QFT and its non-rigorous assumptions

Question

I have been trying to figure out all the non-rigorous assumptions of QFT (as performed in an operator theory) that allow it to function as it currently is. So far, the three big candidates I found are these :

1. The product of distributions is allowed in some sense (this runs afoul of Schwartz's impossibility theorem, although it is possible to define it rigorously but it is rarely done in QFT)
2. The interacting field operator is related to the free field operator by some unitary transformation (Wrong in most cases, as shown by Haag's theorem and other such things)
3. The vacuum of the interacting theory is related to the vacuum of the free theory. In particular, we have that $\langle 0 \vert \Omega \rangle \neq 0$. According to Haag (p. 71), this is also wrong. The usual formula for relating the vacuas fail to converge.

Are those the only common wrong assumptions? I'm also thinking that derivatives of the field operators may be ill defined as derivatives on operators are usually well defined for bounded operators, but I'm not quite sure.

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user Slereah

Comments

@CosmasZachos this has nothing to do with proofs. People are trying to organize their knowledge of issues with QFT by asking questions on PSE. Nothing wrong with this, don't you agree? As an example of a real problem which QFT most likely isn't able to solve, try Quantum Gravity.

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user Solenodon Paradoxus

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@CosmasZachos The fact that something is "working" physically does not mean it is acceptable mathematically. On the other hand it does not also mean that the theory is false, but simply that we do not have yet the proper tools to make it true, by mathematical standards. QFT is a successful physical theory full of mathematical inconsistencies, that nobody has yet been able to overcome (at least in interesting systems).

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user yuggib

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To overcome these inconsistencies would be an extremely important result in mathematical physics (so important that there is a huge money prize to solve it). And the OP is just asking about these inconsistencies.

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user yuggib

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@yuggib I had do clue the rant poses as a portal to solve the Millennium problems and quantizing gravity... The "inconsistencies" undergirding it appear like a mis-statement of QFT as currently practiced. Constructive field theory was initiated to improve and understand successful QFT, not negate it and take it into arid directions. Nobody could ever argue with success, but one must solve something.

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user Cosmas Zachos

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I don't think "lies" would be an appropriate term in this context. I have essentially 0 knowledge of QFT; but I highly doubt that the theory is based off of lies. Assumptions, sure, invalid assumptions, perhaps, but lies; I don't think that is the proper term for it. As far as I'm aware the intent of the theory is not to provide incorrect information.

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user JMac

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I am asking what the assumptions of QFT are, beyond the well known ones. All things related to Hilbert spaces, Poincaré invariance, gauge theory, commutation relations, etc, are well known, but there are a few hidden assumptions that many textbooks do not point out explicitely, since they are not 100% rigorous. How many such assumptions form the basis of most QFT (for instance, the QFT of the standard model)?

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user Slereah

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@CosmasZachos I feel like there's a misunderstanding. Nobody in their mind would argue against using QFT in its domain of applicability. Nobody in their mind would deny the great experimental successes of QFT models. This question is about the subtleties which are related to the regime where QFT approximation is no longer sufficient. It is about replacing QFT with something else, and about understanding the core issues which may or may not help with developing this "something else". And this is highly speculative, there's no doubt about that.

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user Solenodon Paradoxus

Answer 1

I'm not sure there is a definite answer, since nobody seems to agree as to what QFT really is (at least so says Nate Seiberg in his 2015 Breakthrough Prize talk), but the problems you mention are well known and accounted for.

Quantum Mechanics gets loads of issues with infinite degrees of freedom. So in reality what we actually do is to regularize the theory, both infrared (e.g. putting in finite volume) and ultraviolet (e.g. putting in a lattice). When properly regularized QFT has finite degrees of freedom, and therefore all Haag's objections go away. You have a perfectly defined interaction picture and a mapping between free and interacting ground states, so everything goes smoothly.

This has the drawback of breaking Lorentz Symmetry, but the hope is that once we renormalize, by taking appropriate limits, we are left with sensible answers and recover Poincaré invariance.

Since the couplings in QFT are very singular and ill-defined we should have done the regularization in the first place just to know what the hamiltonian is, so this is a key part of QFT.

The problem with Haag's approach, and other axiomatic ones, is that they work directly with already renormalized fields, and therefore run on multiple problems. This is why more recent textbooks, starting with Weinberg (I think), emphasize that regularization and the renormalization group are key concepts one cannot do QFT without, even in the absence of interactions.

This post imported from StackExchange Physics at 2017-10-11 16:30 (UTC), posted by SE-user cesaruliana