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

*Fri, 17 Sep 1999 07:23:30 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 778] Re: [time 775] Re: [time 771] Re: Noumenon and Phenomenon"**Previous message:**Matti Pitkanen: "[time 776] Re: [time 770] Generalizing the concept of inte..."

Hi Stephen,

Thank you for the question about inner product. I added some lines

of text about it to the original text and formulated more precisely

the interpretation of infinite integers as superpositions of

bosocnic Fock states. There are some comments about quaternions and

octonions too.

Best,

MP

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentstyle [10pt]{article}

\begin{document}

\newcommand{\vm}{\vspace{0.2cm}}

\newcommand{\vl}{\vspace{0.4cm}}

\newcommand{\per}{\hspace{.2cm}}

\section{How to generalize the concepts of integer,

rational and real?}

The allowance of infinite primes forces to generalize also the concepts

concepts of integer, rational and real number. It is not

obvious how this could be achieved. The following arguments

lead to a possible generalization which seems practical (yes!) and

elegant.

\subsection{Infinite integers form infinite-dimensional vector space

with integer coefficients}

The first guess is that infinite integers $N$ could be

defined as products of the powers of finite and infinite primes.

\begin{eqnarray}

N&=&\prod_k p_k^{n_k}= nM\per , \per n_k\geq 0\per ,

\label{product}

\end{eqnarray}

\noindent where $n$ is finite integer and $M$ is infinite integer

containing only powers of infinite primes in its product expansion.

\vm

It is not however not clear whether the sums of infinite integers

really allow similar decomposition. Even in the case

that this decomposition exists, there seems to be no way of

deriving it. This would suggest that one

should regard sums

$$ \sum_i n_iM_i$$

\noindent of infinite integers

as infinite-dimensional linear space spanned by $M_i$

so that the set of infinite integers would be analogous

to an infinite-dimensional algebraic extension of say p-adic numbers

such that each coordinate axes in the extension corresponds to single

infinite integer of form $N=mM$.

Thus the most general infinite integer $N$ would have the form

\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. One can interpret the set of integers $N$

as a linear space with integer coefficients $m_0$ and $m_i$:

\begin{eqnarray}

N&=& m_0\vert 1\rangle + \sum m_i\vert M_i\rangle \per .

\end{eqnarray}

\noindent $\vert M_i\rangle $ can

be interpreted as a state basis representing many-particle states formed

from bosons labelled by infinite primes $p_k$ and $\vert 1\rangle$

represents Fock vacuum. Therefore this

representation is analogous to a quantum superposition

of bosonic Fock states with integer, rather than complex valued,

superposition coefficients.

If one interprets $M_i$ as orthogonal state basis

and interprets $m_i$ as p-adic integers, one can define inner

product as

\begin{eqnarray}

\langle N_a,N_b\rangle &=& m_0(a)m_0(b)+ \sum_i m_i(a)m_i(b)\per .

\end{eqnarray}

\noindent This expression is well defined p-adic number

if the sum contains only

enumerable number of terms and is always bounded by p-adic

ultrametricity.

It converges if the p-adic norm of of $m_i$ approaches to zero when

$M_i$ increases.

\subsection{Generalized rationals}

Generalized rationals could be defined

as ratios $R=M/N$ of the generalized integers. This works nicely

when $M$ and $N$ are expressible as products of powers of finite

or infinite primes but for more general integers the definition

does not look attractive. This suggests that one should restrict

the generalized rationals to be numbers having the expansion

as a product of positive and negative primes, finite or infinite:

\begin{eqnarray}

N&=&\prod_k p_k^{n_k}= \frac{n_1M_1}{nM}\per .

\label{rational}

\end{eqnarray}

\subsection{Generalized reals form infinite-dimensional

real vector space}

One could consider the possibility of defining

generalized reals as limiting values of the generalized rationals.

A more practical definition of the generalized reals is

based on the generalization of the pinary expansion of ordinary

real number given by

\begin{eqnarray}

x&=& \sum_{n\geq n_0} x_np^{-n}\per ,\nonumber\\

x_n&\in& \{0,..,p-1\} \per .

\end{eqnarray}

\noindent It is natural to try to generalize this expansion

somehow. The natural requirement is that

sums and products of the generalized reals

and canonical identification map from the generalized

reals to generalized p-adcs are readily calculable.

