[time 156] Re: [time 150] Re: [time 81] Discreteness and p-adics

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 4 Apr 1999 20:41:47 +0300 (EET DST)

On Sun, 4 Apr 1999, Ben Goertzel wrote:

> Hi all,
> to avoid getting several copies of each e-mail, let's please address
> messages only to
> time@kitada.com and not additionally to individuals within the group ;)
> >The point is that k can be also *infinite* as integer: these have finite
> >p-adic norm also. For instance 1/1-p =1+p+p^2+.... is completely well
> >defined p-adic number with p-adic norm one.
> >
> >One could argue that discreteness is only apparent but the fact p-adic
> >Fourier basis obeys Kronecker delta normalization suggests that this is
> >not the case.
> What is your definition of "discrete" if it's not cardinality ;)

My criteria are physics based.
The behaviour of plane waves like discrete planewaves could be one
criterion. Second criterion is also physics based.
 By super Virasoro invariance mass squared eigenvalues of
elementary particle states
are always non-negative integers: M^2= n in suitable units.
With respect to p-adic thermodynamics n behaves as integer.
Thermal mass squared is sum SUM p(n)n where p(n) are thermal masses and
two lowest n:s are important since p(n) behaves as p^n is converges
extremely rapidly with respect to p-adic norm.
On the other hand, the real counterparts of masses in canonical
identification with *infinite n:s allowed* fill the interval
[0,p] so that mass spectum is continuous in this sense.

By the way, the real counterpart for the spectrum of p-adic quantum
harmonic oscillator with spectrum E=n is continuous so that
p-adic quantum oscillator behaves like classical oscillator in this



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