Atiyah and the Fine-Structure Constant

[socrat44]

Atiyah and the Fine-Structure Constant
Posted on September 25, 2018 by Sean Carroll
===
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/
=====

[socrat44]

Richard Feynman wrote about the fine-structure constant:

''It has been a mystery ever since it was discovered more than fifty years ago,
and all good theoretical physicists put this number up on their wall and worry about it.
Immediately you would like to know where this number for a coupling comes from:
is it related to pi or perhaps to the base of natural logarithms?
Nobody knows.
It’s one of the greatest damn mysteries of physics: a magic number that comes
to us with no understanding by man. You might say the “hand of God” wrote
that number, and “we don’t know how He pushed his pencil.
” We know what kind of a dance to do experimentally to measure this number
very accurately, but we don’t know what kind of dance to do on the computer
to make this number come out, without putting it in secretly! ''
–  Richard Feynman (1985).
/ QED: The Strange Theory of Light and Matter.
Princeton University Press. p. 129./
===========

[socrat44]

Nobel laureate Max Born believed that, if the value of the fine-structure
constant were any different, the universe would be very different:

“If alpha [the fine-structure constant] were bigger than it really is,
we should not be able to distinguish matter from ether
[the vacuum, nothingness], and our task to disentangle the natural
laws would be hopelessly difficult. The fact however that alpha has just
its value 1/137 is certainly no chance but itself a law of nature.
It is clear that the explanation of this number must be the central
problem of natural philosophy.”
#
Nobel Laureate Leon M. Lederman intentionally numbered his home 137.
Prof. Lederman explained the significance of the number in his book
“The God Particle: If the Universe Is the Answer, What Is the Question?”,
noting that not only was it the inverse of the fine-structure constant,
but it was also related to the probability that an electron will emit or absorb a photon.
He further noted that it also “contains the crux of electromagnetism (the electron),
relativity (the velocity of light), and quantum theory (Planck’s constant).
It would be less unsettling if the relationship between all these important concepts
turned out to be one or three or maybe a multiple of pi. But 137?”
Lederman wrote that the number 137, “shows up naked all over the place,” i.e.,
scientists on any planet in the universe using whatever units they have for charge
or speed, or Planck’s constant, will all come up with 137, because it is a pure number.
======

[socrat44]

THE MEANING OF THE FINE STRUCTURE CONSTANT
   by  Robert L. Oldershaw , Amherst College

https://arxiv.org/ftp/arxiv/papers/0708/0708.3501.pdf
====