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Gamma matrices and how they operate

  • #1
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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]
 

Answers and Replies

  • #2
167
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Can no one confirm I have done this right?
 
  • #3
dextercioby
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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
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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
167
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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
167
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[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
167
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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
TSny
Homework Helper
Gold Member
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[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
167
0
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
 

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