The Fractal-Quantum Field Theory
Frontier
The Micro-Quantum Realm
DANN PASSOJA
LOS ANGELES CALIFORNIA SPRING
2019
[DOCUMENT SUBTITLE]
MICROSOFT OFFICE USER
QUANTUMFIELD THEORY FRACTAL HORIZIONS | [Company address]
Lessons From Fractals
I had a quite remarkable experience several years ago with
fractals While preparing a print of a fractal by laser printing I
encountered problems with balancing the colors. I had chosen to use
very conventional colors about. Four different ones but, matter what I
tried the I couldn’t solve the problem. The most frustrating aspect of all
was that there were not set of colors that would work. The problem
would appear~ 3-4 minutes after the print had been made. The failures
were similar, due to color in balance, but they never were the same.
Notes on the diagnosis and solution of the color problem at I
used are shown below..
Finally I remembered that I had a famous book in my library that
I referred to constantly while enrolled in art school
M.E.Chevreul” The Principles of Color Harmony and
Contrast of Color
Which invoked total amazement from me when I
learned that he had published it in 1839!!
I realized that I was printing fractal patterns and I was seeing
interactions among the color harmonies that were
exacerbated by Fractals. One possible source of this problem
could start by printing two pixels side-by side that happen to
have a complimentary appearance.
The eye senses certain color combinations that are known as
color complements. Only on the pixel scale could this be seen.
But fractals are self- similar and I’d expect to see a compliment
problem to appear on a grand scale.
I’ve explained this in more detail in an article that was
published in Leonardo
Another Connection to be Remembered
The relationship that’s developed in this work between QFT
and Fractals is based on the on the Binomial Probability Distribution.
Therefore it is It is important that I say something about the
generating function used for the PL set. I cloned the generating function
by using the algorithm for Pascals’ triangle. It represents a new discrete
probability distribution ( I checked).
It has the following properties:
Limit 𝑙𝑖𝑚
∑ 𝑅𝑜𝑤+1
𝑖 ∑ 𝑅𝑜𝑤
==1+√2 the silver number
∑𝑖 𝑅𝑜𝑤 =2 𝑅𝑜𝑤 (𝑖) + 𝑅𝑜𝑤 (𝑖 − 1)= Pell Series
I discovered the relationship for the generating function and used it
for all of my work and only to find out later that the generating function
happened to be the same as the Delannoy numbers.
That’s an isolated fact that has nothing to do with the prime
fractals. However, no one had found any relationships such as the ones
that I had. There has been no mention of Pascal’s triangle either, only
the standard discussion on the number of paths that could be found on
a checkerboard.
I mention this because it might influence Quantum Field Theory in
an important way:
Fermions and Bosons are identified by the Pauli principle but
their partition functions are based on the discrete Binomial
Distribution. I created the generating function used to make the P-L
set by changing the algorithm that’s used in Pascal’s triangle and is
the Binomial Coefficient.
𝑖
𝐶𝑗𝑖 = 𝐶𝑗𝑖−1 +𝐶𝑗−1
Pascal’s
𝑖+1
𝑖
𝑃𝐿𝑗+1
= 𝑃𝐿𝑗𝑖 + 𝑃𝐿𝑗𝑖+1 +𝑃𝑗+1
Passoja Lakahtakia
There’s something that accompanies all of this -a particle's
identity is based on selection rules, however, a new distribution such
as this would imply that a Fractal particle, if it were to be
distinguished would have a probability distribution, would be selfsimilar and would have a “ quantum number”.
This would accomplish one of the project’s goals a particle
such as a Fractal Particle” would present on many scales and would
have a fractal identity distinguished by an intreger
Transformations of the particles could be described by
arithmetic.
Forward
As I was completing this report I realized that I had covered a
variety of topics because it was necessary establish some very basic
ideas from scratch. My purpose was to determine a scientific and
philosophical structure, that could accommodate both the Fractal and
the Quantum Field viewpoints. I might find something that Fractals
provided but too many viewpoints would become too focused and
therefore would overlook the opportunity that Fractals can provide.
The viewpoint of this work stems from presenting some
practical “mile markers” using
to compare the energy density of
the Femi sphere with other energy densities. Fermions and Bosons are
abstractions but we speak about them as though they were real and I
needed to determine just how and where I should begin a development.
I present a viewpoint that maintains constant contact with
structures of many different types. I use geometry based on k the wave
vector, actually inverse length, but a length that can be measured
directly and has matter in terms of meaning behind it.
I want to state that any geometry that I’ll discuss is
(conceptually) different than usual. Geometry is generic, it never has
had need to consider “Specific Geometries” such as Specific Area or
Specific Volume or even Specific Density. All of these items are
associated with real things.
