{
  "schema_version": 1,
  "artifact_id": "qft.artifact.holography-quantum-gravity.quantum-cosmology-and-singularity-resolution-programs.wheeler-dewitt-four-branch-current-map",
  "owner_page_id": "qft.topic.holography-quantum-gravity.wheeler-dewitt-cosmology-and-boundary-conditions",
  "title": "Four Wheeler–DeWitt Branches and Their Superspace Currents",
  "kind": "superspace-current sign schematic",
  "source_revision": 1,
  "source_date": "2026-08-30",
  "creator": "OpenAI Codex, for QFT.org",
  "original_work": true,
  "schematic": true,
  "not_to_scale": true,
  "reader_question": "Why does selecting positive scalar-clock frequency not select an expanding or contracting geometry?",
  "takeaway": "The sign of p_varphi selects scalar-clock frequency, while the independent sign of p_alpha selects expansion or contraction. In the declared convention J^varphi = p_varphi and J^alpha = -p_alpha, so both expanding and contracting currents occur in either frequency sector.",
  "scientific_status": "exact sign dictionary for the declared flat homogeneous massless-scalar minisuperspace convention; geometric lengths and arrow positions are schematic",
  "claim_ceiling": "The diagram classifies four plane-wave branches of one flat homogeneous massless-scalar reduction. It does not make the raw Klein-Gordon current positive, select a physical state or boundary condition, prove singularity avoidance, extend through a WKB turning point, or establish factor-ordering, measure, clock-choice, perturbative, or full-theory independence.",
  "conventions": {
    "logarithmic_scale_variable": "alpha = ln(a/a_star)",
    "rescaled_scalar_clock": "varphi = phi/(sqrt(6) M_Pl)",
    "reduced_constraint": "C = p_varphi^2 - p_alpha^2 = 0",
    "superspace_inverse_metric": "G^(AB) = diag(+1,-1) in coordinates (varphi,alpha)",
    "current": "J^A = -(i/2) Psi* <-> partial^A Psi",
    "plane_wave": "Psi is proportional to exp[i(p_varphi varphi + p_alpha alpha)]",
    "current_components": "J^varphi = p_varphi |Psi|^2 and J^alpha = -p_alpha |Psi|^2; the common positive factor |Psi|^2 is suppressed in the figure labels",
    "positive_frequency_equation": "-i partial_varphi Psi = sqrt(Theta) Psi with Theta = -partial_alpha^2, so p_varphi = +k and exp(+i k varphi) define the selected positive scalar-clock-frequency sector",
    "positive_frequency_modes": "u_(k,s) = exp(i k varphi + i s k alpha)/sqrt(4 pi k), with k > 0 and s = sign(p_alpha)",
    "expansion_sign": "For positive lapse, p_alpha = -6 V_0 M_Pl^2 a^3 H_FLRW; p_alpha < 0 is expanding and p_alpha > 0 is contracting",
    "nullness": "(J^varphi)^2 - (J^alpha)^2 = p_varphi^2 - p_alpha^2 = 0"
  },
  "branches": [
    {
      "quadrant": "upper right",
      "clock_frequency": "positive",
      "geometry": "expanding",
      "p_varphi_over_k": 1,
      "p_alpha_over_k": -1,
      "J_varphi_over_k": 1,
      "J_alpha_over_k": 1,
      "mode": "exp[i k(varphi-alpha)]/sqrt(4 pi k)",
      "chirality_s": -1
    },
    {
      "quadrant": "upper left",
      "clock_frequency": "positive",
      "geometry": "contracting",
      "p_varphi_over_k": 1,
      "p_alpha_over_k": 1,
      "J_varphi_over_k": 1,
      "J_alpha_over_k": -1,
      "mode": "exp[i k(varphi+alpha)]/sqrt(4 pi k)",
      "chirality_s": 1
    },
    {
      "quadrant": "lower right",
      "clock_frequency": "negative",
      "geometry": "expanding",
      "p_varphi_over_k": -1,
      "p_alpha_over_k": -1,
      "J_varphi_over_k": -1,
      "J_alpha_over_k": 1,
      "mode": "exp[-i k(varphi+alpha)]/sqrt(4 pi k)",
      "chirality_s": -1
    },
    {
      "quadrant": "lower left",
      "clock_frequency": "negative",
      "geometry": "contracting",
      "p_varphi_over_k": -1,
      "p_alpha_over_k": 1,
      "J_varphi_over_k": -1,
      "J_alpha_over_k": -1,
      "mode": "exp[-i k(varphi-alpha)]/sqrt(4 pi k)",
      "chirality_s": 1
    }
  ],
  "slice_dictionary": [
    {
      "slice": "Sigma_varphi0: varphi = varphi_0",
      "normal_direction": "varphi",
      "flux_component": "J^varphi",
      "interpretation": "Its sign distinguishes positive from negative scalar-clock frequency; positivity requires a frequency-sector restriction or another positive-inner-product construction."
