Does it really make sense to talk about the color of gluons?
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It is my understanding that by enforcing SU(3) gauge invariance on our lagrangian of 3-colored quark fields, we are forced to accept the existence of 8 new massless vector fields, the gluons. The 8 here comes directly from the dimension of SU(3).
That being said I often see discussions about the gluons in terms of linear combinations of $rbar r$, $bbar b$, etc.
This simply cant be the nature of the gluons though can it? Because it seems to imply that the number of colors and the number of gluon fuelds are not independant, while they clearly are.
Certainly gluons are not singlets in color space and so they must have color, but it doesnt make sense to me that this color of the gluons would be some mapping directly from quark color.
Thanks to anyone with the insight and time to share it!
quantum-chromodynamics gauge-invariance strong-force
asked Nov 10 at 18:03
Craig
779
add a comment |
up vote
4
down vote
favorite
It is my understanding that by enforcing SU(3) gauge invariance on our lagrangian of 3-colored quark fields, we are forced to accept the existence of 8 new massless vector fields, the gluons. The 8 here comes directly from the dimension of SU(3).
That being said I often see discussions about the gluons in terms of linear combinations of $rbar r$, $bbar b$, etc.
This simply cant be the nature of the gluons though can it? Because it seems to imply that the number of colors and the number of gluon fuelds are not independant, while they clearly are.
Certainly gluons are not singlets in color space and so they must have color, but it doesnt make sense to me that this color of the gluons would be some mapping directly from quark color.
Thanks to anyone with the insight and time to share it!
quantum-chromodynamics gauge-invariance strong-force
asked Nov 10 at 18:03
Craig
779
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
It is my understanding that by enforcing SU(3) gauge invariance on our lagrangian of 3-colored quark fields, we are forced to accept the existence of 8 new massless vector fields, the gluons. The 8 here comes directly from the dimension of SU(3).
That being said I often see discussions about the gluons in terms of linear combinations of $rbar r$, $bbar b$, etc.
This simply cant be the nature of the gluons though can it? Because it seems to imply that the number of colors and the number of gluon fuelds are not independant, while they clearly are.
Certainly gluons are not singlets in color space and so they must have color, but it doesnt make sense to me that this color of the gluons would be some mapping directly from quark color.
Thanks to anyone with the insight and time to share it!
quantum-chromodynamics gauge-invariance strong-force
asked Nov 10 at 18:03
Craig
779
It is my understanding that by enforcing SU(3) gauge invariance on our lagrangian of 3-colored quark fields, we are forced to accept the existence of 8 new massless vector fields, the gluons. The 8 here comes directly from the dimension of SU(3).
That being said I often see discussions about the gluons in terms of linear combinations of $rbar r$, $bbar b$, etc.
This simply cant be the nature of the gluons though can it? Because it seems to imply that the number of colors and the number of gluon fuelds are not independant, while they clearly are.
Certainly gluons are not singlets in color space and so they must have color, but it doesnt make sense to me that this color of the gluons would be some mapping directly from quark color.
Thanks to anyone with the insight and time to share it!
quantum-chromodynamics gauge-invariance strong-force
quantum-chromodynamics gauge-invariance strong-force
asked Nov 10 at 18:03
Craig
779
asked Nov 10 at 18:03
Craig
779
edited Nov 10 at 18:55
asked Nov 10 at 18:03
Craig
779
asked Nov 10 at 18:03
Craig
779
asked Nov 10 at 18:03
Craig
779
779
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
6
down vote
The quarks transform according to the fundamental representation $mathbf{3}$ of SU(3), and the antiquarks according to the conjugate representation $mathbf{overline 3}$. The gluons transform according to the adjoint representation $mathbf{8}$.
The adjoint representation is contained in the product of the fundamental representation and its conjugate:
$$mathbf{3}otimes mathbf{overline 3} = mathbf{8}oplus mathbf{1}$$
Therefore gluons are conventionally labeled using color-anticolor combinations, avoiding the color singlet combination $(roverline{r}+boverline{b}+goverline{g})/sqrt{3}$.
answered Nov 10 at 18:49
G. Smith
2,760614
Awesome do you have a any sources/papers/books I could find more about this?
– Craig
Nov 10 at 18:53
As well; in a universe where we had say, 4 or 5 colors and 8 gluons, how would this work?
– Craig
Nov 10 at 18:57
1
@Craig, are you familiar with representation theory? A big concept is that the same group can be "represented" with larger or smaller vector spaces. In this case, the quarks have the minimum number of vectors needed to realize SU(3) symmetry, and the gluons have (very loosely) that maximum number of vectors that can have the SU(3) symmetry, one for each dimension. If we had 4 colors, then we would instead want $4otimes bar 4 = 15oplus 1$, and we would have 15 gluon fields for the 4 colors.
