PhysicsLogic / Nambu 1960

Gauge Independence Is Not State Identification

Inspect when a current is independent of the vector potential and whether it identifies the magnetic field.

Definitions and assumptions

Scope and physical limitations

This is one static complex Fourier mode at the fixed nonzero wavevector (1,1,0), with B=i q cross A and J=K A. The represented physical object is this magnetic amplitude, not a whole electromagnetic state, material, vacuum or measurement record. Only transverse magnetic amplitudes have representatives. The two anisotropic response tensors and probes are synthetic choices. Gauge independence does not by itself establish the Ward identity, a microscopic response calculation, stability, bulk field exclusion or empirical adequacy. No boundary-value problem, negative-wavevector reality condition, dynamical electric field or zero-wavevector limit is encoded here.

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Examples and distinctions

Constructed examples of this model; each answer includes the Lean statement.

Exact root statement

∀ (response : Kernel),
  WardContractionAt response direction →
    ∀ (field : Amplitude),
      Transverse direction field →
        ∃ current,
          (∃ potential, magneticAmplitude potential = field ∧ Matrix.mulVec response potential = current) ∧
            ∀ (potential : Amplitude), magneticAmplitude potential = field → Matrix.mulVec response potential = current
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