{ "schema_version": "qft.hadamard-wavefront-pairing.v1", "artifact_id": "qft.artifact.curved-spacetime.states-hadamard-microlocal-control.hadamard-wavefront-pairing", "title": "Hadamard wavefront pairing and its reversed-orientation control", "generated_by": "figures-src/curved-spacetime/hadamard-wavefront-pairing.mjs", "source_revision": 1, "generated_on": "2026-09-04", "owner_page": { "id": "qft.topic.curved-states.microlocal-spectrum-condition", "file": "src/content/docs/curved-spacetime/states-hadamard-microlocal-control/hadamard-admissibility-two-point-wavefront-criterion.md", "route": "/curved-spacetime/states-hadamard-microlocal-control/hadamard-admissibility-two-point-wavefront-criterion/", "sha256": "3e15bb827b6231df895ffdb9d969576d1a0c9cfa377e56235ba4238738ba7e0b" }, "reader_question": "How are the two cotangent directions in one allowed Hadamard two-point wavefront element related, and what changes when the first-slot time orientation is reversed?", "takeaway": "Parallel transport compares k with k prime along one null geodesic, but the distributional second slot carries minus k prime. Reversing the first slot produces the transposed orientation and fails the declared Hadamard condition for omega two.", "theorem_scope": { "spacetime_dimension": 4, "background": "smooth, time-oriented, globally hyperbolic Lorentzian spacetime without boundary", "metric_signature": "(+---)", "field": "real scalar Klein-Gordon field", "operator": "P=Box_g+m^2+xi R, normally hyperbolic", "two_point_input": "state two-point bidistribution satisfying the Klein-Gordon equation in both slots and the canonical antisymmetric part", "conclusion": "the Hadamard condition is represented by the future-first-slot null wavefront pairing shown here", "exclusions": [ "timelike boundaries or reflected-ray boundary conditions", "gauge or constrained field complexes", "non-globally-hyperbolic backgrounds", "positivity or preferred-state selection inferred from wavefront data alone" ] }, "conventions": { "localized_fourier_phase": "F_+(chi u)(p)=", "positive_frequency_kernel_phase": "exp[-i omega(t-t_prime)+i p.(x-x_prime)]", "causal_propagator": "E=G_ret-G_adv", "canonical_antisymmetric_part": "omega_2-omega_2^T=-i E", "future_null_bundle": "N^+ contains nonzero future-directed null covectors", "parallel_transport": "k_prime=PT_gamma(k)", "null_geodesic_alignment": "k is nonzero and null, and k_sharp is tangent to the same null geodesic gamma that contains x and x_prime", "wavefront_pairing": "WF(omega_2) contains (x,k;x_prime,-k_prime), never (x,k;x_prime,k_prime)", "drawing_note": "arrows depict time orientation in cotangent fibers; the spacetime sketch is schematic and not to scale" }, "accepted_case": { "id": "allowed_future_first_slot", "kernel": "omega_2(x,x_prime)", "relation": "(x,k)~(x_prime,k_prime)", "relation_definition": "x and x_prime lie on one null geodesic gamma, k is a nonzero null covector with k_sharp tangent to gamma at x, and k_prime=PT_gamma(k)", "tuple": "(x,k;x_prime,-k_prime)", "first_slot": { "base_point": "x", "covector": "k", "orientation": "future-directed null" }, "transported_covector": { "base_point": "x_prime", "covector": "k_prime", "relation": "k_prime=PT_gamma(k)" }, "second_slot": { "base_point": "x_prime", "covector": "-k_prime", "orientation": "past-directed null", "reason": "the two-point distribution carries the negative transported covector in its second slot" }, "expected_result": "accept" }, "negative_control": { "id": "reversed_first_slot_orientation", "kernel": "omega_2^T(x,x_prime)=omega_2(x_prime,x)", "relation": "(x,-k)~(x_prime,-k_prime)", "tuple": "(x,-k;x_prime,+k_prime)", "first_slot_orientation": "past-directed null", "expected_result": "reject for the declared WF(omega_2)", "strongest_surviving_statement": "the tuple has the transposed wavefront orientation and belongs to WF(omega_2^T), not the future-first-slot set of omega_2", "control_passed": true }, "exact_minkowski_fixture": { "coordinates": "(t,x,y,z)", "x": [ 0, 0, 0, 0 ], "x_prime": [ 1, 1, 0, 0 ], "null_tangent": [ 1, 1, 0, 0 ], "spatial_momentum": [ 1, 0, 0 ], "positive_frequency_energy": 1, "k": [ 1, -1, 0, 0 ], "k_sharp": [ 1, 1, 0, 0 ], "k_prime": [ 1, -1, 0, 0 ], "accepted_second_covector": [ -1, 1, 0, 0 ], "rejected_first_covector": [ -1, 1, 0, 0 ], "rejected_second_covector": [ 1, -1, 0, 0 ], "covector_sum_after_parallel_transport": [ 0, 0, 0, 0 ] }, "visual_relations": [ { "id": "shared_null_geodesic", "source": "x", "target": "x_prime", "label": "one null geodesic gamma", "line_style": "solid" }, { "id": "future_first_covector", "source": "x", "target": "k", "label": "k in N^+", "line_style": "solid arrow" }, { "id": "parallel_transport", "source": "k", "target": "k_prime", "label": "k_prime=PT_gamma(k)", "line_style": "dashed arrow" }, { "id": "distribution_second_slot", "source": "x_prime", "target": "-k_prime", "label": "slot 2: -k_prime", "line_style": "solid arrow with direct label" }, { "id": "reversed_orientation_failure", "source": "-k", "target": "rejected WF(omega_2)", "label": "past-directed first slot", "line_style": "dotted arrow plus multiplication sign" } ], "accessibility": { "composition": "two vertically stacked panels in accepted-then-rejected reading order", "palette": "black and gray on an explicit white canvas", "redundant_encodings": [ "direct labels", "solid, dashed, and dotted line styles", "accepted and rejected panel headings", "multiplication-sign rejection marker" ], "semantic_equivalent": "/figures/curved-spacetime/hadamard-wavefront-pairing.json", "minimum_viewport_css_pixels": 320, "horizontal_panning_required": false, "schematic": true, "not_to_scale": true }, "checks": { "four_dimensional_fixture": true, "base_points_are_null_related": true, "positive_frequency_phase_gives_future_first_covector": true, "first_covector_is_null": true, "first_covector_is_coparallel_to_connecting_geodesic": true, "parallel_transport_preserves_covector": true, "second_slot_is_negative_transport": true, "transported_slot_covectors_sum_to_zero": true, "reversed_first_slot_is_past_directed": true, "reversed_tuple_is_exact_sign_flip": true } }