{"paper":{"title":"Countable basis for free electromagnetic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Free Maxwell fields admit a countable basis of polychromatic single-photon waves that lie inside the Hilbert space.","cross_cats":["math.MP","physics.optics"],"primary_cat":"math-ph","authors_text":"Ivan Fernandez-Corbaton","submitted_at":"2026-01-19T10:05:08Z","abstract_excerpt":"Polychromatic electromagnetic fields are typically expanded as integrals over monochromatic fields, such as plane waves, multipolar fields, or Bessel beams. However, monochromatic fields do not belong to the Hilbert space of free Maxwell fields, since their norms diverge. Moreover, the continuous frequency integrals involved in such expansions complicate the treatment of light--matter interactions via the scattering operator. Here, we identify and study a polychromatic basis for free Maxwell fields whose basis vectors belong to the Hilbert space. These vectors are defined as simultaneous eigen"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The basis set is countable, and the Hilbert space is separable and isomorphic to ℓ². Each basis vector represents a polychromatic single-photon wave with quantized energy and a wavelet-like temporal dependence.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That four commuting operators exist whose simultaneous eigenstates with integer eigenvalues form a complete basis for the Hilbert space of free Maxwell fields.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A countable basis of polychromatic single-photon waves for free Maxwell fields is constructed as simultaneous eigenstates of four commuting operators with integer eigenvalues, making the space isomorphic to ℓ².","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Free Maxwell fields admit a countable basis of polychromatic single-photon waves that lie inside the Hilbert space.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6b9d946ddc88dd1e86bdaaeafc1f42cc5e19c76d0b1b5987bbdcef8a4e964549"},"source":{"id":"2601.12911","kind":"arxiv","version":1},"verdict":{"id":"c5cf5070-9032-4f20-b859-a1949af8e306","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T13:28:31.811988Z","strongest_claim":"The basis set is countable, and the Hilbert space is separable and isomorphic to ℓ². Each basis vector represents a polychromatic single-photon wave with quantized energy and a wavelet-like temporal dependence.","one_line_summary":"A countable basis of polychromatic single-photon waves for free Maxwell fields is constructed as simultaneous eigenstates of four commuting operators with integer eigenvalues, making the space isomorphic to ℓ².","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That four commuting operators exist whose simultaneous eigenstates with integer eigenvalues form a complete basis for the Hilbert space of free Maxwell fields.","pith_extraction_headline":"Free Maxwell fields admit a countable basis of polychromatic single-photon waves that lie inside the Hilbert space."},"references":{"count":47,"sample":[{"doi":"","year":null,"title":"The second row, which contains Eq","work_id":"3874bbd8-164b-48a2-8d61-f435841164d3","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1964,"title":"J. S. Lomont and H. E. Moses, The representations of the inhomogeneous lorentz group in terms of an angular momentum basis, Journal of Mathematical Physics5, 294 (1964)","work_id":"c810bf38-51f5-4d93-b878-6b1a90c98f10","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"H. A. Kastrup, Conformal group and its connection with an indefinite metric in hilbert space, Phys. Rev.140, B183 (1965)","work_id":"0136df23-dfa7-4a3f-af2f-acc06e91ce9b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"H. E. Moses, Transformation from a linear momentum to an angular momentum basis for particles of zero mass and finite spin, Journal of Mathematical Physics6, 928 (1965)","work_id":"af22da2d-8218-442f-a3ab-ef1360f88f1f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"H. E. Moses, Transformation from a linear momentum to an angular momentum basis for relativistic particles of nonzero mass and any spin, Journal of Mathematical Physics6, 1244 (1965)","work_id":"6907d146-cedc-4d44-b118-9ebf8ef3ffdd","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":47,"snapshot_sha256":"36c1a5bcd51132ddb5e1de710d399e414d4d28b29370ab6b35f432c536bd3551","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"27010414d775e836f9d8e872fde09dde70742820d9edda0cbe811a0d12daae97"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}