Asymptotics, renormalization and scientific theories

In lieu of a post with original material and/or updates on the other posts, here is a nice quote relating to some of the key themes that I’ve started exploring on this blog. Specifically a quote about asymptotics and renormalization (and, by implication, model closure, approximation and invariance), and how these can illuminate some aspects of the nature of scientific theories.

On renormalization
From ‘Intermediate Asymptotics and Renormalization Group Theory’  by Goldenfeld, Martin, Oono (1989).

[a] macroscopic phenomenological description…consists of two parts: the universal structure, i.e., the structure of the equation itself, and phenomenological parameters sensitive to the specific microscopic physics of the system. Any good phenomenological description of a system always has this structure: a universal part and a few detail-sensitive parameters…In this sense, it is [also] possible that there is no good macroscopic phenomenology [for a given system of interest].

Thus if we consider a set of transformations that alters only the microscopic parameters of a model…the macroscopic universal features should remain unchanged. Therefore, if we can absorb the changes caused by modification of microscopic parameters into a few phenomenological parameters, we can obtain universal relations between phenomenological parameters.

If this is possible by introducing a finite number of phenomenological parameters, we say that the model (or the system) is renormalizable. This is the standard method of formulating the problem of extracting macroscopic phenomenology with RG. RG seeks the microscopic detail sensitive parts in the theory and tries to absorb them into macroscopic phenomenological parameters.

…Suppose that the macroscopic phenomenology of a system can be described successfully with a renormalizable microscopic model. The phenomenological parameters must be provided from either experiment or from a description valid at a smaller length scale. Is this a fundamental limitation of the renormalizable theory? If one is a reductionist, the answer is probably yes. However, another point of view is that microscopic models are not more fundamental than macroscopic phenomenology.

In fact, it is inevitable that in constructing models of physical systems, phenomena beyond some energy scale (or on length scales below a threshold) are neglected. In this sense, all present-day theoretical physics is macroscopic phenomenology.

Renormalization group theory has taught us how to extract definite macroscopic conclusions from this vague description. Of course, this is not always possible…However, we clearly recognize general macroscopic features of the world in our daily lives as macroscopic creatures! Thus, we may believe that for many important aspects of the macroscopic world there must be renormalizability. We may say that renormalizability makes physics possible.

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