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%Copyright (c) 1999 by William D. Eshleman
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\title{A Many-Worlds Product Paradigm for\\
Quantum Inertia and Quantum Gravity}
%File name: \fn{testmath.tex}}
\author{By William D. Eshleman\\
(C) 1998-2000 by William D. Eshleman, All Rights Reserved\\
Permission to copy in its entirety granted for non-commercial purposes}
%\date{29 October 1996 \\
%Version 1.02}
\usepackage{amsmath,amsthm,graphicx}
\begin{document}
\maketitle
\small
\noindent
{\bf Abstract.}
The conventional method for specification of a formalism for reality
is to assume and determine for the chosen space, the following
collection of mathematical objects: 1) state as a vector in a vector
space, 2) observable as an operator that acts on the vector space, and
3) an algebra determined by the set of operators and certain
constraints. Usually, this requires a detailed analysis of how vectors
add in the chosen space. That is, the evolution of the state of a vector
is a process whereby differential vectors are added to a vector. The
original many-worlds interpretation took this approach and assumed the
Hilbert space.
Detailed analysis of how vectors could magnify one another in the chosen
space is usually avoided due to a change of the type of a vector upon
multiplication; e.g., the product of two position tensors is not a tensor,
but sums of tensors remain tensors. The report presented here attempts
to investigate hypothetical mathematical products that predict equivalent
results, as compared to conventional results, for relatively low-energy
trajectories, and then to compare the differences for high-energy motion.
The assumed collection of mathematical objects for this report is:
1) state as a product of factors (resembling vectors) in a world of an
observer, 2) observable as the logarithm of a particular factor that acts
on the space of the observer, and 3) the algebra of convergent products,
the notion of the equivalence of products, and the notion of the entropy
of products. That is, the evolution of the state of a vector is a process
whereby vectors magnify to produce change; the Product Paradigm.
\newpage
\medskip
\leftskip=0pt
\rightskip=0pt
\normalsize
\vskip 8pt
\begin{center}
{Preface reprinted from The Many-Worlds Interpretation of Quantum Mechanics}
{(A Fundamental Exposition by HUGH EVERETT, III,
with papers by J. A. WHEELER, B. S. DEWITT, et. al.
and edited by BRYCE S. DEWITT and NEILL GRAHAM, 1973)}\\
\begin{center}
** This is the beginning of Dr. DeWitt's preface of Dr. Everett's work **\\
\end{center}
\vskip 12pt
\textbf{PREFACE}
\end{center}
\indent In 1957, in his Princeton doctoral dissertation, Hugh Everett, III,
proposed a new interpretation of quantum mechanics that denies the existence
of a separate classical realm and asserts that it makes sense to talk about a
state vector for the whole universe. This state vector never collapses and
hence reality as a whole is rigorously deterministic. This reality, which is
described jointly by the dynamical variables and the state vector, is not the
reality we customarily think of, but is a reality composed of many worlds. By
virtue of the temporal development of the dynamical variables the state vector
decomposes naturally into orthogonal vectors, reflecting a continual splitting
of the universe into a multitude of mutually unobservable but equally real
worlds, in each of which every good measurement has yielded a definite result
and in most of which the familiar statistical quantum laws hold.
\indent In addition to his short thesis Everett wrote a much larger exposition
of his ideas, which was never published. The present volume contains both of
these works, together with a handful of papers by others on the same theme.
Looked at in one way, Everett's interpretation calls for return to naive realism
and the old fashioned idea that there can be direct correspondence between
formalism and reality. Because physicists have become more sophisticated than
this, and above all because the implications of his approach appear to them so
bizarre, few have taken Everett seriously. Nevertheless his basic premise
provides such a stimulating framework for discussions of the quantum theory of
measurement that this volume should be on every quantum theoretician's shelf.\\
\noindent ``... a picture, incomplete yet not false, of the universe as Ts'ui
Pen conceived it to be. Differing from Newton and Schopenhauer, ... [he] did
not think of time as absolute and uniform. He believed in an infinite series
of times, in a dizzily growing, ever spreading network of diverging, converging
and parallel times. This web of time -- the strands of which approach one
another, bifurcate, intersect or ignore each other through the centuries
-- embraces every possibility. We do not exist in most of them. In some you
exist and not I, while in others I do, and you do not, and in yet others both
of us exist. In this one, in which chance has favored me, you have come to my
gate. In another, you, crossing the garden, have found me dead. In yet another,
I say these very same words, but am an error, a phantom."
\indent Jorge Luis Borges, The Garden of Forking Paths \\
\noindent ``Actualities seem to float in a wider sea of possibilities from out
of which they were chosen; and somewhere, indeterminism says, such possibilities
exist, and form a part of the truth."\\
\indent William James
\begin{center}
** This is the end of Dr. DeWitt's preface of Dr. Everett's work **\\
\end{center}
\newpage
\begin{center}
\textbf{Einstein on Space}
\end{center}
\indent ``When a smaller box s is situated, relatively at rest, inside the hollow space
of a larger box S, then the hollow space of s is a part of the hollow space of S,
and the same ``space", which contains both of them, belongs to each of the boxes.
When s is in motion with respect to S, however, the concept is less simple.
One is then inclined to think that s encloses always the same space, but a variable
part of the space S. It then becomes necessary to apportion to each box its
particular space, not thought of as bounded, and to assume that these two spaces
are in motion with respect to each other.
