{
  "title": "Objects, hypotheses, licensed conclusions, and stopping conditions",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "axioms-reconstruction-hypothesis-conclusion-table",
  "caption": "Objects, hypotheses, licensed conclusions, and stopping conditions",
  "columns": [
    {
      "text": "Starting object",
      "html": "Starting object"
    },
    {
      "text": "Domain and regularity",
      "html": "Domain and regularity"
    },
    {
      "text": "Essential input",
      "html": "Essential input"
    },
    {
      "text": "Construction or theorem",
      "html": "Construction or theorem"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "What is not implied",
      "html": "What is not implied"
    },
    {
      "text": "Boundary test",
      "html": "Boundary test"
    }
  ],
  "rows": [
    {
      "Starting object": {
        "text": "Smeared fields",
        "html": "Smeared fields"
      },
      "Domain and regularity": {
        "text": "Operator-valued tempered distributions on one common invariant dense domain",
        "html": "Operator-valued tempered distributions on one common invariant dense domain"
      },
      "Essential input": {
        "text": "Vacuum, Poincaré covariance, positive energy, adjoints, and local or graded-local exchange",
        "html": "Vacuum, Poincaré covariance, positive energy, adjoints, and local or graded-local exchange"
      },
      "Construction or theorem": {
        "text": "Take all vacuum expectation values in their stated order",
        "html": "Take all vacuum expectation values in their stated order"
      },
      "Licensed conclusion": {
        "text": "A complete Wightman hierarchy with the inherited distributional and symmetry properties",
        "html": "A complete Wightman hierarchy with the inherited distributional and symmetry properties"
      },
      "What is not implied": {
        "text": "Pointwise field operators or positivity from covariance alone",
        "html": "Pointwise field operators or positivity from covariance alone"
      },
      "Boundary test": {
        "text": "Attempt to multiply unsmeared fields or change domains between factors",
        "html": "Attempt to multiply unsmeared fields or change domains between factors"
      }
    },
    {
      "Starting object": {
        "text": "Vacuum hierarchy",
        "html": "Vacuum hierarchy"
      },
      "Domain and regularity": {
        "text": "All tempered distributions W n , not only the two-point function",
        "html": "All tempered distributions <var>W</var><sub>n</sub>, not only the two-point function"
      },
      "Essential input": {
        "text": "Normalization, Hermiticity, covariance, spectral support, locality, and positivity of every polynomial",
        "html": "Normalization, Hermiticity, covariance, spectral support, locality, and positivity of every polynomial"
      },
      "Construction or theorem": {
        "text": "Quotient the Borchers algebra by null vectors and complete",
        "html": "Quotient the Borchers algebra by null vectors and complete"
      },
      "Licensed conclusion": {
        "text": "A cyclic Hilbert-space theory, vacuum, Poincaré action, and fields, unique up to unitary equivalence",
        "html": "A cyclic Hilbert-space theory, vacuum, Poincaré action, and fields, unique up to unitary equivalence"
      },
      "What is not implied": {
        "text": "Reconstruction from a positive two-point kernel with arbitrary higher functions",
        "html": "Reconstruction from a positive two-point kernel with arbitrary higher functions"
      },
      "Boundary test": {
        "text": "Keep W 2 positive but choose an incompatible W 4",
        "html": "Keep <var>W</var><sub>2</sub> positive but choose an incompatible <var>W</var><sub>4</sub>"
      }
    },
    {
      "Starting object": {
        "text": "Relative-momentum distributions",
        "html": "Relative-momentum distributions"
      },
      "Domain and regularity": {
        "text": "Tempered distributions in translation-invariant difference variables",
        "html": "Tempered distributions in translation-invariant difference variables"
      },
      "Essential input": {
        "text": "Support in the appropriate products of the closed future cone and polynomial growth control",
        "html": "Support in the appropriate products of the closed future cone and polynomial growth control"
      },
      "Construction or theorem": {
        "text": "Fourier–Laplace transformation",
        "html": "Fourier–Laplace transformation"
      },
      "Licensed conclusion": {
        "text": "Holomorphic functions in primitive tubes with the original distributions as tempered boundary values",
        "html": "Holomorphic functions in primitive tubes with the original distributions as tempered boundary values"
      },
      "What is not implied": {
        "text": "The correct tube direction without fixing the Fourier sign, or arbitrary complex continuation",
        "html": "The correct tube direction without fixing the Fourier sign, or arbitrary complex continuation"
