Can Non-Commutative Geometry Describe Gluons?

[casparov]

Hey,

I have a question regarding the gluons. Is it possible for a non-commutative group/geometry to represent them mathematically ? Replacing the Gell-Mann matrices. I read that the frameworks for gluons /gluonic forces are various, depending on the context.

 

[malawi_glenn]

I read that the frameworks for gluons /gluonic forces are various, depending on the context.

Where did you read that?

Are you sure about that A-level tag? (Grad student / professional level)

What commutative groups can you think of to replace SU(3) with? Why do you think one modelled the quark-gluon interactions with SU(3) in the first place?

I think you need to revisit your question. SU(3) is a non-abelian group to start with
So, what is the point of this thread?

 

[Demystifier]

Is it possible for a non-commutative group/geometry to represent them mathematically ?

How else could it be represented? By poetry?

 

[casparov]

Where did you read that?

Are you sure about that A-level tag? (Grad student / professional level)

What commutative groups can you think of to replace SU(3) with? Why do you think one modelled the quark-gluon interactions with SU(3) in the first place?

I think you need to revisit your question. SU(3) is a non-abelian group to start with
So, what is the point of this thread?

My bad. Let I specify my question a bit more.

Yes I wish to talk to someone who knows a lot about gluons.

On Wikipedia under the Gluons, Eight Colors section https://en.wikipedia.org/wiki/Gluon it is stated on the color octet list: "There are many other possible choices, but all are mathematically equivalent, at least equally complicated, and give the same physical results."

What physical results to they mean exactly? The results that the Gell-Mann matrices give, or does it give the Gell-Mann matrices?
Now, it seems they are not meaning the Gell-Mann matrices but the way of the combination of the octets.

I know the SU(3) choice is logical and not coming from nowhere.

I mean to ask , would it be possible for a Dihedral group, possibly extended to 3x3 be able to account for gluons ? Is the Gell-Mann matrices choice the only one? Is it known?

 

[casparov]

How else could it be represented? By poetry?

You know what I meant to say, so thank you I'm not interested in your take, making fun of some wonky sentence I wrote :D

 

[renormalize]

On Wikipedia under the Gluons, Eight Colors section it is stated on the color octet list: "There are many other possible choices, but all are mathematically equivalent, at least equally complicated, and give the same physical results."

Can you please provide the specific link to this statement so we may read it ourselves?

 

[casparov]

https://en.wikipedia.org/wiki/Gluon : Under the Eight Colors section

My main question should also be changed : is it possible for another group/geometry to represent gluons

 

[renormalize]

My main question should also be changed : is it possible for another group/geometry to represent gluons

The statement you referenced is simply saying that the eight $3\times 3$-matrix generators of $SU(3)_{\text{color}}$ are conveniently realized as the set of Gell-Mann matrices (GMm), but that there are also many other possible matrix-representations that are equivalent, i.e., that satisfy the same group commutation relations as the GMm. The physics of QCD is invariant under the choice of any particular realization of $SU(3)_{\text{color}}$, so the GMm are chosen for convenience. Note that this is directly analogous to the three $2\times 2$ Pauli-matrix generators of the 3D rotation group $SU(2)$. Other realizations are possible, but the Pauli matrices are taken as the standard choice.

So the answer to your question is no, there is no other group that can replace $SU(3)_{\text{color}}$ that is consistent with the plethora of experimental evidence for QCD gluons and quarks, just like no group can replace $SU(2)$ to describe rotations in 3D space.

 

[casparov]

The statement you referenced is simply saying that the eight $3\times 3$-matrix generators of $SU(3)_{\text{color}}$ are conveniently realized as the set of Gell-Mann matrices (GMm), but that there are also many other possible matrix-representations that are equivalent, i.e., that satisfy the same group commutation relations as the GMm. The physics of QCD is invariant under the choice of any particular realization of $SU(3)_{\text{color}}$, so the GMm are chosen for convenience. Note that this is directly analogous to the three $2\times 2$ Pauli-matrix generators of the 3D rotation group $SU(2)$. Other realizations are possible, but the Pauli matrices are taken as the standard choice.

So the answer to your question is no, there is no other group that can replace $SU(3)_{\text{color}}$ that is consistent with the plethora of experimental evidence for QCD gluons and quarks, just like no group can replace $SU(2)$ to describe rotations in 3D space.

Thank you very much sir.

Are you sure your last statement is correct?

I believe we can employ a double-cover method on matrix elements of a 2D [dihedral] group to recover 3D rotational dynamics.

 

[Demystifier]

You know what I meant to say

Actually, I do not. Are you asking about other groups, or about other unitarily inequivalent representations of the same group, or about other unitarily equivalent representations of the same group?

 

[renormalize]

I believe we can employ a double-cover method on matrix elements of a 2D [dihedral] group to recover 3D rotational dynamics.

Can you point me to a specific reference for this? I'm no expert, but I thought a dihedral group consists of only a finite number of group elements, unlike the continuous rotation group with an infinite number of elements.

