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  Why is color conserved in QCD?

+ 6 like - 0 dislike
6548 views

According to Noether's theorem, global invariance under $SU(N)$ leads to $N^2-1$ conserved charges. But in QCD gluons are not conserved; color is. There are N colors, not $N^2-1$ colors. Am I misunderstanding Noether's theorem?

My only guess (which is not made clear anywhere I can find) is that there are $N_R^2-1$ conserved charges, where $N_R$ is the dimension of the representation of SU(N) that the matter field transforms under.

EDIT:

I think I can answer my own question by saying that eight color combinations are conserved which do correspond to the colors carried by gluons. Gluon number is obviously not conserved, but the color currents of each gluon type are conserved. An arbitrary number of gluons can be created from the vacuum without violating color conservation because color pair production {$r,\bar{r}$}, {$g,\bar{g}$}, {$b,\bar{b}$} does not affect the overal color flow. Lubos or anyone please correct me if this is wrong, or if you want to clean it up and incorporate it into your answer Lubos I will accept your answer.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user user1247
asked Mar 14, 2013 in Theoretical Physics by user1247 (540 points) [ no revision ]
retagged Apr 11, 2015
Most voted comments show all comments
@Lubos, and besides, I asked you specifically if you would edit your own answer to address what I am trying to understand. I am trying to synthesize your responses to my questions and come up with the conceptual answer to my question that I am looking for, and by asking you to check it or incorporate it into your answer, I am obviously continuing a dialog rather than just "answering my own question."

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user user1247
Dear @user1247, the room for questions on this server is simply not dedicated to "synthetizing" the answers, especially not in the hugely misleading way that you demonstrated. Also, if you're trying to understand these (or other) things, you must be incredibly grateful for my guidance because I am identifying some key conceptual errors in your very general approach to similar technical questions - something that you would fine helpful in many other contexts. If you're not trying to understand but you came here to pretend you're bright, you may be insulted but that's not my fault.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl
@lubos, this behavior is exactly what I'm talking about. I'm not remotely trying to pretend that I'm bright. I'm trying to understand why in almost every QFT resources the authors say "color is conserved" but then don't describe in detail what color combinations are conserved. You have really not addressed this question direclty, as far as I can tell, but instead continue to misrepresent my questions in unfavorable ways and answer questions that I am not really asking. As I pointed out below, wikipedia seems to agree with my edit, even though you seem to think it is "silly."

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user user1247
@lubos, it would be extremely helpful if you would compare what I say in my edit to what wikipedia says (en.wikipedia.org/wiki/Gluon#Eight_gluon_colors), where they They specifically associate the 8 gluon color combinations with the gell-mann matrices. This is common in other texts. Either I'm right, or everybody else is stupid. It would be great if, instead of continuing to call me stupid, if you would actually try to address this discrepancy, and in so doing, my question.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user user1247

This seems to be a rather elementary confusion, it's like asking since Lorentz transformations are 4 by 4 matrices and act on column vectors with 4 independent entries, how can there be more than 4 conserved quantities originating from Lorentz invariance? 

Most recent comments show all comments
Sorry, at schools I attended, including Prague, this kind of thing - how many conserved charges SU(N) implies etc. - was known to all the undergraduate students who were going to particle or theoretical physics although it's questionable whether people have to know it by the sophomore or junior age (it's been taught to the freshmen and sophomores, kind of, in linear algebra and math methods in physics, and cemented in early QM/QFT courses for juniors). I have absolutely no clue how someone could do any particle or theoretical physics research without understanding similar things.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl
@User1247: concerning your update, I find this way of using this website highly bizarre. The questions aren't meant to be the answers at the same moment. Moreover, claiming that you "answered your own question" after a rather long course when people, not just me, were explaining you these things - it really took much more time than it should have - seems dishonest to me. You haven't answered your own question. You have just partly understood what others were telling you and wrote a confused version of these explanations inside your question where it has no business to oxidize.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl

1 Answer

+ 6 like - 0 dislike

Global invariance under $SU(N)$ is equivalent to the conservation of $N^2-1$ charges – these charges are nothing else than the generators of the Lie algebra ${\mathfrak su}(N)$ that mix some components of $SU(N)$ multiplets with other components of the same multiplets. These charges don't commute with each other in general. Instead, their commutators are given by the defining relations of the Lie algebra, $$ [\tau_i,\tau_j] = f_{ij}{}^k \tau_k $$ But these generators $\tau_i$ are symmetries because they commute with the Hamiltonian, $$[\tau_i,H]=0.$$ None of these charges may be interpreted as the "gluon number". This identification is completely unsubstantiated not only in QCD but even in the simpler case of QED. What is conserved in electrodynamics because of the $U(1)$ symmetry is surely not the number of photons! It's the electric charge $Q$ which is something completely different. In particular, photons don't carry any electric charge.

Similarly, this single charge $Q$ – generator of $U(1)$ – is replaced by $N^2-1$ charges $\tau_i$, the generators of the algebra ${\mathfrak su}(N)$, in the case of the $SU(N)$ group.

Also, it's misleading – but somewhat less misleading – to suggest that the conserved charges in the globally $SU(N)$ invariant theories are just the $N$ color charges. What is conserved – what commutes with the Hamiltonian – is the whole multiplet of $N^2-1$ charges, the generators of ${\mathfrak su}(N)$.

