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I was reading this stackexchange question, and found the answer to my question not totally answered. Clearly there is color and anti-color in analogy to electric charge, and color charge clearly cannot vary from color to anti-color. However can color (or anti-color) continuously vary between a red green and blue basis, or is it like wavelengths in atomic orbitals where in order to go from one color to another you have emit the exact amount of color charge required?

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    $\begingroup$ Closely related: physics.stackexchange.com/q/176478/50583 $\endgroup$ – ACuriousMind Dec 21 '18 at 0:11
  • $\begingroup$ @ACuriousMind I recently left a question similar to this one as a comment on your answer to the question you linked. Give it a read if you get the chance $\endgroup$ – Craig Dec 21 '18 at 0:22
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    $\begingroup$ If by quantized you mean that there are only 3, then yes. $\endgroup$ – InertialObserver Dec 21 '18 at 2:00
  • $\begingroup$ No. there are not 8 colors.. those are just labels on the gauge fields.. $\endgroup$ – InertialObserver Dec 21 '18 at 4:40
  • $\begingroup$ @InertialObserver, Yes there are 8 charges associated to the global symmetry, not 3. In naive popular descriptions of QCD people talk about the 3 colors as though they are like charges, but they are not a very a physical thing. ACuriousMind is talking about a different 3 that is coming from center symmetry. I originally deleted my comment to you because I thought he was saying something more interesting than me, but I'll put it back just for clarity. $\endgroup$ – octonion Dec 21 '18 at 12:49
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Naively, color can vary continuously between the colors according to a gauge transformation $\psi\mapsto \mathrm{e}^{\mathrm{i}\epsilon^a T^a}$ for some $\mathfrak{su}(2)$-valued object $\epsilon$, this is precisely the same as saying that a particle with electrical charge $e$ can vary continuously in phase according to $\psi\mapsto \mathrm{e}^{\mathrm{i}e\phi}\psi$.

However, there is a crucial difference: The $\mathrm{U}(1)$ symmetry of electromagnetism is Abelian, and so all transformations with constant $\phi$ are global symmetry transformations that have no gauge character, since the gauge field does not change under such transformations. In contrast, the $\mathrm{SU}(3)$ symmetry group is non-Abelian, and even constant $\epsilon$ change the gauge field, unless they commute with it. The set of elements of a non-Abelian group that commute with all others is called the center, and the center of $\mathrm{SU}(N)$ is the discrete group $\mathbb{Z}_n$.

So while electrically charged matter retains a continuous $\mathrm{U}(1)$ symmetry even after eliminating the gauge, color-charged matter retains only a discrete $\mathbb{Z}_3$ symmetry. That is, if you eliminate the gauge (which, in general, we cannot do: Gribov ambiguities prevent us even in principle from doing so globally, and even then, we will face a loss of covariance) you will end up with a particular set of red/blue/green particles that no longer can transform into each other. In this gauge-fixed world, you could think of color as a fixed property of each object, but this is not a useful intuitive picture to have. We describe the world through gauge theories precisely because the gauge-less description is not tractable.

However, that there is a discrete global $\mathbb{Z}_3$ symmetry is a valuable insight, as this is what is actually broken in the Higgs mechanism, as explained in this answer by Dominic Else.

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    $\begingroup$ That's interesting, as I've always heard this situation described as spontaneous breaking of global $SU(3)$ symmetry, not global $\mathbb{Z}_3$ symmetry. Is this because you're choosing to interpret transformations with constant gauge parameter as redundancies? Is this convention also taken by the usual QFT texts? $\endgroup$ – knzhou Dec 21 '18 at 2:10
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    $\begingroup$ @knzhou It's not a "choice". By the very nature of a gauge theory, configurations of the gauge field related by a gauge transformation (regardless of the space-time dependence of the transformation parameter!) are physically equivalent. Therefore, the only transformations between physically distinct configurations can be those that leave the gauge field invariant. But I'll grant that many texts do not ponder this subtlety at all, since they follow the strange pedagogy of thinking of a gauge theory as arising from "gauging" a global symmetry. $\endgroup$ – ACuriousMind Dec 21 '18 at 2:32
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    $\begingroup$ (As supplementary eivdence that my claims are correct, consider that no text I've ever read spends any time to dwell on the conserved quantities associated to the global variant of a non-Abelian gauge symmetry. That's because the global symmetry is discrete, and Noether's theorem doesn't apply to discrete symmetries) $\endgroup$ – ACuriousMind Dec 21 '18 at 2:35
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    $\begingroup$ Hmm, I'm still not sure. How does your second comment square with your answer here? $\endgroup$ – knzhou Dec 21 '18 at 2:40
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    $\begingroup$ @octonion I was using "observable" in the meaning of "physically measurable" there. However, now that I think about it, this local/global/global continuous distinction is a can of worms and I'm not sure my understanding is exactly right. I might ask a more detailed question soon. $\endgroup$ – ACuriousMind Dec 21 '18 at 12:40

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