# Why does the fine-structure constant $α$ have the value it does?

1,059

## Solution 1

No one knows, and, at the moment, there is no realistic prospect of computing the fine-structure constant from first principles any time soon.

We do know, however, that the fine-structure constant isn't a constant! It in fact depends on the energy of the interaction that we are looking at. This behaviour is known as 'running'. The well-known $$\alpha \simeq 1/137$$ is the low-energy limit of the coupling. At e.g., an energy of the Z-mass, we find $$\alpha(Q=M_Z)\simeq 1/128$$. This suggests that there is nothing fundamental about the low-energy value, since it can be calculated from a high-energy value.

In fact, we know more still. The fine-structure constant is the strength of the electromagnetic force, which is mediated by massless photons. There is another force, the weak force, mediated by massive particles. We know that at high energies, these two forces become one, unified force. Thus, once more, we know that the fine-structure constant isn't fundamental as it results from the breakdown of a unified force.

So, we can calculate the fine-structure constant from a high-energy theory in which electromagnetism and the weak force are unified at high-energy (and perhaps unified with other forces at the grand-unification scale).

This does not mean, however, that we know why it has the value $$1/137$$ at low energies. In practice, $$\alpha \simeq 1/137$$ is a low-scale boundary condition in theories in which the forces unify at high-energy. We know no principled way of setting the high-energy values of the free parameters of our models, so we just tune them until they agree sufficiently with our measurements. In principle it is possible the high-scale boundary condition could be provided by a new theory, perhaps a string theory.

## Solution 2

One theory is that we live in a multiverse where physical constants such as $$\alpha$$ are different in different universes. This theory is speculative but based on plausible physics such as cosmic inflation and the large number of different vacuum states believed to exist in string theory.

We happen to live in a child universe with a small-but-not-too-small value of the fine structure constant, because such a value is compatible with the existence of the periodic table, organic chemistry, and life, while signifcantly different values are not.

Share:
1,059

Author by

### Magix

Forever learning

Updated on January 23, 2021

• Magix almost 2 years

This is a follow-up to this great answer.

All of the other related questions have answers explaining how units come into play when measuring "universal" constants, like the value of the speed of light, $$c$$. But what about the fine-structure constant, $$α$$? Its value seems to come out of nowhere, and to cite the previously-linked question:

And this is, my friend, a real puzzle in physics. Solve it to the bottom, and you will win yourself a Nobel Prize!

I imagine we have clues about the reason of the value of the fine-structure constant. What are they?

• ann marie cœur almost 2 years
But 𝛼∼1/137 is a value we observe in the length scale of our measurable range. If we go to a shorter distance or the larger energy, the 𝛼 will run to a larger coupling because the QED quantum electrodynamics tells us that the 𝛼 runs under renormalization group flow. So by taking this into account, do you think comment still is valuable and valid? See physics.stackexchange.com/q/618719/42982
Pet peeve of mine in this answer: it is not true that the electromagnetic and weak forces unify before the GUT scale. It's not even close. The name "electroweak unification" is a misnomer that refers only to the fact that the SM includes both forces; it does not mean the fields are unified like in grand unification.
Maybe you could emphasize that the $U(1)_{\text{EM}}$ coupling doesn't just split off from some one "unified" electroweak coupling, but rather is a combination of two independent couplings, the $SU(2)_L$ and $U(1)_Y$. I think that actually shows the arbitrariness better.