{
  "title": "Hypotheses, licensed conclusions, and decisive failure tests in relativistic scattering theory",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "scattering-infrared-particle-structure-hypothesis-conclusion-table",
  "caption": "Hypotheses, licensed conclusions, and decisive failure tests in relativistic scattering theory",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Spectral or scattering hypotheses",
      "html": "Spectral or scattering hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or extension",
      "html": "Excluded converse or extension"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Joint spectral projection on a positive-energy mass hyperboloid",
        "html": "Joint spectral projection on a positive-energy mass hyperboloid"
      },
      "Spectral or scattering hypotheses": {
        "text": "The shell is nonzero and isolated in the real translation spectrum; the Poincaré representation is strongly continuous.",
        "html": "The shell is nonzero and isolated in the real translation spectrum; the Poincaré representation is strongly continuous."
      },
      "Licensed conclusion": {
        "text": "A stable one-particle subspace carrying Wigner representations.",
        "html": "A stable one-particle subspace carrying Wigner representations."
      },
      "Excluded converse or extension": {
        "text": "A resonance pole or threshold enhancement does not imply such a projection.",
        "html": "A resonance pole or threshold enhancement does not imply such a projection."
      },
      "Adversarial check": {
        "text": "Replace the shell by a second-sheet pole: the spectral projection is zero, so the construction stops.",
        "html": "Replace the shell by a second-sheet pole: the spectral projection is zero, so the construction stops."
      }
    },
    {
      "Object and domain": {
        "text": "Haag–Ruelle products on a common finite-energy domain",
        "html": "Haag–Ruelle products on a common finite-energy domain"
      },
      "Spectral or scattering hypotheses": {
        "text": "Stable massive shell, locality or controlled almost locality, smooth wave packets, and pairwise disjoint velocity supports.",
        "html": "Stable massive shell, locality or controlled almost locality, smooth wave packets, and pairwise disjoint velocity supports."
      },
      "Licensed conclusion": {
        "text": "Strong in- and out-state limits with Fock inner products for the constructed species.",
        "html": "Strong in- and out-state limits with Fock inner products for the constructed species."
      },
      "Excluded converse or extension": {
        "text": "Existence of these states does not imply that they span the physical Hilbert space.",
        "html": "Existence of these states does not imply that they span the physical Hilbert space."
      },
      "Adversarial check": {
        "text": "Overlap two velocity supports, or remove shell isolation: the commutator estimate no longer proves convergence.",
        "html": "Overlap two velocity supports, or remove shell isolation: the commutator estimate no longer proves convergence."
      }
    },
    {
      "Object and domain": {
        "text": "Møller maps from asymptotic Fock space",
        "html": "Møller maps from asymptotic Fock space"
      },
      "Spectral or scattering hypotheses": {
        "text": "Haag–Ruelle limits exist and their inner products factorize on the finite-particle core.",
        "html": "Haag–Ruelle limits exist and their inner products factorize on the finite-particle core."
      },
      "Licensed conclusion": {
        "text": "Isometric wave operators and a scattering operator on the appropriate asymptotic ranges.",
        "html": "Isometric wave operators and a scattering operator on the appropriate asymptotic ranges."
      },
      "Excluded converse or extension": {
        "text": "An isometry is not automatically unitary; range density is an additional completeness theorem.",
        "html": "An isometry is not automatically unitary; range density is an additional completeness theorem."
      },
      "Adversarial check": {
        "text": "Add a nonzero orthogonal bound-state or topological sector: the constructed map remains isometric but is not onto.",
        "html": "Add a nonzero orthogonal bound-state or topological sector: the constructed map remains isometric but is not onto."
      }
    },
    {
      "Object and domain": {
        "text": "LSZ-smeared time-ordered distributions",
        "html": "LSZ-smeared time-ordered distributions"
      },
      "Spectral or scattering hypotheses": {
        "text": "Isolated stable-particle poles with nonzero residue, asymptotic states, tempered boundary values, and controlled amputation.",
        "html": "Isolated stable-particle poles with nonzero residue, asymptotic states, tempered boundary values, and controlled amputation."
      },
      "Licensed conclusion": {
        "text": "On-shell scattering matrix elements as limits of amputated distributions.",
        "html": "On-shell scattering matrix elements as limits of amputated distributions."
