Research proof index

The potential energy from the interaction of the electromagnetic potential with the free current J. -/ noncomputable def freeCurrentPotential (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) (x : SpaceTime d) : ℝ := ⟪A x, J x⟫ₘ /-! ### A.1. Shifts in the free current potential under shifts in the potential -/ lemma freeCurrentPotential_add...

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.

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...