{
  "$schema": "https://qft.org/data/schemas/qft-many-body-comparison-1.0.0.schema.json",
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.many-body-green-functions-claim-validity-ledger",
  "table_id": "many-body-green-functions-claim-validity-ledger",
  "canonical_source": {
    "route": "/many-body-quantum-matter/many-body-green-functions/",
    "heading": "Comparison of claims and tests",
    "anchor": "comparison-of-claims-and-tests"
  },
  "caption": "Tests and limits of claims made with many-body Green functions.",
  "common_scope": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "qualifications": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "columns": [
    {
      "key": "claim",
      "label": "Claim"
    },
    {
      "key": "object_and_convention",
      "label": "Object and convention"
    },
    {
      "key": "necessary_test",
      "label": "Necessary test"
    },
    {
      "key": "what_the_test_establishes",
      "label": "What the test establishes"
    },
    {
      "key": "what_it_does_not_establish",
      "label": "What it does not establish"
    }
  ],
  "rows": [
    {
      "key": "spectral-weight-is-normalized",
      "row_id": "many-body-green-functions-claim-validity-ledger--spectral-weight-is-normalized",
      "cells": {
        "claim": {
          "markdown": "Spectral weight is normalized",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "Canonical one-particle $A=-2\\operatorname{Im}G^R$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "A=-2\\operatorname{Im}G^R"
            }
          ],
          "links": []
        },
        "necessary_test": {
          "markdown": "Zeroth moment from the equal-time anticommutator",
          "math": [],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Operator normalization and total one-particle weight",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "Correct placement of the weight",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-peak-is-a-quasiparticle",
      "row_id": "many-body-green-functions-claim-validity-ledger--a-peak-is-a-quasiparticle",
      "cells": {
        "claim": {
          "markdown": "A peak is a quasiparticle",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "Retarded pole on the physical continuation",
          "math": [],
          "links": []
        },
        "necessary_test": {
          "markdown": "Small width, finite residue, stable pole, separated scales",
          "math": [],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Long-lived one-particle-like excitation",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "Transport lifetime or Landau-liquid behavior",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-density-approximation-is-consistent",
      "row_id": "many-body-green-functions-claim-validity-ledger--a-density-approximation-is-consistent",
      "cells": {
        "claim": {
          "markdown": "A density approximation is consistent",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "$\\chi^R_{nn}$ with contact terms",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\chi^R_{nn}"
            }
          ],
          "links": []
        },
        "necessary_test": {
          "markdown": "Continuity Ward identity and $f$-sum rule",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "f"
            }
          ],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Conservation and normalization",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "Quantitative accuracy at all $q,\\omega$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "q,\\omega"
            }
          ],
          "links": []
        }
      }
    },
    {
      "key": "rpa-screening-is-controlled",
      "row_id": "many-body-green-functions-claim-validity-ledger--rpa-screening-is-controlled",
      "cells": {
        "claim": {
          "markdown": "RPA screening is controlled",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "$\\varepsilon=1-v\\Pi_0$ with neutral background",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\varepsilon=1-v\\Pi_0"
            }
          ],
          "links": []
        },
        "necessary_test": {
          "markdown": "High-density or large-degeneracy counting plus vertex estimate",
          "math": [],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Parametric reason bubble chains dominate",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "Universal reliability at strong coupling",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-loss-peak-is-a-plasmon",
      "row_id": "many-body-green-functions-claim-validity-ledger--a-loss-peak-is-a-plasmon",
      "cells": {
        "claim": {
          "markdown": "A loss peak is a plasmon",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "Pole of $\\varepsilon^{-1}$ on the retarded sheet",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\varepsilon^{-1}"
            }
          ],
          "links": []
        },
        "necessary_test": {
          "markdown": "Complex pole, residue, and continuum separation",
          "math": [],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Collective charge mode with a damping rate",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "A perfectly sharp excitation",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-bethe-salpeter-eigenvalue-is-large",
      "row_id": "many-body-green-functions-claim-validity-ledger--a-bethe-salpeter-eigenvalue-is-large",
      "cells": {
        "claim": {
          "markdown": "A Bethe–Salpeter eigenvalue is large",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "Named irreducible channel and kernel",
          "math": [],
          "links": []
        },
        "necessary_test": {
          "markdown": "Grid, basis, crossing, and finite-size checks",
          "math": [],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Enhanced susceptibility in that approximation",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "Established broken symmetry",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-scheme-is-phi-derivable",
      "row_id": "many-body-green-functions-claim-validity-ledger--a-scheme-is-phi-derivable",
      "cells": {
        "claim": {
          "markdown": "A scheme is Φ-derivable",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "$\\Sigma=\\delta\\Phi/\\delta G$ solved self-consistently",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\Sigma=\\delta\\Phi/\\delta G"
            }
          ],
          "links": []
        },
        "necessary_test": {
          "markdown": "Functional symmetry and consistent response vertex",
          "math": [],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Named macroscopic conservation laws",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "Crossing, positivity, uniqueness, or accuracy",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "gw-gives-a-useful-spectrum",
      "row_id": "many-body-green-functions-claim-validity-ledger--gw-gives-a-useful-spectrum",
      "cells": {
        "claim": {
          "markdown": "GW gives a useful spectrum",
          "math": [],
          "links": []
        },
        "object_and_convention": {
          "markdown": "Declared $G_0W_0$, eigenvalue-only, quasiparticle self-consistent, or fully self-consistent scheme",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "G_0W_0"
            }
          ],
          "links": []
        },
        "necessary_test": {
          "markdown": "Starting point, self-consistency, vertex, sum-rule, and benchmark tests",
          "math": [],
          "links": []
        },
        "what_the_test_establishes": {
          "markdown": "Performance in the tested regime",
          "math": [],
          "links": []
        },
        "what_it_does_not_establish": {
          "markdown": "A general controlled expansion",
          "math": [],
          "links": []
        }
      }
    }
  ],
  "references": [
    {
      "id": "ref-441de122a0fc180e",
      "markdown": "Baym, Gordon. “Self-Consistent Approximations in Many-Body Systems.” *Physical Review* **127** (1962): 1391–1401. [DOI](https://doi.org/10.1103/PhysRev.127.1391).",
      "urls": [
        "https://doi.org/10.1103/PhysRev.127.1391"
      ]
    }
  ],
  "reference_scope": "The reference catalog reproduces the canonical chapter bibliography. Cell links are associated with references only when their resolved URL matches; uncited chapter references are not asserted as claim-specific support."
}