Michael Duff has released a hep-th preprint
By a more economic parameterization of the physical observables, e.g. the choice of \(1=c=\hbar=\epsilon_0=k=G=\dots \) units, one may completely eliminate the symbols of these constants from the equations describing the laws of Nature. This choice can be made even in situations when some people say that the "constants are evolving in time". To summarize, the number of physical fundamental dimensionful constants that would affect the laws of physics is always zero.
You may compare Duff's paper with some texts of mine such as
Dimensionless Constants in Physics (Physics Stack Exchange, answer, 2011)and elsewhere. I am sure that he would agree that we fully agree. More precisely, Duff would say (2011):
Changes of dimensionful quantities are unphysical (TRF 2009)
Let's fix the value of Planck's constant (TRF 2012)
Parameters of Nature (TRF 2004)
As a fresh member of the Royal Society, I am grateful to my overlords and I am ready to trample on politically incorrect babies in order to be admitted to as many similar societies as possible. So I will happily start by saying that I do not share Lubos Motl’s extreme views on politics, global warming, and sometimes not even string theory. However, he occasionally has some good physics summaries, including a recent one giving a nice history of the triumphs of uniﬁcation Yes, Mike, this introduction of yours was despicable, utterly unethical, and you will be grilled in Hell throughout the infinite asymptotic future. But yes, we agree on the units.
It's remarkable how many authors and papers saying completely stupid things about the dimensionful constants Duff has been able to find.
For example, John Moffat argues that if the dimensionful constants are "evolving", physics is very different, and it would be totally incomprehensible if you tried to switch from some units to others, and so on. Duff correctly replies that switching from one set of units to others is a trivial operation that doesn't affect the form of the fundamental equations at all.
The statement that "these dimensionful constants are varying" isn't a statement about Nature only. It is a statement about the union "Nature plus our conventions". So this statement isn't pure physics. And even if it happens to be true with your choice of conventions, nothing stops you from setting \(1=c=\hbar=\dots\) at each moment of time and at every point of space! It is a very natural choice of units which is why mature conventions always allow us to say that \(c,\hbar\), and others are independent of time, true constants (just like the number one). This statement doesn't restrict the form of the laws of Nature at all; it is not making any assumptions about Nature that could fail. It is always possible to adjust the conventions so that the statement holds.
Paul Davies indefensibly disagrees with Duff's claim that theories with varying dimensionful constants are operationally meaningless and claims that such theories exist and specific experimental tests to confirm that they are true are known, too. Duff disagrees and corrects Davies' claim that Dirac was on Davies' side in his misunderstanding of the role of units.
Duff says many correct things – mostly reiterating the insights that were already mentioned above. But the key always is that Davies and others misunderstand that whatever mechanisms exist that make "something" time-dependent, one may always choose units that eliminate all the numerical values of (a correctly large set of) dimensionful universal constants. With this choice, the "time evolution" of "anything" is inevitably translated into the evolution of some dimensionless parameters. And if the laws of physics are complete, they must of course describe the rules that dictate the evolution of these dimensionless constants. They are evolving dynamical parameters just like any others (e.g. the speed of Venus) which means that their changing values aren't "universal" and shouldn't be called "fundamental", either.
Not too surprisingly, João Magueijo is deluded about all these issues, too. This crank has actually made a living out of spreading these totally self-evidently wrong assertions about the variable speed of light theories and he is currently a colleague of Duff's at the Imperial College in London. Holy crap. Magueijo would scream things like "it must be impossible to allow the choice of \(c=1\) units". He just completely misunderstands that certain values depend on human conventions.
The list of people who are completely confused includes John Barrow, a guy who likes to write superficial books stupidly combining physics and God, and, more disappointingly, Gary Gibbons who is otherwise a deep physicist. It probably doesn't make sense to discuss the quotes separately because all the wrong people are reiterating the same delusion in different words. But those people include R.D. Reasenberg; J.-P. Uzan; C.J. Copi, A.N. Davis, and Lawrence Krauss (who ignore the unit dependence of \(\dot G / G\) in a discussion of the Big Bang Nucleosynthesis); and Terry Quinn (who was, ironically enough, a director of the International Bureau of Weights and Measures but who insanely believes that a theory will once calculate the numerical value of \(G\) in SI units).
I find it remarkable that so many people who have become physics professionals are incapable of understanding and agreeing with these very simple things. Many of those are chronic writers of popular books (e.g. Barrow, Krauss, and others) and I used to assume that what they were writing were just silly simplifications to make their books more understandable to the lay audiences (who don't really understand the difference between dimensionful and dimensionless constants or variables and similar nuances). But they seem to be damn serious.