# Tag Info

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Comments to the question (v1): Indices are raised and lowered vertically by the pertinent metric tensor of the theory. The horizontal position of indices is important for a tensor that is not totally symmetric, e.g., the EM field strength $F_{\mu\nu}$ or the Riemann curvature tensor $R_{\mu\nu\lambda\kappa}$, etc, in order to properly identify which ...

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On a two-index tensor, swapping the two indices is equivalent to transposing a matrix. You may not see many authors spending a lot of effort on this issue simply because an awful lot of the tensors we deal with are symmetric. This includes the metric, Ricci tensor, Einstein tensor, and stress-energy tensor. Therefore there is no special interest in ...

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Comments to the question (v1): As usual, be prepared that different authors use different conventions and notations. E.g. what some authors call a vielbein might be what other authors call a transposed vielbein. A curved index (aka. as coordinate index) is raised and lowered vertically with the curved metric tensor, while a flat index (aka. as vielbein ...

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This graphic on Things Made Thinkable uses the $\bar\Delta^-$ notation, which corroborates rob's prediction.

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As stated in the comment, the "fish" is the mathematical symbol for "proportional to". In the case of the amplitude and energy, $$E\propto A^2$$ means that $$E=CA^2$$ for some constant $C$.

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It seems that "fish" refers to $\propto$. It is means "proportional to". Since @Kyle Kanos has provided useful links about this, I won't repeat them. But I want to add a useful tool for checking unfamilar symbols: http://detexify.kirelabs.org/classify.html You can draw the symbol and get its latex code, and then it's easy to find its meaning (usually the ...

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Here are the two most important tricks: Commutators via derivative If $A$ is an operator which has been written as a normal ordered$^{[1]}$ product of $a$ and $a^\dagger$, then the following is true $$[a, A] = \frac{\partial A}{\partial a^\dagger} \quad \textrm{and} \quad [a^\dagger, A] = -\frac{\partial A}{\partial a} \, .$$ Wick's theorem (boson ...

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