Negative-energy continuum

As noted earlier, for each particle i, there is a discrete spectrum of positive energy bound states and positive and negative energy continuum states. Let us consider a product wave function of the form = i//i(l)i//2(2), a normalizable stationary bound-state eigenfunction of... 

The Breit-Pauli spin-orbit operator has one major drawback. It implicitly contains terms coupling electronic states (with positive energy) and posi-tronic states (in the negative energy continuum) and is thus unbounded from below. It can be employed safely only in first-order perturbation theory. 

Let us first summarize the results of the field-theoretical description of pair creation. For details we may refer to Eichler and Meyerhof (1995) and Strayer et al. (1990). Electron-positron pair creation can be viewed as an excitation of an electron from the (occupied) negative energy continuum into a positive enetgy bound or continuum state... 

The decomposition of the eigenvalue W into a rest mass contribution mc and a remainder E is only meaningful if IT > 0 and IE] energy close to the onset of the positive-energy continuum, in particular for a bound positive-energy state. If we were interested in a state near the negative-energy continuum, we would define E as VT - - mc vide infra). 

As common in relativistic electronic structure theory, one invokes the so-called no-sea approximation where one neglects all vacuum contributions of the filled negative energy continuum [41]. The only remaining effect of the sea is the restriction for electrons to occupy only states of positive energy. Then the density is constructed from DKS one-electron orbitals of a single-determinant A-electron wave function ... 

Dirac was not worried by die faet that both roots appear after an ad hoc deeision to square die expression for the enei y [Eqs. (3.40) and (3.41)]. As we ean see, since the momentum may ehange from 0 to oo (p = mv, and for v c, we have m —> oo), we therefore have the negative energy continuum and symmetrically located positive energy continuum, both of which are separated by the energy gap 2moc (Fig. 3.2). 

When he was 26 years old, Dirac made the absurd assumption that what people call a vacuum is in reality a sea of electrons occupying the negative energy continuum (known as the Dirac electronic sea). The sea was supposed to consist of an infinite number of electrons, which had to imply catastrophic consequences concerning, for example, the infinite mass of the Universe, but Dirac did not feel any doubt about his notion. " We see only those electrons that have positive energy " said Dirac. Why he was so determined Well, Dirac s concept of the sea was proposed to convince us that due to the Pauli exclusion principle, the doubly occupied sea electronic states... 

Paul Dirac factorized the left side of this equation by treating it as the difference of squares. This gave two continua of energy separated by a gap of width 2monegative energy) continuum is fuUy occupied by electrons (i.e., a vacuum), while the upper continuum is occupied by the single electron (our particle). If we managed to excite an electron from the lower continuum to the ujqier one, then in the upper continuum, we would see an electron, while the hole in the lower continuum would have the properties of a positive electron (positron). This corresponds to the creation of the electron-positron pair from the vacuum. 

Lorentz transformation (p. 113) Michelson-Morley experiment (p. 1111 Minkowski space-time (p. 177) negative energy continuum (p. 125) positive energy continuum (p. 125) positron (p. 126)... 

Because of the negative-energy continuum in the spectrum of the one-electron Dirac Hamiltonian already a two-electron equation constructed from... 

The next two sets are mixtures of bound and continuum states. Here, one electron is in a bound state and the other is in either a positive- or negative-energy continuum state. The former set, spans the range... 

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