- #1

Wrichik Basu

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## Main Question or Discussion Point

In Sakurai's book "Advanced QM", he writes the Dirac equation (equation 3.31 to be exact) as: $$\left(\gamma _\mu \ \dfrac{\partial}{\partial\ x_\mu} + \frac{m\ c}{\hbar}\right) \ \psi= 0$$ which we can write as $$\left(\gamma _\mu \ \partial ^\mu \ + \frac{m\ c}{\hbar}\right) \ \psi= 0$$

Next, we go to the section 3-3, and subsection "Free particles at rest".

The author multiples the Dirac equation from the left by ##\gamma _\nu \ \partial ^\nu##. We get the form $$\left[ \partial ^\nu \ \partial ^\mu \ \gamma_\nu \ \gamma_\mu \ - \left(\frac{m\ c}{\hbar}\right)^2 \ \right]\psi = 0$$ If we interchange the indices ##\mu## and ##\nu##, and add to the above equation, we get $$\left[ \partial ^\nu \ \partial ^\mu\ \left( \gamma_\nu \ \gamma_\mu + \gamma_\mu \ \gamma_\nu\right) \ - 2 \ \left(\frac{m\ c}{\hbar}\right)^2 \ \right]\psi = 0$$ which reduces to $$\Box \ \psi - \left(\frac{m\ c}{\hbar}\right)^2 \ \psi = 0$$ using the anticommutation relations of the ##\gamma## - matrices.

Till now, I have just quoted from the book (except changing the ##\partial / \partial x_\mu## to ##\partial ^\mu##).

I knew the D'Alembertian operator to be ##\partial ^\mu \ \partial _\mu##, but Sakurai seems to be suggesting that it is ##\partial ^\mu \ \partial ^\nu## instead. The two can be related: $$\partial _\mu \ \partial ^\mu \ = \ g_{\mu \nu} \ \partial ^\nu \ \partial ^\mu\ ,$$ but that's not what the author has written. What am I missing?

Here is a picture of the page:

Next, we go to the section 3-3, and subsection "Free particles at rest".

The author multiples the Dirac equation from the left by ##\gamma _\nu \ \partial ^\nu##. We get the form $$\left[ \partial ^\nu \ \partial ^\mu \ \gamma_\nu \ \gamma_\mu \ - \left(\frac{m\ c}{\hbar}\right)^2 \ \right]\psi = 0$$ If we interchange the indices ##\mu## and ##\nu##, and add to the above equation, we get $$\left[ \partial ^\nu \ \partial ^\mu\ \left( \gamma_\nu \ \gamma_\mu + \gamma_\mu \ \gamma_\nu\right) \ - 2 \ \left(\frac{m\ c}{\hbar}\right)^2 \ \right]\psi = 0$$ which reduces to $$\Box \ \psi - \left(\frac{m\ c}{\hbar}\right)^2 \ \psi = 0$$ using the anticommutation relations of the ##\gamma## - matrices.

Till now, I have just quoted from the book (except changing the ##\partial / \partial x_\mu## to ##\partial ^\mu##).

I knew the D'Alembertian operator to be ##\partial ^\mu \ \partial _\mu##, but Sakurai seems to be suggesting that it is ##\partial ^\mu \ \partial ^\nu## instead. The two can be related: $$\partial _\mu \ \partial ^\mu \ = \ g_{\mu \nu} \ \partial ^\nu \ \partial ^\mu\ ,$$ but that's not what the author has written. What am I missing?

Here is a picture of the page: