Re: Duplication Thought Experiment Involving Complementarity

From: Russell Standish <>
Date: Wed, 11 Sep 2002 10:58:22 +1000 (EST)

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.


A/Prof Russell Standish Director
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Received on Tue Sep 10 2002 - 18:07:03 PDT

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