# Unitless constants and unitized forces

• enotstrebor
In summary: The fine structure constant is the ratio of two energies, not forces. In summary, coupling constants are either unitized or unitless, with forces being associated with unitized coupling constants such as charge and gravity. Unitless coupling constants, such as the fine structure constant and the Lorentzian \beta, are ratios where the units cancel out. While the weak coupling constant is unitless, it does not necessarily mean that it is a ratio of forces like the fine structure constant, as not all unitless constants are ratios of forces. The value of coupling constants does not depend on the choice of units, making them a convenient mathematical tool.
enotstrebor
Coupling constants are unitized or unitless. Forces are associated to unitized coupling constants like charge (e), gravity (G), etc. All unitless coupling constants that I know of, like the fine structure constant ($$\alpha$$), or the Lorentzian $$\beta$$, etc are ratios where the units cancel out.

As the weak coupling constant is unitless;
Does this mean it is a ratio of forces like the fine structure constant? If not why not.

enotstrebor said:
Coupling constants are unitized or unitless. Forces are associated to unitized coupling constants like charge (e), gravity (G), etc. All unitless coupling constants that I know of, like the fine structure constant ($$\alpha$$), or the Lorentzian $$\beta$$, etc are ratios where the units cancel out.
What do you mean when you refer to the "Lorentzian $$\beta$$?" To me this suggests the relativistic $\beta=v/c$, which isn't a coupling constant.

I don't think it makes sense to classify coupling constants as unitful or unitless. The SI has three basic units, for length, time, and mass. You can pick any three universal constants you like and set them equal to 1, and then you get a system of units where everything is unitless. For instance, if you pick c=1, G=1, and h=1, you get a system of units where everything is unitless, and the Planck distance equals 1. Of c, G, and h, one (G) is clearly a coupling constant, another (c) could be considered to be one, and one (h) is not. Suppose I choose to set c=1, h=1, and the mass of the electron=1. Then I get a system of units where G does not equal 1, but it's unitless, because everything is unitless. I could also just set c=1 and h=1, and then G would have units.

As the weak coupling constant is unitless;
Does this mean it is a ratio of forces like the fine structure constant? If not why not.

Above the electroweak unification energy, the weak interaction is unified with the EM interaction, so aren't the coupling constants the same?

bcrowell said:
What do you mean when you refer to the "Lorentzian $$\beta$$?" To me this suggests the relativistic $\beta=v/c$, which isn't a coupling constant.

I don't think it makes sense to classify coupling constants as unitful or unitless. The SI has three basic units, for length, time, and mass. You can pick any three universal constants you like and set them equal to 1, and then you get a system of units where everything is unitless. For instance, if you pick c=1, G=1, and h=1, you get a system of units where everything is unitless, and the Planck distance equals 1. Of c, G, and h, one (G) is clearly a coupling constant, another (c) could be considered to be one, and one (h) is not. Suppose I choose to set c=1, h=1, and the mass of the electron=1. Then I get a system of units where G does not equal 1, but it's unitless, because everything is unitless. I could also just set c=1 and h=1, and then G would have units.

Above the electroweak unification energy, the weak interaction is unified with the EM interaction, so aren't the coupling constants the same?

But the beauty of the coupling constants is that their value does not depend on your choice of units. So in units where h=c=G=1 the value of alpha, the EM coupling constant, is still 1/137 (approx.), and it stays that way whatever your choice of units is.

No coupling constant does not per se be unitless, look at

$$\lambda \, \phi^3$$

a fully allowed term in a lagrangian.

is $\lambda$

unitless here? (in the choice where c = hbar = 1 and unitless)

bcrowell said:
What do you mean when you refer to the "Lorentzian $$\beta$$?" To me this suggests the relativistic $\beta=v/c$, which isn't a coupling constant.
?
My appologies I meant to say unitless constants in the first part.

bcrowell said:
You can pick any three universal constants you like and set them equal to 1, and then you get a system of units where everything is unitless. For instance, if you pick c=1, G=1, and h=1, you get a system of units where everything is unitless

Using $$c=\hbar=1$$ does not make the value of a force unitless. For instance if you were really setting c=\hbar=1 then the units of the fine structure constant alpha=e^2/(c*h)=e^2/(1*1) would not be unitless but have the units of charge. The use of thetype conventions are just a mathematical convenience. They do not make things unitless.

The question is, if the weak angle is a unitless coupling constant does that imply that it is the ratio of two forces just like the fines structure constant is ther ratio of two forces (charge and spin).

If not why not?

If so, what are the components of weak angle ratio (what are the two forces?).

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enotstrebor said:
The question is, if the weak angle is a unitless coupling constant does that imply that it is the ratio of two forces just like the fines structure constant is ther ratio of two forces (charge and spin).

As was pointed out in https://www.physicsforums.com/showthread.php?t=365156", charge and spin are not forces.

Last edited by a moderator:

## 1. What are unitless constants?

Unitless constants are mathematical values that do not have a unit of measurement. They are used to describe physical or mathematical phenomena and remain constant regardless of the unit system used.

## 2. How are unitless constants different from unitized forces?

Unitized forces are forces that have been converted into a specific unit of measurement, such as newtons or pounds. They are different from unitless constants because they have a specific unit and can be measured in a physical context.

## 3. What are some examples of unitless constants?

Examples of unitless constants include the gravitational constant (G), the speed of light (c), and the fine structure constant (α). These values remain constant regardless of the unit system used and are essential in many scientific calculations.

## 4. How are unitless constants and unitized forces used in scientific research?

Unitless constants and unitized forces are used in scientific research to provide consistency and accuracy in measurements. They are also used in mathematical models and equations to describe physical phenomena and make predictions.

## 5. Can unitless constants and unitized forces change?

Unitless constants, by definition, do not change. However, unitized forces can vary depending on the unit system used. For example, a force of 10 newtons can also be expressed as 2.2480894 pounds-force. It is important to specify the unit when using unitized forces in calculations.

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