How fundamental are fundamental constants?

Abstract

I argue that the laws of physics should be independent of one’s choice of units or measuring apparatus. This is the case if they are framed in terms of dimensionless numbers such as the fine structure constant, $\mathit{\alpha }$. For example, the standard model of particle physics has 19 such dimensionless parameters whose values all observers can agree on, irrespective of what clock, rulers or scales$\dots$ they use to measure them. Dimensional constants, on the other hand, such as $ħ$, c, G, e and k $\dots$, are merely human constructs whose number and values differ from one choice of units to the next. In this sense, only dimensionless constants are ‘fundamental’. Similarly, the possible time variation of dimensionless fundamental ‘constants’ of nature is operationally well defined and a legitimate subject of physical enquiry. By contrast, the time variation of dimensional constants such as $c$ or $G$ on which a good many (in my opinion, confusing) papers have been written, is a unit-dependent phenomenon on which different observers might disagree depending on their apparatus. All these confusions disappear if one asks only unit-independent questions. We provide a selection of opposing opinions in the literature and respond accordingly.

Keywords:

• constants
• units
• dimensionless

Acknowledgements

I would like to thank my ‘adversaries’ in Section 4 (several of whom are personal friends) for their thought-provoking contributions to this debate. If they wish to present a case for the defence of views I have disagreed with in this paper, I would be delighted to hear them. I am also grateful to Lev Okun and Gabriele Veneziano for first arousing my interest in this topic and to Chris Pope, Hong Lu and Constantino Tsallis for stimulating conversations. Section 2.1 reproduced from [2] with permission from SISSA.

Notes

1 Interestingly enough, one independent source with which I am in almost entire agreement is Wikipedia [59]

2 One might argue that seven units demand more than seven conversion factors, but I am reproducing the CGPM presentation [20] as summarised by their diagram in Figure 1.

3 To its credit, Physics World also published the dissenting view [60].

4 If the experiment is performed in a medium, or a time-dependent gravitational field, one would have to factor out the effects of the refractive index, or $\sqrt{g_{xx}/g_{tt}}$. After all, light slows down when passing through a piece of glass, but no one is suggesting that this produces an increase in $\mathit{\alpha }$.

5 It is curious that Nature rejected [3] on the grounds that Davies, Moffat and Magueijo were right, but published [47] on the grounds that they were wrong.

Additional information

Funding

This work is supported by the STFC under rolling [grant number ST/G000743/1].