2018/11/21 by Hideyasu Yamashita, Yamashita, Hideyasu
Mathematics · Medicine · Physics and Astronomy · #22E30 #22E45 #22E46 #22E70 #60H10 #81R30 #81S40 #Advanced Algebra and Geometry #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Medical Imaging Techniques and Applications #math-ph #math.MP #msc:22E30 #msc:22E45 #msc:22E46 #msc:22E70 #msc:60H10 #msc:81R30 #msc:81S40
paper · pdf · doi:10.48550/arxiv.1811.08844
22 pages
openalex publication_date 2018/11/21 · openalex created_date 2018/11/29 · arxiv created 2019/03/15 · arxiv updated 2019/03/18 · openalex updated_date 2026/07/28
In this paper, we consider an arbitrary irreducible unitary representation (πλ,Vλ) of a compact connected, simply connected semisimple Lie group G with highest weight λ, 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--Itô formula on a vector bundle of a Riemannian manifold, due to Güneysu (2010). In the second main theorem, we consider a sequence (μn) 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 (μn). The formulation and the proof of the second main theorem employ rough path theory originated by Lyons (1998).