Negative-energy states interpretation

One avoids such unphysical states if one interprets the Dirac operator as acting in a Fock space, with a vacuum, in which no electrons and no positrons are occupied. Annihilation (creation) operators for negative-energy states are then interpreted as positron creation (annihilation) operators. 

The Dirac-Hartree-Fock iterative process can be interpreted as a method of seeking cancellations of certain one- and two-body diagrams.33,124 The self-consistent field procedure can be regarded as a sequence of rotations of the trial orbital basis into the final Dirac-Hartree-Fock orbital set, each set in this sequence forming a basis for the Furry bound-state interaction picture of quantum electrodynamics. The self-consistent field potential involves contributions from the negative energy states of the unscreened spectrum so that the Dirac-Hartree-Fock method defines a stationary point in the space of possible configurations, rather that a variational minimum, as is the case in non-relativistic theory. 

Interpretation of Negative-Energy States Dirac s Hole Theory 1187... 

Although accurate, first-quantized four-component methods are both computationally demanding and plagued by interpretive problems due to the negative-energy states. From a conceptual point of view it is desirable to decouple the upper and lower components of the Dirac Hamiltonian and to obtain a two-component description for electrons only. 

There are several points to be noted about this operator. First, the second term creates an electron-positron pair, and the third term annihilates an electron-positron pair. This means that the Hamiltonian connects states with different particle numbers, that is, particle number is not conserved, though charge is. The existence of these terms embodies the idea of an infinitely-many-body problem that arose from the filling of the negative-energy states in Dirac s interpretation. Second, the order of the operators in the fourth term means that the vacuum expectation value of this operator is not zero, but... 

Thus the vacuum has an energy that is equal to the sum of the energies of the negative solutions of the Dirac equation, as is expected from Dirac s interpretation. Note that the matrix elements are the same as in the Dirac equation, so the sum is negative and infinite. This Hamiltonian operator therefore represents the first stage of the Dirac reinterpretation with the negative-energy states all filled. 

Both Equations (64) and (65) have the same form and they can be interpreted as Schrodinger equations in circular-like coordinates for harmonic oscillators [33], as indicated by their respective kinetic energy and quadratic potential energy terms. The identification and interpretation are even more convincing if we parametrize the negative energy of the bound states of the hydrogen atom as... 

This interpretation is almost self-evident, if we consider, not the quantum states proper (with discrete, negative energy-values), but the states of positive energy, which correspond to the hyperbolic orbits of Bohr s theory. We have then to solve a wave equation... 

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