Fundamental and More-Fundamental in Modern Physics

Conceptions of fundamentality in modern physics have been heavily shaped by the framework of effective field theory (EFT), and its associated philosophy. The frame­work of EFT is a means of constructing theories—called effective field theories—that are each valid only at a given “level”, i.e., at large distances (corresponding to low energy scales, since energy is inversely proportional to length) compared to a par­ticular short length (high-energy) scale “cutoff”, A.

Because of this, there is a compelling case to be made that what is fundamental depends on the level we are interested in—for each level there is a theory that clearly describes the relevant physics of the system being studied, and is framed in terms of the appropriate parameters [2]. While physicists generally believe that we could in principle use a shorter length scale theory in order to make predictions about the system at some particular larger distance scale (i.e., if we had access to the required computational resources, plus the ability to use them), to do so would not only be very (and needlessly) complicated, but would also hopelessly obscure the picture of the relevant physics at the scale of interest.

This asymmetry is captured in the way we move between theories: It is generally believed that in principle, with full knowledge of the physics of a system at a particular short-length scale (plus, again, the required computational resources and the ability to use them), we could arrive at results valid at any larger scales without requiring any additional information. On the other hand, the large-scale physics is supposed to underdetermine the shorter-scale theory: We could not, even in principle, derive the correct theory of a system at small-length scales from a complete description of its physics at some larger length scale. More information would be required. For this reason, the tower of theories is usually thought to be ordered hierarchically, with the shorter-scale theories being more fundamental, and so lower on the tower, than the larger-scale, “higher-level” ones.¹1 use this notion of relative fundamentality here.^{[24] [25]}

We encounter problems, however, when we attempt to move from relative to absolute terms—what does it mean for a theory to be fundamental rather than just more- or less-fundamental? How do we define “rock bottom” of the tower? We might suppose it would be a theory valid at the tiniest length scales. Yet, according to quantum field theory (QFT)—the framework within which the standard model of particle physics is formulated (and, as I explain below, is also understood in terms of EFT)—there is an arbitrarily large amount of energy available in the vacuum, and so the tower may be endless, “shorter and shorter-length turtles all the way down”, thus implying that there is no final theory [3].

Although stepping outside the framework of QFT may free us from worrying about an arbitrarily large amount of energy available in the vacuum,^[27] however, it does not save us from the possibility that there is still new physics beyond any theory that we arrive at. The recognition that we are not trapped in the framework of any given theory produces an epistemic worry: Even if we reach a theory that yields predictions for the tiniest length scales (or, equivalently, all possible high-energy scales), we cannot be sure that these predictions are actually correct—unless, of course, we have access to experimental data at all possible high energy scales!

A theory (whether a QFT or not) that is formally predictive at all possible high energy scales is said to be UV complete (“UV” referring to the short-wavelength end of the electromagnetic spectrum, the ultraviolet). Although I take it that a theory being UV complete is necessary for its being fundamental,^[28] being UV complete is not sufficient for a theory to be fundamental—it does not guarantee that there is “nothing beyond” [7]. For example, consider Newton’s laws of motion: These are formally predictive in all domains, yielding results that are prima facie mathematically sensi­ble. Yet we know that these laws are not correct at all scales: At small length scales (and under particular conditions) they must be replaced by quantum-mechanical laws, and for large velocities they are replaced by relativistic laws. Another example is quantum chromodynamics (QCD, the theory of the strong nuclear force), which is UV complete and yet, as I discuss below, should not be considered fundamental.^[29]

So, a theory formally being predictive to all high-energy scales, and thus appar­ently being the lowest brick in the tower (or, at least, one of the bricks at the lowest level of the tower), is no guarantee that it is in fact a fundamental theory—UV com­pleteness alone is not enough reason to stop digging. Yet, it is one constraint on a fundamental theory.^[30] In order to understand what kind of theory would motivate physicists to stop digging, and to answer the question of what it means for a theory to be fundamental, I now invert it: Why do we not consider our current best theories of physics to be the final word? Why are we currently digging for a more fundamental theory?

3