Narrow Results

Let $H:O\mapsto H(O)$ be a local net of real Hilbert subspaces of a complex Hilbert space ${\mathscr{H}}$ on the family $𝒪$ of double cones of the spacetime ${ℝ}^{d+1}$ ($d$ odd), covariant with respect to a positive energy, unitary representation $U$ of the Poincaré group ${{𝒫}}_{+}^{\uparrow }$, with the Bisognano–Wichmann property for the wedge modular group. We set an upper bound on the local entropy $S_H(\varphi ||C)$ of a vector $\varphi \in {\mathscr{H}}$ in a given region $C\subset {ℝ}^{d+1}$ that depends only on $U$ and the PCT anti-unitary canonically associated with $H$. A similar result holds for local, Möbius covariant nets of standard subspaces on the circle. We compute the entropy increase with respect to the dual net and illustrate this bound for the nets associated with the $U(1)$-current derivatives.

In the framework of the functional renormalization group and of the perturbative, algebraic approach to quantum field theory (pAQFT), in D’Angelo et al. [Ann. Henri Poinc.25 (2024) 2295–2352] it has been derived a Lorentzian version of a flow equation à la Wetterich, which can be used to study nonlinear, quantum scalar field theories on a globally hyperbolic spacetime. In this work, we show that the realm of validity of this result can be extended to study interacting scalar field theories on globally hyperbolic manifolds with a timelike boundary. By considering the specific examples of half-Minkowski spacetime and of the Poincaré patch of Anti-de Sitter, we show that the form of the Lorentzian Wetterich equation is strongly dependent on the boundary conditions assigned to the underlying field theory. In addition, using a numerical approach, we are able to provide strong evidences that there is a qualitative and not only a quantitative difference in the associated flow and we highlight this feature by considering Dirichlet and Neumann boundary conditions on half-Minkowski spacetime.

An approach to renormalization of scalar fields on the Doplicher–Fredenhagen–Roberts (DFR) quantum spacetime is presented. The effective nonlocal theory obtained through the use of states of optimal localization for the quantum spacetime is reformulated in the language of (perturbative) Algebraic Quantum Field Theory. The structure of the singularities associated to the nonlocal kernel that codifies the effects of non-commutativity is analyzed using the tools of microlocal analysis.