Glauber-Sudarshan-type quantizations and their path integral representations for compact Lie groups

Abstract

In this paper, we consider an arbitrary irreducible unitary representation $(\pi_{\lambda},V_{\lambda})$ of a compact connected, simply connected semisimple Lie group $G$ with highest weight $\lambda$, and apply the idea of Daubechies--Klauder (1985) and Yamashita (2011) on rigorous coherent-state path integrals to this representation, where the orbit of the highest weight vector is interpreted as the manifold of coherent states. Our main theorem is two-fold: the first main theorem is in terms of Brownian motions and stochastic integrals, and proven using the Feynman--Kac--It\^o formula on a vector bundle of a Riemannian manifold, due to G\"uneysu (2010). In the second main theorem, we consider a sequence $(\mu_{n})$ of finite measures on the space of smooth paths, and a `path integral' is defined to be a limit of the integrals with respect to $(\mu_{n})$. The formulation and the proof of the second main theorem employ \emph{rough path theory} originated by Lyons (1998).