Analytic semigroups approaching a Schrödinger group on real foliated metric manifolds
Rudrajit Banerjee, Max Niedermaier
TL;DR
The paper introduces a lapse-Wick-rotated interpolation $\Delta_{\theta}$ between the elliptic and hyperbolic generators on real foliated manifolds and constructs a one-parameter analytic semigroup $e^{\zeta\Delta_{\theta}}$ for $\theta\in(0,\pi)$. It proves sectoriality and existence of a smoothing kernel $K_\zeta^{\theta}$, and derives a diagonal small-time expansion with Seeley–DeWitt coefficients evaluated on the complex metric $g^{\theta}$. In the strict Lorentzian limit $\theta\to0^+$, the semigroups converge (in the ultra-weak topology) to a Schrödinger evolution generated by the closure of ${\cal D}_-$, under appropriate self-adjointness assumptions. The framework yields a mathematically rigorous link between Euclidean heat-kernel techniques and Lorentzian quantum dynamics on foliated spacetimes, including a controlled near-Lorentzian Hadamard-like expansion for the lapse-Wick rotated kernels. These results pave the way for generalizations to non-scalar operators and deeper connections to Lorentzian propagators and state constructions in curved spacetime QFT.
Abstract
On real metric manifolds admitting a co-dimension one foliation, sectorial operators are introduced that interpolate between the generalized Laplacian and the d'Alembertian. This is used to construct a one-parameter family of analytic semigroups that remains well-defined into the near Lorentzian regime. In the strict Lorentzian limit we identify a sense in which a well-defined Schrödinger evolution group arises. For the analytic semigroups we show in addition that: (i) they act as integral operators with kernels that are jointly smooth in the semigroup time and both spacetime arguments. (ii) the diagonal of the kernels admits an asymptotic expansion in (shifted) powers of the semigroup time whose coefficients are the Seeley-DeWitt coefficients evaluated on the complex metrics.