The Theory of Causal Fermion Systems

Why the Dirac Sea?

As it often happens with mathematical frameworks and physical theories, the way they were conceived is not necessarily the easiest/correct way to make sense of them. For completeness, and because most of the existing literature at the moment is based on this approach to causal fermion systems, we will discuss here the original ideas that lead to the development of the setting of causal fermion systems. It is helpful to have these ideas in mind when approaching the currently available academic literature.
At the beginning there was the Dirac sea. This is sometimes considered an antiquated idea, successfully superseded by the more modern Feynman-Stückelberg interpretation underlying Quantum Field Theory. Although this has proven successful as a computational tool, it can neither prove nor disprove Dirac’s original conception resting on the three experimental observations that

The concept of the Dirac sea gives an effective description of these observations. While the concept of the Dirac sea might be a matter of taste, what everyone certainly agrees on, though, is the fact that the Dirac equation gives a good description for fermionic particles at observational scales.

The conceptional problem with the Dirac Sea, which was the reason that the concept was eventually abandoned, does not lie in its application in particle physics, but in its interplay with gravitation in the setting of general relativity. Indeed, from a particle physics perspective, the Dirac sea is perfectly fine as a starting point to develop a new fundamental theory. In General Relativity, however,

the Dirac sea gives rise to an “infinite negative energy density” which, treated naively, would cause the spacetime to collapse. Therefore, the challenge of a theory that takes the Dirac Sea as a starting point is to explain why the states/energy stored in the Dirac sea does not interact gravitationally.

Eventually, this goal was achieved as we already mentioned in several places (see eg. here and here). The point is that the causal fermion system describing the regularized Dirac sea vacuum is a critical point of the causal action principle. The Dirac sea turns out to be an effective description for the physical wave function representation of the Hilbert space underlying the causal fermion system. However, in the early stages of the development of the theory, the focus was on obtaining a new mathematical framework starting from the perturbative Dirac sea in Minkowski space. That means the spacetime was treated as a fixed background and the efforts focused on finding a mathematical structure that is based on the Dirac sea and encodes the physical properties of Minkowski space.
This lead to the study of the kernel of the fermionic projector. The structures that now make up the mathematical framework of the Theory of Causal Fermion Systems emerged from a detailed analysis of these constructions. A trial and error approach, guided by the requirement for mathematical consistency and the aim to obtain the well-known theories in an appropriate limit eventually led to the form of the causal action principle that now lies at the heart of the Theory of Causal Fermion Systems.