The Development of Feynman's Path Integral Thought (I)

The Classical Self-Energy Problem

In the 1930s, the entire physics community was filled with dissatisfaction regarding the state of quantum electrodynamics. Some famous physicists, such as Dirac and Oppenheimer, believed that some brand-new perspective must be introduced. Feynman was then an undergraduate at the Massachusetts Institute of Technology. When he learned of this situation from standard works like Dirac’s The Principles of Quantum Mechanics, he “took it as a challenge.” [5] At that time, Feynman did not have a clear understanding of the problem and attributed the infinities to two points: one was the infinite self-energy of an electron acting on itself, and the other was the infinity associated with the infinite degrees of freedom of the field. For the former, he still had a classical understanding; for the latter, he only saw the zero-point energy problem. However, with youthful passion and innate talent, Feynman designed a grand goal for himself: first solve the divergence difficulties of classical electrodynamics, and then quantize it to obtain a satisfactory theory of quantum electrodynamics. “Since they had not given a satisfactory answer to the problem I wanted to solve, I didn’t have to pay attention to their work.” ([5]) Feynman thought at the time that the idea of an electron producing a field which then acts back on the electron that produced it was too absurd. He believed that an electron could only act on other electrons, which would eliminate both the electron’s self-energy and the infinite degrees of freedom of the field. Carrying this idea, Feynman entered the Princeton Graduate School in 1939 to study under the guidance of the physicist Wheeler, who was seven years his senior.

Shortly after arriving at Princeton, Feynman realized that if an electron does not act on itself, radiation damping could not be explained. According to classical electrodynamics, an accelerating electron must radiate energy, so there must be an additional force to compensate for this energy loss. Where does this additional force come from? At the beginning of this century, Lorentz believed it came from the interaction between different parts of the electron through retarded potentials, and derived its explicit representation for the one-dimensional case:

F=\alpha \frac{e^2}{ac^2}\ddot{X}-\frac{2}{3}\frac{e^2}{c^3}\dddot{X}+\gamma \frac{e^2 a}{c^4}\ddddot{X}+...

Here \alpha and \gamma are of order 1, and a is the electron radius. At that time, the electron was conceived as a small sphere with mass m and charge e. For a sphere with uniformly distributed charge, \alpha=\frac{2}{3}. Lorentz showed that as a approaches zero, the higher-order terms also approach zero. The second term is finite and exactly explains the radiation loss during the accelerated motion of the electron. But the first term diverges, giving an infinite electromagnetic mass \frac{2e^2}{3ac^2}, which is different from the electromagnetic mass of a stationary electron \frac{e^2}{2ac^2}. If the mass of an electron came entirely from electromagnetic mass, then the electrostatic repulsion between the parts of the electron would cause it to fly apart. Therefore, imagining the structure of the electron easily leads one into a fog [6].

Needing to explain radiation damping but unwilling to give up his original idea, Feynman was at a loss. How could the reaction of another electron be used to explain radiation damping? To this end, Feynman consulted Wheeler. Wheeler immediately pointed out three errors in Feynman’s idea: (1) the reaction force depends on the mass of the second electron and the distance r between them, whereas radiation damping is independent of these; (2) the reaction has a time delay, whereas radiation damping does not; (3) if there are a large number of electrons uniformly distributed around the electron, the reaction is proportional to the volume element r^2 dr, and thus proportional to the thickness of the surrounding matter, which would also tend to infinity. Under Wheeler’s guidance, Feynman realized that his reaction was just general light reflection and had nothing to do with radiation damping.

However, Wheeler took up the idea of no self-interaction and introduced Dirac’s advanced wave hypothesis to save Feynman’s idea [7]. Dirac had once imagined that an electron could act on itself through both advanced and retarded potentials, because Maxwell’s equations equally allow advanced solutions. Accordingly, we have:

F_{adv}=\alpha \frac{e^2}{ac^2}\ddot{X}+\frac{2}{3}\frac{e^2}{c^3}\dddot{X}+\gamma \frac{e^2 a}{c^4}\ddddot{X}+...

