The most successful quantum model in terms of generating precise empirical predictions is quantum electrodynamics (QED), which was developed in the late 1940’s (Tomonaga, 1946; Schwinger, 1948; Schwinger, 1948; Feynman, 1948; Feynman, 1949; Feynman, 1949; Dyson, 1949; Dyson, 1949). This model synthesized significantly different conceptual and mathematical frameworks compared to the relatively simple Hilbert space matrix mechanics and Schrodinger wave functions. There are two key conceptual elements to QED: the path integral and quantum field theory, which enable it to accurately represent the detailed interactions between an electron and the EM field.
The path integral method developed by Richard Feynman (originated by Dirac) works by iteratively enumerating all the possible events that might take place in an interaction between electron and photon, and computing the total probability of different outcomes by integrating across all these paths. This method is intuitively illustrated by the famous Feynman diagrams showing the different possible scenarios in each path. Critically, at this level, the key elements of the theory are the coupled Dirac equation for the electron and Maxwell’s equations for the EM field.
A central problem with this path integral framework is that the path integral sums diverge into infinity. A kind of mathematical “hack” (according to Feynman and others) called renormalization was finally able to resolve this divergence. Despite its seemingly arbitrary “pragmatic” basis, renormalization proved successful and allowed the framework to be extended to other domains, involving the weak and strong forces. Clearly, a mathematical framework based on divergent infinite integrals is not a particularly promising basis for a plausible physical model, and is thus clearly a calculational tool.
At higher energy levels, the possible paths include the spontaneous creation of new particles out of raw energy (i.e., pair production). Dealing with these kinds of events in an efficient way mathematically required an entirely new framework called quantum field theory (QFT), which does away with the configuration-space representation based on a specific number of particles, and instead considers an entire spatially-extended field where particles can more easily be created and destroyed as “excitations of the field”.
As discussed in field theory, this framework is based on the convenience of Fourier space, which represents the momentum, but not position, of the relevant particles. Thus, it only provides a partial window into the actual underlying physical phenomena, and is overall quite misleading if adopted as a physical model.
In summary, QED and QFT are amazing calculational tools that have been used to make some of the most accurate predictions in all of physics. But they have a number of obvious problems as physical models for how nature actually operates, especially relative to the assumptions of the pilot-wave framework. Nevertheless, the coupled Dirac and Maxwell fields at the heart of this theory are clearly the key basic ingredients, and it is possible that a more computationally-based approach, with the strong constraint of discrete electron particles in specific spatial locations, could provide the basis for an accurate physical model.