Richard Feynman's approach to quantum electrodynamics (QED) was revolutionary and highly influential in the development of the theory. However, it had one main problem known as the "ultraviolet divergence" or the "Feynman divergence problem."
In Feynman's approach, he used a technique called perturbation theory to calculate the probabilities of different particle interactions. This involved expressing the probability amplitudes as a series expansion, known as a Feynman diagram, in terms of coupling constants and propagators. While this approach was successful in producing accurate predictions for many physical phenomena, it encountered difficulties when dealing with high-energy or high-momentum processes.
The problem arose because the series expansion used by Feynman's approach contained terms that diverged as the momenta or energies involved increased to arbitrarily large values. These divergences occurred in the form of infinities, rendering the theory mathematically inconsistent and physically meaningless.
The ultraviolet divergence problem indicated that Feynman's approach was incomplete and required further development. It suggested that the theory needed a more rigorous mathematical foundation and a proper treatment of quantum field theory. The infinities implied that there were fundamental issues with how the theory was formulated and required the implementation of renormalization techniques to address them.
To overcome the ultraviolet divergence problem, subsequent theoretical work by other physicists, such as Julian Schwinger, Sin-Itiro Tomonaga, and Freeman Dyson, introduced renormalization. Renormalization involves redefining certain parameters in the theory, such as the mass and charge of particles, in a way that cancels out the infinities. This technique allowed for consistent calculations and meaningful physical predictions in quantum electrodynamics. Feynman himself later contributed to the development of renormalization techniques.
In summary, the main problem with Feynman's approach to quantum electrodynamics was the occurrence of ultraviolet divergences, leading to infinities in the calculations. This problem was addressed through the development of renormalization techniques by subsequent physicists, ensuring the mathematical and physical consistency of the theory.