If we assume that the electron’s self-interaction is half the retarded potential minus half the advanced potential, then:

F=-\frac{2}{3}\frac{e^2}{c^3}\dddot{X}+...

The first term in Lorentz’s formula now disappears, and we thus obtain a finite radiation damping. Wheeler first changed Dirac’s half-advanced self-interaction into a half-advanced interaction between different electrons, thereby solving the time delay problem of Feynman’s reaction. Next, Wheeler tentatively assumed that advanced waves have no refraction when passing through a medium, while retarded waves have a refractive index n. With this rather arbitrary assumption, he proved that Feynman’s reaction is independent of the charge and the thickness of the medium. Based on Wheeler’s idea, Feynman quantitatively proved that the half-advanced and half-retarded solutions correctly give the radiation damping term. He also proved that if a test charge is placed near the source, the two half-advanced waves from the source and the medium arrive simultaneously and cancel each other out, while the two half-retarded waves combine into one retarded wave. The advanced wave has no observable effect and only contributes to radiation damping.

This is a very wonderful theory. It not only successfully avoids the infinite electromagnetic mass in classical field theory but also retains the two elegant features of linear Maxwell equations and point charges. It neither requires complex non-linear equations like those of Born and Infeld, nor does it fall into the mire of guessing the electron’s structure. According to traditional theory, if we accelerate electron a at time t, electron b will be affected by electron a at time t' = t + \frac{r}{c}, and electron a will be affected by electron b at a later time. Now, electron a is affected by electron b at time t'' = t - \frac{r}{c}, which means electron a is already affected at time t by the action of other electrons that it will affect. Applying the direct interaction of half-advanced and half-retarded potentials, Feynman discovered a new form of action:

A=\sum_{i} m_i \int (\dot{X}_\mu^i\dot{X}_\mu^i)^{1/2} d\alpha_i + \frac{1}{2}\sum_{i\neq j}\int \int \delta(I_{ij}^2)\dot{X}_\mu^i(\alpha_i)\dot{X}_\mu^j(\alpha_j)d\alpha_i d\alpha_j

Here X_\mu^i(\alpha_i) is the four-vector position of particle i. The first term of the formula is the proper time integral, representing the ordinary action of a relativistic free particle of mass m_i; the second term is the direct interaction between charges through half-retarded and half-advanced Maxwell potentials, excluding self-interaction. Starting from the action A, we can re-derive the Maxwell equations with half-retarded and half-advanced solutions. Thus, Feynman obtained a new form of classical electrodynamics.

Different formulations of the same theory left a deep impression on Feynman from then on and had a major impact on his future research work. “Every good theoretical physicist knows six or seven different formulations of the same theory” [8]. In his 1964 Messenger Lectures at Cornell University, he said: “Scientifically they are equivalent, but psychologically they are very different in two ways. First, you may favor one over the other for philosophical reasons; second, they are very different in their heuristic value when you are trying to guess new laws.” ([8], p.53) The study of classical self-energy made Feynman prefer the particle interpretation starting from the action, rather than the field theory form described by differential equations. In his view, “the field is just a bookkeeping variable for those who insist on using the Hamiltonian method.” [5] Using this new action, Feynman could also easily make certain modifications to Maxwell’s equations. For example, if Bopp’s suggestion were adopted to consider non-local effects at small distances, one would only need to replace the \delta function in action A with f(s^2). Furthermore, if the mass of an electron could be attributed to electromagnetic mass, then action A would have a very simple form:

A=\frac{1}{2} \sum_{i,j} e_i e_j \int \int f(I_{ij}^2) \dot{X}_\mu^i(\alpha_i)\dot{X}_\mu^j(\alpha_j)d\alpha_i d\alpha_j

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