Editing Dirac equation (section)

== Physical interpretation ==
=== 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}}

=== Hole theory ===
The negative {{math|''E''}} solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, they cannot simply be ignored, for once the interaction between the electron and the electromagnetic field is included, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of [[photon]]s.

To cope with this problem, Dirac introduced the hypothesis, known as '''hole theory''', that the [[vacuum]] is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the [[Dirac sea]]. Since the [[Pauli exclusion principle]] forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.

Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a '''hole''' – would behave like a positively charged particle. The hole possesses a ''positive'' energy because energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but [[Hermann Weyl]] pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the [[positron]], experimentally discovered by [[Carl David Anderson|Carl Anderson]] in 1932.<ref>{{cite book |last1=Penrose |first1=Roger |title=The Road to Reality |date=2004 |publisher=Jonathan Cape |isbn=0-224-04447-8 |page=625}}</ref>

It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "[[jellium]]" background so that the net electric charge density of the vacuum is zero. In [[quantum field theory]], a [[Bogoliubov transformation]] on the [[creation and annihilation operators]] (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.

In certain applications of [[condensed matter physics]], however, the underlying concepts of "hole theory" are valid. The sea of [[conduction electron]]s in an [[electrical conductor]], called a [[Composite fermion#Fermi sea|Fermi sea]], contains electrons with energies up to the [[chemical potential]] of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, and although it too is referred to as an "electron hole", it is distinct from a positron. The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.

=== In quantum field theory ===
{{See also|Fermionic field}}
In [[Quantum field theory|quantum field theories]] such as [[quantum electrodynamics]], the Dirac field is subject to a process of [[second quantization]], which resolves some of the paradoxical features of the equation.