One quantum mechanical analogy explored in the literature is that of a de Broglie wave (scaled by a factor of Planck's reduced constant) associated with a momentum four-vector to illustrate how one could imagine a covariant version of a contravariant vector. The inner product of two contravariant vectors could equally well be thought of as the action of the covariant version of one of them on the contravariant version of the other. The inner product is then how many times the arrow pierces the planes. The mathematical reference, Lee (2003), offers the same geometrical view of these objects (but mentions no piercing).