# Action Of Time-Reversal Operator On Spherical Harmonics

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

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