- #1

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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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- Thread starter help1please
- Start date

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

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

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help1please said:

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

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Expanding that, gives back that?

What is mathematica??

What is mathematica??

- #4

Science Advisor

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- #5

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so you don't know how to expand the terms I asked of?

- #6

Science Advisor

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- 204

I thought the Wikipedia article explained it pretty clearly. It gives the explicit form of γso you don't know how to expand the terms I asked of?

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

iħγ

- #7

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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]Bill_K said:I thought the Wikipedia article explained it pretty clearly. It gives the explicit form of γ^{0}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...

iħγ^{0}

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

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Can you show me, in plane mathematical language, in an equation, how to expand it please.

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- #10

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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]

\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

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PhilDSP said:

\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

- 643

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\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

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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.

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.

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.

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.

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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