Research proof index

The (inl 0, inr i) component of the field strength matrix of the bare-gradient potential B^μ := ∂_μ χ for χ(x) = x⁰·xⁱ equals 2. This witnesses that the bare covariant gradient does not produce a gauge-invariant field strength, so the raised-index contraction η^{μν} ∂_ν χ in ofGradient is necessary (see the module overview).

Gauge transformations commute with Lorentz transformations: applying Λ and then performing a gauge transformation by χ equals performing a gauge transformation by χ ∘ (Λ⁻¹ • ·) and then applying Λ.

ofGradient intertwines the Lorentz action on potentials with composition by Λ⁻¹ on the gauge function: Λ • ofGradient χ = ofGradient (χ ∘ (Λ⁻¹ • ·)).

The field strength of the bare-gradient potential B^μ := ∂_μ χ for χ(x) = x⁰·xⁱ is nonzero (follows from fieldStrengthMatrix_bareGradient_inl_inr).

A pure-gauge potential has vanishing field strength.