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

TL;DR

This paper proves the H"older continuity of the integrated causal Lagrangian in Minkowski space by analyzing the $L^4$-integrability of the fermionic projector kernel and its perturbations, with implications for quantum field theory.

Contribution

It establishes the H"older continuity of the integrated causal Lagrangian and analyzes its topological features under perturbations, using a specific regularization scheme.

Findings

Kernel of fermionic projector is $L^4$-integrable.

Causal Lagrangian is $L^1$-integrable.

H"older-like estimates are proved for spacetime variations.

Abstract

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.