In this survey article, we explain a few ideas behind the fermionic projector approach and summarize recent results which clarify the connection to quantum field theory. The fermionic projector is introduced, which describes the physical system by a collection of Dirac states, including the states of the Dirac sea. Formulating the interaction by an action principle for the fermionic projector, we obtain a consistent description of interacting quantum fields which reproduces the results of perturbative quantum field theory. We find a new mechanism for the generation of boson masses and obtain small corrections to the field equations which violate causality.
It was shown in work \cite{vergeles2021note} that in the theory of gravity coupled with the Dirac field, each state $|λ\rangle$ has its own twin $|λ;PT\rangle$, which is obtained by a discrete PT transformation. If in the state $|λ\rangle$ the Dirac sea is filled, then in the state $|λ;PT\rangle$ there is an "anti-Dirac" filling (in terms of the state $|λ\rangle$). It is important that the energies of these states are the same. Therefore, there may be domains with different filling of the Dirac sea. Here we study a domain wall connecting two such adjacent domains.
The quantum fluctuations of the Dirac field in external classical gravitational and electromagnetic fields are studied. A self-consistent equation for torsion is calculated, which is obtained using one-loop fermion diagrams.
We present a "primitive" way of realizing finite-mass Dirac monopoles in $U(1)$ gauge theories involving a single non-minimally interacting scalar field. Typically, the energy density of this type of monopole is not concentrated at its core, but it is distributed in a spherical shell, as we illustrate on several exact solutions in the Bogomol'nyi-Prasad-Sommerfield (BPS) limit. We show that our construction can be interpreted as a limit of infinitely massive $W$ bosons coupled to electromagnetic field-strength via a dipole moment. Combining our approach with ideas of Weinberg and Lee, we prese…
We study quantum effects of recently discovered kink solitons which are constructed self-consistently by coupling to a single, excited fermion bound state. Our studies are based on the observation that in a semi-classical expansion the energies of this single level and of the Dirac sea should be treated equally. For these kink solutions we compute the energy of the Dirac sea as the fermion vacuum polarization energy. We find it to be substantial and to typically outweigh the energy gain from binding the single level.
We revisit the chiral anomaly in the quantum kinetic theory in the Wigner function formalism under the background field approximation. Our results show that the chiral anomaly is actually from the Dirac sea or the vacuum contribution in the un-normal-ordered Wigner function. We also demonstrate that this contribution modifies the chiral kinetic equation for antiparticles.
