Question

Can a non-renormalizable relativistic quantum field theory exist as an exact (rather than effective) theory?

Answer 1

There are theories that are non-renormalizable according to standard perturbation theory but are (at the usual level of rigor in theoretical physics) renormalizable when perturbed around the large N limit. See, e.g.,

G. Parisi, The theory of non-renormalizable interactions: The large N expansion, Nuclear Physics B 100.2 (1975): 368-388.

A fully rigorous construction of such a theory has been given in 2 space-time dimensions. See

K. Gawedski and A. Kupiainen, Renormalization of a non-renormalizable quantum field theory, Nuclear Physics B 262 (1985), 33-48.

In general, the only difference between renormalizable and non-renormalizable theories is that the collection of the former/latter forms a finite/infinite-dimensional manifold of theories. But of course, infinite-dimensional manifolds contain lots of finite-dimensional ones; for example those that set all but finitely many of the parameters to zero. The problem is just identifying the right ones to describe given physics is much harder, especially if one aims at an infinitely accurate (''fundamental'') description of Nature. 

Comments

An " infinitely accurate (''fundamental'')" QFT is, by definition, a Theory of Everything, isn't it? In this sense, it cannot exist since we do not know and may not know everything.

If we do not intend to describe everything, then there may be infinite number of QFT constructions amongst which there may be those who does not need renormalization at all, I guess. I tried to give a toy model of such theories here (so far without any reaction from the physical community). This, third option, is avoided by the physical community as a crackpottery (non-mainstream direction of approximate physics description).

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@VladimirKalitvianski: yours is neither a QFT nor invariant under a relativistic group, hence it is nothing but a toy for you personally.

All relativistic QFT need renormalization, since the covariant interactions one can easily write down are too singular to make sense with a finite coupling constant.

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@ArnoldNeumaier: The "easiness" of writing such an interaction is "compensated" with heaviness of renormalizations and interpretations. You may not exclude the third possibility, you cannot prove its impossibility. And any new direction has started first from toy models, historically. You cannot speak for all physics community. Okay, you underappreciate this direction and my toy model in particular, but it is your personal underappreciation, which you extrapolate implicitly to the whole third direction. Too bad.

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IMHO the moot point is that the correct theory won't need renormalization.

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@JohnDuffeld: You implicitly imply there may only be one "correct theory" whereas there may be many of them.

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 @Vladimir Kalitvianski : I was thinking of one theory for one subject.