What Should A Deeper Theory Of The Fine Structure Constant Look Like?

Sean Carroll has a nice little musing at his blog that explains (along the same lines I raised concerns at Physics Forums in a post when I saw the latest high profile effort to derive a mathematical formula that produces the Fine Structure Constant a.k.a. alpha a.k.a. the physical constants that governs the strength of electromagnetism) the intuitions of people who are familiar with the Standard Model and electroweak theory about what a formula that can determine the value of the Fine Structure Constant from a deeper theory out to look like in general terms.

Obviously, we don't have the deeper theory yet, so we can't be sure. But, his insights explain why, given the many deep interrelationships between the physical constants of the electroweak theory, the Fine Structure Constant doesn't look like it is a constant that should be capable of being deduced from pure mathematics because it is fundamental in a deeper theory from which these mathematical formulas arise.

A good analogy is the width of a particle that decays (either fundamental or composite) which is roughly speaking the inverse of its mean lifetime. It turns out that in the Standard Model, the width of a particle can always be derived by conceptually determining every possible decay chain for the particle in question, giving those decay chains weights based upon some constants of the Standard Model (like coupling constants and the masses of particles in the decay chains). The width of a particle that can decay in the Standard Model is basically the sum of the width of each of the possible decay chains of the particle.

So, even though every fundamental particle and every composite particle that can decay in the Standard Model (which is basically every particle except the composite proton and the fundamental particle electron in the Standard Model), none of those physical constants are fundamental and none of them would be expected to have a derivation that could be derived directly from a formula involving only pure mathematics.

The general consensus is that the proposed formula is the flawed numerology of a once world class mathematical genius who is now past his prime and making embarrassing mistakes without realizing it. But, on the other hand, as even a formerly great professional mathematician, even these flawed proposals can look pretty spectacular and deep compared to the usual fare in physics numerology circles.