scholarly journals The vacuum in Dirac's theory of the positive electron

1934 ◽  
Vol 146 (857) ◽  
pp. 420-441 ◽  
Keyword(s):  
Kinetic Energy ◽  
Infinite Number ◽  
Strong Support ◽  
Negative Energy ◽  
Positive Energy ◽  
Quantum Mechanical ◽  
Normal Density ◽  
Physical Picture ◽  
Negative Energy States ◽  
Energy States

According to a theory proposed by Dirac one has to picture the vacuum as filled with an infinite number of electrons of negative kinetic energy, the electric density of which is, however, unobservable. One can observe only deviations from this "normal" density which either consist of an addition of electrons in states of positive energy or absence of electrons from some of the negative energy states (positive electrons). The discovery of the positive electron and the observed magnitude of the processes involving it give strong support to this view. This theory, as it stands, however, is not complete because it makes use of infinite quantities which are inadmissible in physical equations. It therefore must be understood (and was meant so by Dirac) to be a physical picture showing a way in which the quantum mechanical equations can probably be modified in order to give account of the positive electron and to solve the difficulty connected with the states of negative energy.

scholarly journals BOSON SEA VERSUS DIRAC SEA: GENERAL FORMULATION OF THE BOSON SEA THROUGH SUPERSYMMETRY

2004 ◽  
Vol 19 (32) ◽  
pp. 5561-5583 ◽  
Author(s):  
YOSHINOBU HABARA ◽  
HOLGER B. NIELSEN ◽  
MASAO NINOMIYA
Keyword(s):  
Harmonic Oscillator ◽  
Negative Energy ◽  
Positive Energy ◽  
Second Quantization ◽  
Negative Energy States ◽  
Quantization Formalism ◽  
Dirac Sea ◽  
Energy States ◽  
Physical Result ◽  
Usual Formalism

We consider the long standing problem in field theories of bosons that the boson vacuum does not consist of a "sea," unlike the fermion vacuum. We show with the help of supersymmetry considerations that the boson vacuum indeed does also consist of a sea in which the negative energy states are all "filled," analogous to the Dirac sea of the fermion vacuum, and that a hole produced by the annihilation of one negative energy boson is an antiparticle. Here, we must admit that it is only possible if we allow — as occurs in the usual formalism anyway — that the "Hilbert space" for the single particle bosons is not positive definite. This might be formally coped with by introducing the notion of a double harmonic oscillator, which is obtained by extending the condition imposed on the wave function. This double harmonic oscillator includes not only positive energy states but also negative energy states. We utilize this method to construct a general formalism for a boson sea analogous to the Dirac sea, irrespective of the existence of supersymmetry. The physical result is consistent with that of the ordinary second quantization formalism. We finally suggest applications of our method to the string theories.

Negative-Energy States and Quantum Electrodynamics

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Dirac Equation ◽  
Energy State ◽  
Rest Mass ◽  
Free Particle ◽  
Negative Energy ◽  
Transition Moment ◽  
Energy Solution ◽  
Positive Energy ◽  
Negative Energy States ◽  
Energy States

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 E+ = +√(m2c4 + p2c2) or E− = − √(m2c4 + p2c2) . The minimum absolute values we can have are therefore |E| = mc2 for p = 0 (that is, the particle at rest). As the momentum of the particle increases, we generate a continuum of solutions, either below − mc2 or above + mc2. This is a general feature of all solutions we will obtain from the Dirac equation—we will have continuum solutions on both sides of an energy gap stretching from − mc2 to + mc2, in addition to any discrete solutions. Classically and nonrelativistically we would expect a free particle to have a positive energy. The addition of a rest mass term mc2, which is definitely also positive, should not change this. However, the fact that we now have negative-energy states means that a single particle in a positive-energy state could spontaneously fall to a negative energy state with the emission of a photon. The interaction with the radiation field occurs via the operator α • A. The radiative transition moment therefore connects the large component of the positive-energy solution with the small component of the negative-energy solution. As we have seen, both of these should be large in magnitude, giving a large transition moment. Calculations (Bjorken and Drell 1964) show that the transition rate into the highest mc2 section of the negative continuum is approximately 108 s−1, and the rate of decay into the whole continuum is infinite. Any bound state would therefore immediately dissolve into the negative continuum with the emission of photons. This is clearly an unphysical situation. To resolve this dilemma, Dirac postulated in 1930 that the negative-energy states are fully occupied. The implications of this postulate are significant and wide-ranging.

On the Annihilation of Electrons and Protons

1930 ◽  
Vol 26 (3) ◽  
pp. 361-375 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Quantum Theory ◽  
Kinetic Energy ◽  
Energy Distribution ◽  
Physical Meaning ◽  
Negative Energy ◽  
The Other ◽  
Exclusion Principle ◽  
Positive Energy ◽  
Frequency Of Occurrence ◽  
Negative Energy States

An electron, according to relativity quantum theory, has two different kinds of states of motion, those for which the kinetic energy is positive and those for which it is negative. Only the former, of course, can correspond to actual electrons as observed in the laboratory. The latter, however, must also have a physical meaning, since the theory predicts that transitions will take place from one kind to the other. It has recently been proposed that one should assume that nearly all the possible states of negative energy are occupied, with just one electron in each state in accordance with Pauli's exclusion principle, and that the unoccupied states or ‘holes’ in the negative-energy distribution should be regarded as protons. According to these ideas, when an electron of positive energy makes a transition into one of the unoccupied negative-energy states, we have an electron and proton disappearing simultaneously, their energy being emitted in the form of electromagnetic radiation. The object of the present paper is to calculate the frequency of occurrence of these processes of annihilation of electrons and protons.

Discussion of the infinite distribution of electrons in the theory of the positron

1934 ◽  
Vol 30 (2) ◽  
pp. 150-163 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Quantum Theory ◽  
Kinetic Energy ◽  
Negative Energy ◽  
The Other ◽  
Exclusion Principle ◽  
Negative Energy States ◽  
The World ◽  
Energy States

The quantum theory of the electron allows states of negative kinetic energy as well as the usual states of positive kinetic energy and also allows transitions from one kind of state to the other. Now particles in states of negative kinetic energy are never observed in practice. We can get over this discrepancy between theory and observation by assuming that, in the world as we know it, nearly all the states of negative kinetic energy are occupied, with one electron in each state in accordance with Pauli's exclusion principle, and that the distribution of negative-energy electrons is unobservable to us on account of its uniformity. Any unoccupied negative-energy states would be observable to us, as holes in the distribution of negative-energy electrons, but these holes would appear as particles with positive kinetic energy and thus not as things foreign to all our experience. It seems reasonable and in agreement with all the facts known at present to identify these holes with the recently discovered positrons and thus to obtain a theory of the positron.

scholarly journals Negative energy states in the Reissner–Nordström metric

2021 ◽  
pp. 2150120
Author(s):  
O. B. Zaslavskii
Keyword(s):  
Turning Point ◽  
High Energy ◽  
Negative Energy ◽  
Infinite Chain ◽  
White Hole ◽  
Negative Energy States ◽  
Energy Formula ◽  
High Energy Collisions ◽  
Energy States

We consider electrogeodesics on which the energy [Formula: see text] in the Reissner–Nordström metric. It is shown that outside the horizon there is exactly one turning point inside the ergoregion for such particles. This entails that such a particle passes through an infinite chain of black–white hole regions or terminates in the singularity. These properties are relevant for two scenarios of high energy collisions in which the presence of white holes is essential.

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