Exact Results in Quantum Theory

The heat kernel is a central object for quantum field theory in Euclidean signature, both for one-loop perturbation theory and non-perturbative functional renormalization group methods. On generic curved backgrounds, however, the link between Euclidean and Lorentzian signature QFT via a Wick rotation is not fully understood. In this talk, I will present a proposal for a Wick rotation in the lapse (not in time) on foliated metric geometries, interpolating between Lorentzian and Riemannian metrics on the same underlying smooth real manifold. This construction keeps the coordinates real and passes through admissible complex metrics (in the sense of Kontsevich-Segal) dampening the exponential of the action of a real scalar field. As a consequence, even when specialized to free field theories on near Lorentzian backgrounds one does not obtain the Feynman iε prescription, but a variant due to Zimmermann which entails absolutely convergent Feynman diagrams. On generic foliated backgrounds and strictly away from the Lorentzian metric, the associated Laplace-Beltrami operator generates a “Wick rotated heat semigroup”, an analytic semigroup generalizing the usual heat semigroup on a Riemannian manifold. I will discuss the properties of the Wick rotated heat semigroup, specifically: (i) its existence and uniqueness, (ii) the existence and smoothness of a unique integral kernel, the “Wick rotated heat kernel”, (iii) that the kernel's diagonal admits an asymptotic expansion for small semigroup time, and (iv) that in the strict Lorentzian limit the Wick rotated heat semigroup converges under well-defined traces to the unitary Schrödinger group generated by the d'Alembertian (assumed to be essentially self-adjoint). This talk is based on the papers J. Funct. Anal. 289 (2025) 110898 and Class. Quant. Grav. 42 (2025) 095003, and is joint work with Max Niedermaier.