1. A Question of Style
In 1949, Richard Feynman (1918–1988) published the essentials of his solution to the recalcitrant problems that plagued quantum theories of electrodynamics of his days (Feynman 1949a). The main problem was that the theory, that was considered to be correct and often led to correct observable consequences, also implied that some quantities should be infinite, while by common sense or empirical evidence they were finite. Feynman devised a method of solving the relevant theoretical equations in which particular combinations of elementary solutions yielded empirically adequate results even when applied to complex problems. The combination of the basic solutions was guided by relatively simple graphical considerations as to how the elements that represented elementary interactions of electrons and light quanta could be put together to form a complex diagram.
Much of Feynman’s method relied on a diagrammatic representation of the physical processes as well as, at the same time, of the mathematical expressions used to describe the processes quantitatively. Ever since their first appearance, these Feynman diagrams (Figure 1), as they came to be known, have been extensively used in theoretical particle physics to calculate reaction rates and other observable, or otherwise relevant, quantities. After making their initial “leap” out of Feynman’s head about the year 1948 they took the world in a rapid “dispersion” and were put to a wide variety of uses (Kaiser 2005).
It has been widely acknowledged that Feynman’s approach to the difficulties in finding a quantum theory of electrodynamics was idiosyncratic, although it may be hard to say what the characteristics of his style exactly were (see, e.g., Schweber 1986a). A conspicuous feature of his approach certainly was his prominent use of diagrams in his publications on the topic and the way he used them in a peculiar mix of diagrammatic, physical, and mathematical reasoning.
One goal of the present article is to identify characteristic aspects of the way in which Feynman was developing his version of quantum electrodynamics (QED) at different stages of his early career (i.e., up to 1949). But I will also discuss the question of which circumstances or events might have been responsible for Feynman’s exhibiting different characteristic methods at different times. Finally, I will contrast my findings with similar studies, especially with Peter Galison’s, in which Feynman’s war-related work takes center stage (Galison 1998). To a considerable extent, I will build on my own reconstruction of the genesis of Feynman diagrams, which in turn is based on my selection of documents from the Feynman Papers hosted at the California Institute of Technology (Wüthrich 2010).
2. The Main Problem and Feynman’s First Attempt to Solve It
In the early 1940s, when Richard P. Feynman was a graduate student, one of the most pressing problems facing theoretical physicists was the fact that infinite and, therefore, uninterpretable quantities arose from some of the principles of electrodynamics—in both classical electrodynamics as well as in the early attempts to establish a quantum version of it.1 In classical electrodynamics, the difficulties of divergences had been known for some time, and it had been hoped that quantizing the theory would eliminate them. An alternative strategy was to first remove the infinite quantities in the classical theory before attempting to quantize it. It is in this theoretical context that Feynman wrote his Ph.D. thesis, with the removal of the divergences in electrodynamics being his superordinate objective (Feynman 2005, p. 2). In his thesis, he adopted the second strategy of first trying to establish a divergence-free classical theory and then proceeded to quantize it. Indeed, together with his supervisor John Archibald Wheeler, Feynman had already developed an alternative theory of classical electrodynamics with the desired feature that just awaited quantization.
The standard procedure for quantizing a classical theory was to interpret the classical Hamiltonian function H as an operator in a Hilbert space of state vectors. This operator would then determine the time evolution of the quantized system. The problem with quantizing the Wheeler-Feynman theory of electrodynamics was that it could not be formulated by specifying a Hamiltonian function. Feynman’s first step towards a solution to this problem was to represent the time evolution of the quantum wave function of a system using the classical Lagrangian function L in a way which he borrowed from Paul Dirac (Dirac 1933a). However, since the Hamiltonian function of a system can be constructed from the Lagrangian by Legendre transformations, each system that can be described by a Lagrangian can also be described by a Hamiltonian. By logical contraposition, this means that when a system is not describable by a Hamiltonian, like in the Wheeler-Feynman theory, it is not describable by a Lagrangian either. Dirac’s method did, therefore, not provide a solution to Feynman’s problem.
Here, the q’s are position variables. The t’s are time variables. The g’s are for the volume element in q-space. The A’s are normalization constants.