Only in this manner the representation can

have practical value.

\vm

These requirements suggest the following generalization

\begin{eqnarray}

X&=& x_0+ \sum_{N} x_N p^{-N}\per ,\nonumber\\

N&=& \sum_i m_iM_i\per ,

\end{eqnarray}

\noindent where $x_0$ and $x_N$ are ordinary reals. Note

that $N$ runs over

infinite integers which has {\it vanishing finite part}.

Note that generalized reals can be regarded as infinite-dimensional

linear space such that each infinite integer $N$ corresponds

to one coordinate axis of this space. One could interpret

generalized real as a superposition of bosonic Fock states formed

from single single boson state labelled by prime $p$ such that

occupation number is either $0$ or infinite integer $N$ with a

vanishing finite part:

\begin{eqnarray}

X&=& x_0\vert 0\rangle + \sum_{N} x_N \vert N>\per .

\end{eqnarray}

\noindent The natural inner product is

\begin{eqnarray}

\langle X, Y\rangle &=& x_0y_0 + \sum_{N} x_N y_N\per .

\end{eqnarray}

\noindent The inner product is well defined if the number of

$N$:s in the sum

is enumerable and $x_N$ approaches zero sufficiently rapidly when

$N$ increases. Perhaps the most natural interpretation of the inner

product is as $R_p$ valued inner product.

\vm

The sum of two generalized reals can be

readily calculated by using only sum for reals:

\begin{eqnarray}

X+Y&=& x_0+y_0 + \sum_{N} (x_N +y_N)p^{-N}\per ,\nonumber\\

\end{eqnarray}

\noindent The product $XY$ is expressible in the form

\begin{eqnarray}

XY&=& x_0y_0 +x_0Y+Xy_0 + \sum_{N_1,N_2} x_{N_1} y_{N_2}

p^{-N_1-N_2}\per ,\nonumber\\

\end{eqnarray}

\noindent If one assumes that

infinite integers form infinite-dimensional vector space in the

manner proposed, there are no problems and one can calculate

the sums $N_1+N_2$ by summing component wise manner

the coefficients appearing in the sums defining

$N_1$ and $N_2$ in terms of

infinite integers $M_i$ allowing expression as a product

of infinite integers.

\vm

Canonical identification map from ordinary reals to p-adics

$$x =\sum_k x_kp^{-k}\per \rightarrow \per x_p= \sum_k x_kp^{k}\per ,$$

\noindent generalizes to the form

\begin{eqnarray}

x&=& x_0+ \sum_{N} x_N p^{-N}\per \rightarrow \per

(x_0)_p+ \sum_{N} (x_N)_p p^{N}\per ,

\end{eqnarray}

\noindent so that all the basic requirements

making the concept of generalized real calculationally useful

are satisfied.

\vm

There are several interesting questions related to generalized reals.

a) Are the extensions of

reals defined by various values of p-adic primes mathematically

equivalent or not? One can map

generalized reals associated with various choices of the base $p$

to each other in one-one manner using the mapping

\begin{eqnarray}

X&=& x_0+ \sum_{N} x_N p_1^{-N} \per \rightarrow \per x_0+

\sum_{N} x_N p_2^{-N}\per .\nonumber\\

\end{eqnarray}

\noindent The ordinary real norms of

{\it finite} (this is important!) generalized reals are

identical since the representations

associated with different values of base $p$ differ from each other only

infinitesimally.

This would suggest that the extensions are

physically equivalent.

It these extensions are not mathematically equivalent then p-adic

primes could have a deep role in the definition of the generalized

reals.

b) One can generalize previous formulas for the generalized

reals by replacing the

coefficients $x_0$ and $x_i$ by complex numbers, quaternions or

octonions so as to get generalized complex numbers, quaternions and

octonions. Also inner product generalizes in an obvious manner.

The 8-dimensionality of the imbedding

space provokes the question whether it might be possible to regard

the infinite-dimensional configuration space of 3-surfaces,

or rather, its tangent space, as a Hilbert space realization of the

generalized octonions. This kind of identification could perhaps reduce

TGD based physics to generalized number theory.

\end{document}

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