Editing Dirac equation (section)

=== Identification of observables ===
The critical physical question in a quantum theory is this: what are the physically [[observable]] quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by [[self-adjoint operators]] that act on the [[Hilbert space]] of possible states of a system. The eigenvalues of these operators are then the possible results of [[Measurement problem|measuring]] the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. To maintain this interpretation on passing to the Dirac theory, the Hamiltonian must be taken to be
<math display="block">H = \gamma^0 \left[mc^2 + c \gamma^k \left(p_k - q A_k\right) \right] + c q A^0.</math>
where, as always, there is an [[Einstein notation|implied summation]] over the twice-repeated index {{math|''k'' {{=}} 1, 2, 3}}. This looks promising, because one can see by inspection the rest energy of the particle and, in the case of {{math|'''A''' {{=}} 0}}, the energy of a charge placed in an electric potential {{math|''cqA''<sup>0</sup>}}. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is
<math display="block">H = c\sqrt{\left(\mathbf{p} - q\mathbf{A}\right)^2 + m^2c^2} + qA^0.</math>

Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and one must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables.{{Citation needed|date=January 2020}}