The discussion revolves around the expansion of the gamma matrix, specifically i\hbar \gamma^0, within the context of quantum mechanics and spinor space. Participants seek clarification on how to express this matrix without resorting to a unity matrix, exploring various representations and definitions of the gamma matrices.
i\hbar \gamma^0 and requests assistance.i\hbar \gamma^0 itself and mentions using Mathematica.\gamma^0 in the Dirac representation and shows how to compute i \hbar \gamma^0 using that matrix.\gamma^{0}^2 equals the unity matrix, prompting further clarification on the concept of the unity matrix.Participants do not reach a consensus on the meaning of "expand" in this context, leading to multiple interpretations and ongoing confusion regarding the expansion of the gamma matrix.
There are unresolved assumptions about the definitions and representations of the gamma matrices, as well as the specific nature of the requested expansion.
help1please said:\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0
How do I expand
i\hbar \gamma^0
the matrix in this term, I am a bit lost. All the help would be appreciated!
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?so you don't know how to expand the terms I asked of?
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
PhilDSP said:Do you mean this? \qquad \gamma^0 = <br /> \begin{pmatrix}<br /> 1 & 0 & 0 & 0\\<br /> 0 & 1 & 0 & 0\\<br /> 0 & 0 & -1 & 0\\<br /> 0 & 0 & 0 & -1<br /> \end{pmatrix} \qquad \qquad <br /> (which is the Dirac representation)
so that \ \ \qquad \qquad i \hbar \gamma^0 = <br /> \begin{pmatrix}<br /> i \hbar & 0 & 0 & 0\\<br /> 0 & i \hbar & 0 & 0\\<br /> 0 & 0 & -i \hbar & 0\\<br /> 0 & 0 & 0 & -i \hbar<br /> \end{pmatrix} \qquad \qquad <br />