A canonical construction of the extended Hilbert space for causal fermion systems

It is shown that second variations of the causal action can be decomposed into a sum of three terms, two of which being positive and one being small. This gives rise to an approximate decoupling of the linearized field equations into the dynamical wave equation and bosonic field equations. A concrete construction of homogeneous and inhomogeneous solutions of the dynamical wave equation in time strips is presented. In addition, it is show that the solution space admits a positive definite inner product which is preserved under the time evolution. Based on these findings, a canonical construction of the extended Hilbert space containing these solutions is given.