Moreover, the sum involving the Lagrangian was almost exactly the usual expression for the classical action:
In the limit as we take finer and finer subdivisions of the interval t0 to T and thus make an ever increasing number of successive integrations, the expression on the right side of (47) becomes equal to ψ (Q, T). The sum in the exponential resembles [i.e. the classical action] with the integral written as a Riemann sum. (Feynman 2005, p. 31)
What we have been doing so far is no more than to reexpress ordinary quantum mechanics in a somewhat different language. In the next few pages we shall require this altered language in order to describe the generalization we are to make to systems without a simple Lagrangian function of coordinates and velocities. (Feynman 2005, p. 39)
Even though the method worked very well for simple systems such as the non-relativistic interaction of two particles mediated by an oscillator, Feynman was unable to perform the quantization of the Wheeler-Feynman theory in his thesis. Feynman does not mention the reasons for this—he only tells us that he could not do it. The fact that, at the very end of his thesis where he talks of the limitations of his methods, he does not even mention the Wheeler-Feynman theory any longer, but rather the Dirac equation, suggests that he lost interest in the Wheeler-Feynman theory already at this stage. He did lose it later as testified by a letter to Wheeler dated May 4, 1951 (Brown 2005, p. xvi, fn. 10; Sauer 2008). Only passages from a later publication (Feynman 1950) allow us to identify the major reason for Feynman’s failure to quantize the Wheeler-Feynman theory (see Blum 2017, p. 72). The most natural boundary conditions for the elimination of the field oscillators lead to a quantum electrodynamics with only retarded interactions unlike in the Wheeler-Feynman theory where there also is an advanced part to the interactions between particles.
Thus, the goal which motivated Feynman’s thesis most was not achieved. However, despite its failure, Feynman’s plan was a reasonable one, and some characteristics of his way of proceeding stand out (see also Wüthrich 2013):
By directly using the representation of the quantum dynamics of a system by the action function of the corresponding classical system, Feynman was able to generalize the quantization procedure to systems with neither a Lagrangian nor a Hamiltonian.
More generally speaking, the alternative formulation in terms of the action function allowed him to transgress the domain of application of the standard quantization procedures.
Finding alternative formulations was a central element of Feynman’s heuristics.
Moreover, the action emerged as the central quantity. In Feynman’s formulation, the classical action encoded the time evolution of the corresponding quantum system.
3. War Related Work
One can imagine that such a style suited more the ivory-tower of the Princeton graduate school than the Manhattan Project in Los Alamos. There, in Los Alamos, the theorists needed to produce answers to the questions that the military posed to them, and approximate answers arrived at by rough and ready methods, valued more than an elaboration of the reasons of why the problem at hand could not be exactly solved.
Feynman mainly worked on diffusion problems and, towards the end, became head of the group T4 of the Manhattan Project’s theory division that was dedicated to these kinds of problems.3 There he perfected himself at solving equations with just the right amount of approximation, and in a way that was adaptable to other similar kinds of problems. Galison (1998) speaks of the modular culture at Los Alamos, which apparently suited Feynman just as well as the focus on the foundations of a theory that he had in Princeton. But, according to Galison, the modular culture left a lasting impression on Feynman, and in that culture “Feynman’s style of reasoning altered in powerful ways” (Galison 1998, p. 429).
Feynman’s method of choice in that war-related work was to find elementary solutions, or kernels, of differential equations from which increasingly complex solutions could be built-up, which allowed the application to the real-world problems of war related research (Galison 1998, pp. 416, 421). Those problems included the construction of a hydride bomb, which, for Galison, was the “most original contribution” of Feynman’s group. Galison adds:
[The hydride bomb project’s] importance lay not so much in the contribution it made to the immediate war effort, but in its consolidation of the modular-effective conception of theory that marked Feynman’s postwar approach to quantum electrodynamics. (Galison 1998, p. 414)
Like Galison, Schweber (Schweber 1986b, p. 96) had also arrived at the conclusion that Feynman’s style changed during his work at Los Alamos, in particular due to Hans Bethe’s influence resulting in a focus on “getting the numbers out.” I share, to a large extent, Galison’s and Schweber’s assessments. Also, Galison’s reconstruction of Feynman’s work during the war from formerly classified archival material is essential for a comprehensive reconstruction of Feynman’s scientific biography. However, there is archival evidence that suggests that the characteristic modular approach as we find it in Feynman’s published version of QED after the war was not only the continuation of a conversion of Feynman to such an approach during the war. Rather, the modular approach, in which the kernels are the basic unit of analysis, was also forced upon him by a theoretical difficulty which he encountered after the war and was largely unrelated to his war-time work (cf. Blum 2017, p. 76).
