{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:Y6VQBUFK3FMI72YUMWUSL4FS4H","short_pith_number":"pith:Y6VQBUFK","canonical_record":{"source":{"id":"1806.00980","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-04T06:49:11Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"969fee952a3f7921d0607067420d9b6a3ad93e38cbb99bb997128a747b3bb612","abstract_canon_sha256":"b8cb44403af269c04b7a4bac02576536e8b1f7e57e53af929b6681d714978e0c"},"schema_version":"1.0"},"canonical_sha256":"c7ab00d0aad9588feb1465a925f0b2e1e7ac697b91c05d84d0dd4112db910c73","source":{"kind":"arxiv","id":"1806.00980","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00980","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00980v1","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00980","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"Y6VQBUFK3FMI","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"Y6VQBUFK3FMI72YU","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"Y6VQBUFK","created_at":"2026-05-18T12:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:Y6VQBUFK3FMI72YUMWUSL4FS4H","target":"record","payload":{"canonical_record":{"source":{"id":"1806.00980","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-04T06:49:11Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"969fee952a3f7921d0607067420d9b6a3ad93e38cbb99bb997128a747b3bb612","abstract_canon_sha256":"b8cb44403af269c04b7a4bac02576536e8b1f7e57e53af929b6681d714978e0c"},"schema_version":"1.0"},"canonical_sha256":"c7ab00d0aad9588feb1465a925f0b2e1e7ac697b91c05d84d0dd4112db910c73","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:17.828958Z","signature_b64":"ZzRviXhb9KcLynJ0xLh/UKlPklAYj+j+DsDRbUU5W1yAXKtSiZs1LuCEDoIvIYWIGIm/S8vQI6vZb4zu7sUjBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c7ab00d0aad9588feb1465a925f0b2e1e7ac697b91c05d84d0dd4112db910c73","last_reissued_at":"2026-05-18T00:14:17.828210Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:17.828210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.00980","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"31bfWbkm+g2tiqPotfuUT6JBuN0Us8LffX/yv3naMBV+qkzxW2ZrZmkQkivePmHKaffex71kjgR7t3rhc6xCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T19:11:03.949486Z"},"content_sha256":"dd4a3ea180116c1a4d94bd442f3fb227edfb8e6e96a5a81027e8da4cb7a1f17d","schema_version":"1.0","event_id":"sha256:dd4a3ea180116c1a4d94bd442f3fb227edfb8e6e96a5a81027e8da4cb7a1f17d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:Y6VQBUFK3FMI72YUMWUSL4FS4H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Weyl calculus for group generators satisfying the canonical commutation relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.FA","authors_text":"Jan van Neerven, Pierre Portal","submitted_at":"2018-06-04T06:49:11Z","abstract_excerpt":"Classical pseudo-differential calculus on $\\mathbb{R}^{d}$ can be viewed as a (non-commutative) functional calculus for the standard position and momentum operators $(Q_{1}, \\dots , Q_{d})$ and $(P_{1}, \\dots , P_{d})$. We generalise this calculus to the setting of two $d$-tuples of operators $A=(A_{1}, \\dots , A_{d})$ and $B=(B_{1}, \\dots , B_{d})$ acting on a Banach space $X$ such that $iA_{1}, \\dots , iA_{d}$ and $iB_{1}, \\dots , iB_{d}$ generate bounded $C_0$-groups satisfying the Weyl canonical commutation relations $e^{isA_j}e^{itA_k} = e^{itA_k}e^{isA_j}$, $e^{isB_j}e^{itB_k} = e^{itB_k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00980","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qE0jhYHFIgoGLQwPBpCJ5ezRPKYdhDNQStwXkiM6BsL7Ymc27lD+KMArjYoZ+SaHDE8VqJtbOHVUKFRDYSAzCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T19:11:03.949830Z"},"content_sha256":"3695f260a3e3a3d6c878b2b01e85e998db2f784cbaaa02b59682b5b0fcb8a8b1","schema_version":"1.0","event_id":"sha256:3695f260a3e3a3d6c878b2b01e85e998db2f784cbaaa02b59682b5b0fcb8a8b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y6VQBUFK3FMI72YUMWUSL4FS4H/bundle.json","state_url":"https://pith.science/pith/Y6VQBUFK3FMI72YUMWUSL4FS4H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