# Dimensionless vs. dimensional "constants"

• I
• valenumr
In summary, the conversation discusses the concept of measuring the one-way speed of light and how it is defined as the distance between a source and a detector divided by the difference in time between two synchronized clocks. It is stated that this is not something that can be avoided and is inherent to the definition of the one-way speed of light. The conversation then shifts to a discussion about units and how they are defined and measured, with a focus on the fine structure constant and its relationship to the speed of light. It is clarified that the fine structure constant is not a function of the speed of light, but rather an effective coupling constant that takes renormalization into account. The conversation concludes with a question about how changes in the fine structure constant can impact
valenumr
[Moderator's note: Spin off from another thread due to topic change.]

Dale said:
No, there is 100% unavoidably without reference to any experiment unambiguously without exception no possible way to measure the one-way speed of light without assuming a synchronization convention. It is not a matter of clever experimentation, it is a matter of definition. The one-way speed of light is DEFINED as the distance between a source and a detector divided by the difference in time between two synchronized clocks at the source and the detector. Regardless of HOW you are measuring it that is WHAT you are measuring if you are in fact measuring what is known as the one-way speed of light. There is no way to avoid the issue. It is intrinsic to the definition of the thing that is being measured.
This is a weird argument to me. We can define metric units, but pretty much everything is defined by something else that we measure. Certainly their are unitless scalar "constants", but they are still basically defined by relationships between fixed values that we decide are fixed and measured values that we decide are parameters. Then we assume all the values are unchanging. But when you have a constant that can be stuck in an equation that defines a relationship between, say five other fundamental constants, what's to say say changing more than one variable on the right doesn't keep the left side more or less the same.

How much would your mind explode if planks constant was a function of time? Even the fine structure constant runs with energy level, does it not? What does that even mean? That at different energy levels the speed of light is different, or that the permittivity of free space is different, or that the fundamental charge is different?

Units are pretty circular. We fundamentally think of time and distance, but relativity tells us these are inherintly intertwined. We measure seconds on a process determined by alpha, we measure distance based on how far light travels in so much time. We fix the speed of light. And alpha is a function of multiple variable including the speed of light. Back to square one.

valenumr said:
We fix the speed of light. And alpha is a function of multiple variable including the speed of light.

No. Alpha is a dimensionless physical constant whose value does not depend on any choice of units. You could use the ton-furlong-fortnight system (mass in tons, distance in furlongs, time in fortnights, with appropriate derived units for other quantities) and the value of ##\alpha## would still be the same. So ##\alpha## is not a function of anything else.

The speed of light, OTOH, is a dimensionful quantity whose numerical value depends on your choice of units. So it is not correct to view ##\alpha## as a function of the speed of light.

bhobba and vanhees71
valenumr said:
Even the fine structure constant runs with energy level, does it not? What does that even mean?

It means that what you are thinking of as ##\alpha## is not the "bare" coupling constant that appears in the "bare" equations of quantum electrodynamics (i.e., the equations you would write down if you just took Maxwell's Equations and made them into a quantum field theory in the simplest, straightforward way, without worrying about all the complications that physicists have learned of in the last century or so of studying quantum field theories). What you are thinking of as ##\alpha## is actually the "effective" coupling constant that takes renormalization into account. "Renormalization" here is shorthand for "as the center of mass energy of an interaction increases, more and more virtual processes contribute significant amplitudes to the resulting scattering probabilities, which changes the effective coupling constant".

bhobba and vanhees71
PeterDonis said:
No. Alpha is a dimensionless physical constant whose value does not depend on any choice of units. You could use the ton-furlong-fortnight system (mass in tons, distance in furlongs, time in fortnights, with appropriate derived units for other quantities) and the value of ##\alpha## would still be the same. So ##\alpha## is not a function of anything else.

The speed of light, OTOH, is a dimensionful quantity whose numerical value depends on your choice of units. So it is not correct to view ##\alpha## as a function of the speed of light.
Yes, it is something we can measure very accurately (ridiculously accurately), but my point was the we make several relations among units, so.e we choose to define, some we choose to measure, and some we choose to calculate.

But it isn't even really a constant in the grand scheme of things (is it? Maybe I misunderstand). My point is that units only have relationships, and even scalar constants are circular. Just because something doesn't have units doesn't mean it is fixed or defines other values as constant. If I can write an equation that has one value on the left and five values on the right, that is a relationship that (may be) is constant.

Also, I sort of made this comment in response to the wrong thread of discussion... My apologies.

