Electron-Positron Interpretation

Consider the Dirac ooperator for a charge e in an external field (4 eh ), 

Here the bar denotes a complex conjugation and Uc is a unitary 4x4 matrix with the following property  

The operator C is called charge conjugation. As an antiunitary transformation it is a symmetry transformation, that is, all transition probabilities are left invariant. 

If 4 t) is a solution of the Dirac equation with Hamiltonian H e), then Ci t) is a solution of the Dirac equation with Hamiltonian H —e). More precisely, we have 

the negative-energy subspace of H e) is connected via the symmetry transformation C with the positive-energy subspace of the Dirac operator H —e) for a particle with opposite charge, but with the same mass. Hence if ip describes an electron, then Cxp describes a positron. Moreover, we have 

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Our discussion of the electron-positron interpretation shows that the Dirac equation can have bound states for both attractive and repulsive electric potentials. Consider for example an attractive electrostatic potential well eV(x) < 0. It may support some bound states at energies En in the gap A... 

Of course, Dirac was well aware of the many difficulties arising from the electron-positron interpretation — from the mass-dissymmetry if the negative-energy electron was a proton to the fact that he has actually created a many-particle theory which, as a true many-particle theory, would have been difficult to study. We shall have a closer look into these difficulties below. 

A further term, which has no analogue in hydrogen, arises in the fine structure of positronium. This comes from the possibility of virtual annihilation and re-creation of the electron-positron pair. A virtual process is one in which energy is not conserved. Real annihilation limits the lifetimes of the bound states and broadens the energy levels (section 12.6). Virtual annihilation and re-creation shift the levels. It is essentially a quantum-electrodynamic interaction. The energy operator for the double process of annihilation and re-creation is different from zero only if the particles coincide, and have their spins parallel. There exists, therefore, in the triplet states, a term proportional to y 2(0). It is important only in 3S1 states, and is of the same order of magnitude as the Fermi spin-spin interaction. Humbach [65] has given an interpretation of this annihi-... 

Other selection rules may be derived from parity arguments [145]. The conclusion is reached that double quantum emission is allowed from all even states (except where J = 1) and all odd states pf even J. In interpreting these rules it is to be understood that an electron-positron pair, with both particles at rest, has odd parity. S, D. . . states are therefore odd, while P, F. . . are even. 

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... 

The present volume of the Advances in Quantum Chemistry is the sequel of the first volume, mentioned above, i.e., Unstable States in the Continuous Spectra, Part II Interpretation, Theory and Applications. It contains six chapters with contents varying from a pedagogical introduction to the notion of unstable states to the presence and role of resonances in chemical reactions, from discussions on the foundations of the theory to its relevance and precise limitations in various fields, from electronic and positronic quasi-bound states and their role in certain types of reactions to applications in the field of electronic decay in multiply charged molecules and clusters, as well. 

At Z> 1/a = 137, this solution "dives" into the electron sea of the positron continuum and no real solution exists. This has been interpreted as an unstable state which, if vacant, can be filled by "autoionization" from the positron continuum. [16]... 

Historically, the first Ps formation mechanism was suggested by Ore for the purpose of interpretation of experiments on e+ annihilation in gases [9]. It implies that the hot positron, e+, having some excess kinetic energy, pulls out an electron from molecule M, thereby forming a Ps atom and leaving behind a positively charged radical-cation M+ ... 

For free particles this point of view has indeed some attractive features. There are, however, situations where the sign of the energy does not distinguish between electronic and positronic behavior. Consequently, transitions from electronic to positronic states cannot be excluded. A famous example is the Klein paradox, where a potential step divides space into two regions with a different interpretation of particles and antiparticles. If the step size is larger than twice... 

In a non-relativistic theory we would now continue by adding a second quantized operator for two-body interactions. In the relativistic case we need to step back and first consider the interpretation of the eigenvalues of the Hamiltonian. Dirac stated that positrons could be considered as holes in an infinite sea of electrons . In this interpretation the reference state for a system with neither positrons nor electrons is the state in which all negative energy levels are filled with electrons. This vacuum state... 

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. 

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