Negative-Energy States and Quantum Electrodynamics

We now return to the problem of the negative-energy solutions that appeared when we solved the free particle Dirac equation in the previous chapter. The energy eigenvalues obtained there were either + = or = - /nfic + p c. The 

There is, however, a problem with this approach. The filling of the negative continuum implies that the vacuum state is infinitely charged, and as a consequence we would expect an infinite interaction between any state and the vacuum. We therefore need to take the reinterpretation one step further. What we actually measure is not the absolute properties of a state but the differences in properties between a state and the vacuum. The Dirac equation is therefore only the starting point for a theory that is an infinitely-many-body theory, even for a system with only one electron—or indeed for the vacuum itself  

In fact, there is a more direct connection between the charge of the particle and the mass-energy, so that we only need to solve one Dirac equation to obtain solutions for both signs of the charge. We write the Dirac equation for an electron of charge —e in four-vector form as 

We can take the complex conjugate of this equation, but we have to exercise some care with regard to the four-vector quantities, which represent a mixture of real and complex entities. Showing the space and the time components of the four-vectors explicitly, the equation above takes the form 

Due to the presence of a scalar product between the vectors, we can recover the four-vector quantities and A by changing the sign of the last component in each 

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An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

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. 

However, already in 1930s deviations were observed between the results of precision spectroscopy and the Dirac theory for simple atomic systems, primarily for the hydrogen atom. The existence of negative-energy states in the solutions of Dirac equation is the mathematical but not the physical grounds of the existence of particles and antiparticles (electrons and positrons). Besides, the velocity of light is finite. For an complete model we must turn to quantum field theory and quantum electrodynamics (QED) [4]. 

Since the spectrum of the DCB Hamiltonian is not bounded from below it is not possible to optimize the wave function by minimization of the energy. The unphysical unboundedness is due to the fact that not all possible normalizible antisymmetric wave functions of N coordinates are states of an N-electron system. The set of possible solutions also contains wavefunctions in which one or more negative energy levels are occupied and it is the mixing with such states that gives rise to unphysical arbitrarily low energies. One needs the second quantization formalism of quantum electrodynamics (QED) for a proper treatment of these states. As this is discussed in more depth elsewhere in this... 

Unless carefully implemented the representation of the Dirac spectrum obtained within the algebraic approximation may exhibit undesirable properties which are not encountered in non-relativistic studies. In particular, an inappropriate choice of basis set may obliterate the separation of the spectrum into positive and negative energy branches. So-called intruder states may arise, which are impossible to classify as being of either positive or negative energy character. The Furry bound state interaction picture of quantum electrodynamics is thereby undermined. 

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