# The fine structure constant - can it genuinely be a random variable?

### The question

Does it make sense, and are there physical reasons to think about the fine structure constant as a (very concentrated) probability distribution rather than a single real number?

The same question applies to other physical constants like the mass of a proton or the plank constant if you wish.

### more

The question is not if these constant are not constants but varies with time or from one place of the universe to another (that people sometime discuss and was asked here several times); the question is if these constants are really constants, but they are described not by a single real number but by a probability distribution.

We can also ask if the fine structure constant is a random variable, does it have interesting physical consequences? Could it have consequences for computing quantities where we use it for computation, such as QED computations (say, for energy levels of the Hydrogen atom). Perhaps the effect on such calculation will be more interesting than simple averaging over this random variable.

### Sort-of-related

A related mathematical fact is that in critical lattice systems of statistical mechanics, quantities (like the size of the connected cluster containing the origin) often have distributions which are "smeared" - the variance behave like square root of the expectation. (So the fluctuations do not vanish as $n$ goes to infinity.) I am not aware of such quantities that are directly related to fundamental physical constants though.

### background

This (speculative) question arose from a discussion on my blog as a sharp alternative to the following (speculative, as well) possibility: Quantum computers will allow the computation of more and more digits of the fine structure constant, just like we can compute with a digital computer more and more digits of e and π.

• No. Why? There is no evidence. – CuriousOne Dec 25 '14 at 20:16
• We don't compute things in physics, we measure them. You can express the fine structure constant trough other fundamental constants. Which one you pick as "fundamental" and which ones are "derived" is therefor just a matter of your personal taste. Since the other constants are not computed from first principles, at this moment, you have to accept that they are "god given". Beyond that, of course, there is no evidence that any of these constants has changed in the known universe. That's just where we stand with out knowledge and no amount of computing power gets you around that. – CuriousOne Dec 25 '14 at 21:58
• Dear CuriosOne, my question is not if these constants have changed in the known universe, the question takes for granted that they do not change for time or location. The question is if these constants are genuinely "stochastic objects". – Gil Kalai Dec 25 '14 at 22:19
• And the correct answer was no, they are not because there is no observational evidence for that. Oh, wait... I already said that. – CuriousOne Dec 25 '14 at 22:33
• Curious one, since we can only compute the constant up to some precision unless we study some theoretical consequences/connections related to a (hypothetical) possibility that they are not constant our observations cannot tell the difference. – Gil Kalai Dec 26 '14 at 5:34

There seems to be no empirical motivation to consider this possibility. But if we approach it as a pure "what if" anyway, it opens a big can of worms.

(1) How do we even implement the idea in a principled way? We might take a given calculation in which the fine-structure constant is employed, and substitute a sampling of Gil's "very concentrated probability distribution" for each use of the constant (thus making the outcome of the calculation a random variable too). But there are usually several ways of doing a physical calculation. Do we need to worry that different methods, when "randomized" in this way, will yield different random variables for the final answer?

(2) In applied quantum field theory, one deals with quantities that run with energy scale, as described by renormalization group flow. The experimental data is used to make this framework concrete, by telling us (e.g.) a coupling constant's effective value at a particular energy. Then the RG equations (beta functions, etc) tell us what the effective value would be at other scales, should we need that information.

If we are trying to implement the notion of "physical constant as random variable" in a way that fits the deepest physical principles we know, this framework is probably the one that we want to adhere to.

The best idea I have, would be for the "constant" to actually be the VEV (vacuum expectation value) of a new field. For example, in the standard model, the yukawa couplings are just numbers, but there are beyond-standard-model theories in which they are VEVs of new fields (called flavons in this context).

Perhaps one can do the same for the two electroweak couplings which combine to give us the observed electromagnetic coupling, and then look for effects on the constancy of the W and Z masses, and other sensitive electroweak observables.

(3) Finally... in string theory, the string coupling is a VEV, the VEV of the dilaton field. So there may be, in theory, an (unobservably small?) effect exactly such as Gil proposes. But it would be best to hear about this from someone who actually knows string theory.

• Dear Mitchell, thanks for the answer! Certainly having constants as VEV is very close to what I ask about. Re (1) or how to use an entire distribution rather its expected value in a computation, this is precisely what I am curious about. So in cases where quantities in one theory indeed correspond to VEV of a new field, are there known ways to base computations in that theory on the entire distribution rather only on the expectation? Are there ways to empirically distinguish if a quantity is an absolute constant or a VEV? – Gil Kalai Dec 26 '14 at 9:57
• @Gil Kalai, the answer must be yes, but I need to study / think about the details more, before I can say more. – Mitchell Porter Jan 6 '15 at 10:49