What is a ‘fundamental theory’?

‘Fundamental theory’ is a phrase that is often used by philosophers and scientists alike, but I’m not sure what they mean by that. In this paper, Mathias Frisch writes:

In the case of non-fundamental theories, domain restrictions may also include restrictions to certain length- or energy-scales.

This suggests that fundamental theories are supposed to apply to all length- and energy-scales.

In the paper that Frisch is replying to, Gordon Belot writes:

We will of course need further principles that demarcate the domain of applicability of each theory — but every non-fundamental theory involves such principles.

Do fundamental theories, then, have no such restrictions on their domains of applicability? If so, does that mean that fundamental theories are just ‘theories of everything’? Does it also mean that we have no real examples of fundamental theories for our world, since all of our theories apply only to limited kinds or arrangements of matter?

Elena Castellani gives a quite different account of ‘fundamental physics’:

“fundamental physics” is the physics concerned with the search for the ultimate constituents of the universe and the laws governing their behavior and interactions. Fundamentality, on this view, is the prerogative of the physics of smaller and smaller distances (or higher and higher energies), and, accordingly, particle physics and cosmology are currently identified as the fields where the quest for the “final theory” takes place.

I find the quick move from ‘fundamentality’ to ‘finality’ in her last sentence quite intriguing. But that aside, note that Castellani here takes fundamental physics to refer to the study of the “ultimate constituents” of the universe and their behaviour. This characterisation is quite different from Frisch’s — Castellani’s ‘fundamental physics’ may involve a restriction to certain length scales (the very small). But perhaps she has a notion of ‘fundamental theory’ that is distinct from that of ‘fundamental physics’.

In fact, although they never explicitly state this, the discussion between Frisch and Belot regarding the consistency of classical electrodynamics often refers to a certain subset of the equations of classical electrodynamics as characterising the ‘fundamental theory’, with other subsets being non-fundamental theories of classical electrodynamics. But we know that classical electrodynamics has a limited domain of applicability. So I’m not sure what they mean by ‘fundamental theory’ in this context.

There is also a notion of ‘fundamental’ at work that talks about theories being more or less fundamental than other theories, but may not call any one theory ‘fundamental’ tout court. Stephan Hartmann, for example, characterises one theory as ‘more fundamental’ than another if it covers a broader range of energy scale which includes the energy scales at which the ‘less fundamental’ theory is valid. In the same paper, Hartmann quotes Steven Weinberg as claiming that a more fundamental theory is ‘on a level closer to the source of the arrows of explanation than other areas of physics’. However, this characterisation doesn’t mesh so well with debates in philosophy of science about whether a less fundamental theory can explain phenomena better than a more fundamental theory. If theories are to be defined as more fundamental according to the ‘arrows of explanation’, then it is simply by definition that there are no phenomena that can be explained better by less fundamental theories.

Next, we have Carlo Rovelli, who suggests that a limited domain of validity renders a theory non-fundamental, yet claims that QFT is “the fundamental theory of motion” (p. 257 of the linked book):

General relativity cannot be seen as a ‘fundamental’ theory since it neglects the quantum behavior of the gravitational field, but many of the directions that are explored with the aim of finding a quantum theory of the gravitational field and/or extending the Standard Model — perhaps to a theory of everything — are grounded in QFT. Thus, one may regard QFT as the fundamental theory of motion at our present stage of knowledge — playing the role that Newtonian mechanics and its Lagrangian and Hamiltonian extensions played in the eighteenth and nineteenth centuries.

Rovelli seems to be implying here that a ‘fundamental’ theory is one that proves to be fruitful in a large number of ways — at least that’s what I take his comparison with Newtonian mechanics, and his quip about the current role of quantum field theory, to imply. Yet he disqualifies GR as a fundamental theory on account of its neglect of quantum behavior.

So, what is a fundamental theory?