PeterDonis said:
It means that what you are thinking of as ##\alpha## is not the "bare" coupling constant that appears in the "bare" equations of quantum electrodynamics (i.e., the equations you would write down if you just took Maxwell's Equations and made them into a quantum field theory in the simplest, straightforward way, without worrying about all the complications that physicists have learned of in the last century or so of studying quantum field theories). What you are thinking of as ##\alpha## is actually the "effective" coupling constant that takes renormalization into account. "Renormalization" here is shorthand for "as the center of mass energy of an interaction increases, more and more virtual processes contribute significant amplitudes to the resulting scattering probabilities, which changes the effective coupling constant".
And also this is pretty much coming from my own in ignorance, but if alpha can change based on the energy (at electro weak it is higher by what, 8 percent?), How is that constant? And what other aspects of qed are impacted by that. Do all the fundamental relationships hold?

This isn't from a position if understanding the in depth mathematics, it just seems weird to me based on my understanding of some of the definitions of the fine structure constant and it's relationship to other fundamental units.

valenumr said:
we make several relations among units, so.e we choose to define, some we choose to measure, and some we choose to calculate.

None of this has anything to do with ##\alpha##, since, as I have already pointed out, the numerical value of ##\alpha## is the same regardless of your choice of units.

valenumr said:
My point is that units only have relationships, and even scalar constants are circular.

This is wrong. Scalar constants like ##\alpha## are not "circular". The renormalization issue you refer to (see further comments below) has nothing to do with any choice of units. It has to do with a correct understanding of what a "coupling constant" is (see below).

valenumr said:
if alpha can change based on the energy (at electro weak it is higher by what, 8 percent?), How is that constant?

It isn't. The term "coupling constant" is really a misnomer, but is used for historical reasons.

Physically, a "coupling constant" is a measure of the strength of an interaction. But in quantum field theory, an interaction does not have a single constant "strength". For example, suppose we try to measure ##\alpha## by scattering two electrons off of each other. The value we measure will depend on the energy of the electrons.

One way to think of this is in terms of Feynman diagrams. At low energy, there is just one Feynman diagram that contributes significantly to the scattering process, the one where the two electrons exchange just one virtual photon.

But at somewhat higher energy, we now get significant contributions from diagrams in which two photons are exchanged; and we also get a significant contribution from diagrams in which one virtual photon is exchanged, but while it is traveling from one electron to the other, it temporarily turns into a virtual electron-positron pair, which then turns back into a virtual photon. And we also get contributions from diagrams where one virtual photon is exchanged, but also one of the electrons creates a virtual photon and then re-absorbs it.

At still higher energies, even more complicated diagrams, with more virtual photons, or more virtual electron-positron pairs, make significant contributions. And by "significant contributions", we mean that the measured strength of the interaction changes. And that means the measured value of ##\alpha## changes. So really ##\alpha## is not a coupling "constant"; it's a coupling "function", a function of the energy of the interacting particles. (To be clear, "energy" here means energy in the center of momentum frame of the interaction.)

However, as I've already said, none of this has anything to do with any choice of units. The numerical function that describes how ##\alpha## varies with the energy of the interacting particles is the same no matter what units you choose.

valenumr said:
it just seems weird to me based on my understanding of some of the definitions of the fine structure constant and it's relationship to other fundamental units.

Unfortunately, most textbooks do their best to mislead you on this point, by giving you formulas for ##\alpha## in terms of "constants" that have units and whose numerical values change when you change your choice of units. They then neglect to explain how the numerical value of ##\alpha## somehow magically stays the same even as you change all the other numerical values by changing your system of units.

It would be better, IMO, to start with the things that don't depend on your choice of units, like ##\alpha##, and then show how, for convenience, you can define things with units like "the speed of light", "the charge on the electron", etc., and how different choices of numerical values for things with units can be useful for different purposes.

PeterDonis said:
Unfortunately, most textbooks do their best to mislead you on this point, by giving you formulas for ##\alpha## in terms of "constants" that have units and whose numerical values change when you change your choice of units. They then neglect to explain how the numerical value of ##\alpha## somehow magically stays the same even as you change all the other numerical values by changing your system of units.

It would be better, IMO, to start with the things that don't depend on your choice of units, like ##\alpha##, and then show how, for convenience, you can define things with units like "the speed of light", "the charge on the electron", etc., and how different choices of numerical values for things with units can be useful for different purposes.
Thanks, I genuinely appreciate your guidance. I am still confused in how we can relate so many things as constants based on a mathematical relationship where we chose which variables are fixed.

Anyhow, this is off topic, so I will ponder on what you have said and try and learn more to understand.