\indent{Before one has become aware of this complication, space appears as an unbounded
medium or container in which material objects swim around. But it must now be
remembered that there is an infinite number of spaces, which are in motion with
respect to each other. The concept of space as something existing objectively
and independent of things belongs to pre- scientific thought, but not so the
idea of the existence of an infinite number of spaces in motion relatively to each
other. This latter idea is indeed logically unavoidable, but is far from having
played a considerable role even in scientific thought."\\
\section{Introduction}
\indent {We begin, as a way of entering our subject, by the development of an
infinite product identity for the Lorentz (inertial) factor of Einstein's
Special Relativity. This identity was discovered by fortuitous accident, and
what was initially a curious mathematical object, has resulted in the extension
of infinite product theory and the suggestion that high-energy cosmology need
not proceed toward chaos and mathematical breakdown. The definitions and proofs
that follow this introduction are included to reinforce confidence that the
interpretation is sound in a mathematical sense.}
\indent Since so much of the method of this theory has been taken directly from
the work of Everett, it helps to be exposed to some interpretation of Everett's
perspective (as I see it); Everett's view is probably more of a paradigm shift in
causality. Whereas the Newtonian model requires that the deviation from straight-line
motion in a gravitational field be due to forces ``acting-at-a-distance" under the
control of The Universal Law of Gravitation, the Everett perspective is probably
closer to Einstein's ``warp in the continuum" without the need for space-time to be
that continuum. Motion is accomplished on paths through symmetrical peaks and valleys
of an entropy continuum. ``Action-at-a-distance" does not apply because the entropy
structure of space-time, containing mass and motion, does not possess the information
required to distinguish between meters, seconds, or kilograms. In other words, motion
does not depend on ``invisible-springs", ``acting-at-a-distance", connecting massive
bodies over empty space, but depends instead on ``invisible and symmetrical dents"
in the entropy continuum that can constrain motion to within the ``dents" like a
ball-bearing in a coffee cup.
%\indent The word, paradigm, is used in the title of this small treatise because
%the intent is to show a generally agreeable side-by-side comparison of the
%properties of a very particular mathematical object to the Newton-Einstein
%model of orbital motion. Perhaps the most notable disagreement is that the
%object's so-called event horizon is not located at the expected radius of $2GM/c^2$,
%but shifts closer to $1.4GM/c^2$, inside of which orbital calculations cannot be made.
\indent A consequence of this analysis is that a mathematical structure identical
to the proposed inertial factor is the only candidate for the gravitational
factor that preserves the classical model for orbital motion into the realm
of high-energy astrophysics. A statement of the equivalence of inertial and
gravitational information of these two factors is this theory's General Relativity
of Motion.
\section{A Priori Notions of Relative State}
\indent Our most accepted notion of state is
that it is due to the property of
the exponential function that, for $h$ a constant,
\begin{eqnarray}
\frac {d \psi} {d t}(t) =
h \; \psi(t) \; ,
\nonumber
\end{eqnarray}
\noindent has the solution, for $\psi_0 f_0^0 = \psi(0)$ and $\psi_0f_1^t = \psi(t)$,
\begin{eqnarray}
\psi_0 f_1^t=
\psi_0 f_0^t (e^h)^t=
\psi_0 f_0^t (1 + h + \frac{(h)^2} {2!} + \frac{(h)^3} {3!} + ...)^t \; ,
\nonumber
\end{eqnarray}
\noindent or,
\begin{eqnarray}
\frac {f_1} {f_0}=
e^h=
(1 + h + \frac{(h)^2} {2!} + \frac{(h)^3} {3!} + ...) {\;},
\nonumber
\end{eqnarray}
\noindent the classical notion of subjective relative state
independent of observation and time.
\indent On the other hand, relativity theory predicts what observers
perceive objectively; distorted by the process of observation itself.
For example, inertially, before and after an acceleration,
\begin{eqnarray}
m =
\frac {m_0} {\sqrt{1-v^2/c^2}}
\nonumber
\end{eqnarray}
\noindent or,
\begin{eqnarray}
m^2 =
{m_0}^2 + \frac {v^2} {c^2} m^2
\nonumber
\end{eqnarray}
\noindent where $m_0$ is the rest mass and m is the mass in motion.\\
\noindent And, gravitationally (I suggest), before and after a
displacement from infinity,
\begin{eqnarray}
m =
m_0 + \frac {GM} {Rc^2} m
\nonumber
\end{eqnarray}
\noindent or,
\begin{eqnarray}
m =
\frac{m_0} {1-GM/R/c^2}
\nonumber
\end{eqnarray}
\noindent where M is a central mass separated from m by a
distance of R and $m_0$ is the mass when infinitely distant
from M.\\
\indent Therefore, our relativistic/objective notion of relative state is:
\begin{eqnarray}
\frac {f_1} {f_0}=
1 + x+ x^2 + x^3 + ...\; ,
\nonumber
\end{eqnarray}
\begin{eqnarray}
= \frac {1} {1-x} {\;},{\;} [Model {\;} 4]
\nonumber
\end{eqnarray}
\noindent where $x=v^2/c^2$, or (I suggest) $x=GM/R/c^2$, or (I suggest)
$x=h$.\\
\indent The goal is now to separate the subjective relative state from the
objective relative state to reveal the physics as it truly proceeds. That is,
\begin{eqnarray}
Objective \; Distortion = \frac {1} {(1-x)e^x}
= \frac {1} {e^x - xe^x} \;,
\nonumber
\end{eqnarray}
\noindent a most difficult task, fraught with ambiguity and yielding only
troublesome interpretations. Another method is to invent a ``factorial operator"
that divides every $i^{th}$ term by i!. This operator is also hard to justify on
physical grounds.
\indent There does exist, though, a form for $1/(1-x)$ that yields a separation
of a candidate for subjective relative state, e.g.,
\begin{eqnarray}
\frac {f_1} {f_0} =
\prod_{i=0} ^{\infty}
{(1+x^{2^i})} ^{2^{-i}} , {\;}
[Model {\;} 5]
\nonumber
\end{eqnarray}
\noindent The above relationship is this theory's
assumed notion of subjective relative state. Figure 1. compares
Model 5 (candidate for subjective relative state), Model 4 (relativistic
objective relative state), $e^x$ (classical subjective relative state), and
$1+x$ (simple relative state).\\
\indent The reasoning concerning many-worlds is that without the
effect of many-worlds, the relative state would follow the simple relationship
$(1+x)$. That is, the additional factors would be due to the presence
of many-parallel-worlds. Conversely, if all of the additional factors
beyond $(1+x)$ are observational distortions, then simple relative state
$(1+x)$ is the only true candidate for subjective relative state. This
undecidability and the incompatibility of classical and relativistic
state are at the core of the difficulties in the unification of
quantum mechanics and general relativity.
\begin{center}
\includegraphics[width=5.0in,height=5.0in]{fig1.bmp}\\
Figure 1. Comparison of Clocks
\end{center}
%\begin{center}
\section{Proof of the Mathematical Object}
%\end{center}
\textbf{I. Lemma.}
\begin{eqnarray}
1-x^{2^N} =
(1-x) \prod_{n=0}^{N-1}
(1+x^{2^n}), {\;} N=1,2,...
