# Color charge is not scalar -- do their components have dimensions?

• I
• south
south
TL;DR Summary
It's dimensions
Color charge is not scalar. Still, do their components have dimensions (in metrological terms)?

south said:
Color charge is not scalar.
What do you mean by this statement? Can you state it in mathematical terms?

PeterDonis said:
What do you mean by this statement? Can you state it in mathematical terms?
Thank you Peter Donis for assisting me

.Before writing the question I read an online article that warns about this. Now your message makes me suspect that this is false. My interest in metrological dimensions continues.

south said:
What article? Please give a reference.

PeterDonis said:
What article? Please give a reference.
Article:
https://es.wikipedia.org/wiki/Campo_de_color

Specific part of the content:
El campo de color (o campo gluónico) es un campo físico asociado a la interacción fuerte entre partículas que llevan asociadas una carga de color.

Desde el punto de vista matemático el campo de color se representa por un campo gauge tensorial.

Translated:

The color field (or gluonic field) is a physical field associated with the strong interaction between particles that have a color charge associated with them.

From a mathematical point of view, the color field is represented by a tensor gauge field.

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south said:
Have you tried looking at actual textbooks, papers, or course notes on the Standard Model (there are plenty available for free online)?

south said:
do their components have dimensions (in metrological terms)?
What do you mean by "in metrological terms"? You do realize that we cannot directly measure color charge since all observable particles are color neutral?

The word dimensions has more than one meaning in physics. I simply add the adjective metrological to refer the meaning to the type of magnitude that is not dimensionless and that is quantitatively expressed including some unit of measurement.

PeterDonis said:
Have you tried looking at actual textbooks, papers, or course notes on the Standard Model (there are plenty available for free online)?
I haven't tried it. I'm momentarily interested in what kind of magnitude the color charge is, dimensionless or not.

south said:
I simply add the adjective metrological to refer the meaning to the type of magnitude that is not dimensionless and that is quantitatively expressed including some unit of measurement.
Ok, but then:

south said:
I haven't tried it. I'm momentarily interested in what kind of magnitude the color charge is, dimensionless or not.
There is no point in being "momentarily interested" in physics. Either you want to learn it, or you don't. If you do, it takes time, and is not best done by asking random questions that "momentarily" occur to you.

The only answer that can be given to your question as you ask it is that, as I have already said, color charges are not measurable since all observable particles are color neutral. So it is meaningless to ask "what kind of magnitude" they are. They are abstract objects in a theory.

south said:
TL;DR Summary: It's dimensions

Color charge is not scalar. Still, do their components have dimensions (in metrological terms)?
Fundamentally, neither electric nor color charge has a dimension. In quantum electrodynamics (QED, the theory underlying classical electrodynamics) the dimensionful elementary electric charge ##e## in coulombs is replaced by the dimensionless charge ##\mathscr{\mathbf{e}}=\frac{e}{\sqrt{\varepsilon_{0}\hbar c}}## and then used to define a dimensionless electric coupling strength ##\alpha_{\text{QED}}\equiv\frac{\mathbf{e}^{2}}{4\pi}=\frac{e^{2}}{4\pi\varepsilon_{0}\hbar c}##. This quantity is usually referred to as the fine-structure "constant", but it actually increases slowly as a function of the energy ##Q## at which it is measured:

(from https://arxiv.org/pdf/1102.2380v1)
Similarly, the strongly-interacting theory of quantum chromodynamics (QCD) is characterized by a dimensionless color charge ##\mathscr{\mathbf{g}}## and the corresponding color coupling strength ##\alpha_{\text{QCD}}=\alpha_{\text{S(trong)}}\equiv\frac{\mathbf{g^{2}}}{4\pi}##. This strength is large at low-energies but rapidly diminishes at high-energies ##Q## ("asymptotic freedom"):

(from https://cerncourier.com/a/the-history-of-qcd/)
So the problems with challenges of the "metrology of charge" are include:
• Charges and coupling strengths that are inherently dimensionless parameters.
• Their values are a function of the energy-scales of measurement.
(Edited to state that ##e## denotes the electric charge of the electron.)

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dextercioby and south
renormalize said:
Charges and coupling strengths are inherently dimensionless quantities.
From a theory point of view this is true. However, from a metrological point of view in SI units, electric charge is fundamental and the unit charge is a fundamental defined quantity of measurement. The fine structure constant - or equivalently, ##\epsilon_0## - is of course still a theory parameter to be determined by measurement.

The fact that coupling constants run is not an a priori issue in terms of measurements as the underlying theory predicts the relation between measurements at different scales.

renormalize
Thanks renormalize. I find the note clear and didactic. It contains exactly what I needed to find out. I found it very useful. Best regards.

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