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While studying the Renormalization Group, we found the interesting result that states "every field theory is an effective theory" (every FT is only valid for energies smaller than a particular scale $\Lambda$), because the renormalization flow always goes towards the IR regime and going in the opposite direction would mean turning on an infinite number of couplings. To build a higher-energy-theory, we can't continuously change the coupling constants: we need to take a leap.

Now my questions are:

  • is this valid for every theory that we ever had in the past? Were they field theories too? Example: if we knew the RG at Newton's time, could we have guessed that, in order to understand why Mercury wasn't behaving well, we needed a completely different theory?

edit: one answer correctly points out that gravity as we know it is not-renormalizable, so it isn't a good example. Maybe a better example would be if we knew the RG at Maxerll's time, could we have guessed that, in order to understand the photoelectric effect, we needed a completely different theory?

  • is there an escape? To build a fundamental theory, we would need a non-field theory, from the statement above. While I'm aware of the fact that non-fundamental theories are perfectly fine and a fundamental theory may not even exist, but if there is one, it's not a field theory. Do non-field-theories exist? What's an example of a modern theory that isn't a field theory?

Edit: in the comment, I was rightfully asked to provide my definition of "fundamental theory", since it can be open to interpretations. I mean fundamental as opposed to effective: I'm looking for a theory that, at least in theory, works at every energy scale. If there is such a theory we could obviously find experimental issues at unprobed scales, but that's not what I'm asking.

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We have to be careful here about what we mean by "field." A field is a mathematical object that has been found to be very useful in making physical theories. I don't want to get bogged down in definitions, so I'll take a more conceptual approach.

An example could be basic quantum mechanics, where the state of a quantum system is represented by the abstract vector quantity called a "ket." Paul Dirac's PhD laid down this theory, which he used to show that Schrodinger's wave mechanics and Heisenberg, Born, and Jordan's matrix mechanics were two different algebraic representations of the same abstract object.

Now, a physicist or mathematician trained in field theory will say that ANY theory can be cast in terms of an effective field theory. And this seems to be true. But this is also true about writing any theory in a covariant form, even if the equations themselves are not fully invariant under general coordinate transformations (i.e. Newtonian gravity can be expressed in covariant form too, but only Einstein's equations are fully invariant under general coord. trans.). So it comes down to what perspective do you want to have?

There's a rich history in modern quantum theory about whether it's better to use particles or fields. It traces its origins back to the classic debate between Huygen's wave theory of light vs Newton's corpuscles of light. This article is a nice, short discussion that highlights the main points of contention for the modern debate between Feynman et al. and Julian Schwinger et al. about QED. Together with Tomonaga, they shared the Nobel prize in 1965 for the synthesis of their ideas that Tomonaga arrived at independently.

However, do not be fooled by this synthesis! There are currently something like 9 equivalent formulations of the laws of quantum theory today, some of them do and some of them do not use the concept of "fields."

Along these lines, another example of a purely "non-field" theory could be Feynman and Wheeler's absorber/emitter formulation of electrodynamics. Even though they still talk about electric and magnetic "fields," these fields interact by emitting a kind of particle, rather than having to probe the fields using test particles like in classical electrodynamics. So they're unlike other usual "fields."

Lastly, one more example can be made with Einstein's general theory of relativity. In its classic form, it is considered a "field theory," hence they are called "Einstein's field equations." Feynman and Weinberg successfully created an effective field theory version of Einstein's equations, which has proven useful in quantum gravity research. So again, it comes down to what perspective is most useful for what you're trying to do.

A nice philosophical discussion in the modern context is here, where the author concludes that healthy, active areas of research should involve a mixture of field theoretic approaches with other approaches. I tend to agree with this attitude, since it never hurts to have several perspectives to understand a theory through - it actually almost always helps to advance the theory! Even if its just rewriting it in a different mathematical framework, which is essentially how Dirac discovered his relativistic equation for spin-1/2 fermions.... by rewriting the Schrodinger equation in a different way.

This is my 2 cents, but I bet others have good, and probably more technical, insights.

Your questions are good questions, but they are not well defined. For example, no one knows how to precisely differentiate between "effective field theories" and "fundamental theories." Indeed, we don't really even know what makes something "fundamental" anyway, this is something that is still up for debate.

