George Levy wrote:
> >
> >Complementarity is a property of any two quantum operators that are
> >related by the Fourier transform (x <-> id/dx). The proof is well
> >known, and can be found (eg) in Shankar's book.
> >
> 
> Come on! This is circular reasoning. Conventional QM complementarity 
> requires 2D Fourier. Therefore 2D Fourier must describe complementarity. 
> True for conventional QM. I was talking about other MWs within the 
> Plenitude. Could their complementarity be described by Hadamar 
> transforms for example?
> 

Not sure - the Hadamard transform is defined on a 2D vector space, and
is equivalent to rotations by 45 degrees. This is rather
restrictive. However, you could make your point with other transforms
perhaps a Laplace transform, or wavelets. I suspect that the proper
transform to use depends on what are the natural boundary conditions,
ie if your wavefunctions are elements of L^2, then the Fourier
transform is the only transform that makes sense.

In more general terms, a Heisenberg uncertainty relation of the form 

    \Delta X \Delta Y >= const

must hold if [X,Y]=const

Of course, in general [X,Y] is an operator, not a number. Can there be
3-way complementary structure? What if [X,[Y,Z]]=const? What does it
all mean? The mathematics of 3D rotations (or equivalently
Quarternions) has some interesting properties, which could be
important in all this.


                                                Cheers


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