Group Theory Appearing in Griffith's Elementary Particles (2nd Ed.)

[Jdeloz828]

TL;DR Summary

I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles, and have some specific questions related to this which refer to the text.

Hello,

I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles. I have had a fair amount of exposure to elementary group theory, but no representation theory, and have some specific questions related to this which refer to the text.

• Towards the end of section 4.1 Griffith's defines a Group Representationas a set a matrices which are homomorphic to the underlying group. He then lists a few examples:
  • An ordinary scalar belongs to the one-dimensional representation of the rotation group, SO(3)
  • A three-vector belongs to the three-dimensional representation of SO(3).
  • Four-vectors belong to the four-dimensional representation of the Lorentz group.
  • The curious geometrical arrangements of Gell-Mann's Eightfold Way correspond to representations of SU(3).

First of all, none of these objects are matrices so I don't quite understand how Griffith's is identifying them with group representations as he had defined it. Can we associate a matrix with all of these objects in some way, or is the author just speaking loosely and means to say that these objects transform under the stated group representations?

• Griffith's will frequently make statements like the following, referring to the space of gluon color states:

3 ⊗ 3' = 1 ⊕ 8
​

It seems this is the author's group theoretic way of stating that the color/anti-color states are comprised of a color singlet and color octet. Would a more precise way of stating this be that if you take the direct product of a color state and an anti-color state, you form a nine-dimensional vector space with a one-dimensional subspace and an eight-dimensional subspace?

• The author also makes the following statement referring to the product of two quark color states:​

3 ⊗ 3 = 3' ⊕ 6
​

I don't quite what would lead to the difference in this case. Why is a product space of two quark color states fundamentally different in structure than the gluon color/anti-color state space?

• The author also seems to suggest that color singlet states are invariant under the underlying symmetry group.

Why would this be he case? Wouldn't there always be a transformation matrix that can take, say, a color singlet to a color octet in the case of gluon states?

If anyone can shed some light on these matters it would be much appreciates. Also, if anyone knows of any useful resources that might provide a more detailed discussion of this subject, please let me know. Thanks.

​

 

[Vanadium 50]

First of all, none of these objects are matrices

A scalar can be represented as a 1x1 matrix. A vector as a 3x1. The "eightfold way" can be expressed as a set of 8 matrices

3 ⊗ 3 is not 3 ⊗ 3'. The prime in the second description shouldn't be there, and the first 3 should have a bar over it.

 

[PeroK]

Summary:: I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles, and have some specific questions related to this which refer to the text.

Hello,

I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles. I have had a fair amount of exposure to elementary group theory, but no representation theory, and have some specific questions related to this which refer to the text.

• Towards the end of section 4.1 Griffith's defines a Group Representationas a set a matrices which are homomorphic to the underlying group. He then lists a few examples:
  • An ordinary scalar belongs to the one-dimensional representation of the rotation group, SO(3)
  • A three-vector belongs to the three-dimensional representation of SO(3).
  • Four-vectors belong to the four-dimensional representation of the Lorentz group.
  • The curious geometrical arrangements of Gell-Mann's Eightfold Way correspond to representations of SU(3).

First of all, none of these objects are matrices so I don't quite understand how Griffith's is identifying them with group representations as he had defined it. Can we associate a matrix with all of these objects in some way, or is the author just speaking loosely and means to say that these objects transform under the stated group representations?

• Griffith's will frequently make statements like the following, referring to the space of gluon color states:

3 ⊗ 3' = 1 ⊕ 8
​

It seems this is the author's group theoretic way of stating that the color/anti-color states are comprised of a color singlet and color octet. Would a more precise way of stating this be that if you take the direct product of a color state and an anti-color state, you form a nine-dimensional vector space with a one-dimensional subspace and an eight-dimensional subspace?

• The author also makes the following statement referring to the product of two quark color states:​

3 ⊗ 3 = 3' ⊕ 6
​

I don't quite what would lead to the difference in this case. Why is a product space of two quark color states fundamentally different in structure than the gluon color/anti-color state space?

• The author also seems to suggest that color singlet states are invariant under the underlying symmetry group.

Why would this be he case? Wouldn't there always be a transformation matrix that can take, say, a color singlet to a color octet in the case of gluon states?

If anyone can shed some light on these matters it would be much appreciates. Also, if anyone knows of any useful resources that might provide a more detailed discussion of this subject, please let me know. Thanks.

​

I'm working through this book as well. I must admit that I could make little sense of what Griffiths was saying vis-a-vis group theory. I came to the conclusion that either I should take a time out and learn about group representation theory (which I decided against); or, try to understand the symmetries involved without worrying too much how it relates to group representation theory directly. I know a bit about group theory, but not much.

