Calculating Indices: Solve the Mystery

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Discussion Overview

The discussion revolves around the calculation of indices in a mathematical expression, specifically focusing on the handling of repeated indices and their positions. Participants are seeking clarification on the methodology behind the calculation depicted in a referenced picture, with an emphasis on the implications of index placement and summation conventions.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about the calculation process related to indices and express difficulty in understanding the second equal sign in the equation.
  • There is a discussion about the meaning of repeated indices in products, with one participant suggesting that identical indices indicate summation over coordinates, while differing indices are counted once.
  • Questions are raised regarding the placement of repeated indices, with some participants noting that certain authors prefer to write all indices in one position, implying a contraction with the metric.
  • One participant expresses skepticism about the elegance of using indices in a particular way, referencing a specific author and questioning the handling of different forms of indices.
  • Another participant asserts that free indices can be manipulated freely as long as the same operations are applied to both sides, while emphasizing the need for clarity in summation indices being treated as contravariant and covariant.
  • Repeated requests for clarification on the calculation indicate ongoing confusion about the specific mathematical steps involved.

Areas of Agreement / Disagreement

Participants express varying opinions on the handling of indices, with some agreeing on the principles of index manipulation while others question specific practices. The discussion remains unresolved regarding the best approach to index placement and calculation.

Contextual Notes

Participants reference different conventions and practices in literature, indicating a lack of consensus on the preferred methods for handling indices. There are also indications of missing details in the mathematical steps that are causing confusion.

John Fennie
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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.
 

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John Fennie said:
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?
 
Is there any reason why the repeated indices are not placed in opposite directions (i.e. one „upstairs” and one „downstairs”)?
 
dextercioby said:
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.
 
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?
 
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 said:
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 +$\frac{1}{2}$
 
Could you show your work please?
 

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