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

4. Functional Methods in Quantum Field Theory

When path integrals first appeared, most physicists reacted coldly, even doubting their correctness. On one hand, this was due to unfamiliarity and misunderstanding of the path integral method. At the Pocono Conference, Bohr misunderstood Feynman diagrams as the actual trajectories of particle motion and criticized them sharply ([19], p. 459). On the other hand, Feynman did not use an axiomatic approach to systematically derive Feynman rules from the action or the Lagrangian; instead, he relied on experience, guesswork, testing, and comparison to provide the rules corresponding to various diagrams. Nevertheless, Feynman was able to extend his method to the then-popular meson theory, solving in a single night problems that took others months to resolve using canonical Hamiltonian methods. The effectiveness of Feynman’s method greatly surprised Dyson and led him to believe that path integrals were a "fundamentally correct" ([1], p. 54) theory. Subsequently, Dyson decided to make it his primary work to "understand Feynman (his thoughts) and explain them in a language others could understand" ([1], p. 54). In 1948, Dyson successfully proved the equivalence of the theories of Sin-Itiro Tomonaga, Julian Schwinger, and Richard Feynman "within their common domain of applicability" [25]. Feynman’s particle-picture path integral method was thus transformed into the functional integral method of field theory.

After the 1960s, the path integral method gained increasingly widespread application in statistical physics, quantum field theory, and quantum cosmology. In 1963, Feynman first extended it to non-Abelian gauge theories. In 1967, Faddeev and Popov used the path integral method to obtain quantization rules for SU(2) gauge theory. In 1971, ’t Hooft, based on path integral theory and with the help of dimensional regularization, successfully proved the renormalizability of non-Abelian gauge theories. The great success of gauge theories in particle physics allowed people to see more clearly the profound significance of Feynman’s original work.

The path integral method directly introduces the action into quantum theory, making it most suitable for handling symmetries. Some physicists are inclined to believe that the ultimate design of the universe is considered from the perspective of the action. Regarding this, Anthony Zee provided a penetrating analysis in his popular work Fearful Symmetry [26]. Zee emphasized that the action formulation is superior to the differential equation description both aesthetically and mathematically. First, it is far more concise than the differential equation form; second, it remains invariant under transformation groups, rather than being covariant like differential equations. In quantum theory, the principle of superposition of states allows symmetry to play an even greater role. In addition to Lorentz transformations, we have gauge transformations. In fact, symmetry and renormalizability are the two guiding principles of quantum field theory, with symmetry governing the form of the action. Therefore, in the field of fundamental physics, the action formulation has pushed equations of motion to the side; people rarely deal with equations of motion or corresponding concepts such as force and acceleration.

Feynman himself provided a clearer explanation: "From classical mechanics to quantum mechanics, the importance of many concepts has changed considerably. The concept of force has gradually lost its luster, while the concepts of energy and momentum have become of primary importance... We no longer discuss the motion of particles, but rather deal with probability amplitudes that vary in spacetime. Momentum is associated with the wavelength of the probability amplitude, and energy with its frequency. Momentum and energy determine the phase of the wave function and are thus the most important quantities in quantum mechanics. We no longer talk about various forces, but rather deal with interaction methods that change the wavelength. The concept of force, even if used, is a secondary thing." Indeed, the Aharonov-Bohm (A-B) effect can only be explained using the electromagnetic potentials A and \Phi, rather than the field strengths E and B. Similarly, in quantum theory, we can no longer understand mass simply as inertia. In the interference effects of neutral K-mesons, the mass (energy) difference appears directly in the phase of the probability amplitude. Neutron interference experiments show that the phase shift is proportional to the product of "inertial" mass and gravitational mass. As Chen-Ning Yang stated: "The meaning of probability amplitudes with phases. Since physicists began using them to describe nature, it was not until the 1970s that people clearly recognized that every interaction corresponds to a generalized complex phase, and all fundamental forces are phase fields." [28]

The path integral method of action quantization has greatly deepened our understanding of the quantum mechanical revolution. "In the past 10 to 15 years (referring to 1970), as far as discussions of fundamental physics are concerned, path integrals have basically replaced the old wave mechanics and matrix formulations." ([26], p. 152) Zee further pointed out that the path integral "itself is not entirely a theory; it is merely a method for establishing corresponding theories in the quantum realm. Applying this method to Newtonian mechanics yields quantum mechanics. Applying it to Maxwell’s electromagnetic theory yields quantum electrodynamics, and applying it to Einstein’s theory of gravity yields quantum gravity." ([26], p. 152) As for the specific forms of various actions in the probability amplitude expression, they are determined by guesswork, classical analogy, mathematical verification, and experimental testing. Just as the inverse-square law in Newton’s theory of gravity was established through guesswork and verification, Feynman, through eight years of deep contemplation, did not derive a specific theory like Newton’s gravity or Maxwell’s equations, but instead derived a universal principle or method applicable to the entire quantum realm.

Throughout his life, Feynman never stopped his efforts to pursue universal principles surrounding fundamental questions. "Why does it take an infinite number of logical steps to understand what happens in an extremely tiny region of spacetime?" ([8], p. 57) Where does the observed mass of particles come from? [29] Why does vacuum energy not produce observable gravitational effects? [30] In the final years of his life, these mysteries in quantum field theory still haunted Feynman’s mind. His never-ending, passionate pursuit and his fascinating style of expression will forever inspire us to explore this miraculous world and appreciate the beautiful images depicted by physicists.