Calculating Indices: Solve the Mystery

[John Fennie]

Would anyone explain how the calculation in the picture was carried out? (the second equal sign)
I don't seem to be able to get the indices right.

 

[Charles Link]

Would anyone explain how the calculation in the picture was carried out? (the second equal sign)
I don't seem to be able to get the indices right.

When the indices are the same in both terms in a product, it means the product is summed/repeated over all the coordinates. When they are different, they are only counted once. Is that helpful?

 

[dextercioby]

Is there any reason why the repeated indices are not placed in opposite directions (i.e. one „upstairs” and one „downstairs”)?

 

[Orodruin]

Is there any reason why the repeated indices are not placed in opposite directions (i.e. one „upstairs” and one „downstairs”)?

Some authors (I know Schwartz does and mentions it in the preface of his QFT book) find it so obvious that repeated indices should be contracted using the metric that they resort to writing all indices in one position.

 

[dextercioby]

I was suspecting an $x_4 = ict$ there, so that is why I asked. I think Schwartz had a bad idea. There's elegance in using indices that way. Not to mention there is a difference between $F_{\mu\nu}, F^{\mu\nu}, F_{\mu}^{~\nu}, F^{\nu}_{~\mu}$. How is that handled?

 

[Orodruin]

It is quite clear that you can do whatever you want with free indices as long as you do it on both sides, so that is not a problem. When it comes to summation indices, it is clear that one needs to be taken as contravariant and the other as covariant and it really does not matter which is which so there is no possible misunderstanding there either. I am not saying I approve or that it is a good idea, just that there is no possible confusion if you know what you are doing.

 

[John Fennie]

Would anyone explain how the calculation in the picture was carried out? (the second equal sign)
I don't seem to be able to get the indices right.

Hi yes, i understand that. But I am unable to work the math out, specifically the second +$\frac{1}{2}$

 

[Orodruin]

Could you show your work please?