I'm also struggling with why it matters that something is unitless. I still think of it as a relation where the units cancel and whatever value is a scalar we multiply equations by.

But my trouble is the fact that, whether measured, computer, derived, or whatever, it still goes back to the fact that the inputs go back to values we measure based on fundamental properties that emerge from the value under consideration.

Ill go back to time and distance and the speed of light. We fix two, basically, and compute the other. But that's not entirely true. We fix time based on a property that, at least as far as I understand, is pretty much tied to the value we assign to the speed of light (alpha)...

valenumr said:
a mathematical relationship where we chose which variables are fixed.

You are still missing a key point: we don't choose whether ##\alpha## is fixed in that relationship; it is fixed, no matter what we choose. As I have said multiple times now: the numerical value of ##\alpha## is the same regardless of our choice of units. Our only freedom of choice is in the units of the other quantities.

valenumr said:
I'm also struggling with why it matters that something is unitless.

Because, obviously, a number that has no units can't depend on any choice of units. It has to be telling us something direct about the physics involved.

valenumr said:
I still think of it as a relation where the units cancel and whatever value is a scalar we multiply equations by.

And this is the wrong way to think of it. You have to think of it the other way around: we start with the unitless quantities that tell us directly about the physics, and then, for reasons of convenience, we can choose to "unpack" those unitless quantities into quantities with units. But the latter is not necessary; we could write all of the equations without any of those quantities with units in them. We only do so because we find it convenient.

valenumr said:
We fix time based on a property that, at least as far as I understand, is pretty much tied to the value we assign to the speed of light (alpha)...

No. The speed of light is not ##\alpha##.

In our current system of SI units, we make an arbitrary choice to define the unit of time, the second, based on the properties of photons emitted in a particular transition between energy levels in Cesium atoms. Ultimately those properties are determined, physically, by ##\alpha##. No other quantities are involved. But the choice of how many cycles of such photons to define as one "second" is an arbitrary choice. We could just as easily define our unit of time as one cycle instead of 9,192,631,770 of them. That wouldn't change the value of ##\alpha## at all.

We then make an arbitrary choice to define the unit of length, the meter, such that the speed of light has the numerical value 299,792,458. And we make other arbitrary choices to define the rest of the units such that various other quantities like Planck's constant have particular numerical values. All of those are arbitrary choices and could be made differently.

For example, particle physicists typically use units in which the speed of light and Planck's constant both have the numerical value ##1##, and the primary unit is the electron-volt as the unit of energy, with lengths and times having units of inverse electron-volts. But the value of ##\alpha## is still the same.

bhobba
PeterDonis said:
You are still missing a key point: we don't choose whether ##\alpha## is fixed in that relationship; it is fixed, no matter what we choose. As I have said multiple times now: the numerical value of ##\alpha## is the same regardless of our choice of units. Our only freedom of choice is in the units of the other quantities.
Because, obviously, a number that has no units can't depend on any choice of units. It has to be telling us something direct about the physics involved.
And this is the wrong way to think of it. You have to think of it the other way around: we start with the unitless quantities that tell us directly about the physics, and then, for reasons of convenience, we can choose to "unpack" those unitless quantities into quantities with units. But the latter is not necessary; we could write all of the equations without any of those quantities with units in them. We only do so because we find it convenient.
No. The speed of light is not ##\alpha##.

In our current system of SI units, we make an arbitrary choice to define the unit of time, the second, based on the properties of photons emitted in a particular transition between energy levels in Cesium atoms. Ultimately those properties are determined, physically, by ##\alpha##. No other quantities are involved. But the choice of how many cycles of such photons to define as one "second" is an arbitrary choice. We could just as easily define our unit of time as one cycle instead of 9,192,631,770 of them. That wouldn't change the value of ##\alpha## at all.

We then make an arbitrary choice to define the unit of length, the meter, such that the speed of light has the numerical value 299,792,458. And we make other arbitrary choices to define the rest of the units such that various other quantities like Planck's constant have particular numerical values. All of those are arbitrary choices and could be made differently.

For example, particle physicists typically use units in which the speed of light and Planck's constant both have the numerical value ##1##, and the primary unit is the electron-volt as the unit of energy, with lengths and times having units of inverse electron-volts. But the value of ##\alpha## is still the same.
I understand 100 percent everything you are saying. My stumbling block is how we can write a dozen equations in terms of alpha, and just decide that one is fixed, or more importantly that just because alpha is (some are trying to falsify this), every value on the right hand side of the various "alpha equals" equation must also be constants.

