The Significance of the Distinction between a Necessary Condition and a Sufficient Condition

The distinction between the concepts of a Necessary Condition and a Sufficient Condition is a fundamental element of mathematics. This issue certainly applies to physics which has a mathematical structure. However, this work shows that sometimes physical textbooks ignore this distinction. An analysis of the conserved 4-current of the Noether theorem proves that this issue is extremely important. The analysis uses the dimension of the Lagrangian density and the corresponding dimension of the quantum function of physical theories. It is proved that the QED theory of a Dirac electron yields a coherent expression for the 4-current and the QED interaction term. In construct, the Klein-Gordon theory of a charged particle as well as the electroweak theory of the W± particles violates Maxwellian electrodynamics. Unlike the Dirac electron, these theories have no coherent interaction term between the 4-current of the charged particle and the electromagnetic fields. This result relies on new necessary conditions that are required for the compatibility of the 4-current of a charged quantum particle. The new necessary conditions prove that the continuity equation of the Noether theorem is not a sufficient condition for an acceptable 4-current.