H\"older Continuity of the Integrated Causal Lagrangian in Minkowski Space

It is proven that the kernel of the fermionic projector of regularized Dirac sea vacua in Minkowski Space is $L^4$-integrable. The proof is carried out in the specific setting of a continuous exponentially-decaying cutoff in momentum space. As a direct consequence, the corresponding causal Lagrangian is shown to be $L^1$-integrable. Some topological features of the integrated causal Lagrangian are analyzed. In particular, local H\"older-like estimates are proved for continuous regular variations of spacetime, of which a few examples are discussed. Particular emphasis is placed on first-order perturbations of Dirac sea vacua induced by external electromagnetic fields.