states: $\widetilde{\Phi}=U\Phi$, $\widetilde{\Psi}=U\Psi$. All $U$ and $U^{-1}$ operators exactly cancel in the matrix element, and the transition amplitude is unchanged. This is what leads to the property gauge-invariant formalism sometimes ascribed to the length gauge. However, this procedure amounts to an identity or to a change of *quantum picture*, but not to a gauge transformation.

Fundamental contrasts in the applicability of fields and potentials
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The first example to be presented is the very basic one of the impossibility of expressing the Schrödinger equation directly in terms of electric and magnetic fields, which should be possible if fields are truly more fundamental than potentials. Other direct examples of difficulties posed by the assumption of the fundamental importance of fields are shown, many of them long employed unnoticed within the strong-field community.

Schrödinger equation
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The Schrödinger equation$$i\hslash\partial_{t}\Psi\left(  t\right)  =H\Psi\left(  t\right)  , \label{ak}$$ when viewed as a statement in a Hilbert space (that is, without selecting a representation such as the configuration representation or the momentum representation) states that the effect of rotating a state vector $\Psi\left(
t\right)  $ by the operator $H$ within the Hilbert space produces the same effect as differentiating the vector with respect to time (multiplied by 