\nonumber
\end{eqnarray}
\underline{\textit{Proof}}\\
\indent a) at N = 1,
\begin{eqnarray}
1-x^2 =
(1-x) (1+x)
\nonumber
\end{eqnarray}
\indent b) Inductively, the lemma is true for N = 1,...,M
\begin{eqnarray}
1-x^{2^{M+1}} =
1-(x^{2^M})^2
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
(1-x^{2^M}) (1+x^{2^M})
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
(1-x) \prod_{n=0} ^{M-1} (1+x^{2^n}) (1+x^{2^M})
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
(1-x) \prod_{n=0} ^{M} (1+x^{2^n})
\nonumber
\end{eqnarray}
\textbf{II. Lemma.}
\begin{eqnarray}
\frac {1} {1-x} =
\prod_{n=0} ^{\infty} (1+x^{2^n}), {\;}
for {\;} 0{\le}x<1
\nonumber
\end{eqnarray}
\underline{\textit{Proof}}
\begin{eqnarray}
1-x^{2^{N}} =
(1-x) \prod_{n=0} ^{N-1} (1+x^{2^n})
\nonumber
\end{eqnarray}
or,
\begin{eqnarray}
1 =
(1-x) \prod_{n=0} ^{\infty} (1+x^{2^n})
\nonumber
\end{eqnarray}
\newpage
\textbf{III. Lemma.}
\begin{eqnarray}
\prod_{n=m} ^{\infty} (1+x^{2^n}) =
\frac{1} {1-x^{2^m}}
\nonumber
\end{eqnarray}
\underline{\textit{Proof}}
\begin{eqnarray}
\prod_{n=m} ^{\infty} (1+x^{2^n}) =
\prod_{n=0} ^{\infty} (1+x^{2^n})
[\prod_{n=0}^{m-1} (1+x^{2^n})]^{-1}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
\frac {1} {1-x} [\frac{1-x^{2^m}} {1-x}]^{-1}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
\frac {1} {1-x^{2^m}}
\nonumber
\end{eqnarray}
\textbf{IV. Lemma.}
\begin{eqnarray}
\sum_{n=0} ^{m} y^n =
\frac {1-y^{m+1}} {1-y}
\nonumber
\end{eqnarray}
\underline{\textit{Proof}}
\begin{eqnarray}
\sum_{n=0} ^{m} y^n =
1+ \sum_{n=1} ^{m} y^n
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
1+ y \sum_{n=0} ^{m} y^n
-y^{m+1}
\nonumber
\end{eqnarray}
\begin{eqnarray}
(1-y)\sum_{n=0} ^{m} y^n =
1-y^{m+1}
\nonumber
\end{eqnarray}
\begin{eqnarray}
\sum_{n=0} ^{m} y^n =
\frac{1-y^{m+1}} {1-y}
\nonumber
\end{eqnarray}
\newpage
\underline{\textit{Corollary}}
\begin{eqnarray}
1-2^{-m} =
[\sum_{i=0} ^{m-1} 2^{i}]2^{-m}
\nonumber
\end{eqnarray}
\textbf{V. Lemma.}
\begin{eqnarray}
\prod_{m=1} ^{\infty} (1+x^{2^m})^{1-2^{-m}} =
\prod_{m=1} ^{\infty}
\prod_{n=m} ^{\infty} (1+x^{2^n})^{ 2^{-m}}
\nonumber
\end{eqnarray}
\underline{\textit{Proof}}
\begin{eqnarray}
\prod_{m=1} ^{\infty} (1+x^{2^m})^{1-2^{-m}} =
\prod_{m=1} ^{\infty} (1+x^{2^m})
^{(\sum_{i=0} ^{m-1}2^i)2^{-m}}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
\prod_{m=1} ^{\infty} (1+x^{2^n})
^{(\sum_{i=0} ^{n-1}2^{i-n})}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
\lim_{N \to {\infty}}\prod_{n=1} ^{N}
\prod_{i=1} ^n (1+x^{2^n}) ^{2^{-i}}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
\lim_{N \to {\infty}}
%\prod_{\frac {1 \le n \le N} {1 \le i \le n}}
\prod_{\substack{1 \le n \le N\\{1 \le i \le n}}}
(1+x^{2^n})^{2^{-i}}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
\lim_{N \to {\infty}}
%\prod_{\frac {1 \le n \le N} {1 \le i \le n}}
\prod_{\substack{i \le n \le N\\{1 \le i \le N}}}
(1+x^{2^n})^{2^{-i}}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
\prod_{m=1} ^{\infty}
\prod_{n=m} ^{\infty} (1+x^{2^n})^{ 2^{-m}}
\nonumber
\end{eqnarray}
\textbf{VI. Theorem.}
\begin{eqnarray}
\frac {1} {1-x} =
(1+x) \prod_{n=1} ^{\infty}
[ \frac {(1+x^{2^n})} {(1-x^{2^n})} ]^{2^{-n}} , {\;}
for {\;} 0{\le}x<1
\nonumber
\end{eqnarray}
\underline{\textit{Proof}}
\begin{eqnarray}
\frac {1} {1-x} =
\prod_{n=0} ^{\infty}
(1+x^{2^n})
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
(1+x) \prod_{n=1} ^{\infty}
(1+x^{2^n}) ^{2^{-n}} (1+x^{2^n}) ^{1-2^{-n}}
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
[(1+x) \prod_{n=1} ^{\infty}
(1+x^{2^n}) ^{2^{-n}}] \;
[\prod_{n=1} ^{\infty}
(1+x^{2^n}) ^{1-2^{-n}}]
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
[(1+x) \prod_{n=1} ^{\infty}
(1+x^{2^n}) ^{2^{-n}}] \;
[\prod_{n=1} ^{\infty}
\prod_{i=n} ^{\infty}
(1+x^{2^i}) ^{2^{-n}}]
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
[(1+x) \prod_{n=1} ^{\infty}
(1+x^{2^n}) ^{2^{-n}}] \;
[\prod_{n=1} ^{\infty}
(\frac {1} {1-x^{2^n}}) ^{2^{-n}}]
\nonumber
\end{eqnarray}
