[time 773] Re: [time 769] Re: [time 767] Generalizing the concept of integer, rational and real, etc.. (fwd)

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 16 Sep 1999 19:38:16 +0300 (EET DST)

Hi Stephen and all,

I made an erraneous statement while answering to Stephen's
question about infinite integers. Infinite integers form
infinite-dimensional vector space with *integer* coefficients
having as basis infinite integers expressible as powers of
*only* infinite integers. For the infinite integers interpreted
as p-adic integers inner product exist at least formally. See below.

On Thu, 16 Sep 1999, Stephen P. King wrote:

> Dear Matti et al,
> Just to let you know, there is a plug-in for Netscape and MS Explorer
> available to view LaTeX and Tex at:
> http://www.software.ibm.com/network/techexplorer/downloads/
> I interleaved a few questions and comments...
> Matti Pitkanen wrote:
> >
> > \begin{eqnarray}
> > N&=& m_0+ \sum m_iM_i\per .
> > \end{eqnarray}
> >
> > \noindent This representation of infinite integers
> > indeed looks promising from the point of view of
> > practical calculations. The representation looks also
> > attractive physically.
> > The integers $m_0$ and $m_iM_i$ are formally analogous to a many-boson
> > states consisting of $n_k$ bosons in the mode characterized by finite
> > or infinite prime $p_k$. Therefore this
> > representation is analogous to a quantum superposition
> > of bosonic Fock states with binary, rather than complex valued,
> > superposition coefficients.
> Could you elaborate on this last point a bit more?

Here the original statement was wrong. The correct statement should

"One can simply replace m_0 and m_iM_i with

m_0|1> and m_i|M_i> (the error was here!!)

in above formula to see the analogy. The usual complex coefficients c_m of
basis states are now *integers*. But this is only linear space. I do not
no natural inner product."

This is ok in real case. In p-adic context inner product can be formally
defined as

m_0n_0 + SUM_i m_i*n_i

In p-adic context p-adic ultrametricity guarantees that the p-adic
norm of sum is never larger than the maximum of p-adic norm
for summands so that this number, if it exists, is certainly finite.

Note however that the number of elements in SUM should be
*numerable*: otherwise it is not clear whether the sum makes
sense. Weill, I am not however completely sure whether the requirement
about numerability is necessary in p-adic context.


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