Extend holographic mixing and Fock-space dynamics to non-abelian gauge fields

Develop an extension of the causal fermion systems framework introduced in this paper—namely holographic mixing and the associated unitary time evolution on fermionic and bosonic Fock spaces—from abelian (electromagnetic) potentials to non-abelian gauge fields. Construct non-abelian bosonic field operators coupled to Dirac wave functions that satisfy the appropriate commutation relations and derive their dynamics in the presence of nonlocal potentials within causal fermion systems.

Background

The paper derives quantum electrodynamics in Minkowski space from causal fermion systems under assumptions that lead to abelian gauge interactions modeled by a multitude of nonlocal, stochastic vector potentials and holographic phases. The construction yields bosonic field operators satisfying canonical commutation relations and a unitary time evolution on Fock spaces.

While the analysis successfully covers abelian fields, the authors explicitly note that extending these methods to non-abelian gauge theories remains an important next step. Such an extension would require adapting the holographic mixing mechanism, the stochastic covariance structure, and the operator algebra to gauge groups with non-commuting generators, preserving Lorentz covariance and the causal structure.