\begin{eqnarray}
=
(1+x) \prod_{n=1} ^{\infty}
[ \frac {(1+x^{2^n})} {(1-x^{2^n})} ]^{2^{-n}} , {\;}
for {\;} 0{\le}x<1
\nonumber
\end{eqnarray}
%\begin{center}
\section{Notion of Equivalence of Infinite Products}
%\end{center}
\textbf{VII. Definition.}\\
\noindent For $\alpha_n \ge 0, \; \beta_n \ge 0 \; and \; \rho \ge 0,$
\begin{eqnarray}
Infinite \; product \; \textbf{A} =
\prod_{n=1}^{\infty}(1+\alpha_n)
\equiv
\nonumber
\end{eqnarray}
\begin{eqnarray}
Infinite \; product \; \textbf{B} =
\prod_{n=1}^{\infty}(1+\beta_n)
\nonumber
\end{eqnarray}
\newpage
\noindent If, for value \textit{A} of \textbf{A} and value \textit{B} of \textbf{B},
$\textit{A} = \textit{B}^{\; \rho}$
\noindent and, if there is a permutation \textit{f}, such that,\\
\noindent for,\; $a_n =(1+\alpha_{\textit{f}}),$\\
\noindent and,\; $b_n=(1+\beta_n),$\\
\noindent the relation,\; $a_n = b_n^\rho$, \; holds for all n.\\
\noindent \textbf{Remark.} This is an equivalence relation among a class of sequences of
the form ${(1 + a_n)}$ where $a_n$ approaches $0$ as $n$ approaches infinity and
for which the associated infinite product is finite. \\
\noindent \textbf{VIII. Theorem.} (Properties of equivalent infinite products)
Infinite products may be confirmed equivalent without the need for
the knowledge of the value of $\rho$ . That is, equivalence depends on
the existence of $\rho$, not its value.\\
\underline{\textit{Proof}}\\
\noindent For,
\begin{eqnarray}
C = \sum_{n=1} ^{\infty}log(a_n),
\nonumber
\end{eqnarray}
and,\\
\begin{eqnarray}
D = \sum_{n=1} ^{\infty}log(b_n),
\nonumber
\end{eqnarray}
\indent 1) If \textbf{A} and \textbf{B} are equivalent,
\begin{eqnarray}
\frac {log(a_n)} {C} = \frac {log(b_n)} {D}
\nonumber
\end{eqnarray}
\begin{eqnarray}
= log_{\textit{A}}(a_n)
\nonumber
\end{eqnarray}
\begin{eqnarray}
= log_{\textit{B}}(b_n)
\nonumber
\end{eqnarray}
\indent 2) (The Normal Logarithm)
\begin{eqnarray}
1 = \sum_{n=1} ^ {\infty} log_{\textit{A}} (a_n)
\nonumber
\end{eqnarray}
\noindent and,
\begin{eqnarray}
1 = \sum_{n=1} ^ {\infty} log_{\textit{B}} (b_n)
\nonumber
\end{eqnarray}
\noindent \textbf{Remark} Equivalent infinite products need not be Equal Valued
and Equal Valued infinite products need not be Equivalent.\\
\noindent \textbf{IX. Definition.} Measure, \textbf{S}, of the distribution of Normal Logarithms
is defined for equivalent infinite products \textbf{A} and \textbf{B} such that, if,\\
\begin{eqnarray}
\textbf{P}_n = log_{\textit{A}} (a_n)
\nonumber
\end{eqnarray}
\noindent or,
\begin{eqnarray}
\textbf{P}_n = log_{\textit{B}} (b_n)
\nonumber
\end{eqnarray}
\noindent then by Shannon,
\begin{eqnarray}
\textbf{S} = - \sum_{n=1} ^ {\infty} \textbf{P}_n log(\textbf{P}_n )
\nonumber
\end{eqnarray}
\noindent \textbf{Remark.} \textbf{S} is referred to as the entropy of the distribution (Everett).\\
\noindent \textbf{Example.} Consider $\textbf{P}_n = 1/2^n,$
\begin{eqnarray}
\textbf{S} = - \sum_{n=1} ^ {\infty} \frac{1} {2^n} log(\frac{1} {2^n})
\nonumber
\end{eqnarray}
\begin{eqnarray}
= 2 log(2)
\nonumber
\end{eqnarray}
\noindent And if the base 2 logarithm is used, $\textbf{S} = 2 \; bits$\\
%\begin{center}
\noindent \textbf{Calculation of S(x) for $0 \le x < 1$ .}
%\end{center}
\begin{eqnarray}
\frac {1} {1-x} =
(1+x) \prod_{n=1} ^{\infty}
( \frac {1+x^{2^n}} {1-x^{2^n}} )^{2^{-n}} = B
\nonumber
\end{eqnarray}
\begin{eqnarray}
log \frac {1} {1-x} =
log(1+x) + \sum_{n=1} ^{\infty}
\frac {1} {2^n} log (\frac {1+x^{2^n}} {1-x^{2^n}})
\nonumber
\end{eqnarray}
\begin{eqnarray}
1 = log_B(1+x) + \sum_{n=1} ^{\infty}
\frac {1} {2^n} log_B (\frac {1+x^{2^n}} {1-x^{2^n}}) =
\sum_{n=1} ^{\infty} P_n
\nonumber
\end{eqnarray}
\begin{eqnarray}
\textbf{S(x)} =
-\sum_{n=1} ^{\infty} P_n log_2 (P_n) \; (bits)
\nonumber
\end{eqnarray}
\newpage
Figure 2 is a comparison of the entropy of objective state [Model IV]
and the proposed entropy of subjective state [Model V]. The area
between the curves is the proposed entropy of the objective distortion.
