Electrons and holes vs. Electrons and positrons

Question

Drawing parallels between electrons and holes in semiconductors, and electrons and positrons in Dirac equation is certainly useful in the context of learning/teaching the quantum field theory methods, since it allows drawing parallels between the nearly identical formalisms. I am wondering however, how far/literally this analogy can be taken. 

Here are a few points to consider:

• Free electrons and holes in semiconductors are not real particles, but quasiparticles - excitations of many-body system interacting via Coulomb forces. I am not qualified to judge to what extent Dirac electrons and holes are true particles

• The notion of a filled valence band is beyond doubt in semiconductor theory, whereas the concept of the negative spectrum filled with electrons in Dirac theory is just an interpretation.

• Symmetries are certainly different: Dirac equation follows from continuous Lorentz symmetry transformation, while crystal groups are descrete, the number of valence and conduction bands is not the same, the bands have different shapes and even their minima are not necessarily aligned in k-space.

• What is the equivalent of spin in a semiconductor? I did see some articles where spin-orbit coupling in semiconductors was estimated by resorting to the analogy with Dirac equation, but the viability of such estimates has more grounded in perturbation expansion than in actual equivalence between two pictures (replacing $mc^2$ by the gap energy pretty much guarantees getting correct scale for any interband process.)

I am looking for clarifications regarding the points that I raised, and possibly additional similarities/differences.

Answer 1

The Dirac theory was a starting point for Quantum Field theory . QFT has evolved over the years as a formalism using creation and annihilation operators on plane wave quantum mechanical wave functions. This allows the possibility of calculating crossections and decays in particle physics. But QFT has found uses in other disciplines: back in 1961 I was taught a quantum field theory for nuclear physics (found this by searching) .

In particle physics, the elementary particles in the table of the standard model, with axiomatically given mass and quantum numbers, are considered to cover the whole spacetime each with its plane wave solution. Dirac for fermions, Klein Gordon for bosons, quantized Maxwell for photons, on which creation and annihilation operators work . So as far as QFT for elementary particles the particles are considered real if their vector is on mass shell.

That QFT can be used for condensed matter does not mean more than that :once a mathematical tool exists, various ways of using it can be found.We find integrals and differential equations in all physics fields after all.

Electrons and holes vs. Electrons and positrons

In the standard model QFT positrons (all antiparticles) are treated the same way as electrons. If a particle interacts with its antiparticle Feynman diagrams( the calculation tool of QFT) exists to calculate the crossection. To search for analogies with the use of QFT in other fields is not really meaningful.

Answer 2

Here is a monograph that starts out with the relativistic Dirac equation for electrons/positrons and then discusses modifications necessary for it to be a decent model for electron/holes in condensed matter (aimed at discussion of topological insulators):

• Shun-Qing Shen, Topological Insulators -- Dirac Equation in Condensed Matters, Springer 2012 (doi:10.1007/978-3-642-32858-9)

(See section 2 "Starting from the Dirac equation" around "Why not the Dirac equation?")

However, if I may followup on the original question, I'd be interested in seeing decent discussion of both: Using the relativistic Dirac equation to describe energy bands in solid states, taking account of both the fundamental electron/positron field and its incarnation as effective particles in solids.

One article in this direction that I am aware of is

• C. Hainzl, M. Lewin, É. Séré: "Existence of a Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation", Commun. Math. Phys. 257 515–562 (2005) (doi:10.1007/s00220-005-1343-4)

But I wish there were more.

Answer 3

Indeed, while the electrons and holes picture of semiconductors is completely valid, the interpretation of a positron as a hole in a sea of filled electron states is totally flawed.