{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1995:XVCSEKBGEEKGJBQND7G37BABI2","short_pith_number":"pith:XVCSEKBG","canonical_record":{"source":{"id":"quant-ph/9509002","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"quant-ph","submitted_at":"1995-09-05T04:20:00Z","cross_cats_sorted":[],"title_canon_sha256":"8f995352223855f29f149ca6458f8ac4de8ead1b0155b7bd21b2d8ee2ab39914","abstract_canon_sha256":"247d6ab4cdd4f83354046d1f4e83e31d675fb963086a0f16fe2f0e90727259e9"},"schema_version":"1.0"},"canonical_sha256":"bd45222826211464860d1fcdbf840146877781a568227ac5fb1e5c1bf0250f0d","source":{"kind":"arxiv","id":"quant-ph/9509002","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"quant-ph/9509002","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"arxiv_version","alias_value":"quant-ph/9509002v3","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.quant-ph/9509002","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"pith_short_12","alias_value":"XVCSEKBGEEKG","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"pith_short_16","alias_value":"XVCSEKBGEEKGJBQN","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"pith_short_8","alias_value":"XVCSEKBG","created_at":"2026-07-04T15:59:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1995:XVCSEKBGEEKGJBQND7G37BABI2","target":"record","payload":{"canonical_record":{"source":{"id":"quant-ph/9509002","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"quant-ph","submitted_at":"1995-09-05T04:20:00Z","cross_cats_sorted":[],"title_canon_sha256":"8f995352223855f29f149ca6458f8ac4de8ead1b0155b7bd21b2d8ee2ab39914","abstract_canon_sha256":"247d6ab4cdd4f83354046d1f4e83e31d675fb963086a0f16fe2f0e90727259e9"},"schema_version":"1.0"},"canonical_sha256":"bd45222826211464860d1fcdbf840146877781a568227ac5fb1e5c1bf0250f0d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:59:57.442173Z","signature_b64":"NnTvCtBKUp/apb/OA05OYWLFETCHqFPv7VbnQlUGaYCZE7QbELflzBrxgejjaN1o9qpq/HnU/QADCjEkbcvIDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd45222826211464860d1fcdbf840146877781a568227ac5fb1e5c1bf0250f0d","last_reissued_at":"2026-07-04T15:59:57.441817Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:59:57.441817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"quant-ph/9509002","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:59:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uL4FFFpN+l9Dyf20vMwezNyS5ScaoSSePzqv+ny6ggdbEMJ2uE6Uk1cXc/sRTVDCOxOFJkpLmU+Wv2qRfPoECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T03:13:00.402283Z"},"content_sha256":"561822379b95d34ba73a577ca2d1daeee0f0f0462e891c5cc7d11cb567f6d67d","schema_version":"1.0","event_id":"sha256:561822379b95d34ba73a577ca2d1daeee0f0f0462e891c5cc7d11cb567f6d67d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1995:XVCSEKBGEEKGJBQND7G37BABI2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Real Symplectic Groups in Quantum Mechanics and Optics","license":"","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Arvind, B. Dutta, N. Mukunda, R. Simon","submitted_at":"1995-09-05T04:20:00Z","abstract_excerpt":"text of abstract (We present a utilitarian review of the family of matrix groups $Sp(2n,\\Re)$, in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of $Sp(2n,\\Re)$. Global decomposition theorems, interesting subgroups and their generators are described. Turning to $n$-mode quantum systems, we define and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/9509002","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/quant-ph/9509002/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T15:59:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nO8R04R0p9fDhrpq9qfM2D60Wf/raBJ/LsboBT41+VMqjx8kp9bQ1kWz/FmpnuBLLNmfj7HVoyJP8FSBFc7gDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T03:13:00.402792Z"},"content_sha256":"a7c125de94e248581b5287fea125b2aead63822ffb657d52e514d827b42bbc3e","schema_version":"1.0","event_id":"sha256:a7c125de94e248581b5287fea125b2aead63822ffb657d52e514d827b42bbc3e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XVCSEKBGEEKGJBQND7G37BABI2/bundle.json","state_url":"https://pith.science/pith/XVCSEKBGEEKGJBQND7G37BABI2