\begin {center}
\includegraphics[width=5.0in,height=5.0in]{fig2.bmp}\\
Figure 2. Comparison of State Entropy
\end{center}
\section{The Concept of the Entropy of Relative States}
\indent The following identity is of utmost importance in any discussion
of the information theory of Shannon,\\
\begin{eqnarray}
Entropy = -\sum_{n=1} ^ {states} p_n ln (p_n)
\nonumber
= K \; ln(states)\; ,
\nonumber
\end{eqnarray}
\noindent but only for uniform distributions of $p_n$ does the second equality hold.
But suppose that the $p_n$'s are not equal, how do we then compare the entropy
of the uniform distribution against the entropy for the non-uniform distribution?
One way is to assume that all finite uniform distributions have entropy of unity
and that non-uniform distributions have entropy of less than unity. That is, the
non- uniform cases possess more information than the uniform cases. In such a
context, we must define a more consistent form of entropy that not only allows
for fair comparisons between problems with equal numbers of states, but also
allows for fair comparisons between problems possessing different numbers of
states. Define Relative Entropy:\\
\begin{eqnarray}
\textbf{Entropy} =\frac {Entropy} {ln(states)}
\nonumber
=-\sum_{n=1} ^ {states} p_n log_{states} (p_n) \; .
\nonumber
\end{eqnarray}
\noindent Thus defined, the Relative Entropy may also be interpreted as the Entropy of
Relative States. The following dice example should clarify the concept.\\
\noindent Two die each represent a Relative Entropy of unity. That is:\\
\begin{eqnarray}
\textbf{Entropy}_{die} =log_6(6)
\nonumber
=1 \; .
\nonumber
\end{eqnarray}
If the 36 possible states of a toss were uniformly distributed, then the
Relative Entropy of all possible results would be:\\
\begin{eqnarray}
\textbf{Entropy}_{uniform} =log_{36}(36)
\nonumber
=1 \; .
\nonumber
\end{eqnarray}
\noindent But the distribution of results is not uniform, so $\textbf{Entropy}_{toss} =$\\
\begin{eqnarray}
=\frac{-(\frac{2} {36})ln(\frac{1} {36})
-(\frac{4} {36})ln(\frac{2} {36})
-(\frac{6} {36})ln(\frac{3} {36})
-(\frac{8} {36})ln(\frac{4} {36})
-(\frac{10} {36})ln(\frac{5} {36})
-(\frac{6} {36})ln(\frac{6} {36})} {ln(36)}
\nonumber
\end{eqnarray}
\begin{eqnarray}
= 0.6334...
\nonumber
\end{eqnarray}
\noindent Note that the Notion of Relative Entropy is merely a more precise way to
say that $Entropy = K ln(states)$.
\noindent The distinguishing property of Relative Entropy is the manner by which
it is conserved; e.g., two six state systems, of $Entropy = 1$ each, might be
expected to combine into a uniform distribution of twelve states, of $Entropy = 1$,
because no information has been added to produce a non-uniform distribution.
Due to the rules of dice, thirty-six states result, and likewise we might expect
that the thirty-six states be uniformly distributed and therefore have
$Entropy = 1$. The combinations are actually 12345654321, as above, and the
dice and rules, amount to $-0.3666$ units of Relative Information being added
to the expected value of $Entropy = 1$ for a uniform thirty-six state system.\\
\noindent \textbf{Conjecture.}\\
Information is conserved when previously independent systems combine by
some set of rules to produce a composite system with a new number of total
states. That is, what information was necessary to produce the combination
system, can only be determined from the distribution of states, not from the
new number of total states.
\section{The Concept of Relative Entropy for Finite Products}
\indent The concept of entropy in relation to products starts with the
need to interpret the distribution of natural logarithms as probabilities:\\
$ln(AB) = ln(A)+ ln(B)$\\
\noindent or,\\
\begin{eqnarray}
\frac {ln(A)} {ln(AB)}+ \frac{ln(B)} {ln(AB)}=P_A + P_B=1
\nonumber
\end{eqnarray}
\noindent or,\\
$log_{AB}(AB) = log_{AB}(A) + log_{AB}(B)=1$\\
\noindent $P_A$ and $P_B$ are defined as being intermediate probabilities encountered
on the way to the determination of the relative entropy. If $A$ and $B$ are equal,
$P_A = P_B = 0.5$, and we are done with the determination of relative entropy.
On the other hand, if $P_A$ and $P_B$ are not equal, we are not done and
(base $N = 2$ probability states) must be introduced. The additional
computations are:\\
$Relative \; Entropy = -P_A log_2(P_A) - P_B log_2(P_B)$\\
\begin{eqnarray}
= \frac{-\frac{ln(A)} {ln(AB)} ln(\frac{ln(A)} {ln(AB)})
-\frac{ln(B)} {ln(AB)} ln(\frac{ln(B)} {ln(AB)})}
{ln(2)}
\nonumber
\end{eqnarray}
\noindent and when $A = B$,\\
$Maximum \; Relative \; Entropy = -log_2(\frac{1} {2}) = 1$\\
\noindent or, in general,\\
$Maximum \; Relative \; Entropy = -log_N (\frac{1} {N}) = 1$\\
\noindent for a finite product of N identical factors.\\
\indent Extension of this concept to infinite products must accompany a
constraint that the distribution of factors can never be uniform,
but is limited by the convergence properties of the infinite product.
That is, a finite value for $N$ can be determined that will limit the
Maximum Relative Entropy of Infinite Products to a value of $1$ for the
``most uniform" distribution of factors encountered over the domain of
interest.\\
\section{The Concept of Relative Entropy for Convergent Infinite Products}
\indent Combinations or interactions of infinite products do not involve a
change in the number of states because the number of states is
infinity. This must not be taken as a reason for the entropy to be
infinity as it appears to be in the general case; in particular
cases where the factors converge to unity over the domain of interest,
a uniform distribution of factors is never achieved. Therefore,
in this convergent-infinite case, entropy is conserved with respect
to any base logarithm that is appropriate. The base two logarithm
is chosen because the entropy of the infinite products of this
investigation converge to a value of two (bits) as the maximum of
their domain is approached $(x \to 1)$. The ``unit" of the unitless
entropy in (bits) merely signifies that the base of the chosen
logarithm is two and therefore must only be compared to entropies
computed with the base two logarithm.\\
\section{The Concept of Correlation}
\indent The reader must now make a conceptual leap from the notion of
equivalence of infinite products to the notion of correlation.
