# Could a reformulation of GR and QFT in terms of nonstandard analysis be useful?

Are there any serious reformulations of a quantum field theory of general relativity within a nonstandard analysis framework?  Is it possible that such reformulations (which are possible,  in principle) would create any predictive models related to some major problems in physics (like the black hole singularity problem, or quantum gravity)?

I emphasize that this is a question of principle, an elaborate answer cannot be given here. Nonstandard analysis is non-constructive, and model-theoretic nonstandard analysis is based on superstructures (pushing the level of abstractization one step up). A similar phenomenon can be observed with the emergence of the mathematical framework of quantum mechanics, with the transition towards operator calculus, not frequent or necessary in classical mechanics. So basically, could one level of abstractization higher solve major problems in physics?

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• I think it is a non-trivial claim that such reformulations are, in principle, possible. In particular, it requires some evidence to claim that. Especially, uniqueness of general relativity and Yang-Mills theories at "large" distances is almost unavoidable as long as one follows the rules of quantum mechanics and special relativity. – Dvij Mankad Jul 21 at 3:21
• Nonlinear PDE's, Riemannian geometry, differentiable manifolds, topology, measure theory (probability), Hilbert spaces, can all be studied within a nonstandard framework (basically most domains of mathematics). The question is whether this would be beneficial for physics. – Cristian Dumitrescu Jul 21 at 4:13
• I would say that the typical physicist’s conception of $dx$ is already pretty much that “it’s an infinitesimal”, in the spirit of NSA. I personally never think in terms of epsilons and deltas. So it is hard for me to see how NSA could bring any revelations to physics when physicists have been thinking this way for 400 years. – G. Smith Jul 21 at 5:15

Non-standard analysis (NSA) is more of a way to reformulate the foundations of mathematics, than it is a way to reformulate applied-mathematical theories constructed on those foundations, which is what (the theoretical canon of) physics is. It deals with things like how to formulate tools like the derivative, integral, and limits in an alternative fashion to the usual $$\epsilon$$-$$\delta$$ formalism often seen in books, under the idea that that is not such an intuitive formalism, but rather that it would be better to use an explicit idea of an infinitesimal number so that, say, $$dx$$ becomes an actually "infinitely small increment" (though even here, it doesn't quite manage to work out perfectly.). It doesn't change their mathematical properties in any way, just gives a different way of defining them.

The chief purpose of these formalisms are in making mathematical proofs. Theories of physics are applications of differentials, integrals, limits and other things, and physicists are not typically concerned with proofs. The extent to which you will find $$\epsilon$$-$$\delta$$ proofs in a physics paper is thus, also, roughly (with of course due mindfulness of the possibility to be surprised!) the extent to which you could expect NSA to be useful in physics, i.e. outside of studying it from a mathematician's pov as mathematical constructs. Most physicists care so little about what $$dx$$ "really" is in many cases (perhaps too little, imo), that they use it in ways that are "said to make a mathematician cringe". Though for me, with fair interest and study of both perspectives I find them each to have advantages and disadvantages and no problem in using either when suitable.

That said, and to note my Math.SE answer here:

https://math.stackexchange.com/questions/3241309/what-is-the-intuition-behind-uniform-continuity/3241346#3241346

I'd say that actually the "standard" $$\epsilon$$-$$\delta$$ limit formalism is not that bad at all, nor unintuitive. It's just that as with many things in these areas, the way it's commonly presented really doesn't do it the justice it deserves and serves to obscure. $$\epsilon$$ and $$\delta$$ are approximation tolerances, the former on the dependent variable and the latter on the independent variable, like in empirical measurements and the use of calculators with finite numbers of digits, and $$\epsilon$$-$$\delta$$ formalism thus not only should be with proper exploitation of such intuitions be easily understandable to anyone who has mastered the use of those, but in fact I'd even say it is perhaps closer in vein to scientific praxis than NSA is (something that, again, is a view that sometimes may or may not be advantageous to dealing with any given formalism).

ADD: To heed the comments, I would say that mathematical proofs aren't irrelevant to physics - just maybe to most practitioners. Being able to more easily prove theorems or perhaps gain insight into how to prove a long-standing conjecture by a different view of your mathematical tools can certainly help in practice on the purely theoretical side of things that may then lead to conceptual breakthroughs and reformulations. My answer though is really to try and point out that using NSA does not by itself result, as the titular question suggests, in a "reformulation of GR and QFT" any more than, say, recompiling a computer program for a different processor architecture (that can be so recompiled) merely by doing it rewrites its source code or improves the algorithms used. The lower-level meaning/signification of the terms in the programming language changes perhaps dramatically, but the high-level construct does not see a thing.

• @G. Smith : Possibly. Or maybe, "small enough to not worry" about it is closer to what I hear. But my point is that I don't think emphasizing a specific mathematical formalism underneath differentials and integrals is likely to gain any new surprise results to most physicists who don't typically deal in proofs anyways - though I could be wrong because NSA has been used profitably in some endeavors, IIRC, in making new proofs. But it'd be more one for those who are or slanted more toward being a mathematician, certainly not experimentalists, and also if anything more illustrates that – The_Sympathizer Jul 21 at 5:16
• it is best to have on hand a flexible array of formalisms for doing proofs, as long as you can understand their intuitive base (which as I said is perhaps not conveyed so well for the standard $\epsilon$-$\delta$ formalism), and how they relate to each other. More tools in the tool box. – The_Sympathizer Jul 21 at 5:17
• However, the titular question suggests that somehow using NSA "reformulates" GR and QFT as in to cast them in a new set of mathematical equations. This is not the case because they are "higher level" than where the distinction of NSA operates. It rather provides another view on the existing concepts those theories are built upon. The equations that define each will look identical in either SA or NSA, just that how those mathematical gadgets operate on a lower level will look different. – The_Sympathizer Jul 21 at 5:18
• You are right, for the physicist dealing with finite precision measurements, nonstandard analysis is irrelevant. For the theoretician though, trying to unify QFT and GR (or trying to understand what's happening at the center of a black hole, for example) having a CONSISTENT conversation about divergent series or infinite quantities might be useful (if not for any other reason, just to develop better models). Nonstandard analysis can do that. As an example of a conversation of this type : math.stackexchange.com/q/2649573/101896 – Cristian Dumitrescu Jul 21 at 5:49
• @Cristian Dumitrescu : Thus, why I acknowledged that they may have some use, just not necessarily to most. Moreover, they don't provide any conceptual insight into physics - but they might make a mathematical proof needed in the course of pursuing such insight, easier. Also, divergent sums can be "consistently" treated without NSA, but as said, still NSA can provide a different viewpoint. – The_Sympathizer Jul 21 at 5:52