The Theory of Causal Fermion Systems

A causal fermion system gives rise to a distinguished family of wave functions in spacetime, the physical wave functions. When perturbing the system while preserving the Euler-Lagrange equations, also the physical wave equations are perturbed. In [FKO21] it is shown that, under general assumptions, a class of wave functions obtained in this way satisfy the dynamical wave equation

$${\int }_M{Q}^{\text{dyn}}(x,y)\psi (y)d\rho (y)=0.$$

These wave functions form a Hilbert space ${\mathcal{H}}_{\rho }$ which extends the Hilbert space $\mathcal{H}$ of the causal fermion system. It generalizes the solution space of the Dirac equation to the setting of causal fermion systems. The scalar product on ${\mathcal{H}}_{\rho }$ can be represented at any time by a surface layer integral denoted by $⟨.|.{⟩}_{\rho }^t$.

The conservation law corresponding to the symmetry described by unitary transformations generalizes to a conserved surface layer integral for solutions of the dynamical wave equation. It takes the form (for details see [FKO21])

$$\begin{array}{rl}⟨\psi |\varphi {⟩}_{\rho }^{\mathrm{\Omega }}=-2i({\int }_{\mathrm{\Omega }}d\rho (x){\int }_{M\setminus \mathrm{\Omega }}d\rho (y)& -{\int }_{M\setminus \mathrm{\Omega }}d\rho (x){\int }_{\mathrm{\Omega }}d\rho (y))\\ & \times \prec \psi (x)|Q^{\text{dyn}}(x,y)\varphi (y){\succ }_x.\end{array}$$

→ construction of linear fields and waves
→ surface layer integrals