The fine structure constant is a number of constants rolled into one equation. Brian Cox mentioned in the April edition of Focus magazine that it is possible that the speed of light was once faster, say, in the earlier universe - hence I would conjecture this constant must then have been different.
If any or each of the constants that make up the fine structure constant are said to be changing, then is there anywhere in the universe where we can reliably observe this; and if so, what are the consequences in such a case? What of our known laws of physics in such cases, those that are used so broadly to garner great results in the field of astronomy to this day, do they become questionable or blatantly break down?
For reference, this is an excerpt from the magazine concerning what was said regarding observations of the speed of light:
What the astronomers are seeing in their study of the distant gas clouds is the last cosmic moments of that decline.
He further mentions gravity with hints of possibilities of change:
Others look for variations in big G as this could be used to develop a new understanding of gravity.
These are fitting statements for the source of them, but seemingly fanciful without further detail.
Answer
There are some scientists working on this. Here is a page with some references. From a quick reading, it seems that there is some evidence that the fundamental constants might be time-dependent. From one of the papers:
In modern higher-dimensional extensions of the standard model of particle physics, low-energy fundamental constants like the fine structure constant α, the proton-electron mass ratio μ = mp / me, etc, are expected to be dynamical quantities that show spatio-temporal evolution.
The paper then goes on to review the (then) current state of knowledge, including observations and laboratory experiments used for finding this out. Since the paper was published in September 2010, it should be fairly current.
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