4. After the War: Dissatisfaction with Formal Solutions and Struggling for Understanding
Feynman’s thesis was published only in 2005, edited by Laurie Brown (Feynman 2005). In a publication of 1948, however, Feynman had summarized and elaborated the results of his thesis under the title “Space-time approach to non-relativistic quantum mechanics” (Feynman 1948, RMP48, for short).
As Schweber noticed as regards RMP48, “[i]mportant conceptual advances had also been made since 1942” (Schweber 1994, p. 409). In RMP48 Feynman calculated the velocities of the trajectories of particles subject to forces deriving from a space-dependent potential and realized that the trajectories resembled the ones observed in Brownian motion (Feynman 1948, p. 376). In an interview with Schweber, Feynman even claimed to have “seen” the paths (Schweber 1994, p. 409).
Although Galison (1998) does not discuss this particular postwar article of Feynman’s the mentioned aspects of it fit quite well with Galison’s picture of a Feynman who came out of the war with a somewhat changed attitude towards doing physics.4 But it is not yet the particular modular approach that Galison identifies and emphasizes, and which we find in Feynman’s published version of QED in 1949.
Rather, the attitude which Feynman displays in RMP48 is, first of all, one that is not satisfied by formal solutions but rather was after something like a model or a mechanism of the processes that the formal apparatus was supposed to describe (see also Wüthrich 2012). Present-day philosophy of science provides much literature on these terms and I am afraid I will gloss over many subtleties regarding the issues discussed there. Roughly, I am following Giere (1996), Machamer, Darden, and Craver (2000), Suárez (2004), and Craver (2007) when I use the terms.5
A mechanism is a system consisting of several parts, each of which performs some activity. The parts act in such a way that their activities bring about a certain behavior of the whole system. A mechanistic explanation takes the concerted activities of the parts of the system to be the reason or the cause for the behavior of the system. There can be different levels of mechanism (see Figure 2): An entity’s activity can be part of a mechanism and, at the same time, can be decomposed into other entities whose concerted activities bring about the first entity’s behavior. A mechanistic explanation comes to an end when it refers to entities and activities which a scientific field, at a certain time, regards as basic or fundamental. These are often called the “bottoming out” activities and entities (Machamer, Darden, and Craver 2000, pp. 13–4).
A model is an idealized representation of a real-world system. The representation articulates only the most relevant aspects of the system. The model is similar to the real-world system in the sense that inferences made from some features of the model to other features of the model correspond to the fact that if a real-world system displays the former features, it will also display the latter. My main aim, at any rate, is to use the concepts of mechanism and model in a way that helps us better understand the historical development of Feynman’s version of QED. In the course of trying to apply these notions, which in the case of mechanisms have mostly been applied to cases from the life sciences, we may be able to bring to the fore new aspects of Feynman’s style.
Feynman’s postwar aversion to formal results is corroborated by quite explicit passages in RMP48. Some of the new results that the article contained were presented only tentatively. Feynman claimed to have found action functions which could be used to describe quantum systems of particles with spin and high velocities. This would include important features of a description of the system in accordance with the special theory of relativity. The results of his thesis, by contrast, were all non-relativistic. Yet the new results did not please Feynman. At the end of the article he dismissed them as “purely formal” and hinted at “other ways of obtaining the Dirac equation which offer some promise of giving a clearer physical interpretation to that important and beautiful equation” (Feynman 1948, p. 387).
As Blum (2017, p.74) points out, Feynman was, in fact, unable to get the action function which could be used, in his formulation, to obtain the results that are usually derived from the Dirac equation. He only obtained the action resulting in an equation which contained the square of the characteristic operator of the Dirac equation. But it seems to me that, by the time of the writing of RMP48, Feynman was still confident that, sooner or later, his approach would also yield the results of Dirac’s theory. And the above quote, mentioning “a clearer physical interpretation” indicates that he also wanted to justify the action function by other means than just pointing out that it would yield the Dirac equation.
