Atiyah and the Fine-Structure Constant

Posted on September 25, 2018 by Sean Carroll

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In quantum electrodynamics (QED), α tells us the strength

of the electromagnetic interaction. Numerically it’s approximately 1/137.

If it were larger, electromagnetism would be stronger, atoms would be

smaller, etc; and inversely if it were smaller.

It’s the number that tells us the overall strength of QED interactions

between electrons and photons, . . .

As Atiyah notes, in some sense α is a fundamental dimensionless

numerical quantity like e or π. As such it is tempting to try to “derive”

its value from some deeper principles.

Arthur Eddington famously tried to derive exactly 1/137, but failed;

Atiyah cites him approvingly.

But to a modern physicist, this seems like a misguided quest.

First, because renormalization theory teaches us that α isn’t really

a number at all; it’s a function.

In particular, it’s a function of the total amount of momentum involved

in the interaction you are considering. Essentially, the strength of

electromagnetism is slightly different for processes happening at different

energies.

Atiyah isn’t even trying to derive a function, just a number.

. . .

There is a limit we can take as the momentum goes to zero,

at which point α is a single number.

. . .

More importantly, I think, is the fact that α isn’t really fundamental at all.

. . .

And in fact, the total answer we get depends not only on the properties

of electrons and photons, but on all of the other particles that could appear

as virtual particles in these complicated diagrams.

So what you and I measure as the fine-structure constant actually depends

on things like the mass of the top quark and the coupling of the Higgs boson.

. . .

Most importantly, in my mind, is that not only is α not fundamental,

QED itself is not fundamental. It’s possible that the strong, weak,

and electromagnetic forces are combined into some Grand Unified theory,

but we honestly don’t know at this point.

. . .

In QED, α is related to the “elementary electric charge” e by the simple

formula α = e2/4π. (I’ve set annoying things like Planck’s constant and the speed

of light equal to one. And note that this e has nothing to do with the base of natural

logarithms, e = 2.71828.) So if you’re “deriving” α, you’re really deriving e.

. . .

But e is absolutely not fundamental.

. . . .

The elementary electric charge isn’t one of the basic ingredients of nature;

it’s just something we observe fairly directly at low energies, after a bunch

of complicated stuff happens at higher energies.

. . .

It’s possible that, despite all the reasons why we should expect α to be

a messy combination of many different inputs, some mathematically

elegant formula is secretly behind it all.

. . .

http://www.preposterousuniverse.com/blo ... -constant/

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