Arne Hofmann | Searcharxiv

arXiv2026

Relative trace formulas for obstacle scattering with Neumann and transmission boundary conditions

We consider the case of scattering by several obstacles in $\mathbb{R}^d$ for $d \geq 2$. We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet boundary conditions. In the case of $f(x) = x^{1/2}$ the trace has the interpretation of the Casimir energy of the obstacle configuration. In the one-dimensional case, we recover a rigorous version of the Lifshitz formula for the Casimir energy of parallel plates with frequency-independent electric permittivity and magnetic permeability. We thereby strengthen the mathematical foundations of the Casimir effect and demonstrate the flexibility of the rigorous approach established in arXiv:2104.09763 and arXiv:2002.07291.

math-ph↗

arXiv2025

Multiplicative Renormalization in Causal Perturbation Theory

We construct multiplicative renormalization for the Epstein--Glaser renormalization scheme in perturbative Algebraic Quantum Field Theory: To this end, we fully combine the Connes--Kreimer renormalization framework with the Epstein--Glaser renormalization scheme. In particular, in addition to the already established position-space renormalization Hopf algebra, we also construct the renormalized Feynman rules and the counterterm map via an algebraic Birkhoff decomposition. This includes a discussion about the appropriate target algebra of regularized distributions and the renormalization scheme as a Rota--Baxter operator thereon. In particular, we show that the Hadamard singular part satisfies the Rota--Baxter property and thus relate factorization in Epstein--Glaser with multiplicativity in Connes--Kreimer. Next, we define $Z$-factors as the images of the counterterm map under the corresponding combinatorial Green's functions. This allows us to define the multiplicatively renormalized Lagrange density, for which we show that the corresponding Feynman rules are regular. Finally, we exemplify the developed theory by working out the specific case of $\phi^3_6$-theory.

math-ph↗