Recovering QM from QFT: David Tong Notes

[victorvmotti]

Reading through David Tong lecture notes on QFT.On pages 43-44, he recovers QM from QFT. See below link:

[QFT notes by Tong][1] [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdfFirst the momentum and position operators are defined in terms of "integrals" and after considering states that are again defined in terms of integrals we see that the ket states are indeed eigen states and the eigen values are therefore position and momentum 3-vectors.

What is not clear to me is the intermediate steps of calculations not shown in the lecture notes, in particular, the computation of integrals involving operators as their integrand, to obtain the desired results.

 

[atyy]

The full way is very hard, and maybe may not exist.

In the Wilsonian spirit, one would use the Feynman path integral to argue that Galilean symmetry is emergent at low energies and low speeds. Then one would write down all terms consistent with Galilean symmetry. If one could really do the maths, one would be able to determine which of those terms are important and which are not. I don't know if one can do the maths, or even if one can, whether anyone has done it. So the present use of non-relativistic physics must be treated as a mix of theory (Galilean symmetry emergent at low energies) and guesswork confirmed by experiment (determination of which terms are important).

An analogous case is the derivation of the chiral perturbation theory lagrangian. There are very interesting comments, including the non-relativistic case, in "Foundations of Chiral Perturbation Theory" in http://www.scholarpedia.org/article/Chiral_perturbation_theory.

Another interesting place to look is the chapter on bound states in external fields in the first of Weinbger's 3 volumes on QFT. Then one has to also know the relationship between non-relativistic QM which can be derived from relativistic QM which can be derived from relativistic QFT. Relativistic QM, the intermediate theory, is not a coherent theory, but for some strange reason, non-relativistic QM is a coherent theory.