Correct relation is F^{ij} = - epsilon^{ijk} B^k.

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Discussion Overview

The discussion centers around the derivation of the relation F^{ij} = - epsilon^{ijk} B^k, focusing on the conventions used in the derivation and the implications of different metric signatures on the results. The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant reports deriving the relation with a wrong sign and seeks feedback on their mistakes.
  • Another participant suggests that the correctness of the relation may depend on the conventions used, which were not specified by the original poster.
  • A third participant introduces the west-coast convention, explaining the relationship between derivatives and the magnetic field B^i in terms of the gauge potential A^k.
  • The original poster acknowledges the response but points out that their use of a specific metric (diag(1,-1)) leads to a situation where the signs in the expressions for B^i and A^i cancel each other out, requesting a complete derivation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the derivation of the relation, as differing conventions and metric signatures lead to varying interpretations and results.

Contextual Notes

The discussion highlights the dependence on conventions and metric choices, which may affect the signs in the derived expressions. Specific assumptions regarding the metric and conventions are not fully detailed, leaving some ambiguity in the derivation process.

Who May Find This Useful

Readers interested in theoretical physics, particularly in gauge theories and the implications of different metric conventions in field theory, may find this discussion relevant.

Zohaib_aarfi
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When I tried to derive this relation I got the wrong sign. Please check the pic and tell me my mistakes.
 

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What is correct probably depends on your conventions, which you don't give.
 
If you use the west-coast convention you have ##\partial^l=-\partial_l##, and thus
$$B^i=-\epsilon_{ijk} \partial_j A^k.$$
Note that
$$\partial_j=\frac{\partial}{\partial x^j}.$$
 
Thank you for your response. I am using metric [tex]diag(1,-1)[/tex] and the expression you gave [tex]B^i = - \epsilon_{ijk} \partial_j A^k[/tex] contains also [tex]A^i = - A_i[/tex], so I think it does not make any difference. Could you do it for me in complete and explicit steps?
 

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