# Otimes notation and tau matrices used in definition of gamma matrices?

• Peeter
In summary, Zee explains in his Quantum Field theory book the use of the gamma matrices in terms of the Pauli matrices, denoted by \sigma^i, through the equations \gamma^0=I \otimes \tau_3, \gamma^i=\sigma^i \otimes \tau_2, and \gamma^5=I \otimes \tau_1. These \tau_i matrices can be reverse-engineered using the Kronecker product notation, as seen in Wiki. However, it is noted that Zee's notation for the gamma matrices differs from the notation used for the tensor product. Some authors, like Zee, use \tau_i, while others use \sigma_i.
Peeter
In Zee's Quantum Field theory book he writes

\begin{align}\gamma^0 &= \begin{bmatrix}I & 0 \\ 0 & -I\end{bmatrix}=I \otimes \tau_3 \\ \gamma^i &= \begin{bmatrix}0 & \sigma^i \\ \sigma^i & 0\end{bmatrix}=\sigma^i \otimes \tau_2 \\ \gamma^5 &=\begin{bmatrix}0 & I \\ I & 0\end{bmatrix}=I \otimes \tau_1 \end{align}

The Pauli matrices $\sigma^i$ I've seen. However, I have two questions

1) What are these $\tau_i$ matrices?
2) I'm not familiar with this $\otimes$ notation.

strangerep said:
Try Wiki:

http://en.wikipedia.org/wiki/Tensor_product#Kronecker_product_of_two_matrices

That should also allow you to reverse-engineer the tau matrices... :-)

I find that the tau matrices are just the sigma matrices. Odd that two different notations would be used. One notation when defining the gamma matrices in terms sigmas directly, and an entirely different notation when using the tensor product?

Peeter said:
I find that the tau matrices are just the sigma matrices.
... which is what Zee says beneath his eq(4) on p90.

Odd that two different notations would be used. One notation when defining the gamma matrices in terms sigmas directly, and an entirely different notation when using the tensor product?

I don't use tau's myself.

BTW, your earlier expression for gamma^i doesn't match Zee's.

## 1. What is Otimes notation used for in the definition of gamma matrices?

Otimes notation, represented by the symbol ⊗, is used to represent the tensor product in linear algebra. In the context of gamma matrices, it is used to combine two matrices into a single matrix that represents their combined operations.

## 2. How are tau matrices used in the definition of gamma matrices?

Tau matrices, represented by the Greek letter τ, are used to define the Clifford algebra, which is a mathematical framework for studying the properties of gamma matrices. They are also used to generate the gamma matrices through a specific algorithm.

## 3. What is the significance of gamma matrices in physics and mathematics?

Gamma matrices have significance in both physics and mathematics. In physics, they are used in the Dirac equation to describe the behavior of fermions, such as electrons and quarks. In mathematics, they are fundamental elements in the study of Clifford algebras and have applications in fields such as differential geometry and representation theory.

## 4. How many gamma matrices are there in total?

The number of gamma matrices depends on the dimension of the space in which they are defined. In a three-dimensional space, there are four gamma matrices, while in a four-dimensional space, there are eight gamma matrices. In general, in an n-dimensional space, there are 2^(n/2) gamma matrices.

## 5. How is the notation for gamma matrices related to the concept of spinors?

The notation for gamma matrices is closely related to the concept of spinors, which are mathematical objects that represent the intrinsic angular momentum of particles. Gamma matrices are used to construct spinors and are essential in the study of spinors in physics and mathematics.

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