– Alex Meiburg
Nov 10 at 19:01
The adjoint reprsentation of SU(n) has dimension $n^2-1$. So if you have 4 colors there need to be 15 gluons and if you have 5 colors there need to be 24 gluons.
– G. Smith
Nov 10 at 19:05
I’ll let others suggest the best references. But you need to clarify what you main interest is. The mathematics of representation theory? (For example, how do you figure out how an arbitrary product of irreducible representations decomposes into irreducible representations?) The physics of QCD, or the whole Standard Model? The reason why quantum field theory involves group representations? Etc.
– G. Smith
Nov 10 at 20:25
|
show 7 more comments
up vote
3
down vote
The gluons are generators of the SU(3) gauge group; whatever notation is used to describe the fundamental representation can be extended to higher representations through their embedding in tensor products of the fundamental (and its dual.) [Also, by "sums" of $rbar r$, $bbar b$, etc., do you really mean products?]
answered Nov 10 at 18:42
fs137
2,475815
Oops I mean linear combinations of products of $rbar r$, $bbar b$, etc
– Craig
Nov 10 at 18:54
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
The quarks transform according to the fundamental representation $mathbf{3}$ of SU(3), and the antiquarks according to the conjugate representation $mathbf{overline 3}$. The gluons transform according to the adjoint representation $mathbf{8}$.
The adjoint representation is contained in the product of the fundamental representation and its conjugate:
$$mathbf{3}otimes mathbf{overline 3} = mathbf{8}oplus mathbf{1}$$
Therefore gluons are conventionally labeled using color-anticolor combinations, avoiding the color singlet combination $(roverline{r}+boverline{b}+goverline{g})/sqrt{3}$.
answered Nov 10 at 18:49
G. Smith
2,760614
Awesome do you have a any sources/papers/books I could find more about this?
– Craig
Nov 10 at 18:53
As well; in a universe where we had say, 4 or 5 colors and 8 gluons, how would this work?
– Craig
Nov 10 at 18:57
1
@Craig, are you familiar with representation theory? A big concept is that the same group can be "represented" with larger or smaller vector spaces. In this case, the quarks have the minimum number of vectors needed to realize SU(3) symmetry, and the gluons have (very loosely) that maximum number of vectors that can have the SU(3) symmetry, one for each dimension. If we had 4 colors, then we would instead want $4otimes bar 4 = 15oplus 1$, and we would have 15 gluon fields for the 4 colors.
– Alex Meiburg
Nov 10 at 19:01
The adjoint reprsentation of SU(n) has dimension $n^2-1$. So if you have 4 colors there need to be 15 gluons and if you have 5 colors there need to be 24 gluons.
– G. Smith
Nov 10 at 19:05
I’ll let others suggest the best references. But you need to clarify what you main interest is. The mathematics of representation theory? (For example, how do you figure out how an arbitrary product of irreducible representations decomposes into irreducible representations?) The physics of QCD, or the whole Standard Model? The reason why quantum field theory involves group representations? Etc.
– G. Smith
Nov 10 at 20:25
|
show 7 more comments
up vote
6
down vote
The quarks transform according to the fundamental representation $mathbf{3}$ of SU(3), and the antiquarks according to the conjugate representation $mathbf{overline 3}$. The gluons transform according to the adjoint representation $mathbf{8}$.
The adjoint representation is contained in the product of the fundamental representation and its conjugate:
$$mathbf{3}otimes mathbf{overline 3} = mathbf{8}oplus mathbf{1}$$
Therefore gluons are conventionally labeled using color-anticolor combinations, avoiding the color singlet combination $(roverline{r}+boverline{b}+goverline{g})/sqrt{3}$.
answered Nov 10 at 18:49
G. Smith
2,760614
Awesome do you have a any sources/papers/books I could find more about this?
– Craig
Nov 10 at 18:53
As well; in a universe where we had say, 4 or 5 colors and 8 gluons, how would this work?
– Craig
Nov 10 at 18:57
1
@Craig, are you familiar with representation theory? A big concept is that the same group can be "represented" with larger or smaller vector spaces. In this case, the quarks have the minimum number of vectors needed to realize SU(3) symmetry, and the gluons have (very loosely) that maximum number of vectors that can have the SU(3) symmetry, one for each dimension. If we had 4 colors, then we would instead want $4otimes bar 4 = 15oplus 1$, and we would have 15 gluon fields for the 4 colors.
– Alex Meiburg
Nov 10 at 19:01
The adjoint reprsentation of SU(n) has dimension $n^2-1$. So if you have 4 colors there need to be 15 gluons and if you have 5 colors there need to be 24 gluons.