Equivalent mathematical structures correlate in the sense that,
geometrically, they are like an object container and an object
that exactly fits inside the container. In the present discussion
the object container is the gravitational field surrounding a
central mass $(M)$ and the object contained in it is the collection
of all possible circular orbits. The entropies $(S)$, of the objects,
are said to be equal when the fit is exact for a particular radius $(R)$
and orbital velocity $(v)$. In other words, the objects are said to be
in exact correlation when their gravitational and inertial entropies
are equal $(S_G = S_v)$. The ``degree of correlation" is less for elliptical
orbits trying to fit inside circular gravitational fields, and what
correlates (equates) for elliptical orbits are the average entropies.
That is,\\
\begin{eqnarray}
\frac{S_{Gmax} + S_{Gmin}} {2} =
\frac{S_{vmax} + S_{vmin}} {2} =
\overline{S}_G =
\overline{S}_v
\nonumber
\end{eqnarray}
\section{Orbital Interpretation}
Infinite products \textbf{A} and \textbf{B},
\begin{eqnarray}
A = \sqrt{\frac {1} {1-\frac{v^2} {c^2}}} =
\sqrt{(1+\frac{v^2} {c^2})} \; \prod_{n=1} ^{\infty}
( \frac {1+(\frac{v^2} {c^2})^{2^n}}
{1-(\frac{v^2} {c^2})^{2^n}} )^{2^{-(n+1)}},
\nonumber
\end{eqnarray}
\begin{eqnarray}
B = \frac {1} {1-\frac{GM} {Rc^2}} =
(1+\frac{GM} {Rc^2}) \prod_{n=1} ^{\infty}
( \frac {1+(\frac{GM} {Rc^2})^{2^n}}
{1-(\frac{GM} {Rc^2})^{2^n}} )^{2^{-n}},
\nonumber
\end{eqnarray}
\noindent are equivalent when, $v^2/c^2 = GM/R/c^2$, although
their values are related by $A = \sqrt{B}$. This is a case
of equivalence, defined previously, when $\rho = 0.5$ . That is,
when $v=\sqrt{GM/R}$, entropies, $\textbf{S}_{\textbf{A}}$ and
$\textbf{S}_{\textbf{B}}$ are equal. This means that the
inertial and gravitational entropies are equal for the
circular orbit,
\begin{eqnarray}
v = \sqrt{\frac{GM} {R}} \; ,
\nonumber
\end{eqnarray}
\noindent where $v$ is the orbital velocity,
$G$ is the gravitational constant,
$M$ is the central mass,
and $R$ is the distance from $M$.
\indent The statement that this theory makes concerning General Motion,
is based on the principle of the Equivalence of Average Inertial and
Average Gravitational Entropies such that,
\begin{eqnarray}
\overline{\textbf{S}}_{(\sqrt{\frac {1} {1-\frac{v^2} {c^2}}} )} =
\overline{\textbf{S}}_{(\frac {1} {1-\frac{GM} {Rc^2}} )} \; ,
\nonumber
\end{eqnarray}
\noindent from which derives the distortions of length, time, and
mass due to motion near a gravitating central mass.
Note that the inertial (Lorentz)
and gravitational factors are purely themselves (zero correlation),
when $M$ or $v$ are respectively zero or when $R = \infty$.
At this point, rigorous abstract analysis has failed to predict
elliptical motion because of the lack of theorems to determine the
form of the inverse function $\textbf{S}^{-1}$ of our mathematical object,
but its
linear approximation shows why elliptical orbital motion is calculated
in close agreement with the classical theory of the solar system. In
addition, numerical evaluation of $\textbf{S}$ has shown that:
1) $\textbf{S}$ vs $x$ has a
derivative of zero at $\textbf{S}_{max} = 2.5307 \; (bits)$,
$x = 0.99181$, and
2) orbital calculations cannot be made at or above
$\textbf{S} = 2 \; (bits)$,
$x = 0.7035$ to $x = 1$,
because the inverse of $\textbf{S}$ cannot be defined above
$\textbf{S}_{(2)}^{-1} = 0.7035$.
\indent The generating function for calculation of the entropic ellipse of
the computer simulation is:
\begin{eqnarray}
\textbf{S}_{(x)} =
(\frac{\textbf{S}_{max}+\textbf{S}_{min}} {2})+
(\frac{\textbf{S}_{max}-\textbf{S}_{min}} {2})cos(\theta)
\nonumber
\end{eqnarray}
The generated polar coordinate pair, $(\textbf{S}_{(x)},\theta)$,
is then transformed
into $(x,\theta)$ by an algorithm that converges on the value for $x$ that
corresponds to the desired value of $\textbf{S}_{(x)}$ generated above.
