Gamma matrices out of pauli matrices - symmetry/group theory

In summary: This notation is commonly used in quantum mechanics and should not be confused with the Kronecker product.In summary, the article discusses the use of transformation matrices and gamma matrices in quantum mechanics. The transformation matrix for rotation is given as a Kronecker product of sigma and tau, acting on different bases. The gamma matrices are also defined in a similar way and are used to describe Dirac spinors. This notation may be confusing, but it is commonly used in quantum mechanics.
  • #1
Otterhoofd
9
0
I'm reading an article (http://prb.aps.org/abstract/PRB/v82/i4/e045122) but I have some problems understanding certain definitions. The authors have introduced a basis of certain bands (four) and then continue to give the transformation matrices of the symmetry operators. One (rotation) of them is given as:
[tex]R2=\sigma_1 \otimes\tau_3[\tex]
with "In the above, sigma acts in the spin basis and tau acts in the basis of P1+ and P2− subbands"

What does this product look like? Is it really a kronecker/direct product of the two matrices? I'm confused because they work in different bases. Or can I just do the kronecker product, resulting in i times:
0 0 1 0
0 0 0 -1
1 0 0 0
0 -1 0 0

In the appendix, gamma matrices are also defined in a similar way.

Can anyone point me in the right direction or give some insight on this? Thanks
 
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  • #2
!Yes, the product you describe is indeed a Kronecker product of the two matrices. The Kronecker product is defined as the tensor product of two matrices. In this case, it simply means that you take the first matrix and multiply it by the second matrix element-wise. So, the resulting matrix would be:0 0 1 0 0 0 0 -1 1 0 0 0 0 -1 0 0 This means that each element from the first matrix is multiplied by each element from the second matrix.As for the gamma matrices, they are generally used to describe Dirac spinors, and the notation used in the article is very similar. The gamma matrices are related to the Pauli matrices (σ), and they act on a particular basis. In this case, the gamma matrices act on the spin basis, while the Pauli matrices act on the basis of P1+ and P2− subbands.
 

1. What are Gamma matrices and how are they related to Pauli matrices?

Gamma matrices are matrices used in the theory of spinors, which are mathematical objects used to describe the spin of particles. They are related to Pauli matrices through the Dirac equation, which relates the spin of a particle to its momentum. The gamma matrices are a larger set of matrices that includes the Pauli matrices as a subset.

2. How do Gamma matrices relate to symmetry and group theory?

Gamma matrices are used in the theory of symmetry and group theory because they transform under certain symmetry operations in a predictable way. This allows for the use of group theory in understanding the properties of spinors and particles.

3. What is the significance of the symmetry properties of Gamma matrices?

The symmetry properties of Gamma matrices are significant because they allow for the classification and organization of particles based on their spin and other quantum numbers. This is important in understanding the behavior of particles in different physical systems.

4. Can Gamma matrices be used in other areas of physics besides quantum mechanics?

Yes, Gamma matrices have applications in other areas of physics such as general relativity and particle physics. In general relativity, they are used to describe the spin of particles in curved spacetime, while in particle physics they are used to describe the interactions between particles.

5. How are Gamma matrices represented mathematically?

Gamma matrices are usually represented using the Dirac representation, where each gamma matrix is a 4x4 matrix with complex entries. In this representation, the gamma matrices satisfy certain algebraic properties that are important in the study of spinors and particles.

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