My recent work on fractal fracture made it quite obvious that I
could no longer speak of a fracture surface because the surface was a
specific one that had a history that was defined in space-time. Of course
it always had been this way, but it’s become more and more clear that
almost everything that we discuss is treated in this manner. It’s very
important that everyone starts to realize that real things don’t often find
a place in science. I don’t mean that science is divorced from reality,
rather, the specific detail that is part of everyone’s world isn’t of concern
to Science but Universality is. I don’t see any reason for it to remain this
way, because the very best experiments in Science are often done by
finding examples of quite realistic phenomena.
Next issue; Are someone’s faculties (or absence thereof) to be
considered ? After all there are observations-but sometime we must be
realistic and admit that the observer is not just an unnamed “observer”
because he’s part of the experiment.
“What can be observed and measured by an experimentalist and
how can it be understood ?
Probability always interferes with making observations
or measurements, so statistical mechanics offers the opportunity to deal
with large numbers of particles that are distinguishable or
indistinguishable
The major focus of the following work will be concerned
with the Fermi Dirac Distribution. Since there is a major interest in the
structural details of the Fermi Surface I’ve chosen to focus on this and to
bring some new insights to the topic.
`
`The following work is based on developing a
background that would help to identify a more comprehensive
understanding and integrating it into a more workable structure.
This is the major focus of this project however the project has
more specific goals. To determine new directions where QFT and
Fractals will be able to produce new theoretical and fundamental
knowledge
Beginning Impressions
Avagadro’s Number and the Mole…there’s no practical way to
count the number of atoms in a Mole, but for some it’s possible to hold
Mole your hand. At some level details count, but in the world of “grown
ups” not possible to sense atomic granularity.
There are some things all around us that can be identified by
straight forward observation that can be counted by a set of integers.
Euler made observations of Platonic solids, he counted the
number of surfaces, vertices and edges and developed his rules
empirically..The rules are universal and observable on Platonic regular
solids and many other shapes as well.
His rules are
Vertices-Edges+Faces=2
His rules have withstood test of time. It’s unusual that energy
doesn’t readily come into the picture.
I’ll bring this up later…that the geometry that’s familiar that we
all use is generic. It’s just length, area or volume that have absolutely no
character. Natures’ representations of this most certainly are not. In
fact, the most common ones are attached to matter. I’ve always been
aware
Every real physical volume
is surrounded by one
and only one surface.
Surface
Rule
Volume
This was supported by the work that I had done on Fractals
where I found a relationship between the Fractal dimension and the
fracture energy. Therefore such a relationship exists because there’s one
degree of freedom in existence between a solid and a “generic” surface. I
think that this observation is based on topology, because it’s peculiar
and fits into the category of Euler’s rules This is a consequence of a
phase change but expressing that doesn’t help to explain the
observation
Separability is Re lated to the = 1
If solid particles are suspended in a liquid then their surfaces
that keep them separated. The number that can be counted depends on
the ongoing chemistry. It might be possible to make statements about
the different ways that entropy is involved in this , however, all the
rather amazing statements would just be records of everyone’s
conversations and theories. Because there’s no practical and direct way
to confirm anyone’s t theoretical statements.
My point is that real problems can appear from any direction.
They’re not partial to any philosophy- quantum -nano-universe or
anything else.
There’s a lesson that I learned that accompanies this.. People
who are specialists develop personal problems when they realize that
their specialty isn’t that relevant to so many things that they become
conficted and have a great deal of trouble deciding whether they should
continue as a specialist.
Statistical mechanics that’s familiar doesn’t apply to atoms
that can be accounted for by partitioning, but in order to change from a
mathematical idea into something that’s more tangible , such as
something that’s countable, then there has to be an exchange of energy
that’s accompanied by losing part of it.
Don’t think that partitioning insofar as it creates the
opportunity of countability results in a more orderly condition because
that’s a human based judgment. If it’s possible to prove that something
is gained then energy will need to be lost in order to keep everything in
balance.t expect to see that partitioning something makes it appear to
be more tidy and orderly as we might determine, but that’s a temporary
, incomplete judgment; but it means that there are physical
constraints in operation that that influence how things come to be
subdivided that are part of statistical reality.
