Gamma matrices and how they operate

In summary, the conversation revolves around the conventions and equations related to the Dirac matrices in different representations, and the relationship between them. It is discussed that the matrix form of a^k may not necessarily give a nullified matrix and corrections are made to the calculations.
  • #1
help1please
167
0

Homework Statement



Just a matter of convention (question)

Homework Equations



[tex]\gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/tex]

The Attempt at a Solution



If then,

[tex]\gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}[/tex]

and [tex]\gamma^0[/tex] is just [tex]\beta[/tex] and [tex]\beta \alpha^k = \gamma^k[/tex] is it true then that

[tex]\gamma^k = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\alpha^k[/tex]
 
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  • #2
Can no one confirm I have done this right?
 
  • #3
Isn't the particular form of a Dirac matrix representation dependent ? So then only the general relations will hold, i.e.

βrepαkrepkrep

where i/o <rep> one has the Dirac, Majorana or Weyl/chiral representations.
 
  • #4
dextercioby said:
Isn't the particular form of a Dirac matrix representation dependent ? So then only the general relations will hold, i.e.

βrepαkrepkrep

where i/o <rep> one has the Dirac, Majorana or Weyl/chiral representations.

I think so. I think you have to work with [tex]D(\psi(x,t))[/tex] on the three matrices [tex]\gamma^1,\gamma^2,\gamma^3[/tex] to get back the matrix [tex]i\gamma^0 \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix}[/tex]

which when squared gives you the chirality.
 
  • #5
Now I am really confused: consider the matrix form of [tex]a^k[/tex] and calculate it all out we have

[tex]\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \end{pmatrix}[/tex]

A nullified matrix?

Have I got my [tex]a^k[/tex] matrix right... ?
 
  • #6
[tex]a^k[/tex] is just a submatrix, right? of

0_2 sigma^k

\sigma^k 0_2

k=1,2,3

in my case, 1 and 3
 
  • #7
I just don't understand why the relationship

[tex]\beta \alpha^k = \gamma^k[/tex]

would be important if it spat out a zero matrix, which makes me wonder strongly whether I even have the right conditions down.
 
  • #8
help1please said:
[tex]\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \end{pmatrix}[/tex]

A nullified matrix?

You will not get the null matrix. For example, check the element in the first row, third column of the resultant matrix.
 
  • #9
I'm sorry, I did it all wrong didn't I? I now get

[tex]\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & -1 & 0 & 0 \\-1 & 0 & 0 & 0 \end{pmatrix}[/tex]


I am an idiot some times lol
 

1. What are gamma matrices?

Gamma matrices are a set of mathematical objects used in the field of quantum mechanics to describe the behavior of fermions, which are particles that make up matter.

2. How do gamma matrices operate?

Gamma matrices operate by representing the spin and momentum of particles. They are used in equations such as the Dirac equation to describe the behavior of fermions.

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

Gamma matrices are significant because they allow us to accurately describe the behavior of fermions, which are essential building blocks of matter. They are also important in understanding the nature of particles and their spin.

4. How many gamma matrices are there?

There are a total of 16 gamma matrices, which are represented as 4x4 matrices. Each matrix corresponds to a different direction of spin and momentum.

5. Are there any applications of gamma matrices outside of physics?

While gamma matrices are primarily used in physics, they also have applications in other fields such as computer graphics and signal processing. They are used to represent rotations in 3D space and have been used in algorithms for image and audio compression.

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