{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.nambu-gorkov-propagator-convention-map",
  "title": "Nambu-Gor'kov propagator convention map",
  "classification": "original exact algebraic convention map",
  "reader_question": "In the declared two-component Nambu basis, which correlator occupies each matrix entry, how do constant U(1) rotations and hole-component rephasings change those entries, and which outputs remain invariant?",
  "dominant_point": "Normal and anomalous propagators form one basis-dependent matrix: rotations and rephasings change anomalous phases or signs together with the pair field, while the determinant, poles, diagonal spectral weights, and correctly de-duplicated physics remain unchanged.",
  "status": {
    "schematic": true,
    "exact_algebraic_relations": true,
    "not_a_spacetime_diagram": true,
    "contains_experimental_data": false,
    "contains_simulation_data": false
  },
  "scope": {
    "state": "equilibrium, translation-invariant, balanced, even-parity spin-singlet paired state",
    "basis": "Psi_k = (c_{k up}, c^dagger_{-k down})^T",
    "dispersion_condition": "xi_{k up} = xi_{-k down} = xi_k",
    "frequency": "z is complex; z = i omega_n on the fermionic Matsubara axis",
    "gauge_note": "The displayed integer weights apply to a constant global U(1) rotation. A local bilocal Green function transforms with separate endpoint phases and requires a Wilson line or source completion before it is gauge invariant."
  },
  "matrix_entries": [
    {
      "position": "11",
      "symbol": "G",
      "definition": "-<T_tau c_{k up}(tau) c^dagger_{k up}(0)>",
      "role": "normal particle propagator",
      "constant_u1_weight": 0
    },
    {
      "position": "12",
      "symbol": "F",
      "definition": "-<T_tau c_{k up}(tau) c_{-k down}(0)>",
      "role": "pair-annihilation anomalous propagator",
      "constant_u1_weight": 2
    },
    {
      "position": "21",
      "symbol": "Fbar",
      "definition": "-<T_tau c^dagger_{-k down}(tau) c^dagger_{k up}(0)>",
      "role": "pair-creation anomalous propagator",
      "constant_u1_weight": -2
    },
    {
      "position": "22",
      "symbol": "-G_down(-k,-tau)",
      "definition": "-<T_tau c^dagger_{-k down}(tau) c_{-k down}(0)>",
      "role": "hole normal propagator",
      "constant_u1_weight": 0
    }
  ],
  "kernel": {
    "hamiltonian": "h_k = xi_k tau_3 + Re(Delta) tau_1 - Im(Delta) tau_2",
    "inverse_propagator": "G^{-1}(k,z) = z tau_0 - h_k",
    "determinant": "det G^{-1}(k,z) = z^2 - xi_k^2 - |Delta|^2",
    "poles": "z = plus or minus E_k, with E_k = sqrt(xi_k^2 + |Delta|^2)"
  },
  "transformations": [
    {
      "name": "constant U(1) rotation",
      "matrix": "U_alpha = exp(i alpha tau_3)",
      "propagator_rule": "G -> U_alpha G U_alpha^dagger",
      "entry_rules": [
        "G_11 and G_22 are unchanged",
        "F -> exp(2 i alpha) F",
        "Fbar -> exp(-2 i alpha) Fbar",
        "Delta -> exp(2 i alpha) Delta"
      ]
    },
    {
      "name": "hole-component basis rephasing",
      "matrix": "U = tau_3",
      "propagator_rule": "G_tilde = U G U^dagger and G_tilde^{-1} = U G^{-1} U^dagger",
      "entry_rules": [
        "G_11 and G_22 are unchanged",
        "F_tilde = -F",
        "Fbar_tilde = -Fbar",
        "Delta_tilde = -Delta when the transformed Hamiltonian is written in the same Pauli-matrix convention"
      ]
    }
  ],
  "invariants": [
    "det G^{-1}",
    "the poles z = plus or minus E_k",
    "the diagonal electron and hole spectral weights under the displayed transformations",
    "the canonical unit sum rule for each diagonal spectral function",
    "source-complete gauge-invariant response"
  ],
  "counting_rules": [
    {
      "representation": "fully doubled spin-Nambu basis over the full momentum domain",
      "rule": "Include one factor of 1/2 in the quadratic form or functional trace, together with the corresponding normal-ordering constant."
    },
    {
      "representation": "basis or domain already restricted to independent particle-hole sectors",
      "rule": "Do not insert a second factor of 1/2."
    },
    {
      "representation": "analytic Green function in either convention",
      "rule": "Retain both positive- and negative-frequency poles; de-duplication is a state-counting operation, not deletion of a pole."
    }
  ],
  "nonclaims": [
    "The integer phase weights do not replace the endpoint-dependent transformation law for a bilocal correlator under a local gauge transformation.",
    "The lower-left anomalous function is not asserted to be the pointwise complex conjugate of the upper-right function in a general frequency-dependent or multicomponent state.",
    "The factor of 1/2 is not universal; it depends on the declared Nambu basis and summation domain.",
    "The plus and minus poles are not two independent positive-energy quasiparticle species.",
    "The diagram does not establish phase stiffness, a pairing mechanism, or an observable condensate density."
  ],
  "scientific_checks": [
    "Expand all four matrix entries directly from -<T_tau Psi(tau) Psi^dagger(0)> in the declared basis.",
    "Conjugation by tau_3 leaves tau_0 and tau_3 unchanged and reverses tau_1 and tau_2.",
    "Direct inversion gives det G^{-1} = z^2 - xi_k^2 - |Delta|^2 and poles plus or minus E_k.",
    "The normal entries are unchanged and both anomalous entries reverse sign under the hole-component rephasing.",
    "Under a constant U(1) rotation F and Delta have weight plus two while Fbar has weight minus two.",
    "The counting strip removes particle-hole duplication by exactly one declared mechanism while retaining both analytic poles."
  ],
  "sources": [
    {
      "citation": "L. P. Gor'kov, On the Energy Spectrum of Superconductors, Soviet Physics JETP 7 (1958) 505-508",
      "url": "https://www.jetp.ras.ru/cgi-bin/dn/e_007_03_0505.pdf",
      "use": "normal and anomalous Green functions"
    },
    {
      "citation": "Yoichiro Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Physical Review 117 (1960) 648-663",
      "url": "https://doi.org/10.1103/PhysRev.117.648",
      "use": "particle-hole representation, gauge covariance, and conserving response"
    }
  ],
  "alt_text": "A two-by-two Nambu matrix labels normal propagators on the diagonal and pair amplitudes off diagonal; a phase rotation and tau-three rephasing alter only anomalous phases or signs, while the determinant and positive-and-negative-energy poles remain unchanged.",
  "caption": "In the declared spin-singlet Nambu basis, diagonal entries propagate a particle or hole and off-diagonal entries annihilate or create a pair. A constant U(1) rotation changes the anomalous phases, while the basis rephasing U = tau_3 reverses their printed signs together with Delta. The determinant and plus-or-minus E_k poles are invariant; particle-hole duplication is removed once according to the declared summation domain. Exact algebraic convention map for a uniform block, not a spacetime diagram.",
  "rights": {
    "basis": "Original QFT.org algebraic diagram derived from the declared matrix definitions; no third-party figure, layout, data, or artwork was copied or restyled.",
    "creator": "OpenAI Codex, for QFT.org",
    "date": "2026-08-31"
  }
}