Also, gravity as we understand it is still non-renormalizable, probably for good reasons though. So your first question doesn't seem to be the optimal question to ask. To me, it's more interesting to ask, "could we have explained the precession of Mercury's perihelion with an effective field theory instead of needing Einstein's theory?" My answer is that it's unlikely to happen like that, basically because in practice the effective field theory is made after the more "fundamental" theory is already established. That is, by definition, an "effective theory" does not explicitly claim that the mechanism employed in the theory has a direct counterpart in the actual causes of the observed phenomena to which the theory is fitted.

EDIT: However, this has happened numerously in the past. For instance, before Gamow showed that quantum tunneling can explain how the Coulomb barrier is overcome in stellar nucleosynthesis, stellar models assumed that nuclear processes produced the right amount of energy that was observed without explaining those processes. This was not done using a field theory approach, but it shows that its possible in principle.

One could argue that the early forms of QCD were precisely what you ask for: effective theories that used fields to explain the experiments. But this requires the experimentation to be a bit ahead of the theory, meaning it comes down to historical circumstance and timing. Regarding general relativity, this did happen in some form: various theories of gravity were proposed prior to GR that encapsulated a lot of the same physics, but without the equivalence principle, and didn't explain the relativistic precession of mercury. So I'd say it really just depends on the particular circumstance/example you want to discuss.


EDIT #2: The OP has updated their question:

The existing answer gives interesting insights and perspectives, but I'm looking for actual examples (to be possibly studied in a graduate thesis, or a Ph.D.) of theories that don't involve fields at all (and therefore wouldn't fall in the RG idea of "every field theory is effective").

First, not to be rude, if anyone helps you in your thesis then you should include them in your official acknowledgements. After all, obtaining a PhD is not just about solving a problem - it's also about learning how to find relevant problems and making them well-posed/well-defined. So how would we begin to make your proposition well-posed (so it could then be molded into an appealing proposal)?

I think you'd have to define what a "field" is, then look at a few existing physical theories that do use fields so you can have something to compare to. With that exposition established, you'd need to think hard about the example(s) you wish to explore. Truly, you might have to try and fail a few times to find a good one(s). That's part of the process! ;D

One issue is that you'll have to make some kind of (seemingly arbitrary) distinction between "field theories" and "non-field theories." In pure math, a field is an abstract object that follows the field axioms. This includes the various sets of numbers under addition and multiplication - rather elementary things - but there are more complicated constructions as well.

But you're interested in physics. Physicists typically (at least nowadays) make a distinction between "classical fields" and "quantum fields." Given your questions, I presume that you're decently acquainted with both.

So the first thing I would do is examine many examples of "classical field theories" and "quantum field theories," so that I may separate my proposal into examples of "classical non-field theories" and "quantum non-field theories." And then (after you've worked out some details probably) there may be an overlap region composed of quantum theories that use classical fields, or classical theories that use quantum fields. It's okay to point these things out without exploring them immediately (i.e. leave it for future work as a post-doc or for other authors).

Then, I think the next step is to decide HOW you want to examine any such examples. I suggest that you try to formulate this as abstractly as you possibly can, so as to avoid any weirdness (i.e. you don't want to assume what you're trying to prove). Thus, I recommend abstract graph theory. In principle, you never have to talk about fields using graph theory (aside from the mathematical sense but that's not what you're concerned about) in order to define the structure and use it. Usually, fields of physical quantities (like vector or tensor fields) are imposed onto the graph structure. Another abstract setting (that is fundamentally related to graph representations) is category theory, but that may be a bit too abstract to be useful for your purpose. I think graph theory is promising because you can do physics with graphs without necessarily needing to introduce "classical" or "quantum" fields. It would allow you to examine such non-field formulation in depth.

Here's a paper about classical dynamics with graphs, and a short intro to graph methods in physics. If you'd like more references I can dig some more up. I used to be obsessed with graph theory....

So my answer is basically this: if you dont want to use "fields" in formulating a physical theory, then use a different mathematical structure - whether it be algebraic, graph theoretic, etc...

I think a great example - great for its simplicity - of such a theory without "classical fields" that already exists is Kirchoff's laws of electrostatics on a graph. This may serve as a basis for you to explore more complicated theories. If you look deeply enough, I'm sure you can find others. Another one that comes to mind is the Ising model (and its generalization). Some theories of quantum gravity research use spin foams which, as far as I understand, do not depend principally on the concept of "quantum fields." Another example is the theory known as quantum graphity, but delving into a somewhat controversial theory such as this one might not be best for a PhD thesis, especially if you've not had enough time to think it through thoroughly. As I said earlier, the abstract algebraic formulation of quantum mechanics might also be a good starting point for a quantum theory without quantum fields. Maybe you could do it with graph theory.

