equation cannot be written directly in terms of electric and magnetic fields, even though the fields were conventionally assumed to be basic physical quantities. All attempts to express the Schrödinger equation directly in terms of fields have resulted in nonlocality [@mandelstam; @dewitt; @belinfante; @levy; @rohrlich; @priou]. This apparent anomaly is one of the enduring puzzles of quantum mechanics. A hierarchy of physical quantities also serves to clarify the current confused situation in the strong-laser community, where field-based intensity measures are employed that are inconsistent with energy-based criteria. One important example of this misdirection is the introduction of the concept of the critical electric field [@sauter; @schwinger] into the discussion of strong laser effects, despite the fact that lasers produce transverse fields and the critical field has well-defined meaning only for longitudinal fields. The basic differences between transverse fields and longitudinal fields are also exhibited in the macroscopic world in terms of the properties of extremely-low-frequency radio waves.

The range of applicability of the concepts examined here is very large, since it encompasses classical electromagnetism, and also relativistic and nonrelativistic quantum mechanics in which the electromagnetic field is regarded as an external classical field.

The limitation to external classical fields is 