 

[casparov]

Actually, I do not. Are you asking about other groups, or about other unitarily inequivalent representations of the same group, or about other unitarily equivalent representations of the same group?

my apologies for the vagueness, it is for sure on my part. My question is related to a dihedral group so I would guess the first.

 

[casparov]

Can you point me to a specific reference for this? I'm no expert, but I thought a dihedral group consists of only a finite number of group elements, unlike the continuous rotation group with an infinite number of elements.

No reference at this moment.
You are correct. Would that be an issue for the gluon dynamics ?

 

[Structure seeker]

You're kinda asking for rude answers. Yes I saw on wikipedia you can consider the dihedral group also as subgroup of $SO(3)$ but that is very different from $SU(3)$ (dimensions 3 versus 8). Also a double cover of the dihedral group has $4n$ elenents whereas both $SO(3)$ and $SU(3)$ are continuous and therefore infinite groups.

 

[renormalize]

Would that be an issue ?

Well, I think it would mean that rotations in 3D would be quantized rather than continuous. Is there any evidence for this?

 

[casparov]

Well, I think it would mean that rotations in 3D would quantized rather than continuous. Is there any evidence for this?

Well it is for gluons, right ?

 

[renormalize]

Well it is for gluons, right ?

Not that I know of. Can you cite a reference showing that color gauge transformations are discrete, rather than continuous?

 

[Demystifier]

my apologies for the vagueness, it is for sure on my part. My question is related to a dihedral group so I would guess the first.

The dihedral group is not a Lie group, while SU(3) is a Lie group. Yang-Mills theories, such as QCD, require a Lie group. Hence dihedral group cannot be a substitute for the SU(3) in a Yang-Mills theory of quarks and gluons. Is it an answer to your question?

 

[casparov]

The dihedral group is not a Lie group, while SU(3) is a Lie group. Yang-Mills theories, such as QCD, require a Lie group. Hence dihedral group cannot be a substitute for the SU(3) in a Yang-Mills theory of quarks and gluons. Is it an answer to your question?

Thank you sir, yes it is much more clear now, and thank everyone for your answers. Basically for a dihedral group to be applicable, it would require a different approach altogether than QCD...

CAn you explain why it is a hard requirement for it to be a Lie group ? Or is it built in/basically the theory itself?

 

[casparov]

Not that I know of. Can you cite a reference showing that color gauge transformations are discrete, rather than continuous?

Actually it is quite interesting IMO.
Here are a few papers that relate quark/gluon flavor symmetries to discrete groups
https://arxiv.org/abs/1702.08073
https://arxiv.org/abs/1502.00789
https://arxiv.org/abs/hep-ph/0211323
https://arxiv.org/abs/1004.2211
https://arxiv.org/abs/1812.06235
https://arxiv.org/abs/1804.06644

 

[PeroK]

Actually it is quite interesting IMO.
Here are a few papers that relate quark/gluon flavor symmetries to discrete groups
https://arxiv.org/abs/1702.08073
https://arxiv.org/abs/1502.00789
https://arxiv.org/abs/hep-ph/0211323
https://arxiv.org/abs/1004.2211
https://arxiv.org/abs/1812.06235
https://arxiv.org/abs/1804.06644

That's an ambush if ever I saw one!

 

[malawi_glenn]

Actually it is quite interesting IMO.
Here are a few papers that relate quark/gluon flavor symmetries to discrete groups
https://arxiv.org/abs/1702.08073
https://arxiv.org/abs/1502.00789
https://arxiv.org/abs/hep-ph/0211323
https://arxiv.org/abs/1004.2211
https://arxiv.org/abs/1812.06235
https://arxiv.org/abs/1804.06644

Now we are comparing apples and pears (flavor and color) and are moving away from your original question

 

[casparov]

Now we are comparing apples and pears (flavor and color) and are moving away from your original question

Thank you for noting the distinction. Learning a lot here. The interactions with flavor symmetry is important though

 

[malawi_glenn]

Thank you for noting the distinction. Learning a lot here. The interactions with flavor symmetry is important though

Sure, but you need to make a new thread. Your original question has been answered.

Just a side remark, judging from your recent threads, it seems to me that you are an undergraduate student in physics. Do you have any formal training in qft and particle physics? We could give you some nice readings if you want (so you dont need read Wikipedia)

 

[casparov]

Sure, but you need to make a new thread. Your original question has been answered.

