Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 4 Apr 1999 17:25:40 +0300 (EET DST)
Below an observation stimulated by Ben Goertzel's claim about discreteness
as something fundamental. I think Ben is right in a well defined sense!
p-Adic numbers with norm bounded by some upper bound
are of from x= p^rk , where k is possibly *infinite* integer
having pinary expansion k= SUM(n>=0) k_np^n. These numbers form a
discrete set! The real counterparts of these numbers in canonical
identification SUM x_np^n --> SUM x_np^(-n) however form a continuum!
In fact, Fourier analysis for p-adic planewaves providing representation
for translations is possible and p-adic planewaves are orthonormal with
*Kronecker delta normalization* rather than Dirac delta renormalization.
The real counterparts of p-adic momenta with fixed
upper bound form discrete set whereas their
p-adic counterparts correspond to possibly infinite integers. Thus p-adic
planewaves are more beautiful mathematically since infinities are avoided
in the definition of inner product and there is no need to introduce
distributions.
Since the coordinates of the infinite-dimensional configuration space
are proportional to possibly infinite integers one has discrete in
this sense also at the level of configuration space.
Best,
MP
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