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From: Russell Standish <R.Standish.domain.name.hidden>

Date: Wed, 11 Sep 2002 10:58:22 +1000 (EST)

George Levy wrote:

*> >
*

*> >Complementarity is a property of any two quantum operators that are
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*> >related by the Fourier transform (x <-> id/dx). The proof is well
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*> >known, and can be found (eg) in Shankar's book.
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*> >
*

*>
*

*> Come on! This is circular reasoning. Conventional QM complementarity
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*> 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
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*> transforms for example?
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*>
*

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

----------------------------------------------------------------------------

A/Prof Russell Standish Director

High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile)

UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 (")

Australia R.Standish.domain.name.hidden

Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks

International prefix +612, Interstate prefix 02

----------------------------------------------------------------------------

Received on Tue Sep 10 2002 - 18:07:03 PDT

Date: Wed, 11 Sep 2002 10:58:22 +1000 (EST)

George Levy wrote:

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

----------------------------------------------------------------------------

A/Prof Russell Standish Director

High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile)

UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 (")

Australia R.Standish.domain.name.hidden

Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks

International prefix +612, Interstate prefix 02

----------------------------------------------------------------------------

Received on Tue Sep 10 2002 - 18:07:03 PDT

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