      },
      "Boundary test": {
        "text": "Reverse the Fourier exponential and check that the imaginary cone reverses",
        "html": "Reverse the Fourier exponential and check that the imaginary cone reverses"
      }
    },
    {
      "Starting object": {
        "text": "Ordered analytic functions",
        "html": "Ordered analytic functions"
      },
      "Domain and regularity": {
        "text": "Primitive and extended tubes with real Jost configurations on their boundaries",
        "html": "Primitive and extended tubes with real Jost configurations on their boundaries"
      },
      "Essential input": {
        "text": "Local or graded-local equality on a nonempty real open set, plus the relevant analytic continuation theorem",
        "html": "Local or graded-local equality on a nonempty real open set, plus the relevant analytic continuation theorem"
      },
      "Construction or theorem": {
        "text": "Edge-of-the-wedge continuation and complex Lorentz covariance",
        "html": "Edge-of-the-wedge continuation and complex Lorentz covariance"
      },
      "Licensed conclusion": {
        "text": "Equality of the specified orderings in their joined analytic domain",
        "html": "Equality of the specified orderings in their joined analytic domain"
      },
      "What is not implied": {
        "text": "Equality of arbitrary time-ordered functions or permutations outside the proved domain",
        "html": "Equality of arbitrary time-ordered functions or permutations outside the proved domain"
      },
      "Boundary test": {
        "text": "Choose a spacelike-looking configuration that fails the full Jost cone criterion",
        "html": "Choose a spacelike-looking configuration that fails the full Jost cone criterion"
      }
    },
    {
      "Starting object": {
        "text": "Covariant local field multiplet",
        "html": "Covariant local field multiplet"
      },
      "Domain and regularity": {
        "text": "Positive-metric Wightman theory with the representation and adjoint structure used by the theorem",
        "html": "Positive-metric Wightman theory with the representation and adjoint structure used by the theorem"
      },
      "Essential input": {
        "text": "Spectrum condition, locality, analyticity, and theorem-specific transformation laws",
        "html": "Spectrum condition, locality, analyticity, and theorem-specific transformation laws"
      },
      "Construction or theorem": {
        "text": "CPT or spin–statistics argument",
        "html": "CPT or spin–statistics argument"
      },
      "Licensed conclusion": {
        "text": "An antiunitary CPT implementation or the permitted statistics grading for that field class",
        "html": "An antiunitary CPT implementation or the permitted statistics grading for that field class"
      },
      "What is not implied": {
        "text": "The same formula for nonlocal fields, indefinite-metric gauge potentials, or braid statistics in low dimension",
        "html": "The same formula for nonlocal fields, indefinite-metric gauge potentials, or braid statistics in low dimension"
      },
      "Boundary test": {
        "text": "Remove positive metric or replace four-dimensional exchange by braid-group exchange",
        "html": "Remove positive metric or replace four-dimensional exchange by braid-group exchange"
      }
    },
    {
      "Starting object": {
        "text": "Two time-slice field theories",
        "html": "Two time-slice field theories"
      },
      "Domain and regularity": {
        "text": "Wightman fields satisfying the domain, covariance, irreducibility, and vacuum assumptions of the chosen Haag theorem",
        "html": "Wightman fields satisfying the domain, covariance, irreducibility, and vacuum assumptions of the chosen Haag theorem"
      },
      "Essential input": {
        "text": "A unitary identification at one time together with the theorem's other hypotheses",
        "html": "A unitary identification at one time together with the theorem's other hypotheses"
      },
      "Construction or theorem": {
        "text": "Haag–Hall–Wightman comparison",
        "html": "Haag–Hall–Wightman comparison"
      },
      "Licensed conclusion": {
        "text": "If one field is free, the identified field has the corresponding free vacuum functions under the stated scope",
        "html": "If one field is free, the identified field has the corresponding free vacuum functions under the stated scope"
      },
      "What is not implied": {
        "text": "Nonexistence of interacting QFT, or a no-go result for regulated or merely formal interaction pictures",
        "html": "Nonexistence of interacting QFT, or a no-go result for regulated or merely formal interaction pictures"
      },
      "Boundary test": {
        "text": "Introduce a finite cutoff and identify exactly which Wightman assumption no longer holds",
        "html": "Introduce a finite cutoff and identify exactly which Wightman assumption no longer holds"
      }
    }
  ]
}