesimal times. Several days later, when he was lying in bed, he worked out the next fundamental step. Feynman described it as follows: ‘… I’m picturing this thing and I’m putting more and more lengthy times, I have to do this again and again, and so I’ve got this exponential … times again, times again, integrate it, integrate it. But the product of all the exponentials is the exponential of the sum … which is the action. So I go, AAAAAHHHHH, and I jumped, ‘That’s the action!’ That was a moment of discovery!’” (*ibid*., p. 137). What Feynman had discovered was that, to obtain the wave function after a *finite *time has elapsed, one had to integrate over all possible paths containing two arbitrary space-time points. This, in fact, was the path-integral.

“In his PhD thesis, ‘The Principle of Least Action in Quantum Mechanics,’ Feynman did not develop further the physical interpretation of his new method; rather he greatly developed its mathematical formalism.

“First of all, he derived a new method of calculation of quantum averages. From this, Feynman derived the generalization of Ehrenfest’s theorem, which says that in quantum mechanics the classical equations of motion are fulfilled by the average values of the quantum entities … But the most unexpected result was the fact that Feynman’s relation for quantum averages is fundamental, in that ‘when compared to corresponding expressions in the usual form of quantum mechanics, it contains … in one equation, both the equations of motion and the commutation rules for [momentum] and [position].’ Thus Feynman actually invented a completely new formulation of quantum mechanics in terms of classical notions …As Feynman found later, ‘it turned out that there still remained difficulties, and the thing was not satisfactory, but I didn’t realize it at the time in my excitement though, of course, I had solved the difficulties I had previously seen. But it was only a temporary error, for I thought that everything was all right and I wrote up the thesis. The parts that are right, of course, are still a representation of quantum mechanics without delay and so on. But the generalizations that are contained in the thesis are probably erroneous as written.’

“This was the situation in the spring of 1942 when Feynman wrote up his dissertation at the strong urging of John Wheeler. At that time Wheeler was working on the first atomic pile with Fermi at Chicago, and he wrote a letter to Feynman in which he told him that he (Feynman) ‘had done enough for a thesis,’ and Wheeler urged him ‘very strongly to write up what you have in the remaining weeks before you get into the situation in which I now find myself.’

“Feynman had worked on the OSRD [Office of Scientific Research and Development] atomic bomb project at Princeton from December 1941 to March 1942. Then, in April 1942 the Manhattan Project was established, and Feynman continued working for it, and upon graduation after completing his thesis he went back to doing war-related research” (*ibid*., pp. 138-141).

“As the war was ending and Allied victory looked more and more secure, physics departments across the country began jockeying to hire Feynman. In the end, he turned down several attractive offers and followed his wartime boss, [Hans] Bethe, back to Bethe’s home department at Cornell University in upstate New York. At Cornell, Feynman perfected his approach to quantum theory, melding several of his pre-war insights with the more pragmatic, numbers-driven approach he had honed during the war. One of his first tasks was to publish a long article, based on his dissertation, that presented a brand-new approach to quantum mechanics” (DSB).

“Feynman began his 1948 paper in the *Reviews of Modern Physics*, entitled ‘Space-time approach to non-relativistic quantum mechanics,’ by stating: ‘It is a curious historical fact that modern quantum mechanics began with two quite different mathematical formulations: the differential equation of Schrödinger, and the matrix algebra of Heisenberg. The two, apparently dissimilar approaches, were proved to be mathematically equivalent. These two points of view were destined to complement one another and to be ultimately synthesized in Dirac’s transformation theory.

‘This paper will describe what is essentially a third formulation of non-relativistic quantum theory. This formulation was suggested by some of Dirac’s remarks concerning the relation of classical action to quantum mechanics. A probability amplitude is associated with an entire motion of a particle as a function of time, rather than simply with a position of the particle at a particular time.’

“With these words Richard Feynman introduced one of his now most well known papers. In the spring of 1947 he decided to publish the most important parts of his PhD thesis. Feynman had thought about publishing this work in a regular journal earlier, but World War II intervened …

“In the 1948 paper, Feynman first described that part of his PhD thesis that did not produce any difficulties. ‘All the ideas that appear in the *RMP* (1948) article were written in such a form that if any generalization is possible, they can be translated … The reason I did not publish everything in the thesis is this. I met with a difficulty. An arbitrary action functional *S* produces results which do not conserve probability; for example, the energy values come out complex. I do not know what this means, nor was I able to find that class of action functionals which would be guaranteed to give real [values] for the energy’ …

“As a straightforward consequence of Feynman’s extremely important and completely new viewpoint, … one can answer a three-century-old question about the meaning of the principle of least action. ‘Here is how it works: Suppose that for all paths, *S* is very large compared to [Planck’s constant]. One path contributes a certain amplitude. For a nearby path the phase is quite different, because with an enormous *S* even nearby paths will normally cancel their different phases – because [Planck’s constant] is so tiny. So, nearby paths will normally cancel their effects out in taking the sum – except for one region, and that is when a path and a nearby path all give the same phase in the first approximation … Only those paths will be the important ones. So in the limiting case in which Planck’s constant goes to zero, the correct quantum mechanical laws can be summarized by simply saying: ‘Forget about all these probability amplitudes. The particles does go on a special path, namely, that one for which *S* does not vary in the first approximation.’ That’s the relation between the principle of least action and quantum mechanics.’

“Thus Feynman’s postulate leads to the principle of least action and gives us the right explanation as to where this principle is coming from. As far as the principle of least action, *the* most fundamental principle of physics, is concerned, one which leads to the classical dynamical equations in all the fundamental classical theories, one can truly say that it was Feynman who discovered the deepest import of this principle” (*ibid*., pp. 182-7).

“The Feynman path integral approach to quantum mechanics stands presently on a par, both esthetically and practically, with the original formulations of 1925-26. The path integral formulation of the quantum gauge theories which lie at the heart of the Standard Model of elementary particle interactions turned out to be critical in the VeItman-’t Hooft proof that these theories are renormalizable. However, it has an even wider range of applications than to quantum field theories. Feynman’s book of 1965 with Albert R. Hibbs [*Quantum Mechanics and Path Integrals*] uses path integrals to treat problems other than quantum mechanics and quantum electrodynamics, including statistical mechanics, the variational principle, the polaron problem, Brownian motion, and noise. Other applications have been made to quantum liquids and solids, to macromolecules and polymers, and to problems of propagation in dissipative media. The approach is important to various forms of semiclassical approximations in chemical, atomic, and nuclear problems and basic to the instanton problem (barrier penetration between different vacuum ground states). It can be extended to optics and even to the motion of particles in the strong gravitational fields near a black hole” (*Selected Papers*, p. 173).

Brown, *Feynman’s thesis. A new approach to quantum theory*, 2005. Gribbin, *Q is for Quantum*, 1998. Mehra, *The Beat of a Different Drum*, Clarendon Press, 1994. *Selected Papers of Richard Feynman*:* with commentary*, World Scientific, 2000.

4to (267 x 201 mm), pp. [367], 368-455, [1, blank]. Original printed wrappers.

Item #5061

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Price:
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