.Decomposing a 4x4 Matrix into Dirac 16 Matrices

In summary, The conversation is about decomposing an arbitrary 4x4 matrix into Dirac 16 matrices using the defining characteristic of gamma matrices. However, not all 4x4 matrices can be represented as a sum of gamma matrices. The question is asking about using gamma matrices to form a basis for the 16-dimensional vector space of 4x4 matrices. This vector space is usually written as a direct sum of subspaces with dimensions 1, 4, 6, 4, 1.
  • #1
Neitrino
137
0
Dear All,

Could you pls remind me how do I decompose arbitrary 4x4 matrix into Dirac 16 matrices...

I ve forgotten.

Thank you
 
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  • #2
If your unknown matrix is really a sum of gamma matrices you could try using the defining characteristic of the gamma matrices, {gamma_m,gamma_n}=2*g_mn. But not every 4x4 matrix is a sum of gamma matrices.
 
  • #3
Dick said:
But not every 4x4 matrix is a sum of gamma matrices.

I don't know how to answer Neitrino's question, but I think Neitrino is asking about using the gamma matrices and their products to form a basis for the 16-dimensional vector space of 4x4 matrices. Usually, this space is written as a direct sum of subspaces of dimensions 1, 4, 6, 4, 1.
 

What is a 4x4 matrix?

A 4x4 matrix is a rectangular array of numbers with 4 rows and 4 columns. It is commonly used in linear algebra to represent transformations, such as rotations, translations, and scaling.

What is a Dirac 16 matrix?

A Dirac 16 matrix is a 4x4 matrix that is composed of 16 elements, each of which is a complex number. It is used in quantum mechanics to represent the state of a quantum system.

Why is it important to decompose a 4x4 matrix into Dirac 16 matrices?

Decomposing a 4x4 matrix into Dirac 16 matrices allows us to better understand the underlying structure and properties of the matrix. It also helps us to simplify complex calculations and operations on the matrix.

How do you decompose a 4x4 matrix into Dirac 16 matrices?

To decompose a 4x4 matrix into Dirac 16 matrices, we use the Pauli spin matrices and the identity matrix. The Pauli spin matrices are 2x2 matrices that represent spin states in quantum mechanics, and the identity matrix is a 4x4 matrix with 1s on the main diagonal and 0s everywhere else. By combining these matrices in a specific way, we can decompose any 4x4 matrix into 16 Dirac matrices.

What are some applications of decomposing a 4x4 matrix into Dirac 16 matrices?

Decomposing a 4x4 matrix into Dirac 16 matrices has various applications in physics and engineering. It is used in quantum mechanics to study the behavior of particles and in computer graphics to represent 3D transformations. It is also used in signal processing, control systems, and other areas of science and technology.

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