# A Problem about representations of group and particles

• so2so3su2
In summary, the conversation discusses a paper that computes entanglement entropy for free-field theories on a specific type of Euclidean spacetime. The paper also discusses the spin representations of different particles, such as the D-dim SO(2) gauge field and the graviton. The authors divide the SO(D) group into three parts to analyze the different spin representations. There is a typo in equation (2.28) which can be corrected by explicitly counting the number of components in the graviton representation. The conversation ends with a discussion on breaking the SO(D) group into different representations for studying other theories.
so2so3su2
I'm reading a paper these days,

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?

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.

## 1. What is a group in the context of particle representations?

A group is a mathematical concept that describes a set of elements and a binary operation that combines any two elements in the set to produce another element in the set. In the context of particle representations, a group can represent a symmetry or transformation property of a physical system.

## 2. What are particle representations and why are they important?

Particle representations refer to the way particles and their interactions are described and studied in physics. They are important because they allow us to understand and predict the behavior of particles in different physical systems, such as atoms, molecules, and subatomic particles.

## 3. How do representations of groups and particles relate to each other?

Representations of groups and particles are closely related because groups can represent the symmetries and transformations of particles, while particles can be seen as elements in a larger group. This connection is important in understanding the properties and behaviors of particles.

## 4. What is the significance of studying representations of groups and particles?

Studying representations of groups and particles is significant because it allows us to understand the fundamental principles and symmetries that govern the physical world. It also helps us to make predictions and develop theories about the behavior of particles in different systems.

## 5. Are there any real-world applications of representations of groups and particles?

Yes, there are many real-world applications of representations of groups and particles. For example, understanding the symmetries of particles has led to the development of new materials and technologies, and the study of group representations has contributed to advances in fields such as particle physics, chemistry, and materials science.

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