Just a side remark, judging from your recent threads, it seems to me that you are an undergraduate student in physics. Do you have any formal training in qft and particle physics? We could give you some nice readings if you want (so you dont need read Wikipedia)

Yes please go ahead

 

[Demystifier]

Here are a few papers that relate quark/gluon flavor symmetries to discrete groups

Flavor is the key word. The SU(3) group of QCD is a gauge group, not a flavor group.
https://en.wikipedia.org/wiki/Quantum_chromodynamics#Symmetry_groups

 

[PeroK]

Yes please go ahead

Introduction to Elementary Particles by Griifiths is intended for advanced undergraduate students.

https://en.m.wikipedia.org/wiki/Introduction_to_Elementary_Particles_(book)

 

[malawi_glenn]

https://arxiv.org/abs/0810.3328

 

[vanhees71]

The statement you referenced is simply saying that the eight $3\times 3$-matrix generators of $SU(3)_{\text{color}}$ are conveniently realized as the set of Gell-Mann matrices (GMm), but that there are also many other possible matrix-representations that are equivalent, i.e., that satisfy the same group commutation relations as the GMm. The physics of QCD is invariant under the choice of any particular realization of $SU(3)_{\text{color}}$, so the GMm are chosen for convenience. Note that this is directly analogous to the three $2\times 2$ Pauli-matrix generators of the 3D rotation group $SU(2)$. Other realizations are possible, but the Pauli matrices are taken as the standard choice.

So the answer to your question is no, there is no other group that can replace $SU(3)_{\text{color}}$ that is consistent with the plethora of experimental evidence for QCD gluons and quarks, just like no group can replace $SU(2)$ to describe rotations in 3D space.

In a gauge theory, the only thing that's fixed by the choice of the group is that the gauge fields transform in a specific way such that covariant derivatives, $\mathcal{D}_{\mu}=\partial_{\mu} + \mathrm{i} g A_a^{\mu} \hat{T}^a$, applied to a field that transforms under the gauge group under a given representation of this group, the $\hat{T}^a$ are a basis of the corresponding representation of the group's Lie algebra. The gauge connection, $$F_{\mu \nu}=\frac{1}{\mathrm{i} g} [\mathcal{D}_{\nu},\mathcal{D}_{\nu}]$$ then transforms necessarily with the adjoint representation of the gauge group.

You are still free to choose any representations for the "matter fields", and which representations are useful (and also which gauge group is useful) for a specific set of physical phenomena, must be deduced from observations. The Standard Model is the result of a fascinating interplay between theory building and experiments, which themselves were often also inspired by predictions of the theory.

As it turned out, in the case of the strong interaction, the gauge group SU(3) ("color gauge group") was the right choice. This was inspired by the fact that in the old flavor model by Gell-Mann and Zweig for the Hadrons, in order to allow for a particle like the $\Omega$ baryon, which was predicted by this model as being a bound state of three strange quarks and part of a specific representation of another SU(3) group ("flavor SU(3)"), which was introduced as a global symmetry to bring order into the growing zoo of hadrons, which was very successful (and an extension of the corresponding SU(2) version, invented as the "isospin formalism" by Heisenberg with the proton and neutron transforming as a doublet under this SU(2) symmetry group). The problem with the $\Omega$ was that it couldn't be built from three quarks, which necessarily had to be fermions to give the right phenomenology for the baryons (fermions) and mesons (bosons) described as bound states of three quarks or of a quark and an antiquark, respectively. The solution was to assume that each type ("flavor") of quark (then only up, down, and strange) comes in three colors as an additional intrinsic degree of freedom. Then one could build all the hadrons in the proper way with fermionic quarks and antiquarks, if each quarks come in precisely three colors, and that means that the color symmetry SU(3) must be realized for quarks as the fundamental representation (called simply 3), i.e., with a three-dimensional SU(3)-color spinor field. The anti-quarks then necessarily must transform with the conjugate complex representation, $\bar{3}$, which is of course also three-dimensional but not isomorphic to the 3-representation.

Another rule was that observable particles are color neutral, and with this rule all the hadrons could be built as color singlets and various representations of the flavor SU(3).

Finally it turned out that the strong interaction can be described by describing the strong interaction between quarks by making the color-charge symmetry local, i.e., use it as the gauge group of quantum chromodynamics. It also turned out that such non-Abelian gauge theories are asymptotic free, i.e., that the running coupling constant (in the sense of the renormalization group paradigm applied to this QFT) becomes small in the UV, i.e., at large collision energies. This gave a hint that these theories are also "confining", i.e., that no colored fields can be asymptotically free and thus that within such a model there are no observable color-charged particles, and indeed nobody has ever seen a free quark or gluon but only colorless hadrons. Also with help of lattice-QCD calculations, it has been demonstrated that QCD indeed leads to the correct quantitative description of the hadrons as color-neutral bound states of quarks and gluons, including the correct prediction of the hadronic mass spectrum.

 

[malawi_glenn]

@casparov here are two excellent textbooks on QFT with particle physics in mind:

https://www.amazon.com/dp/0521864496/?tag=pfamazon01-20 you can find a draft version here (I have two copies of this textbook, that is how good it is) https://web.physics.ucsb.edu/~mark/qft.html

https://www.amazon.com/dp/1107034736/?tag=pfamazon01-20 you really need a new edition of this text, the older ones have myriads of typos - which is a reason for why I have two copies of this book too :)

Note, you might want to get some more books on group theory in particle physics, like the book by Georgi which is free on Amazon Kindle https://www.amazon.com/dp/B07CYZ2FP7/?tag=pfamazon01-20