On 10-Sep-02, Russell Standish wrote:
> 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.

Vic's working on a book, tentatively called "Why There is
Something Rather Than Nothing" in which he shows that all
fundamental physics, i.e. QM, GR, and the Standard Model, can
be derived from gauge+broken symmetry.  From that standpoint
physics is described by 2nd order PDE's because the two
symmetries we observe are space-time translation and
space-time rotation (i.e. Lorentz boosts).  Vic argues that
these are inherent symmetries of the void, but as to why there
aren't other symmetries I think must be left to empirical
determination.

I do urge you to join Vic's avoid-l mailing list.  I'm sure he
would appreciate your well informed comments.


Brent Meeker
Ms Schroedinger: What did you do to that poor cat? It looks
half  dead.
Erwin: I don't know. Ask Wigner's friend.

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