A Problem about representations of group and particles

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Discussion Overview

The discussion revolves around the representation of particles in the context of a D-dimensional SO(2) gauge field, particularly focusing on the computation related to equation (2.28) from a referenced paper. Participants explore the implications of this computation for different particle spins and representations within the framework of gauge theory and graviton properties.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants clarify that equation (2.28) pertains to a graviton rather than an SO(2) gauge field, emphasizing the need to understand prior discussions in the paper.
  • One participant outlines the representation of a D-dimensional U(1) gauge field, detailing how different components correspond to various spins under SO(D) and SO(2) actions.
  • Another participant questions the existence of a typo in the paper regarding the traceless constraint, suggesting that the symmetric non-traceless representations of SO(D) can be reducible.
  • There is a proposal to categorize the components of the graviton into different representations of SO(2) and SO(D-2), raising questions about the validity of breaking the SO(D) group into blocks.
  • One participant expresses interest in the possibility of studying other theories that might require different representations, such as those involving multiple SO(2) groups.

Areas of Agreement / Disagreement

Participants express differing views on the presence of a typo in the paper and the necessity of a traceless constraint. There is no consensus on whether the SO(D) group can be divided as proposed, indicating ongoing debate regarding the representations and their implications.

Contextual Notes

The discussion includes assumptions about the representations of particles and the structure of the gauge fields, which may not be universally accepted. The implications of breaking the SO(D) group into parts remain unresolved.

so2so3su2
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I'm reading a paper these days,
upload_2015-7-17_1-57-44.png

How can I get 2.28?
It seems for a D-dim SO(2) gauge field, we have spin2, spin1, as well as spin0 particles?
 
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For reference, the paper in question is http://arxiv.org/abs/1412.5606. Eq. (2.28) is a computation done for a graviton, not a ##SO(2)## gauge field.

To understand the computation, you need to understand the discussion starting prior to equation (2.2). The authors want to compute a particular entanglement entropy using the partition function for free-field theories on Euclidean spacetimes of the form ##M= \mathbb{C}/\mathbb{Z}_N\times \mathbb{R}^{D-2}##. The coordinates are ##(x_0,x_1,x_2,\ldots, x_{D-1})##, where ##(x_0,x_1)## refer to ##\mathbb{C}## and ##(x_2,\ldots, x_{D-1})## to ##\mathbb{R}^{D-2}##. The orbifold group ##\mathbb{Z}_N## is a subgroup of the ##SO(2)\subset SO(D)## rotations that act only on ##(x_0,x_1)##.

The ##D##-dimensional ##U(1)## gauge field is in the spin 1 representation of ##SO(D)##. In the paper, the components of the gauge field ##A_a## are numbered from ##1## to ##D## rather than from ##0## to ##D-1##. The ##SO(2)## subgroup acts on the ##A_1## and ##A_2## components, but not on ##A_3,\ldots A_D##. Hence, we determine that ##A_1## has ##SO(2)## spin ##s_1=1##, ##A_2## has ##s_2=-1##, while ##A_3,\ldots A_D## have ##s_a =0##.

The graviton has components ##g_{ab}## (we'll use the Latin indices to match the index chosen on ##s_a##) and is in the spin 2 representation of ##SO(D)##. The ##SO(2)## subgroup acts as follows:
$$ \begin{split}
& g_{11}~~~ \text{with} ~~~ s=2,~~~1~\text{component}, \\
& g_{22}~~~ \text{with} ~~~ s=-2,~~~1~\text{component}, \\
& g_{1a} = g_{a1}, a\neq 1,2~~~ \text{with} ~~~ s=1,~~~D-2~\text{components}, \\
& g_{2a} = g_{a2}, a\neq 1,2~~~ \text{with} ~~~ s=-1,~~~D-2~\text{components}, \\
& g_{ab} , a,b\neq 1,2~~~ \text{with} ~~~ s=0,~~~\frac{D(D-3)}{2}~\text{components}.
\end{split}$$
Note, there is a typo in the last line of (2.28), so I should do the counting explicitly. Since the graviton is a symmetric, traceless tensor, the number of components of ##g_{ab}## with ##a,b\neq 1,2## is
$$ \frac{(D-2)^2 - (D-2)}{2} + (D-2) -1 = \frac{D(D-3)}{2}.$$
The calculation is as follows: the first term counts the number of independent off-diagonal elements, the second is the diagonal elements and the last removes the trace. You should also verify that the above counting gives the RHS of (2.29) correctly.
 
fzero, Thank you. I got the idea. But I don't think the typo is there since they never introduce a traceless constraint.

The other thing is why we can do that. Here by that I mean we divide the SO(D) into 3 parts.
fzero said:
$$ \begin{split}
& g_{11}~~~ \text{with} ~~~ s=2,~~~1~\text{component}, \\
& g_{22}~~~ \text{with} ~~~ s=-2,~~~1~\text{component}, \\
& g_{1a} = g_{a1}, a\neq 1,2~~~ \text{with} ~~~ s=1,~~~D-2~\text{components}, \\
& g_{2a} = g_{a2}, a\neq 1,2~~~ \text{with} ~~~ s=-1,~~~D-2~\text{components}, \\
& g_{ab} , a,b\neq 1,2~~~ \text{with} ~~~ s=0,~~~\frac{D(D-3)}{2}~\text{components}.
\end{split}$$

Can I say g11,g22 part is a representation of SO(2)
g1a, g2a part is a vector representation of SO(D-2)
gab part is a representation of SO(D-2) totally broken?

Can I break the SO(D) group as I wish? For example, I'd like to have two diagonal blocks of SO(2)?
 
so2so3su2 said:
fzero, Thank you. I got the idea. But I don't think the typo is there since they never introduce a traceless constraint.

The symmetric non-traceless representations of ##SO(D)## are reducible into the trace (singlet) and traceless irreducible representations. The elementary particle states always correspond to irreducible representations.

Besides, the extra term is ##4/2=2## which can't be explained away by the trace.

The other thing is why we can do that. Here by that I mean we divide the SO(D) into 3 parts.Can I say g11,g22 part is a representation of SO(2)
g1a, g2a part is a vector representation of SO(D-2)
gab part is a representation of SO(D-2) totally broken?

Can I break the SO(D) group as I wish? For example, I'd like to have two diagonal blocks of SO(2)?

If you wanted to study some other theory, like the spacetime ##\mathbb{C}/{\mathbb{Z}_{N_1}}\times \mathbb{C}/{\mathbb{Z}_{N_2}} \times \mathbb{R}^{D-4}##, then you would have to consider ##SO(2)_1\times SO(2)_2## representations.
 
Got it, thank you.
 

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