    },
    {
      "slice": "Sigma_alpha0: alpha = alpha_0",
      "normal_direction": "alpha",
      "flux_component": "J^alpha",
      "interpretation": "Its sign distinguishes the expanding and contracting chiral currents in this exactly free model."
    }
  ],
  "visual_encoding": {
    "solid_black_rays": "p_varphi > 0 positive scalar-clock-frequency sector",
    "dashed_gray_rays": "p_varphi < 0 negative scalar-clock-frequency sector",
    "open_circle": "p_alpha < 0 expanding geometry",
    "filled_square": "p_alpha > 0 contracting geometry",
    "horizontal_axis": "representative constant-varphi slice with its varphi normal",
    "vertical_axis": "representative constant-alpha slice with its alpha normal",
    "origin": "k = 0 cone tip, explicitly excluded from the four k > 0 branches",
    "direct_labels": "each quadrant states p_varphi, p_alpha, J^varphi, J^alpha, clock-frequency sign, and geometry label",
    "canvas": "explicit white background; no physical distinction is encoded by color alone"
  },
  "independent_checks": [
    "For every branch, J^varphi equals p_varphi and J^alpha equals minus p_alpha.",
    "For every branch, (J^varphi)^2 - (J^alpha)^2 = 0.",
    "Both upper branches have J^varphi > 0 but opposite J^alpha signs, so positive clock frequency alone does not choose expansion or contraction.",
    "Both right branches have p_alpha < 0 and are expanding despite opposite clock-frequency signs.",
    "Both left branches have p_alpha > 0 and are contracting despite opposite clock-frequency signs.",
    "Complex conjugation reverses both canonical momenta and both current components; it therefore maps diagonally opposite quadrants.",
    "The expanding and contracting labels use positive lapse through p_alpha = -6 V_0 M_Pl^2 a^3 H_FLRW.",
    "The k = 0 cone tip is labeled as excluded and is not counted as a branch.",
    "The diagram labels the current through constant-varphi and constant-alpha slices but does not call the indefinite current a probability before a positive-sector or induced-product construction."
  ],
  "accessibility": {
    "reading_order": [
      "title, constraint, metric, and current convention",
      "constant-coordinate slices and their normals",
      "upper positive-frequency expanding and contracting branches",
      "lower negative-frequency expanding and contracting branches",
      "current-component, nullness, expansion-sign, and encoding summary"
    ],
    "non_color_encoding": "Frequency sectors use solid versus dashed rays; geometry branches use open circles versus filled squares; every branch is also directly labeled.",
    "canvas": "explicit white background for light, dark, monochrome, and print contrast",
    "motion": "none",
    "structured_equivalent": "This semantic JSON records all conventions, branch signs, slice meanings, controls, caption, text alternative, and claim ceiling."
  },
  "caption": "Four null Wheeler-DeWitt currents in the flat homogeneous massless-scalar model. Solid upper rays have p_varphi > 0 and dashed lower rays have p_varphi < 0; independently, open-circle right rays have p_alpha < 0 and are expanding, whereas filled-square left rays have p_alpha > 0 and are contracting. All four rays have k > 0, and the k = 0 cone tip is excluded. With G^(AB) = diag(+,-), J^varphi = p_varphi and J^alpha = -p_alpha, so a constant-varphi flux distinguishes clock-frequency sign while a constant-alpha flux distinguishes chirality. Schematic and not to scale; this sign map does not by itself supply a positive probability interpretation.",
  "alt_text": "An alpha-horizontal and varphi-vertical plane contains four diagonal null current arrows with k > 0; the k = 0 cone tip is marked excluded. The two solid upper arrows have positive scalar-clock frequency but split into an open-circle expanding ray to the right and a filled-square contracting ray to the left. The two dashed lower arrows have negative clock frequency and repeat the independent expanding-right and contracting-left split. Labels state J varphi equals p varphi and J alpha equals minus p alpha; normals mark constant-varphi and constant-alpha slices.",
  "scientific_sources": [
    {
      "citation": "Craig, David A., and Parampreet Singh. Consistent Probabilities in Wheeler-DeWitt Quantum Cosmology. Physical Review D 82 (2010): 123526.",
      "identifier": "DOI:10.1103/PhysRevD.82.123526; arXiv:1006.3837",
      "locators": "section III.B, equations (3.19)-(3.31)",
      "use": "flat-FLRW massless-scalar frequency reduction, left-right branches, physical inner product, and positive-frequency sector"
    },
    {
      "citation": "DeWitt, Bryce S. Quantum Theory of Gravity. I. The Canonical Theory. Physical Review 160 (1967): 1113-1148.",
      "identifier": "DOI:10.1103/PhysRev.160.1113",
      "locators": "sections 4-7",
      "use": "canonical gravitational constraint, indefinite superspace structure, and Wheeler-DeWitt framework"
    }
  ]
}