The function will generate a conic ellipse only if $\textbf{S}_{(x)}$
is linear between
$xmin$ and $xmax$; an observation that is consistent with the plot of
$\textbf{S}$ vs $x$ in Figure 1.(for Model IV), but only over portions
of the curve that can be assumed
linear. This fact can be shown by first converting the polar equation
for the $conic \; ellipse$,
\begin{eqnarray}
\frac{1} {R}=(\frac{1} {p})+(\frac{e} {p})cos(\theta),
\nonumber
\end{eqnarray}
\noindent into a representation in terms of $R_{min}$ (perihelion distance)
and $R_{max}$ (aphelion distance) instead of $a$ and $b$, the semi-major and the
semi-minor axes. That is, replacing the semi-latus rectum $(p)$ and the
eccentricity $(e)$ by their equivalent expressions gives,
\begin{eqnarray}
\frac{1} {R}=
\frac{(R_{max}+R_{min})} {(2R_{max}R_{min})}+
\frac{(R_{max}-R_{min})} {(R_{max}+R_{min})}
\frac{(R_{max}+R_{min})} {(2R_{max}R_{min})}cos(\theta)
\nonumber
\end{eqnarray}
\noindent And for the $entropic \; ellipse$,
\begin{eqnarray}
\textbf{S}_{(\frac{1} {R})}=
\frac{\textbf{S}_{(\frac{1} {R_{min}})}+
\textbf{S}_{(\frac{1} {R_{max}})}} {2}+
\frac{\textbf{S}_{(\frac{1} {R_{min}})}-
\textbf{S}_{(\frac{1} {R_{max}})}} {2}
cos(\theta)
\nonumber
\end{eqnarray}
so that, if \textbf{S} is linear over the sub-domain
$R_{min}$ to $R_{max}$,
\begin{eqnarray}
\frac{1} {R}=
\frac{(\frac{1} {R_{min}})+
(\frac{1} {R_{max}})} {2}+
\frac{(\frac{1} {R_{min}})-
(\frac{1} {R_{max}})} {2}
cos(\theta)
\nonumber
\end{eqnarray}
or,\\
\begin{eqnarray}
\frac{1} {R}=
\frac{(R_{max}+R_{min})} {(2R_{max}R_{min})}+
\frac{(R_{max}-R_{min})} {(2R_{max}R_{min})}cos(\theta),
\nonumber
\end{eqnarray}
\noindent which is exactly the same as for the $conic \; ellipse$.\\
\indent To fill the void for theorems concerning the properties
of infinite products, an interactive program (orbsim98) has been prepared
and can be downloaded from,
\begin{center}
$http://EshlemanW.tripod.com/$
\end{center}
\section{Probability Interpretation}
\indent The concept of probability, established earlier, was presented as
if it is merely an intermediate step in the determination of the
relative entropy of a product representation of the value of a number.
I would prefer to interpret the probabilities as observable; showing
their presence in a manner that the overall effect obscures their
rigorously deterministic values. In general, the probabilities are:
\begin{eqnarray}
\textbf{P} =1=
\sum_{n=1}^ {\infty} \textbf{P}_n =
\sum_{n=1}^ {\infty} log_A(a_n)=
\sum_{n=1}^ {\infty}
\frac{ln(a_n)} {ln(A)}=
\sum_{n=1}^ {\infty} \sum_{m=1}^ {\infty}
\frac{ln(a_n)} {ln(a_m)}
\nonumber
\end{eqnarray}
\noindent Or, each probability may be said to be the ratio of the
logarithm of a part to the logarithm of the whole (a change of base).
These probabilities, although deterministic, are indistinguishable upon
observation. That the gravitational and inertial probabilities are
indistinguishable, is this theory's uncertainty principle governing how
much information can be obtained about positions and velocities of
interactions involving energies from zero to infinity.
\indent The general properties do not put upper limits on either the value
or the entropy of an infinite product, but it is a property of the
particular infinite product of this theory that although it is limitless
in value (energy), the probabilities converge to a unique non-uniform
distribution (entropy) as the energy heads toward infinity. This order
out of chaos property, while un-noticed at low inertial and gravitational
energies where first pair approximations for the infinite products are
accurate and Newtonian, has a profound effect on the structure of this
theory's degenerate object, the black-hole of deep- space astronomy.
\section{Interpretation of the Degenerate Form}
The degenerate form of,
\begin{eqnarray}
\frac {1} {1-x} =
(1+x) \prod_{n=1} ^{\infty}
(\frac {1+x^{2^n}} {1-x^{2^n}} )^{2^{-n}} \; ,
\nonumber
\end{eqnarray}
\newpage
\noindent as $x$ approaches $1$ is,
\begin{eqnarray}
\frac {1} {1-x} =
\prod_{n=1} ^{\infty}
( \frac {1} {1-x})^{2^{-n}} \; ,
\nonumber
\end{eqnarray}
\noindent with probabilities,
\begin{eqnarray}
\sum_{n=1}^{\infty} \textbf{P}_n =
1=
\sum_{n=1}^{\infty} \frac{1} {2^n}\; ,
\nonumber
\end{eqnarray}
\noindent and entropy (using the base $2$ logarithm),
\begin{eqnarray}
\textbf{S}_{(1)} =
-\sum_{n=1}^{\infty}
\frac{1} {2^n} log_2(\frac{1} {2^n})=
\sum_{n=1}^{\infty}
\frac{n} {2^n} = 2 \; (bits) \; .
\nonumber
\end{eqnarray}
\noindent Therefore, the degenerate form of this theory is a determinate
fixed structure possessing order. Now, looking at the plot of
\textbf{S} vs $x$, we see that in addition to (point 3) at \textbf{S}$_1$
$= 2$ determined
above, there are two other points of special interest on the curve;
(point 1) at \textbf{S}$_{0.7035}= 2$, and (point 2) at
\textbf{S}$_{0.99181} = 2.5307$.
That is, the inverse function, \textbf{S}$^{-1}$, does not define at or above
$x = 0.7035$ and there is a maximum of $2.5307$ (and a zero derivative)
at $x = 0.99181$.
\indent Interpreting $x$ as equal to $GM/R/c^2$, and $M$ as the mass of our
Sun, we get a sphere of radius $R = 1,476$ meters enclosing all of $M$,
the one-sol black-hole of this theory. Just above the surface of
this black-hole there is a $12$ meter thick ``boundary-layer" inside
which $dS/dx$ is negative and above the ``boundary-layer" there is a
$610$ meter thick layer that fills the majority of the $622$ meters where
the inverse of its entropy function, \textbf{S}$^{-1}$, cannot be defined.\\
\indent The region where \textbf{S}$^{-1}$ cannot be defined is interpreted
as a region where orbital motion cannot exist.
This reasoning is based on the
observation that since the numerical simulation generates the entropy
polar coordinate pairs, (\textbf{S}$_x$,$\theta$), and needs (but is only
numerically capable),
\textbf{S}$^{-1}$ to get the orbital
polar pairs, ($x,\theta$), then analytical methods would likely need
\textbf{S}$^{-1}$ to determine orbits as well. That is,
the analytical problem concerning the lack of a defined inverse
for a function over some subdomain is mirrored by its numerical simulation.
