# Where does the fine structure constant come from?

I have this question: Where does the fine structure constant come from? Is it derived? Is it assumed? I will be most thankful if you will also include other detailed info that you think is also good to know, or just suggest a reading on it.

• Feynman said in the The Douglas Robb Memorial Lectures (fourth video, about 19:45) (and probably the book QED: The Strange Theory of Light and Matter, which is basicly the same) that through the ages, many physisists have reasoned why it must be what it is, starting with why it has to be exactly 1/137, and getting increasingly complicated from there as experiments improved. Of course, these videos are 40 years old, so progress might have been made, but a fundamental reason has yet to be uncovered, I think. – Arthur Jan 2 '18 at 8:22

The fine structure constant $\alpha=e^2/\hbar c$ is the only dimensionless quantity that can be built from the quantities $e$ (the electric charge), $\hbar$ and $c$ (the speed of light).

The latter quantities are physically important to understand the interaction of charged particles interacting with (quantized) radiation, hence the need to construct this dimensionless quantity from $e$, $\hbar$ and $c$.

Because of its small size, it is very useful as an expansion parameter in a perturbative approach as series will converge quickly.

• It's not the only such dimensionless quantity - you could take its inverse, square, square root or some constant multiple. It is in a way true though that, except possibly for its inverse, it's the "simplest" such constant. – Wojowu Jan 1 '18 at 20:34
• @Wojowu of course any function of this would do but it is the “simplest” monomial, as per assumptions of dimensional analysis. – ZeroTheHero Jan 1 '18 at 20:46

The fine structure constant is one of the fundamental constants in nature, just like the speed of light or Planck's constant. It is there, and that's all we know for sure. We don't really have a compelling theory on its origin, nor a mechanism that explains its value.

In short, the fine structure constant is not a derived quantity, it is fundamental. You may want to read more about the following aspects of $\alpha$:

• Its origin. Some modern theories, like String theory or AdS/CFT, propose mechanisms on how this constant emerges from more fundamental objects, but -- in practical terms -- they are not really able to predict its value. One could also argue that the anthropic principle partially fixes this object to its observed scale, but to a wide community, this explanation is just as unconvincing and useless as it gets.

• Its measurement. This constant is measured in ion (Penning) traps (by Gabrielse et al.) and by means of the so-called electron anomalous magnetic moment. It is one of the fundamental constants that have been measured to a highest precision. Truly marvellous if you ask me.

• Its constancy. Finally, it bears mentioning that some people have suggested that this constant is not really a constant, that it varies from place to place (and, consequently, from time to time) in the universe. This has been tested not to be the case, to an astonishing precision.

As stressed by other answers, you may write $\alpha$ in terms of $e$, the charge of the electron. In this case, you may want to argue that $\alpha$ is fundamental and $e\sim\sqrt{4\pi\alpha}$ is derived from it, or that $e$ is fundamental and $\alpha\sim e^2/4\pi$ is derived from it. Needless to say, both interpretations are perfectly valid.

• The fine structure constant is one of the fundamental constants in nature, just like the speed of light or Planck's constant. This is really not a good way of looking at it. Dimensionful constants like $c$ and $h$ have the values they do simply because of our system of units. In fact, $c$ currently has a defined value in the SI. $\sqrt{\alpha}$ is simply the charge of the electron, expressed in a system of units that is appropriate for quantum mechanics. People have searched for changes in $\alpha$ over time, and have even claimed non-null results. You can't do that with $c$ or $h$. – Ben Crowell Jan 1 '18 at 18:51
• @BenCrowell $\alpha$ is not completely equivalent to $c$ or $\hbar$. Agree. But that doesn't make it any less fundamental. To make a complete prediction in QED, you need the value of $m,\hbar,c,\alpha$. Nothing more, but nothing less. One of these parameters is independent of the system of units, which makes it somewhat special. But it is just as important as the rest, and you need all of them to properly specify the model. The four parameters are fundamental, even if one of them is dimensionless. – AccidentalFourierTransform Jan 1 '18 at 21:06

The electrostatic force between two point charges $q_1,\,q_2$ separated by a distance $r$ is proportional to $q_1 q_2 r^{-2}$, but it has the same dimension as $\hbar c r^{-2}$. Therefore, a dimensionless value $\alpha$ exists for which the charge between two "unit" charges (e.g. electrons) is $\alpha \hbar cr^{-2}$. Equating this to $ke^2 r^{-2}$ with $k:=(4\pi\varepsilon_0)^{-1}$ gives $\alpha = ke^2(\hbar c)^{-1}$.

There's a further subtlety. A charged particle rotates pairs of charged virtual particles due to attracting one and repelling the other. For example, virtual positive charges end up slightly closer to an electron than their partner negative charges too. This shields bare charges, and empirical charges depend on the probed length scale and hence the probing energy scale. Thus $\alpha\propto e^2$ is a "running" coupling parameter, approximating $1/137$ at low energies. If you're looking for a theoretical answer as to why that value arises, we don't have one yet.

The two foundational theories of physics are relativity and quantum mechanics. For that reason, it makes sense to pick a system of units in which $c=1$ (for relativity) and $\hbar=1$ (for quantum mechanics). In such a system of units, the strength of electromagnetic interactions (the interaction of charges with other charges) is parametrized by the unitless constant $e^2$. We don't know why this constant has the numerical value it does, about 1/137.

The fine structure constant is not at all analogous to other constants such as $c$ or $\hbar$. The reason for the value of the fine structure constant is a mystery. The reasons for the values of dimensionful constants like $c$ and $\hbar$ are not mysterious. They have the values they do simply because of our system of units. In fact, $c$ currently has a defined value in the SI.

People have searched for changes in the fine structure constant over time, and have even claimed non-null results. You can't do that with $c$ or $h$, even in principle, because they have units.

• You can search for changes in $c$ and $\hbar$. In fact, a dependence of the form $\hbar(t)=\hbar(t_0)\mathrm e^{-(t-t_0)/\tau}$, with $\hbar(t_0)$ its value today, is perfectly meaningful (and consistent with observations, if $\tau\gg t_0$). See also Variable speed of light. – AccidentalFourierTransform Jan 1 '18 at 21:10
• If the value of constants like $c$ are not mysterious, can you explain why, given the average height of a human (or any other reference object of equivalent size), it happens to be the case that in $1/299,792,458$ of a fraction of the time required for $9,192,631,770$ cesium-133 transitions, light will travel $1/1.8$ the length of a human? For the same distance traveled, why doesn't light spend $1/250,000,000$ of that reference timespan, or $1/350,000,000$? I was under the impression that the origin of the value is mysterious, we've just chosen a fixed value to base our metrics on. – Ethan Kaminski Jan 2 '18 at 7:03
• @AccidentalFourierTransform: For a detailed explanation, see the paper linked to from the answer (hyperlinked from the words "You can't do that"). – Ben Crowell Jan 3 '18 at 1:27

## protected by Qmechanic♦Jan 1 '18 at 18:32

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