{
  "title": "Three comparison axes for one free scalar",
  "creator": "OpenAI Codex for QFT.org",
  "created_on": "2026-08-09",
  "revised_on": "2026-09-08",
  "license": {
    "spdx_id": "CC-BY-4.0",
    "url": "https://creativecommons.org/licenses/by/4.0/",
    "scope": "This original diagram and its editable TikZ source; cited third-party works retain their own terms."
  },
  "source": {
    "url": "/downloads/foundations/formulation-map.tex",
    "sha256": "3833e6d4a8d9dfb1b18a5dbfc116edffbcbc9cb4302663a789a753b02a2891b9"
  },
  "figure": {
    "url": "/figures/foundations/formulation-map.svg",
    "sha256": "8914082bb89514d3c5de78212550ce72507dbc62e767cca5ac0dd1fad6611bc7"
  },
  "changes": "Three vertical panels replace the wide comparison lanes. Larger labels and separate arrow gaps make exact, conditional and barred relations visible. Their scientific hypotheses are preserved.",
  "reproduction": {
    "toolchain": "TeX Live 2025, pdfTeX 3.141592653-2.6-1.40.28; dvisvgm 3.4.3",
    "instructions": "Download the editable source into an empty working directory and run these commands. They reproduce the visible vector artwork; glyph identifiers and their definition order can vary between installations.",
    "commands": [
      "latex -interaction=nonstopmode -halt-on-error formulation-map.tex",
      "dvisvgm --no-fonts --exact-bbox -o formulation-map.svg formulation-map.dvi"
    ],
    "published_svg_normalization": "The published SVG uses LF newlines and canonical glyph definitions. Identical path definitions are deduplicated and sorted by path data; glyph-use aliases are deduplicated and sorted by their recursively resolved target and transform. Path and use identifiers are assigned sequentially within their respective groups, and all references are rewritten. Geometry, drawing order and viewBox are preserved. The figure checksum identifies the canonical published file, not arbitrary raw compiler output."
  },
  "references": [
    {
      "citation": "Fewster, Christopher J., and Kasia Rejzner. Algebraic Quantum Field Theory—An Introduction. In Progress and Visions in Quantum Theory in View of Gravity, pp. 1–61. Birkhäuser, 2020.",
      "doi": "https://doi.org/10.1007/978-3-030-38941-3_1",
      "open_pdf": "https://arxiv.org/pdf/1904.04051v2",
      "locator": "2019 arXiv v2 preprint: sections 2.2–2.3, pp. 5–9; sections 4.1–4.3, pp. 13–23.",
      "used_for": "Algebra, state and representation distinctions; the neutral free field, Weyl relations, local algebras and regular states. No graphical material was used."
    },
    {
      "citation": "Hollands, Stefan, and Robert M. Wald. Quantum Fields in Curved Spacetime. Physics Reports 574 (2015): 1–35.",
      "doi": "https://doi.org/10.1016/j.physrep.2015.02.001",
      "open_pdf": "https://arxiv.org/pdf/1401.2026v2",
      "locator": "2014 arXiv v2 preprint: section 1, pp. 2–9.",
      "used_for": "The scope and state dependence of particle, Euclidean, functional, perturbative and algebraic descriptions. This introduction is not an unconditional reconstruction theorem. No graphical material was used."
    },
    {
      "citation": "Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014.",
      "doi": "https://doi.org/10.1017/9781139540940",
      "locator": "Sections 14.2–14.3, printed pp. 254–264.",
      "used_for": "The finite time-slicing, Gaussian kernel and source-correlator comparison. No graphical material was used."
    },
    {
      "citation": "Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995.",
      "doi": "https://doi.org/10.1017/CBO9781139644167",
      "locator": "Sections 2.1–2.5, printed pp. 49–73; section 5.1, printed pp. 191–200.",
      "used_for": "State, symmetry and free-field background after translating the metric convention. No graphical material was used."
    }
  ]
}