Subdivisions of space and matter are essential parts of the
“game “ of scientists. But it’ s still difficult for many of them to
understand that knowledge of microphenomena has never seemed
sufficient to bring a great deal of understanding about
macrophenomena. J. Willard Gibbs spoke of the viability of such
relationships about 100 years ago
100 years after he stated this it has gone into the wind. The first
time that I encountered his statement I was stunned into silence. It was
such a profound and important statement that I knew it would have an
important place in my memory because I’d have to develop some new
theoretical ideas resolve it
Some further thoughts about these thoughts led me to the
definition of a “macrostate” (as defined in Statistical Mechanics) which
has an undeniably and important property -entropy. Over time
structures of macrostates remain stable and are replaced by
microstates that reproduce. I had very vague thoughts about this but
nothing more.
I’ve always had wonderful experiences when I read the work of
E.T.James. He commented on many things such as those mentioned
above.
ET James
if any .microphenomena or macrophenomena
are found to be reproducible then it follows that all
microscopic details that were not reproduced must be
irrelevant for understanding and predicting it. In particular,
all circumstances that were not under an experimenter’s
control are not likely to be reproduced and are very likely to
be irrelevant.
His comments stated something that was relevant to Gibbs’s
comment. …
A Brief discussion that follows:
James; Microstates that don’t meet the right standards don’t matter.
Gibbs : Expect to find colonies that do.
GEOMETRY
Geometry is so essential to mathematics and physics
that I had to review it and demonstrate that there are some very
fundamental aspects of it that I should make an effort to review.
I began with the most elementary Euclidean laws to
demonstrate that there are other ways that don’t require proofs and in
order to present new and understandable ideas. In fact, there are many
important ideas that are known that are transmitted by movement and
light. I use the most important set of geometric rules that have survived
for over 2500 years along with some other organizing features to see
what would happen. And I ended up in a curious place.
The next few pages were constructed to show how a pattern
such as the one that’s basic to Pascals triangle that is hidden but
responsible for other patterns but are based on a different sets of rules.
I hope that you can identify Euclid’s rules in various forms:
Points® 2 Points Lines….
4 Points+4 LinesSurface…
2 Surfaces+ 4 Line+ Surface  Volume
Volume Physical Volume
Volume+Surface
AN ALGORITHM FOR EUCLIDEAN GEOMETRY
I’d like to present an example that although equations and
proofs, seem to satisfy mathematicians they are a traditional way to
introduce premise in order to begin a debate.
There are other avenues that everyone uses that are different
and produce different results. In many cases the “premise” direction is
useless. There’s a different audience and the ideas will be rejected if
there’s an attempt to present them that way.
I wanted to present the most elementary patterns (I guess
patterns)) that I’ve used over and over to produce almost all of the
figures in this work. Not only that, I’ve found myself entering equations
that represent these patterns over and over again. Although these
patterns seem to be simple because everyone recognizes them
immediately as simple patterns. I don’t think that they are because it’s
possible to trace an almost infinite number of connections with them.
Numbers, equations, images and so much more it’s not possible to
capture the meaning of all of this in an equation.
I’ve used algorithms in the next few pages as an example of an
interplay between Euclidean Geometry and the form of the Binomial
coefficient. I did this because I kept finding forms of Pascal’s Triangle in
unexpected places. I’m certain that these examples are just due to
chance but the structure fit in nicely with the “rules”that I devised to
give the proper meaning and symmetry to the figure.
.
This are symmetries and structures for points lines surfaces
This is just some guidance for going beyond two dimensions. are
j
I continued further, and decided to work in phase space , why
not? Everything makes as much sense there as it does anywhere else.
It’s just a matter of becoming accommodated to Euclidean Geometry
with vectors and k space! By using a surface having two sides but having
no thicknesss – one that is based on a geometric definition I was in need
of a something like a “ ground state
.No thickness, no matter, merely a definition. Then I recognized
that there were particles associated with it that had spins all pointing in
the same direction.
Then I separated the surface and ended up with two one sided
surfaces. I thought that there would be a state of lower entropy if I
reorganized the particles and put  on different surfaces. That left me
with two one sided surfaces. So, I glued two sides together
As I finished I wanted to end up with a “device” of some sort
something that developed a meaning from the phase space level that
ended up becoming a device
And made a one sided cylinder. I had to just imagine that the
next step might be possible. This would result in a (theoretical) one sided
torus.
This is the final result separated particles with fields on the final
single sided surface. Ready to accept a field running through its middle.
Every once in a while I encounter a paradox, they’re all different.
I found that Lebesgue had a problem with this one and so did I. It’s
related to the properties of a fractal.
It’s an equilateral triangle that starts out with sides =1. The next
path is constructed by dividing each of the paths that are to be followed
in half every time. The objective is to keep track of how many paths
you’ve used to reach your destination
DOWN TO BASICS
The next topic is so essential that every has learned it many
years ago. But things have changed and the meaning of a number has
changed. Infinitesmals have been introduced into the number system
and that has caused zero and infinity to disappear!
