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.

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.

Unitary Transformations of the Dirac Hamiltonian

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Unitary Transformation ◽  
Free Particle ◽  
Algebraic Manipulation ◽  
Negative Energy ◽  
Positive Energy ◽  
Small Component ◽  
The Matrix ◽  
Component Wave ◽  
Energy States ◽  
Dirac Hamiltonian

The separation of the spin-dependent terms in the Dirac Hamiltonian enables us to make an approximation in which the spin-free terms are included in the orbital optimization and the spin-dependent terms may be treated later as a perturbation. In this process, the parameter space required to treat the large and small components has not changed. Even with the extraction of (σ ·p) from the small component, we still have to calculate integrals involving pfL, which essentially regenerates the original small component space and so the integral work has not really changed. What has been achieved is the ability to use the machinery of spin algebra from nonrelativistic theory, but we are left with a large and a small component. The obvious next step is to separate the large and small components, or the positiveand negative-energy states. The small component can be eliminated from the Dirac equation by algebraic manipulation, but this leaves the energy in the denominator. It would be preferable to obtain an energy-independent Hamiltonian that acted only on positive-energy states and that could therefore be represented as two-component spinors. If, following this separation, it were possible to separate out the spin-free and spin-dependent terms, we would have a spin-free Hamiltonian that would operate on a one-component wave function, and we would then be able to use all the machinery of nonrelativistic quantum chemistry but with modified one- and two-electron integrals. The matrix form of the Dirac Hamiltonian suggests that we should seek a unitary transformation that will make it diagonal with respect to the large- and small-component spinor spaces. Such a transformation is called a Foldy–Wouthuysen transformation (Foldy and Wouthuysen 1950). Although in their original paper only the free-particle transformation was derived, together with an iterative decoupling procedure that will be described later in this chapter, the term Foldy–Wouthuysen transformation has come to mean any unitary transformation that decouples the large and small components, either exactly or approximately, and we will use it in this sense.

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.

scholarly journals Quantum backflow in solutions to the Dirac equation of the spin-1 2 free particle

2018 ◽  
Vol 33 (32) ◽  
pp. 1850186 ◽  
Author(s):  
Hong-Yi Su ◽  
Jing-Ling Chen
Keyword(s):  
Quantum Mechanics ◽  
Dirac Equation ◽  
Relativistic Particle ◽  
Relativistic Quantum ◽  
Free Particle ◽  
Negative Energy ◽  
Dirac Particle ◽  
Relativistic Quantum Mechanics ◽  
Probability Current ◽  
Negative Probability

It was known that a free, non-relativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current — hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different signs of both momenta and energies. Neither of these cases has a counterpart in non-relativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.

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