      },
      "Excluded converse or extension": {
        "text": "The formula does not extend unchanged to unstable particles or charged infraparticles without a sharp pole.",
        "html": "The formula does not extend unchanged to unstable particles or charged infraparticles without a sharp pole."
      },
      "Adversarial check": {
        "text": "Apply electron LSZ in charged QED after the pole has become a continuous threshold: the external-state limit is absent.",
        "html": "Apply electron LSZ in charged QED after the pole has become a continuous threshold: the external-state limit is absent."
      }
    },
    {
      "Object and domain": {
        "text": "Almost-local energy-decreasing detectors and particle weights",
        "html": "Almost-local energy-decreasing detectors and particle weights"
      },
      "Spectral or scattering hypotheses": {
        "text": "Bounded-energy states, energy–momentum transfer away from the forward cone, propagation estimates, and an appropriate large-time topology.",
        "html": "Bounded-energy states, energy–momentum transfer away from the forward cone, propagation estimates, and an appropriate large-time topology."
      },
      "Licensed conclusion": {
        "text": "Asymptotic velocity-sensitive functionals or weights, sometimes without normalizable particle vectors.",
        "html": "Asymptotic velocity-sensitive functionals or weights, sometimes without normalizable particle vectors."
      },
      "Excluded converse or extension": {
        "text": "A detector limit need not produce a sharp-mass state or a complete particle basis.",
        "html": "A detector limit need not produce a sharp-mass state or a complete particle basis."
      },
      "Adversarial check": {
        "text": "Use a strictly local positive operator with nonzero vacuum response: persistent vacuum contamination defeats particle selectivity.",
        "html": "Use a strictly local positive operator with nonzero vacuum response: persistent vacuum contamination defeats particle selectivity."
      }
    },
    {
      "Object and domain": {
        "text": "Null radiation fields and charged Gauss-law sectors",
        "html": "Null radiation fields and charged Gauss-law sectors"
      },
      "Spectral or scattering hypotheses": {
        "text": "For neutral radiation, suitable light-cone propagation and decay; for charge, the Gauss-law flux relation and noncompact localization are retained.",
        "html": "For neutral radiation, suitable light-cone propagation and decay; for charge, the Gauss-law flux relation and noncompact localization are retained."
      },
      "Licensed conclusion": {
        "text": "Neutral asymptotic radiation fields, or charged infrared sectors distinguished by long-range data.",
        "html": "Neutral asymptotic radiation fields, or charged infrared sectors distinguished by long-range data."
      },
      "Excluded converse or extension": {
        "text": "Neither conclusion supplies an ordinary compactly localized charged Fock particle.",
        "html": "Neither conclusion supplies an ordinary compactly localized charged Fock particle."
      },
      "Adversarial check": {
        "text": "Insert a long-range charged field into a neutral radiation theorem: its localization and decay estimates fail.",
        "html": "Insert a long-range charged field into a neutral radiation theorem: its localization and decay estimates fail."
      }
    },
    {
      "Object and domain": {
        "text": "Massive scattering amplitudes as analytic boundary values",
        "html": "Massive scattering amplitudes as analytic boundary values"
      },
      "Spectral or scattering hypotheses": {
        "text": "Locality, positive energy, stable LSZ particles, a mass gap, unitarity, a specified complex domain, and polynomial growth where a bound requires it.",
        "html": "Locality, positive energy, stable LSZ particles, a mass gap, unitarity, a specified complex domain, and polynomial growth where a bound requires it."
      },
      "Licensed conclusion": {
        "text": "Qualified crossing continuations, dispersion relations, and rigorous bounds inside the proved domain.",
        "html": "Qualified crossing continuations, dispersion relations, and rigorous bounds inside the proved domain."
      },
      "Excluded converse or extension": {
        "text": "There is no automatic global crossing identity or massless forward Froissart bound.",
        "html": "There is no automatic global crossing identity or massless forward Froissart bound."
      },
      "Adversarial check": {
        "text": "Add an unsubtracted massless exchange pole at zero momentum transfer: the gap and bounded-domain argument fail.",
        "html": "Add an unsubtracted massless exchange pole at zero momentum transfer: the gap and bounded-domain argument fail."
      }
    }
  ]
}