What is the deep difference Between a hole in Dirac sea and an empty orbit around nucleus?

Question

Dirac sea is proposed to reasoning stability of electrons in positive energies. holes that could be occupied by electrons or not (but almost all of them are occupied). Then Dirac propose such a hole must be something like a positron. I think an empty orbit around a nucleus have somehow same properties as Dirac hole: an electron by more energy simply falls to it and emit photons. So why we don't interpret them as Dirac holes or why we don't think of Dirac holes simply as empty orbits and nothing more?

Answer 1

To begin with:

The Dirac sea is no longer mainstream in physics because it is connected with at least 2 problems. It is based on Pauli's principle and as such does not work for bosons. The second main problem with the Dirac sea is that it creates an infinite negative charge of the vacuum which some kind of has to be subtracted in order to get a neutral vacuum. So negative energy particle (i.e. electron) solutions of the Dirac-equation are now considered as running backward in time, a picture which works for fermions as well as for bosons. One could of course wonder which picture is more "weird", having a Dirac sea or having particles running backward in time. But for the mentioned reasons above the considering negative energy solutions of the Dirac equation as running backward in time is now the preferred picture.

But nevertheless, the Dirac sea is still considered as a rather intuitive picture and this is probably the reason why it is still referred to it. Let's assume it is an acceptable description. The main difference between an electron in the Dirac sea and an electron in an atom is that the Dirac-sea electron has really negative energy $E=-\sqrt{(m_ec^2)^2 + \mathbf{p}^2}$, so for rather small $\mathbf{p}$ it is $E\approx -m_e c^2 =-511$keV, whereas an electron in a normal atom has positive energy $E\approx m_e c^2 =+511$keV, per definition it is not part of the Dirac sea. The potential well of the atom is in usual cases too shallow in order to provide the electron with a negative energy. For this the potential well should have a depth of at least $-511$keV which is usually very difficult to achieve. Auger-electrons can fall a couple of keV in the potential well of an atom emitting X-rays, but they don't fall around or more of $-511$keV.

For completeness one should mention that the question what happens to electrons falling into a potential well of that depth was actually studied. In order to get such a deep potential well, 2 very heavy nuclei have to be fusionated in order to reach a $Z$ for the new nucleus (for a very short time) which is high enough in order to get energy levels which lie below zero energy. Such experiments were carried out and led to spontaneous electron-positron creation, if the potential well is deep enough, probably it would need a potential well of $2 \times -511$keV. Nevertheless, an electron which originally has its rest energy of +511keV, even if it reached very negative energies would be always be considered as an individual electron which had originally positive energy and is not part of the Dirac sea.

But coming to the key of your question and to the beginning: Why should one consider a hole in the shell of an atom as hole in the Dirac sea? Such a consideration should have some purpose. The purpose would be adopting a picture of a rather common electron state which comes along with complications, why forcedly adopting a description with complications ? In the previous paragraph it is shown that even under very particular circumstances one would never consider an electron in the potential well of an atom as a Dirac sea electron. Anyway, coming back to the very beginning, the physics community would not adopt such a picture because it is simply out-dated.

Answer 2

I would guess, that the reason is just how the subject evolved historically:

First QM (Schroedinger eq.) was as a theory describing atoms pretty well. Only later when bringing together QM and special relativity, people found the relativistic equations (Dirac and Klein-Gordon eqs.), which brought with them the problem of negative energies, which Dirac tried to solve using the Dirac sea.

So first of all atomic physics (up to relativistic corrections) were already well understood before Dirac came up with his idea. Another, maybe even better argument is that empty "orbits" are not as densely packed as Dirac holes or holes in solid state physics, where the holes form kind of an continuum.

In the end I don't see any reason why it would help us to interpret one as the other. "Hole theory" is a nice tool for solid state physics, but for fundamental physics it has only historical/didactic importance as far as I know. It is a concept that may make it easier for students to accept the first steps in relativistic quantum theory, but in the end it isn't needed for QFT.

Answer 3

The difference is that the atomic orbitals are bounded from below. There is a maximum energy transition from $r$ tending to infinity and the ground state of hidrogen for example. That is related to a photon of a given frequency.

But a hole in the Dirac sea lacks that bound. An electron decaying to it can generate a photon of any energy, depending on how deep the hole is.

In the same way, a photon of great energy can release an electron from deep inside the sea, leaving a hole.

What is behind that notion (of Dirac sea) was maybe a kind of conservation principle. All the observed particles existed somewhere before the experiment.

If we accept that they can be created and annihilated (photons, electrons and positrons for example), it is no longer necessary.

Answer 4

Excellent question ! Indeed, the empty orbitals around nuclei behave like holes in the Dirac sea. This analogy is widely used in the condensed matter physics, when treating electronic excitations from the valence to the conduction band. The empty orbitals in the valence band are called holes and indeed behave in many ways as positively charged particles: e.g , carrying electric current or forming hydrogen-like bound states with electrons.

Answer 5

I focus on the question in the title. The difference between a hole in a Dirac sea and an empty orbital around a nucleus is that in the latter case there is no Dirac sea.