The Theory of Causal Fermion Systems

The Linearized Field Equations

The Linearized Field Equations

Mimicking the usual procedure for linearizing physical equations, we consider a family $(\tilde{\rho})_\tau$ of measures parametrized by $\tau \in (-\delta, \delta)$ which all satisfy the Euler-Lagrange equations and linearize in the parameter $\tau$. Assuming that the measures are obtained by a push-forward and a change of weight, i.e.

\[ \tilde{\rho}_\tau = (F_\tau)_* \big( f_\tau \,\rho \big) \]

with $F_\tau \in C^\infty(M, \F)$ and $f_\tau \in C^\infty(M, \R^+)$, these variations can be described linearly in $\tau$ by the jet

\[ \mathfrak{v} := \frac{d}{d\tau} \big( f_\tau, F_\tau \big) \big|_{\tau=0} \]

(for details see [jet16]; more general variations allowing for a so-called fragmentation of the measure are considered in [perturb17, Section 5]). The fact that the variation described infinitesimally by the jet preserves the Euler-Lagrange equations gives rise to the linearized field equations

\[ \la \mathfrak{u}, \Delta \mathfrak{v} \ra|_M = 0 \:, \]

to be satisfied for all test jets $\mathfrak{u}$, where the Laplacian $\Delta$ is defined by  

\[ \la \mathfrak{u}, \Delta \mathfrak{v} \ra(x) := \nabla_{\mathfrak{u}} \bigg( \int_M \big( \nabla_{1, \mathfrak{v}} + \nabla_{2, \mathfrak{v}} \big) \L(x,y)\: d\rho(y) – \nabla_\mathfrak{v} \mathfrak{s} \bigg) \:.\]

(for details see [jet16] or the summary in [perturb17, Section 3.3]).

existence theory for linearized field equations
surface layer integrals