Conditions on a Fundamental Theory

According to the above discussion, a fundamental theory must be:

• UV complete (“nothing beyond” formally);

• Non-perturbative (exactly solvable);

• Natural (no sensitive dependence on high-energy parameters);

• Unified;

• Single;

• Internally consistent (well-defined formally, with no problematic singularities).

Several of these ideas reflect a more general principle: That a fundamental theory not leave anything apparently in need of explanation. For instance, if a theory is not UV complete, or otherwise not well-defined everywhere, then we are led to ask what happens in the domains that the theory does not describe. More generally, if a theory is not internally consistent, or relies on approximations, physicists tend to believe that this is a symptom of there being something missing—some physics that the theory fails to take into account.

Consider if we did not have a single (i.e., lone) theory, but a “patchwork” of several (UV-complete, and otherwise apparently fundamental) theories. These would have to fit together in an especially particular way, such that there were neither any gaps, nor any overlap in the domains of the world covered by these theories. Otherwise, if there were gaps, we would ask about the phenomena not described by any of the theories, or the physics “between” the theories—we would search for a description of this. And, if there were overlap, with a particular phenomenon described by more than one theory, then we would ask which (if any) provides the more fundamental description. Thus, if the patchwork of several apparently fundamental theories did not match up perfectly, we would be led to search for a more fundamental theory.

However, if the patchwork did match up perfectly—such that, at the smallest dis­tance scales, all physical phenomena were covered, and there was only one descrip­tion of each phenomenon—we could “stitch” these all together to form a single theory.¹¹ In this case, the theory would be single, and satisfy all conditions for fun- damentality, except for unification.

The requirement of unification is hard to justify. Given that our manifest experi­ence of the world is of diversity rather than a sameness of phenomena, seeking an explanation of heterogeneity seems counter-intuitive—surely a unified description would be more striking than a disunified one, and cry out for explanation? I return to discuss this below. For now, though, I add two more criteria to the list that are also motivated by the need for explanation. These criteria are not drawn from just from high-energy (particle) physics, however, but from other areas, including quantum mechanics and relativity. Here, it is believed that a fundamental theory must also be:

• Level comprehensive (“no gaps and no overlap” in description at the scales that the theory is required^{[34] [35] in order to describe);}

• Background independent (no fixed structures across all models of the theory);

• “Definite” (it should be clear how the theory yields definite measurement results)

The first of these additional criteria—which I call level comprehensiveness— stems from the need for a complete and non-overlapping description of the physics at the most fundamental level. The notion of background independence has several aspects (including that the theory be non-perturbative), but the general idea is that

there are no fixed (“background”) structures in the theory—nothing that has to be specified for the theory “by hand” instead of being determined by the theory itself [14, 15]. Otherwise, the appearance of such structures requires explanation, and implores us to seek a more fundamental theory that provides this.

Finally, the last requirement, “definiteness”, is inspired by the measurement prob­lem in quantum theory: According to quantum mechanics, the wave function describ­ing a system evolves as a superposition of different states, but any measurement we make on a system always finds the system in a definite state. And, in spite of the wave function evolution being deterministic, quantum theory yields only probabilities of particular measurement outcomes.

All nine conditions above assume a lot about the world—for instance, why should it be everywhere amenable to physical description, and why should this description be within our ability to formulate? Why should the world be such that our theories of it are formally neat and mathematically tractable, rather than messy and unus­able? And, as I asked above, why does manifest diversity and disunity, rather than covert underlying unity, require explanation? Each criterion needs further justifica­tion, especially if its standardly-cited motivation is an imagined trajectory of physics based on a particular reading of the history of physics—as, for instance, tending towards greater unification [13], or, as Weinberg [19] believes, a “convergence of the arrows of explanation”.

These are all good philosophical questions for which I have no answers. What I argue, however, is that each condition—while it may not be precisely definable nor rigorously philosophically justifiable—nevertheless captures something central to the enterprise of physics itself. Physics does and must, by its nature, assume that we are able to formulate a physical description of all phenomena, and that this description is useful to us as far as it can be.

The list of conditions has been compiled from the perspective of physics itself: It represents the criteria that any theory must satisfy if it is to be counted, by physics, as a fundamental theory. Thus, I argue that, given that this list is complete, a physical theory’s fulfilment of these conditions is sufficient for that theory to be treated as fundamental according to physics. Note, however, that this list was drawn up based on current theories, and it is possible that these views change; for instance, QG may very well force us to reconsider our requirements.

Before turning to QG, however, I must emphasise that the above conditions are just those that a theory of physics—i.e., a scientific theory—must satisfy in order to be considered fundamental. In other words, I am presupposing that any candidate fundamental theory of physics already satisfies some further conditions such that it is acceptable as a scientific theory. I do not consider these further conditions (which may include requirements relating to predictions, experiment, falsifiability, etc.)— suffice to say they are even more controversial, especially given the current state of QG research.

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