We use quantum field theory in condensed matter physics regularly. Let us focus on bosons. Usually, the field theory picture is motivated using a trotterization of the Hamiltonian using the coherent state basis for the bosons, and we und up with a path integral over complex fields. There is an equivalent motivation for fermions where the complex field is replaced by Grassmann fields.
There have been a few papers in the late 1990s / early 2000s (such as this review [Douglas, Nekrasov]) on non-commutative field theory (NCFT) (where the fields that we sum over live not in Minkowski space, but in a non-commutative space where the coordinates do not commute $[x^\mu,x^\nu] = i\theta^{\mu\nu}$) and that the low energy description of the lowest Landau level must be such a non-commutative field theory. This idea has picked up recently with non-commutative field theoretic descriptions of the $\nu=1$ bosonic HLR state [Dong, Senthil] and the Tkachenko mode in a rotating superfluid [Du et al], both of which claim to be a manifestly lowest Landau level theory.
This is more transparent in the first paper, as they use an exact parton (Pasquier-Haldane) construction to develop the non-commutative field theoretic description. In the second paper, the justification seems more handwavy, using the fact that at low energies, the Eulerian coordinate map $\boldsymbol{X}(\boldsymbol{x})$ describing the vortex lattice configuration has Jacobian $=1$, which means that the classical Poisson bracket $$ \frac{\partial X}{\partial x}\frac{\partial Y}{\partial y} - \frac{\partial X}{\partial y}\frac{\partial Y}{\partial x} = 1$$ and "upgrading" that to claim that $[X,Y] = [x,y]=-il_B^2$. Naively, there seems to be an issue with this construction which is that the non-commutative field that they write down later has more degrees of freedom than the superfluid LLL wave function, but that is just an aside.
The confusion I have is this: in many talks on this subject I have come across the sentiment that an effective field theory restricted to the lowest Landau level must necessarily be a NCFT ($\ast$). I understand this is anecdotal, but I was confused by such statements. Is that always true? If so, it seems weird that one needs complicated constructions (such as Pasquier-Haldane) to get there.
The best "explanation" I know for ($\ast$) is that in the lowest Landau level, the projected position operators (called the guiding center coordinates) satisfy the commutation relation $[\hat{R}_x,\hat{R}_y] = -i l_B^2$, therefore we must write a field theory where the fields live on the space formed by these operators which are non-commuting. This is not very convincing to me, so I thought of the following concrete and elementary question.
Question: Can we start from a generic Hamiltonian (for bosons) which is explicitly in the lowest Landau level and argue that, using ideas similar to the first paragraph, the low-energy description must be a non-commutative field theory?
