How do I expand gamma matrices without adding a unity matrix?

In summary, the gamma matrices are 4 x 4 matrices whose values depend on the spinor space basis, and the form of gamma 0 in the Dirac representation is given by the equation iħγ0 = \begin{pmatrix} iħ & 0 & 0 & 0\\ 0 & iħ & 0 & 0\\ 0 & 0 & -iħ & 0\\ 0 & 0 & 0 & -iħ \end{pmatrix}.
  • #1
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!
 
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  • #2
help1please said:
[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
Expanding that, gives back that?

What is mathematica??
 
  • #4
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
so you don't know how to expand the terms I asked of?
 
  • #6
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
Bill_K said:
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
Can you show me, in plane mathematical language, in an equation, how to expand it please.
 
  • #9
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
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
PhilDSP said:
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
[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
Ah sorry, unity matrix.I don't know what I read earlier but this didn't immediately pop out to me! ty
 

1. What are gamma matrices?

Gamma matrices are a set of mathematical tools used in the field of quantum mechanics to describe the behavior of particles and their interactions. They are represented by matrices with complex numbers as entries.

2. Why is it necessary to expand gamma matrices?

Expanding gamma matrices allows for a more comprehensive understanding of particle behavior and interactions. It also helps to simplify calculations and make predictions about quantum systems.

3. How are gamma matrices expanded?

Gamma matrices are expanded by adding additional rows and columns to the original matrix. This expansion is often done in a systematic way, such as doubling the size of the matrix, to maintain the desired properties of the gamma matrices.

4. What is the significance of expanding gamma matrices?

The expansion of gamma matrices plays a crucial role in theoretical physics, particularly in the study of quantum field theory. It allows for a more accurate description of particle interactions and enables physicists to make predictions about the behavior of particles at high energies.

5. Are there any limitations to expanding gamma matrices?

While expanding gamma matrices has proven to be a useful tool in theoretical physics, it does have its limitations. The expansion process can become increasingly complex as more rows and columns are added, making it difficult to apply in practical applications. Additionally, some theories may require a different approach to describe particle interactions, making the expansion of gamma matrices less applicable.

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