Dirac hole theory

The hole theory was perceived as a Active mathematical construction and was initially rejected by prominent contemporary physicists such as Pauli and Bohr. The physical reality of antiparticles was not taken seriously even by Dirac himself. In 1931 he wrote about his anti-electron we should not expect to find it in Nature [2]. Surprisingly, the first anti-electrons were discovered already in 1932 by Anderson, who studied cosmic rays in Caltech s magnet cloud chamber. Anderson noticed abnormally bending trajectories indicating the presence of light positively charged particles and, as related by Fowler [3], "could not resist the devastating conclusion that they are caused by positive electrons The first piece of antimatter, a positron, made its physical appearance. 

The first anti-particle discovered was the anti-electron, the so-called positron, in 1933 by Anderson [3] in the cloud chamber due to cosmic radiation. The existence of the anti-electron (positron) was described by Dirac s hole theory in 1930 [4], The result of positron—electron annihilation was detected in the form of electromagnetic radiation [5]. The number and event of radiation photons is governed by the electrodynamics [6, 7]. The most common annihilation is via two- and three-photon annihilation, which do not require a third body to initiate the process. These are two of the commonly detected types of radiation from positron annihilation in condensed matter. The cross section of three-photon annihilation is much smaller than that of two-photon annihilation, by a factor on the order of the fine structure constant, a [8], The annihilation cross section for two and three photons is greater for the lower energy of the positron—electron pair it varies with the reciprocal of their relative velocity (v). In condensed matter, the positron—electron pair lives for only the order of a few tenths to a few nanoseconds against the annihilation process. 

In spite of its successes, Dirac s hole theory of the positron is provisional rather than final. If a serious attempt is made to connect the theory with electrodynamics, while postulating that only free electrons and the holes, but not the great body of electrons of negative energy, shall act as generators of a field, the resulting formalism is extremely complicated, and seems far from satisfactory. Here theoretical physics is confronted with a serious problem. 

The no virtual pair approximation. - According to Dirac s hole theory, those states lying below —me2 are taken to be filled according to the... 

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

Figure 5.1 Sketch of the spectrum of the Dirac Hamiltonian for a free electron (a) and a bound electron in a Coulombic potential attractive for electrons [see chapter 6] (b). In (c) creation of an electron-positron pair is illustrated according to Dirac s hole theory, where all negative-energy states are assumed to be occupied for the vacuum state.

Although Dirac s hole theory could account for an acceptable interpretation of the negative continuum states for the very first time, it is still plagued by a series of inconsistencies incompatible with a fundamental physical theory from today s point of view. First of all, it crucially demands the fermionic nature of the electrons since it heavily relies on the Pauli principle in order to... 

Furthermore, starting from the Dirac equation, the assumptions of hole theory lead automatically to a formulation for an infinite number of particles occupying the Dirac sea, and the actual number of electrons and positrons cannot be deduced from hole theory. This is a most disturbing aspect of Dirac s hole theory, namely that it is actually a flni/-particle theory. So far we have considered only a single fermion in the universe and set up an equation to describe its motion. Now, we face a conceptual generalization, which in turn requires a description of the motion of infinitely many fermions. This leads to the inclusion of additional interaction operators in the Dirac equation, and it is this ineraction of electrons which makes life difficult for molecular scientists, as we shall discuss in parts III and IV of this book. 

L. Tomio. Dirac s hole theory versus quantum field theory. Can. J. Phys., 80 (2002) 837-840. 

Awkward questions about the electromagnetic and gravitational fields of infinitely many particles in the vacuum remain unanswered. Also, the Dirac theory, amended by the hole proposition is certainly not a one-particle theory, and hence not a relativistic generalization of Schrodinger s equation. 

Four decades ago, Bell [3] introduced a particle-hole conjugation operator CB into nuclear shell theory. Its operator algebra is essentially isomorphic to that of Cq (for example, CB is unitary), the filled Dirac sea now corresponding to systems with half-filled shells. This was later extended to other areas of physics. For example,... 

Dirac thought at first that these holes corresponded to the protons, though he was conscious of the difficulty of the difference in the masses. But when the first indications of the existence of positrons were found by Aiiderson (see p. 43), Dirac at once saw in this a confirmation of his theory, and by prediction of phenomena took a hand in the direction of experimental research. 

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... 

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

P.A.M. Dirac, who shared the 1933 Nobel prize for physics with Schrodinger (the 1932 price went to Heisenberg), was one of the greatest pioneers of quantum mechanics. Most of his achievements entered textbooks so fast that his original papers are hardly cited. Nobody, who uses Dirac s bra-ket notation or his function would cite the original references [1]. The same is true of Dirac s time-dependent perturbation theory [2] or of the Dirac equation [3], the basis of relativistic quantum mechanics or of his subsequent work on positrons and holes [4]. 

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