In the definitions of the TDVP and TDPES, we indicate with the symbol • x an integration over x, and we use the convention that such variable does not appear explicitly in the term in the bra-ket as it is integrated over. The TDVP encodes information about the momentum field of the particles whose coordinates are given by X, since gauge potentials, namely Ãν (X,t) = A ν (X,t) + ∇ ν θ (X,t) and ε(X,t) = ε(X,t) + ∂ t θ (X,t), and Eqs. ( ) and ( ) are form-invariant.