Is quantization chart-dependent?

Question

I have a bit of confusion because when doing QFT and QFT in curved spaces this particular issue seems to be avoided.

I have this feeling that when we quantize a theory, we somehow choose a chart and we stick to it. This feeling comes from, for example, the way we deal with Lorentz transformations in QFT, namely via unitary representations. In my head, change of coordinates is something more geometrical rather than algebraic as is done in QFT.

I also asked a professor of mine and he told me that the usual way of quantizing things is chart-dependent and then suggested I read TQFT and AQFT papers for which I'm not ready yet. Can someone help me understand? I am searching for a mathematically rigorous construction of the quantization process (in canonical quantization) and if it can be done in a coordinate free way.

Hope my question does make sense.

EDIT:

I think my question was misunderstood: I do believe that, of course, the physics in QFT is Lorentz invariant. But in my understanding of the process of quantization what we are doing mathematically is the following: pick a chart, construct Fock space/Quantize and then model in that Fock space Lorentz transformations via Unitary transformations. In this process, if I take another chart I construct a different (but canonically isomorphic) Fock space. So you see: in QFT (I believe) you don't treat change of chart in a more geometric way, but you model it algebraically.

I think that what I'm saying can be seen in Wightman axioms: there is no reference on the spacetime manifold (of course the choice of the Lorentz Group comes from the isometries of the Minkowski metric but one can avoid talking about the metric completely) , it's purely algebraic. So are Lorentz transformations.

Answer 1

A fully rigorous formulation of the quantization process in QFT is as of yet unknown. Or, rather, to the extent that we have rigorous formulations of QFT we don't quite know how to apply them to QFT as practiced, and to the extent that we have QFT in practice (such as the Standard Model), it is not rigorous. The problem of formulating QFT in 4 dimensions rigorously so that it can cover e.g. the Standard Model is the unsolved Yang-Mills millenium problem.

It is nevertheless true that during "canonical quantization" in QFT, we usually choose a fixed time coordinate along the way - so indeed, a priori this is a "chart-dependent" process. Physics texts usually phrase this as the loss of "manifest Lorentz invariance", and that phrase already contains the solution: This is merely the loss of manifest independence from the choice of coordinates, but not the loss of actual independence. Specifically, the QFT scattering amplitudes that the canonical derivation ends up with have no dependence on the coordinate choice, and neither do any of the other results that we actually want to use.

Think of this like any other computation in differential geometry: Sometimes you need to choose nice coordinates (like Riemann normal coordinates) to do a computation; this does not necessarily mean the result of that computation is somehow only valid for that particular choice of coordinates - proper tensors that are equal in one chart are equal in all charts, after all.

Other approaches to QFT. like the path integral formulation, do not choose particular coordinates in such an obvious manner, yet end up with the same results, another indication that the loss of manifest Lorentz invariance should not worry us all too much.