From some of Feynman’s papers hosted at the Caltech Archives we can get an idea of what Feynman was after. He was looking for an action function that would yield the Dirac equation and for which he could come up with some sort of physical interpretation. He wanted to derive the desired action function from a physical model of the electron.6
Such a model had been proposed by Dirac, Erwin Schrödinger, and Gregory Breit almost twenty years before (Breit 1928; Schrödinger 1930; Dirac 1933b, pp. 322–323, 1935, p. 260). It is possible that the model was brought to Feynman’s attention by Breit in the early phase of the Manhattan project. However, Breit was mainly active in the predecessor organizations and in Chicago whereas Feynman worked most of the time in Los Alamos.7 In this model, at any rate, the electron is zig-zagging, or quivering, back and forth at the speed of light and only its average trajectory, with its lower speed, is observable.
The document displayed in Figure 3 shows Feynman dealing with a one dimensional version of the situation. In this case, Dirac’s equation is, in fact, a system of two equations for the two spin components of the wave-function of the system. Among other things, Feynman here derived the result that the instantaneous velocity of a particle obeying the Dirac equation is always the speed of light, which in the units chosen by Feynman has the value 1. The possible values of the velocity of the particle are given by the eigenvalues of the matrix . Feynman showed that this matrix is equal to the unit matrix α (see the lowest of the highlighted areas in Figure 3). I’ve reconstructed the details of Feynman’s “struggle” with the Dirac equation in Chapter 4 of the Genesis of Feynman Diagrams (Wüthrich 2010). A discussion of some of the documents on which my reconstruction is based has also been discussed in the seminal work on the history of QED by Silvan S. Schweber (Schweber 1994, especially p. 406–07, 1986a). Here, I hope, a summary of what I take to be the most important of Feynman’s steps will suffice to support my claims.
5. Giving up the Model, and Why
5.1. The Stink of a Theoretical Problem
Other unpublished manuscripts testify to how strong a grip the desire to find a mechanism for the time-evolution of a system of one or more electrons had on Feynman after the War. As mentioned, he first tried to incorporate positrons into his model, and then also interactions (see Figures 5 and 6). While he was successful with positrons, dealing with interactions posed insurmountable difficulties. In order to take interactions into account, he even reconsidered the standard quantization method of promoting a Hamiltonian function to an operator (see Wüthrich 2010, pp. 104–08). For this, he first had to find such a function in the cases in which, in the usual sense of the word, there is none. These were the cases for which he had devised his alternative method of quantization by using the action function.
If the archival material on which I base myself here is sufficiently complete and representative, his attempts at incorporating interactions into the model of the zig-zagging electron were very serious but eventually unsuccessful. At the bottom of the page displayed in Figure 6 (lowest highlighted area) Feynman laconically remarked “N.G.,” which I interpret as meaning “no good.” On another page of his manuscripts (Figure 7) we can better see the reasons that made him give up his attempts:
It is a bit hard to see how to define Φ for path pair AB and CD, since there are some terms from interaction at x from y which is unspecified. However if the interaction is zero beyond P we are OK. Hence, at present, I can only specify Φ for paths which are long enough that they go beyond the time of interaction (this stinks).
5.2. Feynman’s Publications on Quantum Electrodynamics 1949
Feynman’s use, in his notes, of the term “Green’s function” (see, e.g., Figure 4) may be evidence of his exposure to the specific types of problems during World War II such as the quantitative description of the diffusion of neutrons as it was relevant for the construction of nuclear weapons (Galison 1998). Also, as far as mathematical details are concerned, Green’s functions, kernels, and Feynman’s peculiar action integral may need to be carefully distinguished. However, I do not see much of a conceptual difference in the transition from the action integral to the use of Green’s functions. As mentioned, the latter is basically the peculiar integral, known from Feynman’s thesis, of the former (see Section 4). Accordingly, Feynman could use either of them to determine the time evolution of a quantum system.
A more interesting difference is that in the subsequent publications, “The Theory of Positrons” (Feynman 1949b) and “Space-Time Approach to Quantum Electrodynamics” (Feynman 1949a) Feynman did not attempt any more to justify the Green’s function, or the action function behind it, by a mechanistic model. Remember that this had bothered him a great deal in his manuscript calculations after the War, and also in his publication RMP48, in which the insight that the electron’s trajectory was of a similar type to the one observed in Brownian motion had been of considerable importance.
