{
  "schema_version": 1,
  "artifact_id": "qft.artifact.curved-spacetime.quantum-fields-cosmology.initial-density-matrix-closed-time-path",
  "title": "Initial density data on a closed time path",
  "generated_by": "figures-src/curved-spacetime/initial-density-matrix-closed-time-path.mjs",
  "source_revision": 1,
  "generated_on": "2026-08-23",
  "figure_kind": "original monochrome schematic; not to scale",
  "reader_question": "Where do the initial density matrix, unitary and adjoint evolution, final trace, bulk interaction vertices, and state-dependent boundary counterterms live on a closed time path?",
  "takeaway": "The initial kernel rho_0 joins the plus and minus fields only on eta_0; U and U-dagger traverse oppositely oriented contour branches that meet in the final trace, while state-dependent counterterms remain localized on eta_0.",
  "alt_text": "Closed-time-path diagram with a solid plus branch running from the initial slice eta_0 to eta_f and a dashed minus branch returning to eta_0. The density kernel connects phi_0 plus and phi_0 minus only at eta_0, boundary counterterms are marked there, bulk vertices lie between the endpoints, and the two final fields are identified by the trace.",
  "caption_semantics": "Schematic, not-to-scale closed time path. The solid plus branch carries U[phi_+] from eta_0 to eta_f; the dashed minus branch carries U-dagger[phi_-] back along the contour. The trace identifies phi_f^+=phi_f^-, whereas rho_0 and its state-dependent counterterms have support only on the initial slice.",
  "conventions": {
    "initial_slice": "eta=eta_0",
    "final_slice": "eta=eta_f",
    "physical_time_direction": "eta_0 to eta_f, left to right",
    "contour_orientation": "the plus branch is traversed eta_0 to eta_f and the minus branch eta_f to eta_0",
    "initial_kernel": "<phi_0^+|rho_0|phi_0^->=N exp(i S_0[phi_0^+,phi_0^-])",
    "hermiticity_condition": "S_0[phi_0^-,phi_0^+]=-conjugate(S_0[phi_0^+,phi_0^-]) up to an irrelevant normalization phase",
    "normalized_closed_path": "Z[0,0]=Tr(rho_0)=1",
    "trace_identification": "phi_f^+=phi_f^-",
    "arrow_meaning": "arrows encode contour traversal and operator ordering, not causal signal propagation"
  },
  "nodes": [
    {
      "id": "initial_plus",
      "label": "phi_0^+",
      "location": "upper endpoint on eta_0",
      "role": "plus-branch field entering U"
    },
    {
      "id": "initial_minus",
      "label": "phi_0^-",
      "location": "lower endpoint on eta_0",
      "role": "minus-branch field leaving U-dagger in contour order"
    },
    {
      "id": "initial_kernel",
      "label": "rho_0[phi_0^+,phi_0^-]",
      "location": "eta_0 only",
      "role": "joins the two initial boundary values and supplies the initial state"
    },
    {
      "id": "boundary_counterterms",
      "label": "S_ct^(eta_0)",
      "location": "eta_0 only",
      "role": "renormalizes state-dependent ultraviolet divergences localized on the initial slice"
    },
    {
      "id": "bulk_vertices",
      "label": "bulk vertices",
      "location": "eta_0<eta<eta_f on both branches",
      "role": "represents interaction insertions generated by the bulk action"
    },
    {
      "id": "final_trace",
      "label": "phi_f^+=phi_f^-",
      "location": "eta_f",
      "role": "identifies the two final boundary values and performs the trace"
    }
  ],
  "branches": [
    {
      "id": "plus",
      "label": "+ branch: U[phi_+]",
      "from": "initial_plus",
      "to": "final_trace",
      "direction": "eta_0 to eta_f",
      "line_style": "solid with right-pointing arrow"
    },
    {
      "id": "minus",
      "label": "- branch: U-dagger[phi_-]",
      "from": "final_trace",
      "to": "initial_minus",
      "direction": "eta_f to eta_0 in contour order",
      "line_style": "dashed with left-pointing arrow"
    }
  ],
  "relationships": [
    {
      "subject": "initial_kernel",
      "relation": "connects",
      "objects": [
        "initial_plus",
        "initial_minus"
      ],
      "support": "eta_0 only"
    },
    {
      "subject": "final_trace",
      "relation": "joins",
      "objects": [
        "plus",
        "minus"
      ],
      "support": "eta_f only"
    },
    {
      "subject": "bulk_vertices",
      "relation": "lie_on",
      "objects": [
        "plus",
        "minus"
      ],
      "support": "open interval eta_0<eta<eta_f"
    },
    {
      "subject": "boundary_counterterms",
      "relation": "renormalize_initial_state_terms_at",
      "objects": [
        "initial_kernel"
      ],
      "support": "eta_0 only"
    }
  ],
  "assumptions_and_limits": [
    "The diagram is schematic and has no metric scale or quantitative time spacing.",
    "The closed system evolves unitarily between eta_0 and eta_f before the final trace.",
    "The initial density operator is normalized; positivity is an additional constraint on S_0 and is not encoded geometrically.",
    "The drawing does not choose a background, interaction, regulator, vacuum, or Gaussian truncation.",
    "The marked boundary counterterms are state-dependent terms supported at eta_0, not bulk counterterms displaced toward the initial slice.",
    "The contour arrows express ordering; they do not assert that influences propagate backward in physical time."
