**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Thu, 16 Sep 1999 19:38:16 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**WDEshleman@aol.com: "[time 774] Re: [time 770] Generalizing the concept of inte..."**Previous message:**Hitoshi Kitada: "[time 772] Re: [time 771] Re: [time 768] Re: Noumenon and Phenomenon"**In reply to:**WDEshleman@aol.com: "[time 768] Re: Noumenon and Phenomenon"

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

read

"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.

Best,

MP

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