The spherical harmonics functions are denoted as $Y_l^m$ or $Y_{lm} $in https://en.wikipedia.org/wiki/Spherical_harmonics

e.g., \begin{align} Y_{lm} &= \dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^m\, Y_\ell^{-m}\right) \text{if}\ m < 0 \end{align}

My question is, does the super and lower indices related to tensor properties as covariant/contravariant? E.g., $Y_{lm} = \sum_n g_{mn} Y_l^n$. I may guess $g_{mn} = \frac{i}{\sqrt{2}}, -(-1)^m \frac{i}{\sqrt{2}}, n=m \text{ or} -m; m<0 $. If yes, is there any advantage to utilize the tensor properties?


1 Answer 1


No. The distinction being made here is between real functions and complex functions. The Wikipedia $Y_{lm}$ are real functions forming an orthonormal basis on the sphere, while the $Y_l^m$ are a basis of complex functions. For example for $l=1$ the $Y_{1m}$ are $\cos\theta$, $\sin\theta\cos\phi$ and $\sin\theta\sin\phi$, while the $Y_1^m$ are $\cos\theta$, $\sin\theta e^{\pm i\phi}$.

Another significant distinction between them is in their eigenvalue properties. The $Y_l^m$ are eingenfunctions of $L_z$ with eigenvalues $m$, while the $Y_{lm}$ are eigenfunctions of $L_x$, $L_y$ and $L_z$ with eigenvalue zero.

That's not to say that there aren't covariant and contravariant versions of the spherical harmonics, although for reasons I've never understood they are usually referred to as cogredient and contragredient. The classic reference for this is Fano Irreducible Tensorial Sets, but there must be more recent texts.


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