Brent Meeker wrote:
> 
> On 10-Sep-02, 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?
> 
> Observables come in complementary pairs (instead of triples or
> something else) because the laws of physics are 2nd order
> (partial) differential equations.  Hence a position has a
> canonically conjugate momentum and vice versa.  The reason
> they are related by a Fourier transform is that the action of
> a wave in the Hamilton-Jacobi form of classical mechanics has
> the products of the conjugate variables in the exponent.  See
> Goldstein, section 10-8.
> 
> ...
> 
> Brent Meeker
> "Pluralitas non sunt ponenda sine necessitate"
>       --- William of Ockham
> 

But that just begs the question of why classical dynamics is 2nd
order (or iow why Newtons 2nd law). Vic Stengar seems to have an
answer to this.

Indeed classical dynamics is 1st order in phase space (2nd order only
in real space). So it comes down to extra meaning attached to the
variable x & p.

                                                Cheers

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