Just because a relation holds, it doesn't mean everything is constant. I can write '5z = 2x + y', and it has infinite solutions. Or I can divide both sides by z, and it still had infinite solutions, but 5 is a constant.

valenumr said:
I understand 100 percent everything you are saying.

I'm not sure you do since you keep making the same errors. See below.

valenumr said:
My stumbling block is how we can write a dozen equations in terms of alpha, and just decide that one is fixed

We are doing no such thing. If you think we are, please give specific examples--by which I mean, give me some links to textbooks or peer-reviewed papers that are actually doing the things you are talking about.

valenumr said:
just because alpha is (some are trying to falsify this)

There are efforts to see if ##\alpha## (more precisely, the limiting value of ##\alpha## at low energy) has changed over time. So far no such change has been detected. But none of that affects what we are discussing in this thread.

valenumr said:
every value on the right hand side of the various "alpha equals" equation must also be constants.

We are not doing that either. Every value on the RHS of the various "alpha equals" equation is a value that you can change by changing your system of units. (Obviously this doesn't apply to things like ##\pi##, or integers like ##4##, but it does to things like the charge on the electron, Planck's constant, and the speed of light.) So they're not "constants" the way ##\alpha## is a constant. As I have been saying for several posts now.

PeterDonis said:
I'm not sure you do since you keep making the same errors. See below.
We are doing no such thing. If you think we are, please give specific examples--by which I mean, give me some links to textbooks or peer-reviewed papers that are actually doing the things you are talking about.
There are efforts to see if ##\alpha## (more precisely, the limiting value of ##\alpha## at low energy) has changed over time. So far no such change has been detected. But none of that affects what we are discussing in this thread.
We are not doing that either. Every value on the RHS of the various "alpha equals" equation is a value that you can change by changing your system of units. (Obviously this doesn't apply to things like ##\pi##, or integers like ##4##, but it does to things like the charge on the electron, Planck's constant, and the speed of light.) So they're not "constants" the way ##\alpha## is a constant. As I have been saying for several posts now.
I articulated that poorly. Try this: given the relation 'x = ay / (bz + cw)'. The left hand can stay constant all day given constraints on the three "variables", or "constants" on the right. I'm struggling to identify a clear independent relationship over what we define as fundamental constants, what we measure as constants, and how those can be computationally independent from one another.

valenumr said:
I articulated that poorly. Try this: given the relation 'x = ay / (bz + cw)'. The left hand can stay constant all day given constraints on the three "variables", or "constants" on the right. I'm struggling to identify a clear independent relationship over what we define as fundamental constants, what we measure as constants, and how those can be computationally independent from one another.
Because I can still replace x and several rhs components with arbitrary values, and the relation can still have infinite solutions.

valenumr said:
Because I can still replace x and several rhs components with arbitrary values, and the relation can still have infinite solutions.
And again, I'm off on a tangent... It's more wondering how "constant" constants are. We choose them to maximize value in terms of measurement accuracy and reducing dependencies. I just wonder if the underlying relationships between fundamental values aren't more significant, and if perhaps while the relations may stay static, the "constants" could evolve over time. I am picking on alpha, because it has so many non-trivial relations to other things, and there is a pretty good basis that it is unchanging.

PeterDonis said:
I'm not sure you do since you keep making the same errors. See below.
We are doing no such thing. If you think we are, please give specific examples--by which I mean, give me some links to textbooks or peer-reviewed papers that are actually doing the things you are talking about.
There are efforts to see if ##\alpha## (more precisely, the limiting value of ##\alpha## at low energy) has changed over time. So far no such change has been detected. But none of that affects what we are discussing in this thread.
We are not doing that either. Every value on the RHS of the various "alpha equals" equation is a value that you can change by changing your system of units. (Obviously this doesn't apply to things like ##\pi##, or integers like ##4##, but it does to things like the charge on the electron, Planck's constant, and the speed of light.) So they're not "constants" the way ##\alpha## is a constant. As I have been saying for several posts now.
But why can't the lhs be true while some values on the rhs vary as long as the relation holds?

valenumr said:
But why can't the lhs be true while some values on the rhs vary as long as the relation holds?

How do you plan to measure the distance, that light moves in 1 second?

• With the current definition of "1 meter", you can by definition not measure a "change of c".
• If you use a ruler, you also cannot measure a "change of c" with unchanged ##\alpha##. Reason: The ruler length is propotional to the Bohr radius of
$$a_0 = \frac{\hbar}{m_e c \alpha } = \frac{\hbar c^2}{E_{0e} c \alpha} = \frac{\hbar }{E_{0e} \alpha} * c$$
Source:

You can regard the electron as created by pair production with photons ##W=hf##. So it's rest energy ##E_{0e}## is proportional to ##\hbar##. Also, your reference time "1 second" is based on the frequency of photons (from Caesium).

