What is a singlet under SU(3)?

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In summary, a singlet under SU(3) is a state with zero eigenvalue under the Casimir operators, and SU(3)xSU(3) refers to a direct product of two SU(3) groups. The expression 3 x 3(bar) = 8 + 1 represents a decomposition of the direct product of two representations, where the resulting representation is a direct sum of the singlet and octet representations. To compute products of irreps for unitary groups, one can use Young tableaux or tables of products. In the specific example of 3 x 3 x 3, we can use the antisymmetric and symmetric products to find that the resulting representation is a direct sum of the singlet, two
  • #1
jinbaw
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what is a singlet under SU(3)?
 
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and what is SU(3)xSU(3)?
What do we mean by 3 x 3(bar) = 8 + 1?
 
  • #3


A singlet of any algebra is the state that has zero eigenvalue under the Casimir operators. These are the generators of the center of the algebra and are simultaneously diagonalizable. The singlet state is a one-dimensional representation. For instance, for [tex]SU(2)[/tex], the singlet state satisfies

[tex]\vec{S}^2 |0\rangle = 0,[/tex]

i.e., it is the [tex]s=0[/tex] state. It satisfies

[tex] S_{\pm} |0\rangle = 0 [/tex]

as well, so there are no other states in this representation.

[tex]SU(3)\times SU(3)[/tex] refers to a direct product of two [tex]SU(3)[/tex] groups. States or particles will simultaneously be in some representation of both groups.

[tex]\mathbf{3} \otimes \bar{\mathbf{3}} = \mathbf{1} \oplus \mathbf{8}[/tex] refers to a direct product of the [tex] \mathbf{3} [/tex] representation with the [tex]\bar{\mathbf{3}}[/tex]. The resulting representation is what's known as reducible and can be written as a direct sum of the singlet [tex]\mathbf{1}[/tex] and octet [tex]\mathbf{8}[/tex] representations.

This relationship is analogous to the way we decompose two-particle spin states in QM. There, for [tex]SU(2)[/tex], the decomposition of specific product states into states of irreducible representations is summed up by the Clebsch-Gordan coefficients.
 
  • #4


Okay, that explains a singlet. Thanks.
But what I still can't get is how we know that it is 1 + 8. For example i need to find what 3 x 3 x 3 is. How can I do that?
 
  • #5


jinbaw said:
Okay, that explains a singlet. Thanks.
But what I still can't get is how we know that it is 1 + 8. For example i need to find what 3 x 3 x 3 is. How can I do that?

You might want to look up Young tableaux, which are very useful for computing products of irreps for unitary groups. They are a pictorial way of keeping track of the symmetries of irreducible tensor representations. Alternatively, there are various places to find tables of products.

In your case, we should first compute [tex] \mathbf{3}\otimes\mathbf{3}[/tex]. Thinking of the fundamental irrep [tex]\mathbf{3}[/tex] like a vector, this product is a rank 2 tensor. There is an antisymmetric product which is the [tex]\bar{\mathbf{3}}[/tex] and the symmetric product which is the [tex]\mathbf{6}[/tex]. So we have

[tex]\mathbf{3}\otimes\mathbf{3} = \bar{\mathbf{3}}\oplus \mathbf{6}.[/tex]

Now we already know that [tex]\mathbf{3} \otimes \bar{\mathbf{3}} = \mathbf{1} \oplus \mathbf{8}[/tex], so it remains to compute [tex]\mathbf{3} \otimes \mathbf{6} [/tex]. We can either antisymmetrize the new index with the indices on the symmetric tensor, which gives us the [tex]\mathbf{8}[/tex], or we can totally symmetrize the rank 3 tensor, which gives us the [tex]\mathbf{10}[/tex]. Therefore

[tex] \mathbf{3} \otimes \mathbf{6} = \mathbf{8}\oplus \mathbf{10}. [/tex]

Putting it all together, we have

[tex]\mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3} = \mathbf{1} \oplus \mathbf{8}\oplus\mathbf{8}\oplus \mathbf{10}.[/tex]

As a sanity check, you can compare the total dimensions of the representations on both sides.
 

What is a singlet under SU(3)?

A singlet under SU(3) refers to a representation of the special unitary group SU(3) that transforms as a single unit under the group's symmetry operations. In simpler terms, it is a specific type of mathematical object used to describe the behavior of particles in quantum physics.

What is SU(3)?

SU(3) is a special unitary group that consists of 3x3 unitary matrices with determinant 1. It is a fundamental symmetry group in quantum physics and is used to describe the strong nuclear force between particles.

How is a singlet under SU(3) different from other representations?

A singlet under SU(3) is unique because it transforms as a single unit, while other representations may transform as a combination of multiple units. This makes it a simpler and more fundamental representation in describing the behavior of particles.

What are some examples of singlets under SU(3)?

Examples of singlets under SU(3) include the flavor singlets of quarks and leptons. These are particles that do not participate in strong interactions and are therefore unaffected by the strong nuclear force.

Why is understanding singlets under SU(3) important in physics?

Understanding singlets under SU(3) is important in physics because it helps us better understand the fundamental forces and interactions between particles. It also plays a crucial role in our understanding of the structure and behavior of matter at the subatomic level.

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