Obtaining NRQM from QFT: Issues, Folklores and Facts by Padmanabhan

[gentzen]

Padmanabhan's paper "Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts" (https://arxiv.org/abs/1712.06605) is long (58 pages) and substantial, therefore it deserves its own thread.

Here is a recent discussion about Padmanabhan's paper in an unrelated thread:

Most think that QFT in the non-relativistic limit reduces to ordinary QM. A careful analysis in the paper I posted shows the limit is not ordinary QM.

The non-relativistic limit of relativistic QFT is non-relativistic QFT. Non-relativistic QFT, also known as "second quantization", is widely used in condensed matter physics. In general, states in non-relativistic QFT do not have a definite number of particles. However, when non-relativistic QFT is applied to states with a definite number of particles, the resulting theory is equivalent to non-relativistic QM.

The Padmanabhan's point is that the NR limit of (relativistic) QFT is not merely NRQM of particles, but NRQM of particles and antiparticles. While this is correct, I find it a bit trivial. For instance, it means that the NR limit of QFT based on the Dirac equation is not merely NRQM of electrons, but NRQM of electrons and positrons. True, but so what? It doesn't make the usual NRQM of electrons wrong, it only means that a similar theory can also be applied to positrons. As long as we study processes in which the number of electrons and positrons does not change, there is no much difference between NRQM and NR limit of relativistic QFT. Of course, when we study processes in which the number of particles and/or antiparticles changes, then we must take into account the full relativistic QFT, nobody denies that.

So loosely speaking, it is still true that NR limit of relativistic QFT is NRQM, with the only caveat that the latter describes particles and antiparticles.

He also points out that you get two Schrodinger's equations (one for the particle, and another for the antiparticle) in the Heisenberg picture, but not operating on the quantum state, but on the quantum field.

 

[Peter Morgan]

A quick note that the published version in Eur Phys J C 2018 is OA, https://doi.org/10.1140/epjc/s10052-018-6039-y