Expanding gamma matrices

  • #1
help1please
167
0
[tex]\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0[/tex]

How do I expand

[tex]i\hbar \gamma^0[/tex]

the matrix in this term, I am a bit lost. All the help would be appreciated!
 

Answers and Replies

  • #2
zje2009
12
0
[tex]\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0[/tex]

How do I expand

[tex]i\hbar \gamma^0[/tex]

the matrix in this term, I am a bit lost. All the help would be appreciated!

It's itself.Using Mathematica.
 
  • #3
help1please
167
0
Expanding that, gives back that?

What is mathematica??
 
  • #4
Bill_K
Science Advisor
Insights Author
4,157
203
The gamma matrices are 4 x 4 matrices whose values depend on the basis ("representation") you decide to use in spinor space. For a list of possibilities, see "gamma matrix" in Wikipedia.
 
  • #5
help1please
167
0
so you don't know how to expand the terms I asked of?
 
  • #6
Bill_K
Science Advisor
Insights Author
4,157
203
so you don't know how to expand the terms I asked of?
I thought the Wikipedia article explained it pretty clearly. It gives the explicit form of γ0 in the Dirac, Weyl and Majorana representations. Isn't that what you want?

But if that's not what you mean by "expanding" it, the only other thing to do is this...

iħγ0
 
  • #7
help1please
167
0
I thought the Wikipedia article explained it pretty clearly. It gives the explicit form of γ0 in the Dirac, Weyl and Majorana representations. Isn't that what you want?

But if that's not what you mean by "expanding" it, the only other thing to do is this...

iħγ0



Why have that when I started off with that... my equation I really thought was simple. HOW do you expand [tex]i \hbar \gamma^0[/tex]

The answer I was looking for was not a go to wiki one! And no wiki does not explain it well for me, I am new at this stuff.
 
  • #8
help1please
167
0
Can you show me, in plane mathematical language, in an equation, how to expand it please.
 
  • #9
kloptok
188
0
You will have to define what you mean by "expand". So far we have only been able to guess, and apparently this was not what you intended. So define "expand" please.
 
  • #10
PhilDSP
643
15
Do you mean this? [itex]\qquad \gamma^0 =
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{pmatrix} \qquad \qquad
[/itex] (which is the Dirac representation)

so that [itex]\ \ \qquad \qquad i \hbar \gamma^0 =
\begin{pmatrix}
i \hbar & 0 & 0 & 0\\
0 & i \hbar & 0 & 0\\
0 & 0 & -i \hbar & 0\\
0 & 0 & 0 & -i \hbar
\end{pmatrix} \qquad \qquad
[/itex]
 
Last edited:
  • #11
help1please
167
0
Do you mean this? [itex]\qquad \gamma^0 =
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{pmatrix} \qquad \qquad
[/itex] (which is the Dirac representation)

so that [itex]\ \ \qquad \qquad i \hbar \gamma^0 =
\begin{pmatrix}
i \hbar & 0 & 0 & 0\\
0 & i \hbar & 0 & 0\\
0 & 0 & -i \hbar & 0\\
0 & 0 & 0 & -i \hbar
\end{pmatrix} \qquad \qquad
[/itex]

Why is then when [tex]\gamma^{0}^2[/tex] is equal to 1?
 
  • #12
PhilDSP
643
15
[tex]\qquad (\gamma^0)^2 =
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1
\end{pmatrix} =
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix}
[/tex]
 
  • #13
help1please
167
0
Ah sorry, unity matrix.


I don't know what I read earlier but this didn't immediately pop out to me! ty
 

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