But there’s definitely something to say: the representation theory of SO(3) dictates the decomposition of functions on a sphere into spherical harmonics.

I’m still trying to work out the right way to approach finite and infinite dimensional non-abelian groups. ]]>

Then, yes, I define clockwise rotations in the x-p plane, which the operator x^2+p^2 is invariant under.

Then I look at infinitesimal transformations that generate the linear transformation. On the positive x-axis a small clockwise rotation is a displacement along the -p axis (it lies tangent to the unit circle), and on the positive p-axis a vanishingly small clockwise rotation is a displacement along the x-axis. In fact the generators of the transformation map x to -p and map p to x. It’s only when we bring back the algebraic structure of that it looks like a fourier transform: multiplication by x is mapped to and is mapped to ix which is a defining property of the Fourier transform. [x and p as an algebra, that is adding compositions like xp, generate all the self-adjoint linear operators in this space – this is more of a definition of our linear operators and our vector space than a consequence].

It sounds a bit artificial really; all I am really saying at the end is that is invariant under a Fourier transform (or more generally a map , , which can be given by where F is the Fourier transform and powers denote composition – to see why this works substitute F^2=- Id into the power series.)

This symmetry then in some way carries on to the solutions – in particular a Fourier transform must map a solution to another solution. I find the eigenvectors in x-p space, and then play around with them to get the eigenvalues of x^2+p^2. There’s something missing; I think it may boil down to a simple representation theory calculation, but I don’t know what group I would try to represent.

]]>Then you define a linear transformation say which maps elements of this new two-dimensional vector space to itself, and applying this map to the operators ‘x’ and ‘p’ defined earlier, gives us some new operators such that 1/2(x’^2 + p’^2) = 1/2(x^2 + p^2),

Then you look at the infinitesimal transformations that generate the linear transformation, and apparently this gives us the Fourier transform, but at this point I can’t see how to relate the idea of a transformation of operators to a transformation such as the Fourier transform?

]]>It should be fixed now.

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