If the distance, that light moves in 1 second, would "double", then also the ruler length would "double". Then you would measure no change.

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valenumr said:
It's more wondering how "constant" constants are.
Actually(even thoug this is not exactly what you were discussing in this thread) the alpha constant has been detected to have changed both in time and spatially in experiments by Webb et al. from the university of South Wales in Australia, and to this day no failure or error has been found on those experiments. You can find references at
https://en.wikipedia.org/wiki/Fine-...he_fine-structure_constant_actually_constant?

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vanhees71
valenumr said:
This is a weird argument to me.
I think that you missed the point of the argument and went off on a big tangent. The point of the argument is that definitions matter.

When you say “I am measuring X” then the definition of X matters. Whatever you measure must conform to the definition of X. If your measurement violates the definition of X then it was not in fact a measurement of X at all. Do you understand that?

Any statement in the definition of X is a statement that will hold for any measurement of X. If a defining statement does not hold for a particular measurement then that measurement is not a measurement of X, by definition.

Trying to avoid a particular statement in the definition through clever experimental design is pure folly. Either you will avoid the defining statement and thereby fail to measure X, or you will measure X and thereby fail to avoid the defining statement. A quantity can only be a measure of X if it satisfies all of the statements in the definition of X.

That is my argument. Hopefully is seems less weird to you now, but regardless of whether it seems weird or not, it is completely solid. There is no way around this argument.

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bhobba, vanhees71 and weirdoguy
That's of course true: If you claim to measure quantity ##X## you must be sure that your measurement device really measures this quantity.

Then it's also the other way around: If you measure something in an experimentally well-defined reproducible way you may ask which quantity you measure, using a given model or theory to describe the situation.

Take the above discussion about the "running couplings" in QFT. What you really measure are transition probabilities for scattering reactions between particles, e.g., elastic electron-electron scattering. The electric elementary charge (or equivalently the fine structure constant ##\alpha##, which has the advantage to be dimensionless and independent of the electromagnetic units used) is a measure for the electromagnetic coupling strength as measured when comparing the theoretical prediction of the corresponding transition probabilities (aka cross sections) with the renormalization scale at low energies and in an on-shell renormalization scheme. That's mostly for historical reasons. You can also predict the cross sections at higher energies and calculate more and more radiative corrections (Feynman diagrams at higher and higher order in the coupling constant), and one part can be subsumed in the "running of the coupling constant" when changing the renormalization scale. Then you can use, e.g., tree-level cross sections with a running coupling constant at a higher renormalization scale. In QED the running coupling gets larger at larger renormalization scales, and ##\alpha## at a renormalizatino scale of the ##Z##-boson mass is about 1/128 (as compared to the value at small renormalization scales, where it is about 1/137).

bhobba
valenumr said:
why can't the lhs be true while some values on the rhs vary as long as the relation holds?

The LHS is "true" (##\alpha## has the same numerical value) while some values on the RHS vary. That's the whole point of saying you can change your system of units--which changes values on the RHS--without changing the value of ##\alpha##.

## 1. What is the difference between dimensionless and dimensional constants?

Dimensionless constants are numerical values that do not have units, while dimensional constants have units associated with them. Dimensionless constants are often used in mathematical equations and do not change with different units of measurement, while dimensional constants vary depending on the units used.

## 2. Can you give an example of a dimensionless constant?

One example of a dimensionless constant is the fine structure constant, which is a fundamental constant in physics that describes the strength of the electromagnetic interaction between elementary particles. It is a pure number with no units and has a value of approximately 1/137.

## 3. How are dimensionless and dimensional constants used in scientific calculations?

Dimensionless constants are often used in equations to simplify calculations and remove the need for unit conversions. Dimensional constants, on the other hand, are used to convert between different units of measurement and are crucial in many scientific calculations.

## 4. Are all physical constants dimensionless?

No, not all physical constants are dimensionless. Some examples of dimensional constants include the speed of light, gravitational constant, and Planck's constant. These constants have units associated with them and play a crucial role in many physical theories and calculations.

## 5. How do dimensionless and dimensional constants relate to dimensional analysis?

Dimensional analysis is a mathematical technique used to check the validity of equations and to derive relationships between physical quantities. Dimensionless constants play a significant role in dimensional analysis as they help to simplify equations and remove unnecessary units, making it easier to identify the relationships between different variables.

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