y6VQBUFK3FMI72YUMWUSL4FS4H/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T19:11:03Z","links":{"resolver":"https://pith.science/pith/Y6VQBUFK3FMI72YUMWUSL4FS4H","bundle":"https://pith.science/pith/Y6VQBUFK3FMI72YUMWUSL4FS4H/bundle.json","state":"https://pith.science/pith/Y6VQBUFK3FMI72YUMWUSL4FS4H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y6VQBUFK3FMI72YUMWUSL4FS4H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:Y6VQBUFK3FMI72YUMWUSL4FS4H","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b8cb44403af269c04b7a4bac02576536e8b1f7e57e53af929b6681d714978e0c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-04T06:49:11Z","title_canon_sha256":"969fee952a3f7921d0607067420d9b6a3ad93e38cbb99bb997128a747b3bb612"},"schema_version":"1.0","source":{"id":"1806.00980","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.00980","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1806.00980v1","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.00980","created_at":"2026-05-18T00:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"Y6VQBUFK3FMI","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"Y6VQBUFK3FMI72YU","created_at":"2026-05-18T12:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"Y6VQBUFK","created_at":"2026-05-18T12:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:3695f260a3e3a3d6c878b2b01e85e998db2f784cbaaa02b59682b5b0fcb8a8b1","target":"graph","created_at":"2026-05-18T00:14:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Classical pseudo-differential calculus on $\\mathbb{R}^{d}$ can be viewed as a (non-commutative) functional calculus for the standard position and momentum operators $(Q_{1}, \\dots , Q_{d})$ and $(P_{1}, \\dots , P_{d})$. We generalise this calculus to the setting of two $d$-tuples of operators $A=(A_{1}, \\dots , A_{d})$ and $B=(B_{1}, \\dots , B_{d})$ acting on a Banach space $X$ such that $iA_{1}, \\dots , iA_{d}$ and $iB_{1}, \\dots , iB_{d}$ generate bounded $C_0$-groups satisfying the Weyl canonical commutation relations $e^{isA_j}e^{itA_k} = e^{itA_k}e^{isA_j}$, $e^{isB_j}e^{itB_k} = e^{itB_k","authors_text":"Jan van Neerven, Pierre Portal","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-04T06:49:11Z","title":"The Weyl calculus for group generators satisfying the canonical commutation relations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.00980","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dd4a3ea180116c1a4d94bd442f3fb227edfb8e6e96a5a81027e8da4cb7a1f17d","target":"record","created_at":"2026-05-18T00:14:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b8cb44403af269c04b7a4bac02576536e8b1f7e57e53af929b6681d714978e0c","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-04T06:49:11Z","title_canon_sha256":"969fee952a3f7921d0607067420d9b6a3ad93e38cbb99bb997128a747b3bb612"},"schema_version":"1.0","source":{"id":"1806.00980","kind":"arxiv","version":1}},"canonical_sha256":"c7ab00d0aad9588feb1465a925f0b2e1e7ac697b91c05d84d0dd4112db910c73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c7ab00d0aad9588feb1465a925f0b2e1e7ac697b91c05d84d0dd4112db910c73","first_computed_at":"2026-05-18T00:14:17.828210Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:17.828210Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZzRviXhb9KcLynJ0xLh/UKlPklAYj+j+DsDRbUU5W1yAXKtSiZs1LuCEDoIvIYWIGIm/S8vQI6vZb4zu7sUjBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:17.828958Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.00980","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd4a3ea180116c1a4d94bd442f3fb227edfb8e6e96a5a81027e8da4cb7a1f17d","sha256:3695f260a3e3a3d6c878b2b01e85e998db2f784cbaaa02b59682b5b0fcb8a8b1"],"state_sha256":"ca6294438bd2da567df6bbdff29f46eaf55591304147e5db7223da97c0aec1e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JS9C9e9WNmR/wOfjiGfCXMepRgMxA+v5XnJWCWkVvPqbPZqHl5TuQTJQYymQTl4h/hTjqMWEYmeJ5GCj7PdwCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T19:11:03.951835Z","bundle_sha256":"a0753dbda4e32ad232393d96fc9b11021b54b8b09388a5db35a0906d5215a1a8"}}