{"paper":{"title":"The Complex-Time Segal-Bargmann Transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Brian Hall, Bruce Driver, Todd Kemp","submitted_at":"2016-10-01T05:29:45Z","abstract_excerpt":"We introduce a new form of the Segal--Bargmann transform for a Lie group $K$ of compact type. We show that the heat kernel $(\\rho_{t}(x))_{t>0,x\\in K}$ has a space-time analytic continuation to a holomorphic function \\[ (\\rho_{\\mathbb{C}}(\\tau,z))_{\\mathrm{Re}\\,\\tau>0,z\\in K_{\\mathbb{C}}} \\] where $K_{\\mathbb{C}}$ is the complexification of $K$. The new transform is defined by the integral \\[ (B_{\\tau}f)(z)=\\int_{K}\\rho_{\\mathbb{C}}(\\tau,zk^{-1})f(k)\\,dk,\\quad z\\in K_{\\mathbb{C}}. \\] If $s>0$ and $\\tau\\in\\mathbb{D}(s,s)$ (the disk of radius $s$ centered at $s$), this integral defines a holomor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00090","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}