# Recovering nonrelativistic quantum mechanics from quantum field theory

In quantum field theory -- specially when applied to high energy physics -- we see that the requirements of Lorentz invariance, gauge invariance, and renormalizability strongly limit the kinds of interactions that can appear on the Lagrangian. On contexts other than high energy physics, however -- say condensed matter physics, or even nonrelativistic quantum mechanics -- we treat the interaction term (essentially the potential $$V(\mathbf{r})$$ in quantum mechanics) as practically arbitrary. I would like to know how this symmetry-breaking "phenomenon" comes about, i.e., how one can start from a fundamental theory which possesses all these restrictions on the kinds of interactions that can appear, and end up with a virtually infinite freedom for the "effective" theory in low energies and low speeds. If there is any literature on that, I would highly appreciate it.

• Are you aware that quantum field theory in particle physics emerges from the underlying simple quantum mechanical relativistic equations? The operators of QFT operate on the plane wave ( no potential) solutions of dirac equations for fermions, klein gordon for bosons and quantized maxwell for photons.. So the question is backward, imo. One should be asking how the coulomb potential is built up by the operator algebra of the QFT see page 147 here damtp.cam.ac.uk/user/tong/qft/qft.pdf – anna v Oct 14 '18 at 17:02

As an example, consider quantum electrodynamics (QED) in four-dimensional space-time. The only known mathematically legitimate way to construct this model involves replacing continuous space-time (or at least space) with a discrete lattice. The use of a discrete lattice is not strictly consistent with Lorentz invariance, but we can choose the lattice scale to be much finer than any experimentally resolvable scale. Then, by tuning the coefficients in the Lagrangian (or Hamiltonian), the model's experimentally-accessible predictions can be made rotation- or Lorentz- invariant as far as any practical experiment can tell. This would not be a satisfying way to formulate a fundamental theory-of-everything, but QED is not a theory of everything. It's scope is already limited even without the artifact of a discrete lattice, so resorting to a discrete lattice is an acceptable (though messy) way to define it.

The key to making this work is tuning the coefficients in the Lagrangian appropriately. If we change the scale of the lattice (that is, the spacing between neighboring sites in the lattice), then we must re-tune the coefficients in order to keep the model's low-resolution predictions unchanged. ("Low resolution" is compared to the lattice scale.) This is renormalization. This is possible as long as we don't make the lattice spacing too small. If we make it too small, then we presumably reach a point where the required values of the coefficients in the Lagrangian diverge. This is what people mean when they say "QED does not exist." What they really mean is that QED by itself (without any additional fields) does not have a strict continuum limit in which electrons and photons still interact with each other. However, there is a broad range of lattice spacings that are much finer than the finest experimentally resolvable scale but still safely coarser than the Landau-pole scale, and any such lattice may be used to define QED.

Now, I'll begin to answer your question. We could reproduce the same low-resolution predictions using a Lagrangian with many more terms than just the usual "renormalizable" terms, even if we use only gauge-invariant terms that are built from the usual fields of QED. There are infinitely many such terms, and we may use them to build infinitely many different Lagrangians whose predictions are indistinguishable from each other at sufficiently low resolution. This is called "universality." Among these infinitely many different options, there is one option that uses only the usual renormalizable terms, which are few in number as you pointed out. We are not really limited to using only these renormalizable interactions in the construction of the model, but we might as well use only these terms as long as experiments are limited to sufficiently low resolution compared to the lattice scale.

Now, suppose that we only want to consider situations in which all electrons are moving much more slowly than the speed of light. (I'm thinking of the simplest version of QED here, in which the electron field and the electromagnetic field are the only two fields.) In other words, we only want to consider situations in which all electrons have energies much lower than the mass of an electron. We could use the Lorentz-symmetric version of QED to address these situations, but we also have the option to use a different model that has the non-relativistic approximation built into it. We can call this model non-relativistic QED (NRQED). Or, if we don't need to consider dynamic electromagnetic effects like photons, then we can even use non-relativistic quantum mechanics. In either case, we can expect something like "universality" to occur again in this context as long as we consider only predictions involving energies that are sufficiently low compared to the mass of the electron, which is the scale that we're using to define the "non-relativistic" approximation. As in the relativistic case, where the artificial lattice spacing was the reference-scale, there are infinitely many different non-relativistic models that all make the same predictions at sufficiently low energy compared to the mass of the electron. Among these models, we may choose the one that uses the fewest and simplest terms, just like we normally do in relativistic QED.

A similar comment applies with respect to condensed matter physics. In this case, we are only sometimes doing experiments close to a "critical point," at which the correlation length becomes much longer than the spacing between the lattice of atoms or molecules of which the material is composed. This occurs, for example, near the phase transition between the magnetized and unmagnetized phases of a ferromagnetic material. In those circumstances, we can indeed get by with a model built from only a few relatively simple terms, and these terms are again called renormalizable. This is analogous to the situation in relativistic QED, except that (1) the models are built using different fields and with different symmetry requirements, and (2) in the case of relativistic QED, we are always restricted to scales far coarser than the (artificial) lattice scale, but we are not restricted to scales far coarser the atomic scale (which is condensed matter's analog of the lattice scale) or to energies far below the electron mass (which would be nonrelativistic QM's analog of the lattice scale in the present analogy).

In summary, as far as your question is concerned, main difference between relativistic QED and nonrelativistic quantum mechanics is that in relativistic QED we are always working at scales far below the artificial lattice scale at which the model is defined, so we can always get by with only a few relatively simple terms in the model's construction, namely the "renormalizable" terms. But in nonrelativistic applications, we are only sometimes working with energies far enough below the mass of the electron to get by with only a few relatively simple terms in the model's construction.

Here are a few references that address these points in more detail:

• This non-introductory article studies a model that is easier than QED but that is still Lorentz-symmetric at sufficiently low resolution: Polchinski (1984), "Renormalization and effective Lagrangians," Nuclear Physics B 231: 269-295, http://max2.physics.sunysb.edu/~rastelli/2016/Polchinski.pdf, accessed 2018-10-14.

• This introductory article studies the same idea in a condensed-matter context: Polchinski (1992), "Effective Field Theory and the Fermi Surface," https://arxiv.org/abs/hep-th/9210046.

• This pedagogical article explains how the same idea works in NRQED: Lepage (1989), "What is renormalization?" Boulder ASI, pages 483-508, https://arxiv.org/abs/hep-ph/0506330.

Hope this helps!

Some time after posting my answer here, I came across this post: Does QED really break down at the Landau pole? That post has an interesting discussion about what goes wrong when we try to take a strict continuum limit in QED.