Field Action Integral

The Field Action Integral is the action of the collective field, the Lagrangian density integrated over the spatial domain and over time, from which the field's evolution follows by the variational principle. It brings classical field theory onto the discrete substrate: the field built from pulses evolves so as to make this action stationary, subject to the initial condition Ψ(x,0) = Ψ_0(x) and the boundary condition Ψ|∂Ω = 0. The Lagrangian density is the standard one modified for the discrete substrate, L = (ħ²/(2m𝒞→²))|∂Ψ/∂t|² − (ħ²/(2m))|∇Ψ|² − V(x)|Ψ|², kinetic minus gradient minus potential energy density, the factors ħ²/(2m𝒞→²) and ħ²/(2m) making each term an energy per volume for Ψ [𝕃⁻³/²]; a self-interaction such as a term in |Ψ|⁴ would add to the potential term. Varying the action gives the Euler-Lagrange field equation for Ψ. The formal statement is S⟐ = ∫_0^T ∫_Ω L[Ψ, ∂Ψ/∂t, ∇Ψ] dx dt, where S⟐ is the Field Action Integral, L the Lagrangian density, Ψ the collective field function, Ω the spatial domain, T the temporal domain, dx the volume element and dt the time element. Dimensions as the book gives them: S⟐ [J·s], L [J·m⁻³], Ψ [𝕃⁻³/²], ∂Ψ/∂t [𝕃⁻³/²·𝕋⁻¹], ∇Ψ [𝕃⁻⁵/²], dx and Ω [𝕃³], dt and T [𝕋]. The book's check is consistent. It connects to the Euler-Lagrange Equation, Collective Field Function and Energy Density Distribution.