Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 17 Sep 1999 07:23:30 +0300 (EET DST)
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
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\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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