I don't believe that Griffiths says enough about GRT for the reader to work things out from that. Instead, I just focused on the specific symmetries in each case and worked out what was going on.

To take an example. For baryons composed of the lightweight quarks $u, d$ and $s$, we have a $3 \times 3 \times 3 = 27$ dimensional space. That breaks down according to symmetry into:

A $10D$ space of completely symmetric states, a $1D$ space of completely asymmetric states and then any two from three of: an $8D$ space of states symmetric in particles $1$ and $2$; an $8D$ space of states symmetric in particles $2$ and $3$; an $8D$ space of states symmetric in particles $1$ and $3$. In the notation of representation theory this is:
$$3 \otimes 3 \otimes 3 = 10 \oplus 1 \oplus 8 \oplus 8$$

Actually, I've just written an Insight (which is in the pipeline) summarising the eightfold way and decuplet. Let me know if you would like to see a draft.

 

[Orodruin]

A group representation $\rho$ is a homomorphism from the group $G$ to linear operators on some vector space $V$. If $V$ is finite-dimensional of dimension $n$, then the representation $\rho$ may be seen as a map to $n\times n$ matrices. For a vector $v \in V$, we say that the group acts on $v$ according to $v \to \rho(a) v$ and that $v$ transforms according to the representation $\rho$ of the group $G$.

The examples given by Griffiths regards the vector space $V$, i.e., the vector space that the linear operators you use to represent $G$ act on, rather than the representation $\rho$ itself. For example, the ordinary three-vectors transform under the fundamental representation of SO(3), i.e., where the matrices of SO(3) are represented by themselves as 3x3 matrices - $\rho(a) = a$.

 

[vanhees71]

To define gluons the prime should be there (I guess the prime stands for the conjugate fundamental representation, usually written rather as $\bar{3}$. Group-theoretically the gluons are the "non-trivial" part of the 9 combinations of a color and an anticolor fundamental representation. Indeed
$$3 \otimes 3'=1 \oplus 8,$$
und thus you have 8 gluons. You just omit the trivial (colorless) piece.

To answer the OP: You should note that in SU(3) there are two distinct 3D representations. I.e., if you define $\psi=(\psi_r,\psi_g,\psi_b)^{\text{T}}$ to transform with the unitary matrix $U \in \mathrm{SU}(3)$, then the conjugate-complex $\psi$ transforms with the conjugate complex matrix $U^*$. This defines also a 3D representation of SU(3), but it's not isomorphic to the original one. Thus you have two distinct 3D representations of SU(3).

That's different in SU(2), where you have only one 2D representation (corresponding to spin 1/2 spinors). Indeed here any SU(2) matrix can be written as $\hat{U}=\exp(\mathrm{i} \vec{\phi} \cdot \hat{\vec{\sigma}}/2)$ with $\vec{\phi} \in \mathbb{R}^3$. Now
$$\hat{U}^* = \exp(-\mathrm{i} \vec{\phi} \cdot \hat{\vec{\sigma}}^*/2).$$
For the Pauli matrices,
$$-\hat{\sigma}_j^*=\hat{\sigma}_2^{-1} \hat{\sigma}_j \hat{\sigma}_2=\hat{\sigma_2} \hat{\sigma}_j \sigma_2.$$
Indeed (now skipping the hats for convenience)
$$\sigma_2 \sigma_1 \sigma_2=-\sigma_1 \sigma_2^2 =-\sigma_1 = -\sigma_1^*,$$
$$\sigma_2 \sigma_2 \sigma_2=\sigma_2=-\sigma_2^*,$$
$$\sigma_2 \sigma_3 \sigma_2=-\sigma_2^2 \sigma_3=-\sigma_3=-\sigma_3^*.$$
Thus
$$U^*=\sigma_2 U \sigma_2=\sigma_2^{-1} U \sigma_2,$$
i.e., the representation $\bar{2}$ of SU(2) is equivalent to the representation $2$ and thus no new representation.

This doesn't work for SU(n) with $n \geq 3$, i.e., for the SU(3) the $\bar{3}$-representation is inequivalent to the $3$-representation.

 

[PeroK]

A group representation $\rho$ is a homomorphism from the group $G$ to linear operators on some vector space $V$. If $V$ is finite-dimensional of dimension $n$, then the representation $\rho$ may be seen as a map to $n\times n$ matrices. For a vector $v \in V$, we say that the group acts on $v$ according to $v \to \rho(a) v$ and that $v$ transforms according to the representation $\rho$ of the group $G$.

