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.