In the two publications from 1949, Feynman analyzed the processes allowed by the Dirac and Schrödinger equation only at the level of propagation of electrons and positrons from one space-time point to another. How this propagation came about was left out of the description. This shift can be interpreted as modelling the same phenomena (e.g., the interaction of two electrons) at a different level of mechanism albeit in a slightly non-standard fashion.
For instance, at the top level (see Figure 8), we have the quantum system, consisting of two electrons (represented each by an incoming and an outgoing arrow), evolving in time (flowing from bottom to top in the Figure). At an intermediate level, we have the electrons and a photon (represented by the wiggly line) propagating from one space-time point to another. This is the level at which Feynman worked when he used the diagram shown in Figure 1.
At the bottom level, we zoom into the propagation of the electron, which is there described as a zig-zagging, or even curly, motion (cf. Figure 5). This last level does not introduce new entities and is therefore not a different level of mechanism in the strict sense (cf. Craver 2007, pp. 188–95). Rather, what changes between the intermediate and the bottom level is what is accepted as the bottoming out activities of the relevant entities. In his notes (see Figure 4, for instance) Feynman worked at the bottom level and tried to reduce the time evolution of the quantum system to the quivering motions of the electrons. In the publications from 1949, he was ready to accept the propagation from space-time point to space-time point as the bottoming-out activity.
For Galison, Feynman’s approach was “rule-governed,” “modular,” and “approximate” (Galison 1998, p. 394). A plausible explanation for Feynman’s attitude, and of others at the time, is readily available: Like many theoretical physicists, especially the leading and most promising ones, he was engaged in the Manhattan Project during World War II and worked on the design of an atomic bomb and on safe storage of the required raw material such as Uranium-235. This context required of him to put forward applicable solutions to practical problems in the shortest amount of time. There was neither time nor resources for pondering on fundamental difficulties of the current theories as he would have had during his Ph.D. thesis with Wheeler at Princeton a couple of years before. Galison’s analysis of previously classified material confirms this picture to a large extent.
Taking into account, however, some lesser known and admittedly sparse archival material suggests that at least one significant shift in Feynman’s style was brought about by a specific theoretical difficulty which was largely unrelated to the war time research. It was the shift from attempts to provide a microstructural explanation of the time-evolution of quantum systems to an explanation in terms of Green’s functions, which left open many details about the electron’s behavior. The shift can, at least to some extent, be interpreted as a transition from a lower to an intermediate level of mechanism. Such an interpretation may help spell out as yet underappreciated aspects of Feynman’s “modular” style.
Some passages in the present text are reprinted from (Wüthrich 2010).
http://history.aip.org/history/acap/institutions/manhattan.jsp#metlab (accessed 2017-01-03).
By the way the article by Machamer, Darden and Craver (2000) is among the most cited papers in philosophy journals in recent decades, see, e.g., http://tar.weatherson.org/2013/06/26/most-cited-articles-from-philosophy-journals/ (accessed 2017-03-01).
Note, however, Blum’s caveats concerning the relation between action and Green’s functions (Blum 2017, p. 75n74).
I thank Richard Staley for pointing out the possible connnection. An overview on the Manhattan project and the people involved in it can be gained from http://history.aip.org/history/acap/institutions/manhattan.jsp (accessed 2017-01-10).
The focus on this particular aspect of the development of Feynman’s version of quantum electrodynamics has been prompted by an invitation by Michael Stöltzner and Craig Fraser to participate in a symposium at the 24th International Congress of History of Science, Technology and Medicine in Manchester 2013. I also profited much from comments by Raphael Scholl and by audiences at the 18th Seven Pines Symposium (2014, Stillwater MN), at the History of Science Colloquium (2012, LMU Munich) and at the 2017 meeting of the History Section of the German Physical Society. The text was written during two stays at the Max-Planck-Institute for the History of Science (2015–2017, Berlin), where I received many substantial comments from Alexander Blum, in particular. The stay was funded by the Swiss National Science Foundation for a related project (grant no. 145409). The DFG-funded Research Unit “The Epistemology of the LHC” (FOR 2063) provided the opportunity for helpful discussions.