The linear dynamics of wave functions in causal fermion systems

A mechanism of baryogenesis for causal fermion systems

“…Then, by applying the above proposition in two steps for N and N and subsequently for N and N , one sees that it suffices to consider the case that Ω ⊂ Ω (this case is shown in figure 7). In this case, by rearranging the integration domains, one obtains We remark that the form of the nonlocal modification of this conservation law is inspired by surface layer integrals for causal fermion systems; in particular the commutator inner product in [30]. We also point out that if B is a local potential (i.e.…”

“…As a result of this minimization process, the physical wave functions satisfy dynamical equations in spacetime, the so-called dynamical wave equation. In this paper, we do not need to enter the analysis of the dynamical wave equation (for details see [30]). But it is important to keep the following points in mind: Our mechanism of fermiogenesis is based on a specific correction to the Dirac equation obtained by analyzing the nonlinear dynamics as described by the causal action principle.…”

“…Here 'conservation law' means that this inner product is independent of t. The inner product (3.10) is referred to as the commutator inner product (the name comes from the fact that the unitary invariance can be expressed in terms of commutators; see [30, section 3] for details). The kernel Q(x, y) appearing in this formula is the first variational derivative of the Lagrangian, where we identified the spinor spaces S x M with corresponding spin spaces of the causal fermion system (see [20, section 1.4.1 and proposition 1.6] or [30] for more details). The wave functions ψ and φ in (3.10) are wave functions in Minkowski space.…”

A mechanism for dark matter and dark energy in the theory of causal fermion systems

“…These two requirements can be understood from the fact that the corresponding objects and quantities have a direct relation to inherent structures of the causal fermion system. Indeed, the number of occupied states corresponds to the dimension of the Hilbert space H, whereas the normalization of the states and the corresponding unitary time evolution are related to a conservation law resulting from a Noether-like theorem for the causal action principle (more precisely, the conservation law for the commutator inner product; for details see [17,20]). Keeping the number of occupied states fixed leads us to choose the width δω independent of ⃗ k as…”