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Qmechanic ♦
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Qmechanic ♦
Qmechanic ♦
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In particle Physics it's usual to write the physical content of a Theory in adjoint representations of the Gauge group. For example:

$24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}$ (Source: SU(5) GUT Wikipedia article)

While I do understand the Basics in representation theory from a mathematical viewpoint, as well as Gauge Theory (up to this point), I've been looking High and Low for some good article on how to understand what the above formula means physically?

Specifically I don't understand the following:

  • I'm having a bit of a problem with the notation. $(1,1)$ denotes the tensor product of a 1 and 1 of $SU(3) \times SU(2)$ in this case, does the subscript $()_0$ belong to the $\times U(1)$$U(1)$ part? Or did I completely misunderstand something?

  • How to arrive at the above transformation? How to choose the right hand side of the 24 transformation, it seems random to me

  • The physical content. $(8,1)_0$ looks to me like gluons, because of the 8, $(1,3)_0$ like W and Z bosons and $(1,1)_0$ like the photon. But these are all guesses I made according to the numbers I see and the fact that the SM should arise from $SU(5)$ breaking. How would one know this? And what are the other 2 components?

Any reference is also greatly appreciated, especially one that focuses on precisely this.

In particle Physics it's usual to write the physical content of a Theory in adjoint representations of the Gauge group. For example:

$24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}$ (Source: SU(5) GUT Wikipedia article)

While I do understand the Basics in representation theory from a mathematical viewpoint, as well as Gauge Theory (up to this point), I've been looking High and Low for some good article on how to understand what the above formula means physically?

Specifically I don't understand the following:

  • I'm having a bit of a problem with the notation. $(1,1)$ denotes the tensor product of a 1 and 1 of $SU(3) \times SU(2)$ in this case, does the subscript $()_0$ belong to the $\times U(1)$ part? Or did I completely misunderstand something?

  • How to arrive at the above transformation? How to choose the right hand side of the 24 transformation, it seems random to me

  • The physical content. $(8,1)_0$ looks to me like gluons, because of the 8, $(1,3)_0$ like W and Z bosons and $(1,1)_0$ like the photon. But these are all guesses I made according to the numbers I see and the fact that the SM should arise from $SU(5)$ breaking. How would one know this? And what are the other 2 components?

Any reference is also greatly appreciated, especially one that focuses on precisely this.

In particle Physics it's usual to write the physical content of a Theory in adjoint representations of the Gauge group. For example:

$24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}$ (Source: SU(5) GUT Wikipedia article)

While I do understand the Basics in representation theory from a mathematical viewpoint, as well as Gauge Theory (up to this point), I've been looking High and Low for some good article on how to understand what the above formula means physically?

Specifically I don't understand the following:

  • I'm having a bit of a problem with the notation. $(1,1)$ denotes the tensor product of a 1 and 1 of $SU(3) \times SU(2)$ in this case, does the subscript $()_0$ belong to the $U(1)$ part? Or did I completely misunderstand something?

  • How to arrive at the above transformation? How to choose the right hand side of the 24 transformation, it seems random to me

  • The physical content. $(8,1)_0$ looks to me like gluons, because of the 8, $(1,3)_0$ like W and Z bosons and $(1,1)_0$ like the photon. But these are all guesses I made according to the numbers I see and the fact that the SM should arise from $SU(5)$ breaking. How would one know this? And what are the other 2 components?

Any reference is also greatly appreciated, especially one that focuses on precisely this.

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Frank Müpe
Frank Müpe
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Introduction to Physical Content from Adjoint Representations

In particle Physics it's usual to write the physical content of a Theory in adjoint representations of the Gauge group. For example:

$24\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}$ (Source: SU(5) GUT Wikipedia article)

While I do understand the Basics in representation theory from a mathematical viewpoint, as well as Gauge Theory (up to this point), I've been looking High and Low for some good article on how to understand what the above formula means physically?

Specifically I don't understand the following:

  • I'm having a bit of a problem with the notation. $(1,1)$ denotes the tensor product of a 1 and 1 of $SU(3) \times SU(2)$ in this case, does the subscript $()_0$ belong to the $\times U(1)$ part? Or did I completely misunderstand something?

  • How to arrive at the above transformation? How to choose the right hand side of the 24 transformation, it seems random to me

  • The physical content. $(8,1)_0$ looks to me like gluons, because of the 8, $(1,3)_0$ like W and Z bosons and $(1,1)_0$ like the photon. But these are all guesses I made according to the numbers I see and the fact that the SM should arise from $SU(5)$ breaking. How would one know this? And what are the other 2 components?

Any reference is also greatly appreciated, especially one that focuses on precisely this.