# Why do universal constants have the values they do?

This is meant to be a generic question of the type that we get repeatedly on this site, in different versions:

Why do universal constants have the values they do? Can we predict their values theoretically? Do they change over time? How would the world be different if a particular constant had a different value?

Bob: Alice, tell me, why do the fundamental constants have the value they have? Why is the speed of light what it is?

Alice: That is not a very meaningful question.

Bob: What do you mean?

Alice: Physics is the art of mathematically quantifying the universe we live in. So physicists map their observations to numbers. Dimensionless numbers. And as a consequence, all fundamental constants in physics are represented by dimensionless numbers.

Bob: Wo, wo, woo... stop! How can you maintain that in experiments we deal solely with dimensionless numbers? If, for instance, I measure my own length, surely I express the result in some length unit! Length measurements come with the dimension of length, duration measurements come with the dimension of time, and so on. Virtually all measurements in physics are expressed in dimensionfull numbers.

Alice: Indeed, expressing measurements in dimensionfull numbers is a common way of communicating physics results. But we should not forget that this represents nothing more than a useful abbreviation. If I make the statement "my length is 1.7 m" what I really mean is that the dimensionless ratio of my length to the length travelled by light in vacuum during 9,192,631,770 periods of the transition between the two hyperfine levels of the ground state of the caesium 133 atom, equals 1.7 divided by 299,792,458. Really, if you give it some thought, only dimensionless measurements make operational sense.

Bob: But surely the fundamental constants $c$, $G$ and $\hbar$ are all three dimensionfull, and a lot of effort goes into accurately measuring their values.

Alice: If you think about it, also these measurements boil down to quantifying dimensionless ratios.

Bob: How can that be? No matter how you take ratios between these constants such ratios end up being dimensionfull. And you should not forget that these are our most fundamental constants, we have nothing more fundamental that we can use to try and build dimensionless ratios.

Alice: You don't need anything 'more fundamental'. If you are quantifying the three parameters $c$, $G$ and $\hbar$, really what you are doing is specifying units. You are specifying the way you abbreviate the results of physical measurements. There is nothing fundamental associated with such a units specification.

Bob: But the fundamental constants are fundamental. They have an intrinsic meaning and knowing their values represents fundamental knowledge.

Alice: I beg to differ. The values for the three parameters $c$, $G$ and $\hbar$ are purely conventional constructs. Their values act as conversion factors. The term 'fundamental constants' is hardly appropriate here. The only fundamental aspect associated with these conversion factors is the fact that their values are finite. Look at it like this: you can set $c$, $G$ and $\hbar$ all equal to unity. It is very common for physicists to make such substitution. This does not change any of the physics.

Bob: That is not true. If you change the fundamental constants, you change everything. If the speed of light would change, all of physics would change. Suppose the speed of light would be 300,000 mm/s instead of 300,000 km/s. This would cause us to live in a relativistic world. A window seat in an airplane would give a spectacular experience of the laws of relativity.

Alice: If the physics has changed, that means you have changed some dimensionless constants. You have done more than just changing units. Again, physics is all about quantifying dimensionless ratios. There is no other quantification that can be operationalized.

Bob: So you are saying that if I would change $c$, $G$ and $\hbar$, such that no dimensionless ratio changes, there would be no observable consequences?

Alice: try it.

• Duff's more recent take on "How Fundamental Are Fundamental Constants?": arxiv.org/abs/1412.2040 – Johannes Dec 8 '14 at 17:13
• This leaves the question why the various dimensionless ratios have the values that they do. Please write a cute allegory explaining that. For example, why does the fine structure constant has the value it does? – DanielSank Jul 20 '16 at 9:15
• A shorter way of saying this, is to say: When you use a ruler, you compare the thing you are measuring to the ruler. And then you use that same ruler to compare it to something else. Note how it does not matter at all, how you marked the ruler. As long as you use the same ruler or at least markings. Units are just rulers. By comparing and reading off the number, you are reading off the ratio of the thing you are measuring to one ruler marking. 1.2 of those meter rulers. 3 of those Planck thingies. Etc. – Evi1M4chine Sep 16 '17 at 6:20

The sample questions above referred to $c$, $G$, and $h$, all of which have units. A dimensionful constant has the value it does because of our system of units. Therefore none of the questions is meaningful.

Examples:

No theory can predict the value of $G$, because $G$ has to be expressed in some units. If we express it in SI units, then we're relating it to properties of the earth, since, e.g., the second was originally defined in terms of the rotation and orbit of the earth. There is no theory that can predict the properties of the earth, which are an accident of the formation of the solar system. However, it is conceivable that a theory of everything could predict some unitless measure of the strength of gravity, such as the ratio between the gravitational attraction of two electrons and their electrical repulsion.

There have been attempts to determine through astronomical observations whether the fine structure constant has changed over time. Webb et al. claimed a positive result, but later work seems to show that they were wrong. This is sometimes described as a search for variation in $c$ over time, but that's wrong, because $c$ has a defined value in the SI.[Duff 2002] Relativists do most of their work in a system of units in which $c=1$; obviously we can't let 1 vary over time!

There is a cute series of fantasy stories by George Gamow about a character named Mr. Tompkins. In these stories, we see the consequences if $c$, $h$, Boltzmann's constant $k$, etc., had different values. For example, when $k$ gets bigger, Mr. Tompkins starts to notice thermal fluctuations that we would normally not be able to sense. But although the stories are entertaining and educational, they are not rigorously valid, even if we are willing to assume that a person could be transported into an alternate universe. An alternate universe in which a single dimensionful constant has a different value could actually be the same universe, simply described in different units. To make the stories rigorous, we would have to have an alternate universe in which what differs is some dimensionless constant such as the fine structure constant.

Duff, 2002, "Comment on time-variation of fundamental constants," http://arxiv.org/abs/hep-th/0208093

• I didn't downvote, but I can tell you that I don't agree with your answer, for what it's worth. I think the questions are meaningful; changing the units will change the constant's value, but that doesn't mean that value is meaningless. What if someone asked why the earth is $12000 km$ wide? Would you tell them that the question is meaningless because it depends on the units? – Javier Nov 1 '14 at 16:13
• @BenCrowell I didn't downvote either, but I don't agree with your answer. There are three dimensionful constants: $c$, $G$ and $\hbar$. They define the Planck units. But what about lots of other parameters? For example, the radius of the Earth has a fixed value in terms of Planck lengths and this value should be predicted (at least its probability distribution). And there are lots of much more fundamental parameters: properties of elementary particles, etc. – Prof. Legolasov Nov 1 '14 at 19:56
• @BenCrowell You say that the relation between the Earth's radius and the meter is chosen by humans, therefore there is no sense in their numerical value. But there are some 'God-given' length units (e.g. the Planck length) which are dimensionful, but physical. So there is indeed no sence in the relation between the Earth's radius and the meter, however, the (Earth's) radius is physical and therefore has to be explained. – Prof. Legolasov Nov 1 '14 at 20:16
• @BenCrowell what people mean by asking such questions is why the Earth's radius is $a$ meters given that the meter is fixed. It is just much more simple to measure distances in meters than in Planck lengths. – Prof. Legolasov Nov 1 '14 at 20:18
• So, technically, you are right. But these questions do have physical meaning if the person asking them understands what I stated above. You just have to forgive them for the jargon. – Prof. Legolasov Nov 1 '14 at 20:20