Wick rotation in the lapse: admissible complex metrics and the Wick rotated heat kernel

The speaker: Rudrajit Banerjee (OIST, Gravity, Quantum Geometry and Field Theory Unit (Reiko Toriumi))

Title: Wick rotation in the lapse: admissible complex metrics and the Wick rotated heat kernel

Abstract: We introduce a Wick rotation in the lapse function of the Arnowitt-Deser-Misner decomposition for foliated metric geometries. This interpolates between Lorentzian and Riemannian metrics on the same underlying smooth real manifold, passing through admissible complex metrics dampening the exponential of the action of a real scalar field. Motivated by the ubiquity of heat kernel methods in Euclidean signature quantum field theory, we show that 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 and sketch proofs of the properties of the Wick rotated heat semigroup: (i) existence and uniqueness, (ii) the existence and uniqueness of an integral kernel, the ``Wick rotated heat kernel'', (iii) the kernel's diagonal admits a small semigroup time asymptotic expansion, and (iv) when the d'Alembertian of the Lorentzian metric is essentially self-adjoint, the Wick rotated heat semigroup converges to the unitary Schrödinger group generated by the d'Alembertian in the strict Lorentzian limit. This talk is based on arXiv:2406.06047.

Date and time: 15^th August Thursday at 13:00

Location: L4F01