– G. Smith
Nov 10 at 19:05
I’ll let others suggest the best references. But you need to clarify what you main interest is. The mathematics of representation theory? (For example, how do you figure out how an arbitrary product of irreducible representations decomposes into irreducible representations?) The physics of QCD, or the whole Standard Model? The reason why quantum field theory involves group representations? Etc.
– G. Smith
Nov 10 at 20:25
|
show 7 more comments
up vote
6
down vote
up vote
6
down vote
The quarks transform according to the fundamental representation $mathbf{3}$ of SU(3), and the antiquarks according to the conjugate representation $mathbf{overline 3}$. The gluons transform according to the adjoint representation $mathbf{8}$.
The adjoint representation is contained in the product of the fundamental representation and its conjugate:
$$mathbf{3}otimes mathbf{overline 3} = mathbf{8}oplus mathbf{1}$$
Therefore gluons are conventionally labeled using color-anticolor combinations, avoiding the color singlet combination $(roverline{r}+boverline{b}+goverline{g})/sqrt{3}$.
answered Nov 10 at 18:49
G. Smith
2,760614
The quarks transform according to the fundamental representation $mathbf{3}$ of SU(3), and the antiquarks according to the conjugate representation $mathbf{overline 3}$. The gluons transform according to the adjoint representation $mathbf{8}$.
The adjoint representation is contained in the product of the fundamental representation and its conjugate:
$$mathbf{3}otimes mathbf{overline 3} = mathbf{8}oplus mathbf{1}$$
Therefore gluons are conventionally labeled using color-anticolor combinations, avoiding the color singlet combination $(roverline{r}+boverline{b}+goverline{g})/sqrt{3}$.
answered Nov 10 at 18:49
G. Smith
2,760614
edited Nov 11 at 20:17
answered Nov 10 at 18:49
G. Smith
2,760614
answered Nov 10 at 18:49
G. Smith
2,760614
answered Nov 10 at 18:49
G. Smith
2,760614
2,760614
Awesome do you have a any sources/papers/books I could find more about this?
– Craig
Nov 10 at 18:53
As well; in a universe where we had say, 4 or 5 colors and 8 gluons, how would this work?
– Craig
Nov 10 at 18:57
1
@Craig, are you familiar with representation theory? A big concept is that the same group can be "represented" with larger or smaller vector spaces. In this case, the quarks have the minimum number of vectors needed to realize SU(3) symmetry, and the gluons have (very loosely) that maximum number of vectors that can have the SU(3) symmetry, one for each dimension. If we had 4 colors, then we would instead want $4otimes bar 4 = 15oplus 1$, and we would have 15 gluon fields for the 4 colors.
– Alex Meiburg
Nov 10 at 19:01
The adjoint reprsentation of SU(n) has dimension $n^2-1$. So if you have 4 colors there need to be 15 gluons and if you have 5 colors there need to be 24 gluons.
– G. Smith
Nov 10 at 19:05
I’ll let others suggest the best references. But you need to clarify what you main interest is. The mathematics of representation theory? (For example, how do you figure out how an arbitrary product of irreducible representations decomposes into irreducible representations?) The physics of QCD, or the whole Standard Model? The reason why quantum field theory involves group representations? Etc.
– G. Smith
Nov 10 at 20:25
|
show 7 more comments
Awesome do you have a any sources/papers/books I could find more about this?
– Craig
Nov 10 at 18:53
As well; in a universe where we had say, 4 or 5 colors and 8 gluons, how would this work?
– Craig
Nov 10 at 18:57
1
@Craig, are you familiar with representation theory? A big concept is that the same group can be "represented" with larger or smaller vector spaces. In this case, the quarks have the minimum number of vectors needed to realize SU(3) symmetry, and the gluons have (very loosely) that maximum number of vectors that can have the SU(3) symmetry, one for each dimension. If we had 4 colors, then we would instead want $4otimes bar 4 = 15oplus 1$, and we would have 15 gluon fields for the 4 colors.
– Alex Meiburg
Nov 10 at 19:01
The adjoint reprsentation of SU(n) has dimension $n^2-1$. So if you have 4 colors there need to be 15 gluons and if you have 5 colors there need to be 24 gluons.
– G. Smith
Nov 10 at 19:05
I’ll let others suggest the best references. But you need to clarify what you main interest is. The mathematics of representation theory? (For example, how do you figure out how an arbitrary product of irreducible representations decomposes into irreducible representations?) The physics of QCD, or the whole Standard Model? The reason why quantum field theory involves group representations? Etc.
– G. Smith
Nov 10 at 20:25
Awesome do you have a any sources/papers/books I could find more about this?
– Craig
Nov 10 at 18:53
Awesome do you have a any sources/papers/books I could find more about this?
– Craig
Nov 10 at 18:53
As well; in a universe where we had say, 4 or 5 colors and 8 gluons, how would this work?
– Craig
Nov 10 at 18:57
As well; in a universe where we had say, 4 or 5 colors and 8 gluons, how would this work?
– Craig
Nov 10 at 18:57
1
1
@Craig, are you familiar with representation theory? A big concept is that the same group can be "represented" with larger or smaller vector spaces. In this case, the quarks have the minimum number of vectors needed to realize SU(3) symmetry, and the gluons have (very loosely) that maximum number of vectors that can have the SU(3) symmetry, one for each dimension. If we had 4 colors, then we would instead want $4otimes bar 4 = 15oplus 1$, and we would have 15 gluon fields for the 4 colors.
– Alex Meiburg
Nov 10 at 19:01
@Craig, are you familiar with representation theory? A big concept is that the same group can be "represented" with larger or smaller vector spaces. In this case, the quarks have the minimum number of vectors needed to realize SU(3) symmetry, and the gluons have (very loosely) that maximum number of vectors that can have the SU(3) symmetry, one for each dimension. If we had 4 colors, then we would instead want $4otimes bar 4 = 15oplus 1$, and we would have 15 gluon fields for the 4 colors.
– Alex Meiburg
Nov 10 at 19:01
The adjoint reprsentation of SU(n) has dimension $n^2-1$. So if you have 4 colors there need to be 15 gluons and if you have 5 colors there need to be 24 gluons.
– G. Smith
Nov 10 at 19:05
The adjoint reprsentation of SU(n) has dimension $n^2-1$. So if you have 4 colors there need to be 15 gluons and if you have 5 colors there need to be 24 gluons.
– G. Smith
Nov 10 at 19:05
I’ll let others suggest the best references. But you need to clarify what you main interest is. The mathematics of representation theory? (For example, how do you figure out how an arbitrary product of irreducible representations decomposes into irreducible representations?) The physics of QCD, or the whole Standard Model? The reason why quantum field theory involves group representations? Etc.
– G. Smith
Nov 10 at 20:25
I’ll let others suggest the best references. But you need to clarify what you main interest is. The mathematics of representation theory? (For example, how do you figure out how an arbitrary product of irreducible representations decomposes into irreducible representations?) The physics of QCD, or the whole Standard Model? The reason why quantum field theory involves group representations? Etc.
– G. Smith
Nov 10 at 20:25
|
show 7 more comments
up vote
3
down vote
The gluons are generators of the SU(3) gauge group; whatever notation is used to describe the fundamental representation can be extended to higher representations through their embedding in tensor products of the fundamental (and its dual.) [Also, by "sums" of $rbar r$, $bbar b$, etc., do you really mean products?]
answered Nov 10 at 18:42
fs137
2,475815
Oops I mean linear combinations of products of $rbar r$, $bbar b$, etc
– Craig
Nov 10 at 18:54
add a comment |
up vote
3
down vote
The gluons are generators of the SU(3) gauge group; whatever notation is used to describe the fundamental representation can be extended to higher representations through their embedding in tensor products of the fundamental (and its dual.) [Also, by "sums" of $rbar r$, $bbar b$, etc., do you really mean products?]
answered Nov 10 at 18:42
fs137
2,475815
Oops I mean linear combinations of products of $rbar r$, $bbar b$, etc
– Craig
Nov 10 at 18:54
add a comment |
up vote
3
down vote
up vote
3
down vote
The gluons are generators of the SU(3) gauge group; whatever notation is used to describe the fundamental representation can be extended to higher representations through their embedding in tensor products of the fundamental (and its dual.) [Also, by "sums" of $rbar r$, $bbar b$, etc., do you really mean products?]
answered Nov 10 at 18:42
fs137
2,475815
The gluons are generators of the SU(3) gauge group; whatever notation is used to describe the fundamental representation can be extended to higher representations through their embedding in tensor products of the fundamental (and its dual.) [Also, by "sums" of $rbar r$, $bbar b$, etc., do you really mean products?]
answered Nov 10 at 18:42
fs137
2,475815
answered Nov 10 at 18:42
fs137
2,475815
answered Nov 10 at 18:42
fs137
2,475815
answered Nov 10 at 18:42
fs137
2,475815
2,475815
Oops I mean linear combinations of products of $rbar r$, $bbar b$, etc
– Craig
Nov 10 at 18:54
add a comment |
Oops I mean linear combinations of products of $rbar r$, $bbar b$, etc
– Craig
Nov 10 at 18:54
Oops I mean linear combinations of products of $rbar r$, $bbar b$, etc
– Craig
Nov 10 at 18:54
Oops I mean linear combinations of products of $rbar r$, $bbar b$, etc
– Craig
Nov 10 at 18:54
add a comment |
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