{
  "title": "Free-scalar mode normalization from Klein-Gordon pairing to later quantum checks",
  "creator": "OpenAI Codex for QFT.org",
  "created_on": "2026-08-09",
  "revised_on": "2026-09-08",
  "license": {
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    "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."
  },
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    "commands": [
      "latex -interaction=nonstopmode -halt-on-error scalar-mode-normalization.tex",
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  },
  "references": [
    {
      "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 2.1, pp. 9–12; pp. 16–17, equations (22)–(25); p. 46, equation (85). The torus two-point weight is 1/(2 omega V) in three spatial dimensions.",
      "used_for": "Classical solution pairing, later field algebra, and the torus 1/sqrt(2 omega V) normalization. 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 2.2 and 3.2, printed pp. 17-19 and 31-32",
      "used_for": "Oscillator reduction and massive scalar action/equation. No graphical material was used."
    },
    {
      "citation": "Srednicki, Mark. Quantum Field Theory. Author-hosted manuscript, 2006; Cambridge University Press, 2007.",
      "stable_url": "https://web.physics.ucsb.edu/~mark/qft.html",
      "locator": "Section 3, printed pp. 37-40",
      "used_for": "Dispersion, real-field mode expansion, invariant measure, and coefficient recovery after signature translation. 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": "Section 5.2, printed p. 201",
      "used_for": "Positive-energy mode normalization in the quantum scalar construction. No graphical material was used."
    }
  ]
}