  ],
  "reading_order": [
    "title",
    "initial slice eta_0",
    "initial plus and minus fields",
    "rho_0 initial kernel",
    "state counterterms on eta_0",
    "solid plus branch and its direction",
    "dashed minus branch and its direction",
    "bulk vertices",
    "final trace at eta_f",
    "physical-time axis"
  ],
  "accessibility_encoding": {
    "color_independence": "plus and minus branches differ by direct labels, line style, sign, and arrow direction; no relation is encoded by color alone",
    "canvas": "explicit white background",
    "structured_equivalence": "this JSON preserves every node, branch, direction, equation, relationship, assumption, caption claim, and alt-text claim in the SVG"
  },
  "scientific_checks": [
    {
      "check": "closed-contour topology",
      "result": "rho_0 connects the two eta_0 endpoints and the trace connects the two eta_f endpoints",
      "pass": true
    },
    {
      "check": "branch orientation",
      "result": "the plus arrow follows increasing physical time and the minus arrow returns in contour order",
      "pass": true
    },
    {
      "check": "unitary normalization limit",
      "result": "with identical sources on both branches, U-dagger U cancels inside Tr(rho_0), yielding Z[0,0]=1 for normalized rho_0",
      "pass": true
    },
    {
      "check": "support separation",
      "result": "bulk vertices appear strictly between eta_0 and eta_f, while rho_0 and S_ct^(eta_0) appear only on eta_0",
      "pass": true
    },
    {
      "check": "wrong-orientation negative control",
      "rejected_fixture": "draw both branch arrows from eta_0 to eta_f while retaining the labels U and U-dagger",
      "reason": "this would erase the return leg of the contour and no longer depict the operator product U rho_0 U-dagger inside the trace",
      "pass": true
    },
    {
      "check": "wrong-support negative control",
      "rejected_fixture": "place S_ct^(eta_0) on an interior bulk vertex at eta_0<eta<eta_f",
      "reason": "an initial-state counterterm is supported on the initial hypersurface; moving it into the bulk changes the operator class",
      "pass": true
    }
  ],
  "scientific_references": [
    {
      "authors": "J. Schwinger",
      "title": "Brownian Motion of a Quantum Oscillator",
      "journal": "Journal of Mathematical Physics",
      "volume": "2",
      "pages": "407-432",
      "year": 1961,
      "doi": "10.1063/1.1703727",
      "url": "https://doi.org/10.1063/1.1703727",
      "use": "closed-time-path expectation values"
    },
    {
      "authors": "H. Collins and R. Holman",
      "title": "Renormalization of initial conditions and the trans-Planckian problem of inflation",
      "journal": "Physical Review D",
      "volume": "71",
      "article": "085009",
      "year": 2005,
      "doi": "10.1103/PhysRevD.71.085009",
      "url": "https://doi.org/10.1103/PhysRevD.71.085009",
      "use": "initial-surface divergences and boundary counterterms"
    },
    {
      "authors": "N. Agarwal, R. Holman, A. J. Tolley, and J. Lin",
      "title": "Effective field theory and non-Gaussianity from general inflationary states",
      "journal": "Journal of High Energy Physics",
      "volume": "2013",
      "article": "085",
      "year": 2013,
      "doi": "10.1007/JHEP05(2013)085",
      "url": "https://doi.org/10.1007/JHEP05(2013)085",
      "use": "density-matrix and boundary-action formulation of general initial states"
    }
  ]
}