A New Quantum Paradox

A gauge transformation of a simple electromagnetic system is analyzed. The Hamiltonian which is derived from the Dirac Lagrangian density is used for determining the state of an electron. The fact that this Hamiltonian is free of time differential operators plays a key role in the analysis and proves that this Hamiltonian is not invariant under a general gauge transformation. An application of a specific gauge transformation illustrates this fact. These results call for a further analysis of the role of gauge transformations in the theoretical structure of electrodynamics.