I D'Alembertian in Sakurai's Advanced QM book

Wrichik Basu

Gold Member
2018 Award
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?

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Gaussian97

One of the properties of $\gamma$ matrices is $$\left\{\gamma^\mu,\gamma^\nu\right\}=2g^{\mu\nu}$$

Wrichik Basu

Gold Member
2018 Award
One of the properties of $\gamma$ matrices is $$\left\{\gamma^\mu,\gamma^\nu\right\}=2g^{\mu\nu}$$
Thank you, that perfectly solves the problem.

I don't know why Sakurai is reluctant to use the metric tensor. He just writes $\{ \gamma_\mu , \ \gamma_\nu \} = 2 \delta _{\mu \nu}$. In fact, in chapter 1, he says,
Note that we make no distinction between a covariant and a contravariant vector, nor do we define the metric tensor. These complications are absolutely unnecessary in the special theory of relativity. (It is regrettable that many textbook writers do not emphasize this elementary point.)

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Gaussian97

Well, this is possible by defining 4-vectors as $x^\mu=(ct,ix,iy,iz)$. Then $$x^\mu x_\mu=\delta_{\mu\nu}x^{\mu}x^{\nu}=c^2t^2-x^2-y^2-z^2$$

Wrichik Basu

Gold Member
2018 Award
Well, this is possible by defining 4-vectors as $x^\mu=(ct,ix,iy,iz)$. Then $$x^\mu x_\mu=\delta_{\mu\nu}x^{\mu}x^{\nu}=c^2t^2-x^2-y^2-z^2$$
OK, but that's not what most other books use. At least I have learnt till date $x^\mu=(ct,x,y,z)$. I am really getting confused with the notation in this book.

Gaussian97

OK, but that's not what most other books use. At least I have learnt till date $x^\mu=(ct,x,y,z)$. I am really getting confused with the notation in this book.
Yes, I think nowadays using $(t,x,y,z)$ with $c=1$ is the most common way to proceed, so maybe you want to change your book... Sakurai is from 1967.

Wrichik Basu

Gold Member
2018 Award
so maybe you want to change your book... Sakurai is from 1967.
Sakurai is quite a recommended book, that's why I started reading it. But I'll not continue with this book any longer.

Gaussian97

Well, I've never read Sakurai's book, but it's true it has a lot of popularity. You need to think if it's worth for you. Maybe you can start with something else and then return to Sakurai when you think you'll be able to be comfortable with the notation.

George Jones

Staff Emeritus
Gold Member
Sakurai is quite a recommended book, that's why I started reading it. But I'll not continue with this book any longer.
Sakurai has written (at least) two different books, "Advanced Quantum Mechanics" and "Modern Quantum Mechanics". I have seen both books recommended, but I have seen "Modern Quantum Mechanics" recommended much more often than "Advanced Quantum Mechanics".

"Modern Quantum Mechanics" is probably the text used most often in North America for graduate quantum mechanics courses.

atyy

OK, but that's not what most other books use. At least I have learnt till date $x^\mu=(ct,x,y,z)$. I am really getting confused with the notation in this book.
It is old notation for special relativity. It doesn't carry over (as far as I know) into general relativity. You can see an example of the change of notation from SR to GR also in 't Hooft's notes http://www.staff.science.uu.nl/~hooft101/lectures/genrel_2013.pdf.

vanhees71

Gold Member
Advanced Quantum Mechanics is a bit old-fashioned. One reason is the use of imaginary components to describe four-vectors.

"D'Alembertian in Sakurai's Advanced QM book"

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