The examples given by Griffiths regards the vector space $V$, i.e., the vector space that the linear operators you use to represent $G$ act on, rather than the representation $\rho$ itself. For example, the ordinary three-vectors transform under the fundamental representation of SO(3), i.e., where the matrices of SO(3) are represented by themselves as 3x3 matrices - $\rho(a) = a$.

What would you say is the best reference to learn this?

The problem with Griffiths is that there is not enough on this (and how it applies to the material) to make any practical use of it.

 

[Orodruin]

What would you say is the best reference to learn this?

As usual, it depends on how deep you want to go and how formal you want to be about it. There are many good (and bad resources out there).

 

[Orodruin]

To define gluons the prime should be there (I guess the prime stands for the conjugate fundamental representation, usually written rather as $\bar{3}$. Group-theoretically the gluons are the "non-trivial" part of the 9 combinations of a color and an anticolor fundamental representation. Indeed
$$3 \otimes 3'=1 \oplus 8,$$
und thus you have 8 gluons. You just omit the trivial (colorless) piece.

I do not fully agree here. Yes, $3\otimes \bar 3 = 1 \oplus 8$, however the 1 was never there in the first place because gluons by definition belong to the adjoint representation of SU(3), which is just 8 as the Lie algebra is 8-dimensional, not to the $3\otimes \bar 3$ representation. The $3\otimes \bar 3 = 1 \oplus 8$ is important when discussing how you can couple the quark fields to the gluon fields - the 8 of the quark-antiquark representation can be coupled to the gluon field to make an overall singlet (as $8\oplus 8$ contains the trivial representation) whereas the 1 cannot.

 

[PeroK]

As usual, it depends on how deep you want to go and how formal you want to be about it. There are many good (and bad resources out there).

That's the problem I faced. I found material that a) obviously required significantly more group theory than I know and b) wasn't in a form that I could easily relate to particle physics!

At this stage, I guess the answer is "enough to get through the book".

Personally, I feel I got a complete understanding of chapters 4 & 5 without using group theory directly. I just looked at the symmetries that were inherent in the material and took it from there.

 

[Orodruin]

That's the problem I faced. I found material that a) obviously required significantly more group theory than I know and b) wasn't in a form that I could easily relate to particle physics!

I would actually suggest starting from a non-particle physics perspective. Start with a perspective that discusses symmetries more related to examples you already know and that are more hands-on. Now, I know of a book that contains a chapter like that on group theory, but I would be biased in pointing it out ...

 

[PeroK]

Summary:: I'm trying to make sense of some of the group theoretic discussion found in Griffith's

• Griffith's will frequently make statements like the following, referring to the space of gluon color states:

3 ⊗ 3' = 1 ⊕ 8
​

It seems this is the author's group theoretic way of stating that the color/anti-color states are comprised of a color singlet and color octet. Would a more precise way of stating this be that if you take the direct product of a color state and an anti-color state, you form a nine-dimensional vector space with a one-dimensional subspace and an eight-dimensional subspace?

• The author also makes the following statement referring to the product of two quark color states:​

3 ⊗ 3 = 3' ⊕ 6
​

I don't quite what would lead to the difference in this case. Why is a product space of two quark color states fundamentally different in structure than the gluon color/anti-color state space?

• The author also seems to suggest that color singlet states are invariant under the underlying symmetry group.

Why would this be he case? Wouldn't there always be a transformation matrix that can take, say, a color singlet to a color octet in the case of gluon states?

Do you have some page references for these?

 

[Jdeloz828]

PeroK, look at sections 8.3-4. The top two bullet points are contained in an asterisk note on the bottom of pg. 293. As, for the third bullet, this seems to be a crucial part of the argument made in problem 8.22, although I don't quite understand what Griffith's is getting at here.

 

[Jdeloz828]

To take an example. For baryons composed of the lightweight quarks $u, d$ and $s$, we have a $3 \times 3 \times 3 = 27$ dimensional space. That breaks down according to symmetry into:

A $10D$ space of completely symmetric states, a $1D$ space of completely asymmetric states and then any two from three of: an $8D$ space of states symmetric in particles $1$ and $2$; an $8D$ space of states symmetric in particles $2$ and $3$; an $8D$ space of states symmetric in particles $1$ and $3$. In the notation of representation theory this is:
$$3 \otimes 3 \otimes 3 = 10 \oplus 1 \oplus 8 \oplus 8$$

Actually, I've just written an Insight (which is in the pipeline) summarising the eightfold way and decuplet. Let me know if you would like to see a draft.

PeroK, what exactly do you mean by:

That breaks down according to symmetry into:

How can we show this?