This abstract approach would have the benefit of avoiding the arbitrariness of your original proposition regarding whether any field theory is effective, since otherwise you would have to define what you really mean.

EDITED: Your definition of a "fundamental theory" as not being "effective" does not appear immediately as being very useful. "Effective" theories, which explain some range of experiments without necessarily claiming the underlying mechanism has a physical counterpart in reality, in principle can "work" at any scale. But then again, what do we really mean by "any scale?" Do we mean to include below the Planck scale? But we don't know what physics even looks like in principle there! So this is like making map of a place that you don't even know exists.

So, I recommend that you stay away from these notions of "fundamental theory" or "theory of everything" because they will likely present as dead-ends for your actual proposition, which is to study theories that don't use fields. Maybe this study can lead to something interesting about "fundamental" physics, but it doesn't seem obvious right now. You must be careful about what you mean by these things, since being sloppy is an easy way for skeptics to reject your conclusions. There are great essays about this, the first prize essay from 2017 especially. You want to avoid philosophical black-holes for your Phd thesis, unless you're trying to do a PhD in philosophy ;)

If you're looking for a golden, shining goose egg, then you'll have to do some digging! It's hard to intuit such an example without wading through various possibilities for a while. Best of luck! And of course, if you have further questions, I'll still be here too ;D

EDIT: I've always been fascinated by the work of Lee Smolin, the main architect of loop quantum gravity. He has recently proposed a theory which he describes as "fundamental" because it relies on no "background" and only depends on the relationships between the constituents of the framework which is, guess what, a dynamical graph (the underlying structure of the causal sets)! Perhaps his work could give you some deeper insights into the difficulties of making "fundamental" theories and how one can spend their entire life chasing that dragon.


LAST EDIT (copied from chat room):

I know that we have no experimental proofs (or even indications) of string theory being correct, I don't care about that for this specific question. I'm asking if there is something, on some physicist table, that isn't automatically bound to fail after some energy limit.

Okay, I see. Sorry, that was not clear to me. I think I agree about SUSY. String theory (ST), however, I think is different. It take an entirely different approach, so that renormalization is not even a thing in ST. That's why if you're really trying to get to something "fundamental", relying on renormalization flow might not be optimal.

That's rather interesting

And ST seems to be a non-field theory that attempts to obtain "fundamentality." ST can be cast into a field theory form though. So, I think I did answer this question: "I'm asking if there is something, on some physicist table, that isn't automatically bound to fail after some energy limit." In principle, an answer to this is classical general relativity which is scale-free. But that's the problem with it precisely: it produces singular points where we expect quantum theory to dominate.

But pointing to classical GR is probably not satisfying at all, understandably. That's why I suggest graph theory as a way to forget about "fields" without going the string theory route. Consider the correspondence principle, which states that a more general theory subsumes the ones that preceded it. So then, the most fundamental theory is one that contains every other theory in some appropriate limit? But that seems to be a never ending game in an infinite universe. Even if we did come about such a thing, it might only be an illusion due to the finicity of our ability to measure phenomena.

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    $\begingroup$ Note: Natural numbers don’t form a field. $\endgroup$ – user76284 Nov 10 '20 at 19:21
  • $\begingroup$ @user76284 I do not think I ever said that it was. However, the set $\aleph \cup \{0\}$ is a field I think (the issue with just $\aleph$ is it is missing the identity element). $\endgroup$ – Daddy Kropotkin Nov 10 '20 at 19:23
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    $\begingroup$ Oh thanks, I've fixed that. $\endgroup$ – Daddy Kropotkin Nov 10 '20 at 19:29
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    $\begingroup$ Interestingly, the term "field" is used for completely different objects: 1. Sets with two group operations plus some further axioms defined on them (field of rational or real numbers for example) and 2. Functions that map points in space to any physical quantity (scalar, vector, tensor...). In some languages the two objects have different names, in some, like English, the names coincide. $\endgroup$ – Photon Nov 10 '20 at 19:50
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    $\begingroup$ @MauroGiliberti Thank you, Mauro! It was a stimulating discussion. $\endgroup$ – Daddy Kropotkin Nov 12 '20 at 15:11

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