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XVCSEKBGEEKGJBQND7G37BABI2/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":"2026-08-13T03:13:00Z","links":{"resolver":"https://pith.science/pith/XVCSEKBGEEKGJBQND7G37BABI2","bundle":"https://pith.science/pith/XVCSEKBGEEKGJBQND7G37BABI2/bundle.json","state":"https://pith.science/pith/XVCSEKBGEEKGJBQND7G37BABI2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XVCSEKBGEEKGJBQND7G37BABI2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1995:XVCSEKBGEEKGJBQND7G37BABI2","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":"247d6ab4cdd4f83354046d1f4e83e31d675fb963086a0f16fe2f0e90727259e9","cross_cats_sorted":[],"license":"","primary_cat":"quant-ph","submitted_at":"1995-09-05T04:20:00Z","title_canon_sha256":"8f995352223855f29f149ca6458f8ac4de8ead1b0155b7bd21b2d8ee2ab39914"},"schema_version":"1.0","source":{"id":"quant-ph/9509002","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"quant-ph/9509002","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"arxiv_version","alias_value":"quant-ph/9509002v3","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.quant-ph/9509002","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"pith_short_12","alias_value":"XVCSEKBGEEKG","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"pith_short_16","alias_value":"XVCSEKBGEEKGJBQN","created_at":"2026-07-04T15:59:57Z"},{"alias_kind":"pith_short_8","alias_value":"XVCSEKBG","created_at":"2026-07-04T15:59:57Z"}],"graph_snapshots":[{"event_id":"sha256:a7c125de94e248581b5287fea125b2aead63822ffb657d52e514d827b42bbc3e","target":"graph","created_at":"2026-07-04T15:59:57Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/quant-ph/9509002/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"text of abstract (We present a utilitarian review of the family of matrix groups $Sp(2n,\\Re)$, in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of $Sp(2n,\\Re)$. Global decomposition theorems, interesting subgroups and their generators are described. Turning to $n$-mode quantum systems, we define and ","authors_text":"Arvind, B. Dutta, N. Mukunda, R. Simon","cross_cats":[],"headline":"","license":"","primary_cat":"quant-ph","submitted_at":"1995-09-05T04:20:00Z","title":"The Real Symplectic Groups in Quantum Mechanics and Optics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"quant-ph/9509002","kind":"arxiv","version":3},"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":"sha256:561822379b95d34ba73a577ca2d1daeee0f0f0462e891c5cc7d11cb567f6d67d","target":"record","created_at":"2026-07-04T15:59:57Z","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":"247d6ab4cdd4f83354046d1f4e83e31d675fb963086a0f16fe2f0e90727259e9","cross_cats_sorted":[],"license":"","primary_cat":"quant-ph","submitted_at":"1995-09-05T04:20:00Z","title_canon_sha256":"8f995352223855f29f149ca6458f8ac4de8ead1b0155b7bd21b2d8ee2ab39914"},"schema_version":"1.0","source":{"id":"quant-ph/9509002","kind":"arxiv","version":3}},"canonical_sha256":"bd45222826211464860d1fcdbf840146877781a568227ac5fb1e5c1bf0250f0d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd45222826211464860d1fcdbf840146877781a568227ac5fb1e5c1bf0250f0d","first_computed_at":"2026-07-04T15:59:57.441817Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:59:57.441817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NnTvCtBKUp/apb/OA05OYWLFETCHqFPv7VbnQlUGaYCZE7QbELflzBrxgejjaN1o9qpq/HnU/QADCjEkbcvIDw==","signature_status":"signed_v1","signed_at":"2026-07-04T15:59:57.442173Z","signed_message":"canonical_sha256_bytes"},"source_id":"quant-ph/9509002","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:561822379b95d34ba73a577ca2d1daeee0f0f0462e891c5cc7d11cb567f6d67d","sha256:a7c125de94e248581b5287fea125b2aead63822ffb657d52e514d827b42bbc3e"],"state_sha256":"2ee56152de03ffe4fcde663a5ae30dd1ef0afa4cecac3bb1f176e15502ae21e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/ygCwIb7UTo0QuUbHpzaRlJ/JbKtA4yORGvyVGi8hXCYns8VAG+DFwKo/e3Ec+LtJD2a56fC3tdbkxo1AyM8AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T03:13:00.407974Z","bundle_sha256":"23a6bdfe7f1fafcd4baea89a853a05429f8e0a6a4d2aecfdf4b51f6366d858ef"}}