As gauge group let's consider the popular $SO(10)$ group.

The fundamental representation $\pi$ of the corresponding Lie algebra $\mathfrak{so}(10)$ is $10$ dimensional

$$ \pi: \mathfrak{so}(10) \rightarrow \Bbb R^{10}$$

This is a real representation, because $\Bbb R^{10}$ is a real vector space.

In such theories in physics when we say particles are in the $10$ dimensional representation, we mean our particles, for example the Higgs fields, are $\in \Bbb R^{10}$. Thus we have $10$ real particles here. A convenient way to label these $10$ states is by using the eigenvalues of the Cartan generators. $$ \left( \begin{array}{c} \{1,0,0,0,0\} \\ \{-1,1,0,0,0\} \\ \{0,-1,1,0,0\} \\ \{0,0,-1,1,1\} \\ \{0,0,0,-1,1\} \\ \{0,0,0,1,-1\} \\ \{0,0,1,-1,-1\} \\ \{0,1,-1,0,0\} \\ \{1,-1,0,0,0\} \\ \{-1,0,0,0,0\} \\ \end{array} \right) $$

This is known as the weight system of the $10$-dimensional representation.

This can be interpreted that we have $5$ particles and their $5$ corresponding antiparticles in this representation. For example,

$$\phi_1 \hat= \{1,0,0,0,0\} \leftrightarrow \phi_1^c \hat= \{-1,0,0,0,0\} .$$

Now a popular trick, because with a real $10$ one gets very wrong mass relations, is to complexify the $10$. This means we then look at

$$ \pi: \mathfrak{so}(10) \rightarrow \Bbb C^{10}$$

Our weight system does not change, because the weight system of a real representation is defined as the weight system of the corresponding complex representation. This definition is necessary, because only for complex representation the existence of the eigenvalues is given.

Now our particles are $\in \Bbb C^{10}$. If we view $ \Bbb C^{10}$ as a vector space over $ \Bbb C$, we have the same $10$ basis vectors as for $ \Bbb R^{10}$, but now complex linear combinations are allowed.

Instead, we can view $ \Bbb C^{10}$ as a vector space over $ \Bbb R$. Then we have 20 basis vectors. These are our physical $20$ degrees of freedom for a complex 10.

My problem is interpreting these degrees of freedom in terms of particles and antiparticles.

In addition to the $10$ basis vectors that we had for a real $10$, for example

$$ \phi_1 \hat= \{1,0,0,0,0\} \hat = \begin{pmatrix} 1 \\0 \\0\\\vdots \end{pmatrix} \quad , \quad \phi_2 \hat= \{-1,1,0,0,0\} \hat = \begin{pmatrix} 0 \\1 \\0 \\\vdots \end{pmatrix}$$

we have the basis vectors $$ ?\hat= ? \hat = \begin{pmatrix} i \\0 \\0\\\vdots \end{pmatrix} \quad , \quad ? \hat= ? \hat = \begin{pmatrix} 0 \\i \\0 \\\vdots \end{pmatrix}$$

What are the weights and particles that correspond to these basis vectors?

In addition: which of these 20 degrees of freedom are now particle and antiparticle pairs, because the identification from above, I think, no longer holds?


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