How to construct gamma matrices with two lower spinor indices for any dimension?

In summary: Essentially, the matrix is composed of the product of all the gamma matrices in the theory. In summary, we discussed the general construction of gamma matrices with two lower indices. These matrices can be constructed using two methods, either by using a charge conjugation matrix or by using the inner product. In even dimensions, we can use the complex Grassmann algebra to specify a representation of the Clifford algebra. This results in the construction of the gamma matrices using the operators defined in the algebra. Finally, we mentioned that the charge conjugation matrix can be found in Appendix B of Polchinski's "String Theory Volume 2".
  • #1
Osiris
20
0
Generally, Gamma matrices with one lower and one upper indices could be constructed based on the Clifford algebra.
\begin{equation}
\gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij},
\end{equation}

My question is how to generally construct gamma matrices with two lower indices. There should be at least two ways.
1). construct and use the charge conjugation matrix to lower one spinor index in the gamma matrix;
2). use inner product to directly obtain the gamma matrices with two lower spinor indices, something like $$<\Gamma e_{\alpha}, e_{\beta}>=\Gamma_{\alpha,\beta}$$, where $$e_{\alpha}$$ are the basis element.

In even dimensions (D=2m), consider
complex Grassmann algebra $$\Lambda_{m}[\alpha^{1},...,\alpha^{m}]$$ with
generators $$\alpha^{1},...,\alpha^{m}.$$) Namely, we define $$\widehat{\alpha
}^{i}$$ and $$\widehat{\beta}_{i}$$ as multiplication and differentiation
operators:
\begin{equation}
\widehat{\alpha}^{i}\psi=\alpha^{i}\psi,
\end{equation}
\begin{equation}
\widehat{\beta}_{i}\psi=\frac{\partial}{\partial \alpha ^{i}}\psi.
\end{equation}

According to the Grassmann algebra, we have
\begin{equation}
\widehat{\alpha}^{i}\widehat{\alpha}^{j}+\widehat{\alpha}^{j}\widehat{\alpha}^{i}=0,
\end{equation}

\begin{equation}
\widehat{\beta}_{i}\widehat{\beta}_{j}+\widehat{\beta}_{j} \widehat{\beta}_{i}=0
\end{equation}

\begin{equation}
\widehat{\alpha}^{i}\widehat{\beta}_{j}+\widehat{\beta}_{j} \widehat{\alpha}^{i}=delta_{j}^{i}
\end{equation}

This means that $$\widehat{\alpha}^{1},...,\widehat{\alpha}^{m}, \widehat{\beta}_{1},...,\widehat{\beta}_{m}$$ specify a representation of Clifford algebra
for some choice of $h$ (namely, for $h$ corresponding to quadratic form
$$\frac{1}{2}(x^{1}x^{m+1}+x^{2}x^{m+2}+...+x^{m}x^{2m})$$). It follows that
operators
\begin{equation}
\Gamma^{j}=\widehat{\alpha}^{j}+\widehat{\beta}_{j},1\leq j\leq m,
\end{equation}
\begin{equation}
\Gamma^{j}=\widehat{\alpha}^{j-m}-\widehat{\beta}_{j-m},m<j\leq2m,
\end{equation}
determine a representation of $Cl(m,m,\mathbb{C})$


For example, in $D=4$, we can obtain
$$\Gamma^{1}=\begin{pmatrix}0&
1&
0&
0\\
1&
0&
0&
0\\
0&
0&
0&
1\\
0&
0&
1&
0\\
\end{pmatrix}$$,
$$\Gamma^{2}=\begin{pmatrix}0&
0&
0&
1\\
0&
0&
{-1}&
0\\
0&
{-1}&
0&
0\\
1&
0&
0&
0\\
\end{pmatrix}$$,
$$\Gamma^{3}=\begin{pmatrix}0&
{-1}&
0&
0\\
1&
0&
0&
0\\
0&
0&
0&
1\\
0&
0&
{-1}&
0\\
\end{pmatrix}$$,
$$\Gamma^{4}=\begin{pmatrix}0&
0&
0&
{-1}\\
0&
0&
1&
0\\
0&
{-1}&
0&
0\\
1&
0&
0&
0\\
\end{pmatrix}$$


My question is how to generally construct the charge conjugation matrix C, so that we could have
$$C\Gamma C^{-1}=+/-\Gamma^T$$
 
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  • #2
Any tips will be greatly appreciated.
:)
 
  • #3
If you are looking for the charge conjugation matrix in various dimensions then check out the back of Polchinski's "String Theory Volume 2". Appendix B has everything you'll need.
 

What are gamma matrices and why are they important in physics?

Gamma matrices are a set of mathematical tools used in quantum field theory to represent spin and angular momentum of particles. They are important because they help us understand the properties of particles and their interactions.

What is the significance of having two lower spinor indices in gamma matrices?

The two lower spinor indices in gamma matrices represent the spinor space, which is a mathematical representation of the spin of a particle. Having two indices allows us to construct higher-dimensional gamma matrices, which are necessary for studying particles in higher dimensions.

How do you construct gamma matrices with two lower spinor indices for any dimension?

The construction of gamma matrices with two lower spinor indices follows a specific pattern, where the gamma matrices are constructed from a combination of Dirac matrices and Pauli matrices. The number of matrices needed depends on the dimension of the space, and the resulting matrices must satisfy certain algebraic conditions.

What are some applications of gamma matrices with two lower spinor indices?

Gamma matrices with two lower spinor indices are used in various areas of theoretical physics, such as quantum field theory, string theory, and supersymmetry. They are also used in particle physics experiments to study the properties of particles and their interactions.

Are there any challenges or limitations in constructing gamma matrices with two lower spinor indices?

One of the main challenges in constructing gamma matrices with two lower spinor indices is the mathematical complexity involved, especially in higher dimensions. Additionally, there are limitations in the dimensions for which these matrices can be constructed, as certain dimensions do not allow for the formations of these matrices.

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