In this case I believe a two-component isospinor is used to represent each quark state (u, d, s), with underlying symmetry group SU(2). Does that mean in this case that the matrices which transform states in the product space of the quark states is a 27 dimensional representation of SU(2)?

Maybe we can break this down according to the precise definition of a representation provided by Orodruin.

 

[PeroK]

PeroK, look at sections 8.3-4. The top two bullet points are contained in an asterisk note on the bottom of pg. 293. As, for the third bullet, this seems to be a crucial part of the argument made in problem 8.22, although I don't quite understand what Griffith's is getting at here.

First, we are dealing with basis states of the form $c_1c_2$, where $c_1, c_2$ are any colours $r, g, b$. So, we have $3 \times 3 = 9$ basis states. But, we want symmetric and antisymmetric states. So, we need to establish which linear combinations (superpositions) of these basis states have the property of symmetry or antisymmetry. We find, therefore, that we have three antisymmetric states (the triplet on page 292) and six symmetric states (the sextet on page 293).

You can check that these are all linearly independent, so form an alternative basis of colour states.

The case where we are dealing with states of the form $c_1\bar c_2$ is different. The relevant basis states there are shown on page 285.

 

[PeroK]

How can we show this?

In this case I believe a two-component isospinor is used to represent each quark state (u, d, s), with underlying symmetry group SU(2). Does that mean in this case that the matrices which transform states in the product space of the quark states is a 27 dimensional representation of SU(2)?

Maybe we can break this down according to the precise definition of a representation provided by Orodruin.

These symmetry groups apply primarily to the basic permutations of $u, d, s$ combined with quantum spin. Isopsin is an additional factor, but I don't think it directly affects the group representations.

Yes, but if you want to use GRT to prove these things, you'll need to go learn it. It's not in Griffiths book!

 

[Jdeloz828]

First, we are dealing with basis states of the form $c_1c_2$, where $c_1, c_2$ are any colours $r, g, b$. So, we have $3 \times 3 = 9$ basis states. But, we want symmetric and antisymmetric states. So, we need to establish which linear combinations (superpositions) of these basis states have the property of symmetry or antisymmetry. We find, therefore, that we have three antisymmetric states (the triplet on page 292) and six symmetric states (the sextet on page 293).

You can check that these are all linearly independent, so form an alternative basis of colour states.

The case where we are dealing with states of the form $c_1\bar c_2$ is different. The relevant basis states there are shown on page 285.

Are you saying that we're deeming the states that are not symmetric or antisymmetric as non-physical and throwing them away? Is that why we look for a symmetric and antisymmetric basis in the first place?

Also, I'm not sure what types of symmetry are present in the basis states for $c_1\bar c_2$. For example, if you look at the octet (equation 8.29 on pg. 285), some of these states are symmetric and others are antisymmetric under swapping of the color and anticolor.

 

[PeroK]

1) Are you saying that we're deeming the states that are not symmetric or antisymmetric as non-physical and throwing them away? Is that why we look for a symmetric and antisymmetric basis in the first place?

2) Also, I'm not sure what types of symmetry are present in the basis states for $c_1\bar c_2$. For example, if you look at the octet (equation 8.29 on pg. 285), some of these states are symmetric and others are antisymmetric under swapping of the color and anticolor.

1) Not throwing them away as such. The only one that gets "thrown away" in the baryons is the totally antisymmetric state because there is no corresponding antisymmetric spin state for three particles (you need to read my Insight!). But, 16 states get reduced to 8 in the baryon octet. So, rather than 27 baryons, we have 10 + 8.

2) Neither do I. I'm only on chapter 6!

 

[nrqed]

That's the problem I faced. I found material that a) obviously required significantly more group theory than I know and b) wasn't in a form that I could easily relate to particle physics!

At this stage, I guess the answer is "enough to get through the book".

Personally, I feel I got a complete understanding of chapters 4 & 5 without using group theory directly. I just looked at the symmetries that were inherent in the material and took it from there.

I would recommend Howard Georgi's book and Cahn's book http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf.

 

[PeroK]

@Jdeloz828 I found this today:

https://www.hep.phy.cam.ac.uk/~thomson/lectures/partIIIparticles/Handout8_2009.pdf

It explains a lot.

 

[George Jones]

@Jdeloz828 I found this today:

https://www.hep.phy.cam.ac.uk/~thomson/lectures/partIIIparticles/Handout8_2009.pdf

It explains a lot.

These interesting 2009 handouts (1 through 15) evolved into the interesting book

https://www.hep.phy.cam.ac.uk/~thomson/MPP/ModernParticlePhysics.html

 

[Jdeloz828]

@nrqed, @PeroK, @George Jones Thanks a lot! These seem like great resources.