Given some spherical harmonic of the form

$ \textbf{Y}_l^m = (i)^lY_l^m$

Where $Y_l^m$ is a standard spherical harmonic, I would like to find the action of the time-reversal operator $T$. My attempt is as follows:

The action of the time reversal operator on a regular spherical harmonic is given by

$TY_l^m = Y_l^{m*} = (-1)^mY_l^{-m}$

So the action on $\textit{our}$ spherical harmonic is then

$T\textbf{Y}_l^m = [(i)^l]^*(-1)^mY_l^{-m}$

So somehow this should simplify to

$T\textbf{Y}_l^m = (-1)^{l-m}\textbf{Y}_l^m = (-1)^{l-m}(i)^lY_l^m$


From the second to last equation,

$$T\textbf{Y}_l^m = [(i)^l]^*(-1)^mY_l^{-m}$$

you need to show that

$$[(i)^l]^*=(-1)^l i^l$$

If you use $i=e^{i\pi/2}$, then

$$[i^l]^* = (e^{i\pi l/2})^* = e^{-i\pi l/2} = e^{i\pi l/2}e^{-i\pi l} = i^l (-1)^l$$

by substituting this in on the right and using the definition, $\textbf{Y}_l^m = (i)^lY_l^m $, you can get the desired expression:

$$T\textbf{Y}_l^m = (-1)^{l-m}\textbf{Y}_l^m = (-1)^{l-m}(i)^lY_l^m$$


You can't reduce the time reversed equation $(1)$ to what appears to resemble the time forward equation $(2)$

$$T\textbf{Y}_l^m = [(i)^l]^*(-1)^mY_l^{-m} \tag 1$$

$$T\textbf{Y}_l^m = (-1)^{l-m}\textbf{Y}_l^m = (-1)^{l-m}(i)^lY_l^m \tag 2$$

unless you apply the inverse time reversal operator $T^{-1}$ operator to equation $(1)$ from the right hand side.

Also, when you applied the $T$ operator to equation $(2)$ it should have produced the complex conjugate spherical harmonics as in equation $(1)$ - but it didn't - so you need to loose the $T$ operator.

The spherical harmonics in equation $(2)$ are the same as in your first equation

$$\textbf{Y}_l^m = (i)^lY_l^m$$


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