Find linear combination of 16 Γ matrices

In summary, the conversation discusses how any spinor matrix can be expressed in a set of 16 linearly independent matrices, and the lecture gave the 16 Γ_J matrices. The task at hand is to express M, given by (σ_μυ)(γ_5), (σ_μυ)(σ^μυ), (γ^α)(σ_μυ)(γ_α), in terms of the 16 given Γ matrices. The coefficients of the linear combination of the 16 matrices can be found using the formula c_J = (1/4) Tr (M (inverse of Γ_J)). The conversation also addresses the difficulty in finding the explicit forms of σ^μυ and the fact that (Γ
  • #1
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Homework Statement


Any spinor matrix can be expressed in a set of 16 linearly independent matrices. In the lecture the 16 Γ_J matrices (J=1 to 16) given are I, γ^0,1,2,3, σ^μυ, (γ^μ)(γ_5), iγ_5. I was asked to express
M = (σ_μυ)(γ_5), (σ_μυ)(σ^μυ), (γ^α)(σ_μυ)(γ_α)
in terms of the 16 given Γ matrices.


Homework Equations


The coefficients of the linear combination of the 16 matrices is
c_J = (1/4) Tr (M (inverse of Γ_J))



The Attempt at a Solution


I'd been trying to find the explicit forms σ^μυ but did not find any table. What I found is that the 16 linearly independent matrices can also be the products of γ^0,1,2,3. But I think I was not supposed to find σ^μυ in terms of products of γ^0,1,2,3 then get the inverse of σ^μυ explicitly.

I'm really bad in linear algebra and have no idea about the inverses of the 16 matrices and the properties of the traces of their inverses. Can anyone please help me? Thank you.

I did not use LaTeX because it did not look right when previewing. Sorry about that.
 
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  • #2
I didn't know that (Γ_J)^2 = ±I. So now the inverse of Γ_J can be found.
 

1. What are linear combinations of Γ matrices?

A linear combination of Γ matrices is an expression where the matrices are multiplied by a set of coefficients and added together. It is a way of combining multiple Γ matrices to create a new matrix.

2. How many possible linear combinations of 16 Γ matrices are there?

There are a total of 2^16 possible linear combinations of 16 Γ matrices. This is because each matrix can be either included or excluded from the combination, resulting in 2 possible options for each of the 16 matrices.

3. What is the purpose of finding linear combinations of Γ matrices?

The purpose of finding linear combinations of Γ matrices is to simplify and manipulate mathematical expressions involving these matrices. This can help in solving complex equations and understanding certain physical phenomena in fields such as quantum mechanics and particle physics.

4. Can linear combinations of Γ matrices be used in practical applications?

Yes, linear combinations of Γ matrices have practical applications in various fields such as computer graphics, cryptography, and signal processing. They are also used in theoretical physics to describe the behavior of particles and their interactions.

5. Are there any rules or properties that govern linear combinations of Γ matrices?

Yes, there are certain rules and properties that govern linear combinations of Γ matrices, such as the distributive property and the commutative property of matrix multiplication. These properties help in simplifying and manipulating these combinations to solve equations and understand physical phenomena.

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