A mathematical approach to Wick rotations

Abstract

In this thesis we define Wick-rotations mathematically using pseudo-Riemannian geometry, and relate Wick-rotations to real geometric invariant theory (GIT). We discover some new results concerning the existence of Wick rotations (of various signatures). For instance we show that a Wick-rotation of a pseudo-Riemannian space (at a fixed point p) to a Riemannian space forces the space to be Riemann purely electric (RPE). We also define compatibility among representations and relate them to real GIT and Wick-rotations. The polynomial curvature invariants of pseudo-Riemannian spaces are also considered and related to Wick-rotations. Wick-rotations of a special class of pseudo-Riemannian manifolds (M; g) are also studied; namely Lie groups G equipped with left-invariant metrics. We prove some new results concerning the existence of real slices (of Lie algebras) of certain signatures of a holomorphic inner product space (gC; gC) (on a complex Lie algebra). The definition of a Cartan involution for a semisimple Lie algebra is defined for a general Lie algebra equipped with a pseudo-inner product: (g; g), and the theorems of Cartan (concerning Cartan involutions) are generalised and proved. For instance we prove that a pseudo-Riemannian Lie group (G; g) can be Wick-rotated to a Riemannian Lie group ( ~ G; ~g) if and only if there exist a Cartan involution of the Lie algebra g.