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Its action on the operator-valued angle $qp_tq$ between the projections induces a flow on the corresponding spectral measures $\\mu_t$; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure $\\mu_t$ possesses a piecewise analytic density for any $t>0$ and any initial projections of trac"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.6037","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-26T17:49:28Z","cross_cats_sorted":["math.AP","math.CV","math.OA","math.PR"],"title_canon_sha256":"25255f494970e19cad8f12cf5f120da34af4ccf617c5377f62827111612a2b32","abstract_canon_sha256":"7eeef8ba93ef870182923ccb5fbd97ce9bf0024c78e48c6b7219d0489839fe9e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:45.440480Z","signature_b64":"hDCnSX3alXImJfP8XR9nqdIXZ2vn9ZXl8DXmlxRizwFg2QwLe+JNWYCTBhJ3B5of+d5xlWPOPHYdChcpmQIxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32216e10a4c44571e0daa547635a276dd55df57b5012516af1d603620a05a170","last_reissued_at":"2026-05-18T01:21:45.439853Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:45.439853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Liberation of Projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CV","math.OA","math.PR"],"primary_cat":"math.FA","authors_text":"Benoit Collins, Todd Kemp","submitted_at":"2012-11-26T17:49:28Z","abstract_excerpt":"We study the liberation process for projections: $(p,q)\\mapsto (p_t,q)= (u_tpu_t^\\ast,q)$ where $u_t$ is a free unitary Brownian motion freely independent from $\\{p,q\\}$. 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