\indent Of course, this is the structure of perception for a
black-hole, distorted by the nature of observation; it is the
objective view of the black-hole. Subjectively
(independent of observation), neither the event-horizon (as an object)
nor the maximum speed of light (as a limit) need apply when the classical
notion of subjective state ($e^x$) is the model. When this theory's
Model 5 is used for subjective state, matter is allowed to attain the
maximum speed of light, but is denied faster than light (FTL) motion due
to the lack of convergence of Model 5 for values of $x>1$.
\section{Remarks Concerning Quantum Mechanics}
The notion of evolution for classical quantum mechanics is,
\begin{eqnarray}
[\frac {f_1} {f_0}]^t =
{[e^H]}^t =
e^{tH} \; ,
\nonumber
\end{eqnarray}
\noindent where $H$
is a constant Hamiltonian operator that is independent of time ($t$).
Furthermore, interpretation of $H$ as a sum gives,
\begin{eqnarray}
[\frac {f_1} {f_0}]^t=
{[e^{\sum_n H_n}]}^t =
e^{t \sum_n H_n} \; .
\nonumber
\end{eqnarray}
\noindent Now, assuming a product form, $e^H = \prod_n h_n$ for
the relative state,
\begin{eqnarray}
[\frac {f_1} {f_0}]^t =
{[\prod_n h_n ]}^t =
{[\prod_ne^{ln(h_n)}]}^t =
{[e^{\sum_n ln(h_n)}]}^t =
e^{t \sum_n ln(h_n)} \; ,
\nonumber
\end{eqnarray}
\noindent an unremarkable result in that it merely demonstrates a longer
way to the same answers if $h_n = e^{H_n}$; or so it would seem.
What the above relation does show
is that if a candidate for a product version of the relative state
is to be tested, then $H_n = ln(h_n)$ must agree with experimental results.
For example, the special relativistic mass increase with velocity
%increase is represented as an additive increase to ${m_0}^2$,
requires that the relative state must be,
%\begin{eqnarray}
%m^2 =
%{m_0}^2 + \frac {v^2} {c^2} m^2 \; ,
%\nonumber
%\end{eqnarray}
%\noindent then the relative state must be assumed to be,
\begin{eqnarray}
\frac {f_1} {f_0} =
%\frac {m^2} {{m_0}^2} =
\sqrt{\frac {1} {1-v^2/c^2}} =
{\sqrt{{1+v^2/c^2}}\prod_{n=1}^\infty
[\frac{1+(v^2/c^2)^{2^n}} {1-(v^2/c^2)^{2^n}} ]^{2^{-(n+1)}}} =
e^{H_v}
\; ,
\nonumber
\end{eqnarray}
\noindent where the special relativistic Hamiltonian is,
\begin{eqnarray}
H_v = \frac{1} {2} ln(1+v^2/c^2) +
\sum_{n=1}^\infty [\frac{1} {2^{n+1}} ln({1+(v^2/c^2)^{2^n}})]
- \sum_{n=1}^\infty [\frac{1} {2^{n+1}} ln({1- (v^2/c^2)^{2^n}})]
\; ,
\nonumber
\end{eqnarray}
\noindent and likewise for gravity,
\begin{eqnarray}
H_g = ln(1+GM/R/c^2) +
\sum_{n=1}^\infty [\frac{1} {2^{n}} ln({1+(GM/R/c^2)^{2^n}})]
- \sum_{n=1}^\infty [\frac{1} {2^{n}} ln({1- (GM/R/c^2)^{2^n}})]
\; .
\nonumber
\end{eqnarray}
%\noindent That circular orbits, $v = \sqrt{GM/R}$, should be
%eigenstates, suggests that $e^{tH_v} = e^{tH_g}$ or,
\newpage
%\noindent That the Hamiltonians do not contain $m$, the satelite mass,
%is because $M>>m$, but $m$ could be multiplied by these velocity
%Hamiltonians to get momentum Hamiltonians without altering the
%following reasoning.
\noindent $H_v$ and $H_g$ are energy operators of the mass $m$,
so the evolution of state is now,
\begin{eqnarray}
e^{tHm} = e^{t(H_v-H_g)m}
\; .
\nonumber
\end{eqnarray}
\noindent One might think that the above Hamiltonian, $H = H_v-H_g$, is
insufficient in that the momentum displacement seems to be missing, but
the reasoning here is that momentum displacement is fictitious. That is,
instead of a priori assuming momentum to place the planets in
their proper orbits, we here assume that stationary
states (closed orbits) are due to the equivalence (correlation)
of the average distributions of the inertial and gravitational
Hamiltonians above. Therefore, there objectively appears to be
a real property of matter called momentum, but it is fictitious in that
it is merely the result of a correlation between motion and gravity
that actually causes the ``momentum" displacement. The fiction is that
momentum is a cause; the suggested truth is that momentum is an effect.
If one is so bold as to fictionalize momentum, then, to be consistent,
magnetism must also be fictionalized. That is, the orbit of an electron
is a relationship between Hamiltonian distributions; the magnetism is but
a result of this relationship, not a partial cause of the relationship.
\indent According to the method of this report we suggest that
Hamiltonians containing momentum and/or magnetism terms are overdetermined
because sufficient information for prediction of these phenomena are
contained in the distributions of motion and gravitational/coulomb
displacements. Furthermore, it is suggested that it is the fictionalized
displacements that are subjectively quantized, whereas what the
fundamental causes describe is an objectively continuous coordinate change.
In other words, the proposed Hamiltonians for the motion and for the
attraction, together with the correlation machinery described above,
form a sufficient and logically self-consistent mathematical
description of the evolution of states in a universe in which many
observers are at work.
\indent Since relativistic inertial and gravitational Hamiltonians
have been determined, a similar approach to quantum mechanics could
lead to a unified formulation.
This method of factorization of the relative state is not
exhausted by the infinite products presented in this report as there are
at least as many infinite product identities as there are number system
bases.
\indent Warm thanks are given to those individuals who have aided my
understanding of this subject, but are in no way concerned with
objections to this report. These people are, in order of appearance,
Glen Smerage,
David C. McLeod, David R. Fischer, Kali, Dryden W. Cope,
Alex McWill, Jim Dars and Hitoshi Kitada.
\end{document}