Non-abelian algebras may be a bit counterintuitive and the hidden motivation behind the OP's misleading claim may be an attempt to represent $SU(N)$ as a $U(1)^k$ because you may want the charges to be commuting – and therefore to admit simultaneous eigenstates (the values of the charges are well-defined at the same moment). But $SU(N)$ isn't isomorphic to any $U(1)^k$; the former is a non-Abelian group, the latter is an Abelian group.

At most, you may embed a $U(1)^k$ group into $SU(N)$. There's no canonically preferred way to do so but all the choices are equivalent up to conjugation. But the largest commuting group one may embed into $SU(N)$ isn't $U(1)^N$. Instead, it is $U(1)^{N-1}$. The subtraction of one arises because of $S$ (special, determinant equals one), a condition restricting a larger group $U(N)$ whose Cartan subalgebra would indeed be $U(1)^N$.

For example, in the case of $SU(3)$ of real-world QCD, the maximal commuting (Cartan) subalgebra of the group is $U(1)^2$. It describes a two-dimensional space of "colors" that can't be visualized on a black-and-white TV, to use the analogy with the red-green-blue colors of human vision. Imagine a plane with hexagons and triangles with red-green-blue and cyan-purple-yellow on the vertices.

But grey, i.e. color-neutral, objects don't carry any charges under the Cartan subalgebra of $SU(N)$. For example, the neutron is composed of one red, one green, one blue valence quark. So you could say that it has charges $(+1,+1,+1)$ under the "three colors". But that would be totally invalid. A neutron (much like a proton) actually carries no conserved QCD "color" charges. It is neutral under the Cartan subalgebra $U(1)^2$ of $SU(3)$ because the colors of the three quarks are contracted with the antisymmetric tensor $\epsilon_{abc}$ to produce a singlet. In fact, it is invariant under all eight generators of $SU(3)$. It has to be so. All particles that are allowed to appear in isolation must be color singlets – i.e. carry vanishing values of all conserved charges in $SU(3)$ – because of confinement!

So as far as the $SU(3)$ charges go, nothing prevents a neutron from decaying to completely neutral final products such as photons. It's only the (half-integral) spin $J$ and the (highly approximately) conserved baryon number $B$ that only allow the neutron to decay into a proton, an electron, and an antineutrino and that make the proton stable (so far) although the proton's decay to completely quark-free final products such as $e^+\gamma$ is almost certainly possible even if very rare.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl
answered Mar 14, 2013 by Luboš Motl (10,278 points) [ no revision ]
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Thanks Lubos, this last answer helps. Can you check my edit to my question and let me know if it is right?

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user user1247
I don't know what to say about your "own answer", @user1247. It is a layman's caricature of a part of the right answer. For example, you call the generators by labels like {red,greenbar} etc. That's silly and omitting some key information. Just look at the first en.wikipedia.org/wiki/Gell-Mann_matrices Gell-Mann matrices $\lambda_1,\lambda_2$. Both are generators of SU(3), both are {red,greenbar} of a sort, but they're totally different - in fact, orthogonal to each other in the space of matrices.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl
It's also wrong to say that each of the r-rbar, g-gbar, b-bbar is inconsequential for "color flow". There are 2 generators of SU(3) - the Cartan subalgebra - that are as good and as nontrivial generators as all other generators (and they affect everything) even though they're composed purely of r-rbar, g-gbar, b-bbar. Only the sum of these three is eliminated to get SU(3) from U(3). It's not clear whether you really want to understand these things or just invent some new misleading layman's caricature. If you ask if you have understood it at the technical level, my answer is clearly No.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl
Lubos, you say I "call the generators by labels like {red,greenbar} etc. That's silly and omitting some key information". But I am doing what everybody does. Is wikipedia (en.wikipedia.org/wiki/Gluon#Eight_gluon_colors) wrong then? They specifically associate the 8 gluon color combinations with the gell-mann matrices. Look, I'm trying to clarify my understanding of a hueristic here, one that is used perhaps sloppily by countless physicists, talking vaguely about "color conservation." Maybe they're all idiots, or maybe it could be a useful hueristic if used properly?

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user user1247

@LubošMotl,@Manishearth, currently Lubos seems to call into question my motives rather than address my doubts. I am an experimental physicist genuinely trying to understand a heuristic that is described in almost every modern QFT textbook, and one described in wikipedia here (en.wikipedia.org/wiki/Gluon#Eight_gluon_colors). Instead of addressing this and fleshing it out, he prefers to tell me I'm trying to "invent some new misleading layman's caricature".

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user user1247

Most recent comments show all comments
"wait, are you saying that in SU(2) weak theory particles must be "color neutral", where now there are two colors?" - Nope, I haven't made any statement about the electroweak SU(2). The claim doesn't directly hold for the SU(2) in the weak interactions because that group isn't confining; it is spontaneously broken. Well, actually, in any gauge theory, all allowed physical states must be gauge-invariant (singlets) when the gauge generators are correctly defined with everything that belongs to them, but for the electroweak theory, this fact has a less direct implications for the spectrum.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl
"But the gauge fields are not conserved, the charges are. How can they both correspond to the generators, but only one is conserved?" - There is no meaningful interpretation of the phrase "gauge fields are conserved". What does it mean for fields to be conserved? Makes absolutely no sense.

This post imported from StackExchange Physics at 2015-04-11 10:31 (UTC), posted by SE-user Luboš Motl

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