The structure of the number line has been debated for centuries
The continuity- discontinuity disagreements have continued to influence
( some unanticipated) areas of mathematics- especially when it came to
understanding a the nature of a derivative.
After, all what exactly is a limit?
Then, there were some other interesting things, such as when
someone added
”…look maybe there is more than one
.continuous line that’s present superimposed
over the one that we’ve been discussing. So
every one was introduced to the idea of density
for the first time.
The idea of closing a gap in some manner was first described by
William of Ockham 1280-1349
While describing a continuum. He proposed that between any
two points arbitrarily chosen on a line there is a third region where there
an infinite number of successions ,more than enough to support a
continuum.
Johann Kepler 1571-1630
Johann Kepler is best known for his astronomical observations
but he is also granted to be the father of optics. He has a theorem,
“Kepler’s Theorem” having to do with the optimal structure that can be
constructed from sphere packing. It has yet to be proven.
Kepler was interested in many things, some contained
remarkable insights
Several were very interesting….he used infinitesimals to
calculate volumes of wine casks etc since he regarded solid bodies to be
composed of infinitesimally solid cones or infinitesimally thin disks solid.
It was customary for him to use the same dimensions as the figures they
constitute.
His next statement was prescient. His grasp on what was to
become a reigning principle in geometry for many years was:
….that .a geometric figure is something that provides a
continuous change of a mathematical object
All too often I’ve noticed that scientists follow well- trodden
paths, only hesitantly try to expand their horizons (an incremental
amount at best) and tend to avoid discussing some very basic issues
because that’s something that they learned in kindergarten (commonly
known as graduate school). I thought that it would be useful to start
with some today’s basics and, of course, to keep them from being boring,
maybe even to bring things up to date.
THE MOST BASIC IDEAS
Numbers are so fundamental to everything that we know of and
are of practical use that they transcend most of our concerns and relate
to basics.
I’ve focused on how simple concepts such as continuity and
discontinuity have evolved because the concepts that emerged from
them are important in the quantum mechanics of today. This topic is
essential but one whose details are often are disregarded everyone
knows about these things.
. However something has changed the meaning of a number-.
Infinitesmals. They’ve have been introduced into the number system
and that has caused zero and infinity to disappear!
THE MODERN NUMBER LINE
I’m certain that everyone knows that number line is where
everything started, probably in Sumeria or Babylon ~ 3500 BCE.
maturing later in Egypt Greece.
structure of the number line has been debated for centuries The
continuity- discontinuity disagreements have continued to influence (
some unanticipated) areas of mathematics- especially when it came to
understanding a the nature of a derivative.
After, all what exactly is a limit?
Then, there were some other interesting things, such as when
someone added
”…look maybe there is more than one
.continuous line that’s present superimposed
over the one that we’ve been discussing. So
every one was introduced to the idea of density
for the first time.
Reviewing a small bit of history has brought me to a point that I
can begin to introduce an idea that’s been around for long time
infinitesimals or in the past continuity and discreteness.. Recently it’s
had a rebirth of the concept of the infinitely small and the infinitely large
in our number system. This offers some new intuitive ways to approach
the concepts of zero and infinity especially how they cause our equations
to blow up or reach zero.
INFINITESIMALS
The necessary rigor has been established by Arthur Robinson in
the 1960 and has had an influence on several areas of mathematics. It’s
known as non- standard mathematics or hyperreal numbers (as named
by the most famous John Conway) .
The hyperreal numbers include the standard number system
and are known as ℕ* when combined with the natural numbers.
The non-standard numbers include infinitesimals, 𝛆 that are
smaller than any other number in the system but is so small that there is
nothing smaller in the system.
Number theorists regard infinity simply as a number defined by
convention it is nevertheless a number. Therefore, it’s always possible to
add (or subtract) more numbers onto it and remember that the since
two other numbers can be added above and below it ± k and that
includes 𝛆 also k+ 𝛆 and k- 𝛆 as shown below
This the derivative in terms on infinitesimals
Positive infinity is up here (and below) but it’s just a number by
convention. So that  an infinitesimal can be added to it.
The funnel shaped object above is known as a monad. Within it
are all the ε that have the property- .. they are clustered around whole
numbers being as close to it as possible. But only infinitesimally close to
it ,but never reaching it.
Thoughts about discontinuous and continuous space are still
part of our mathematical heritage but by observing the microscopic
world I’ve always been left with the feeling that it’s better to accept this
dichotomy as a fact because everything makes more sense if you do.
So….the Number Line is Something That Everyone
Knows
I’m certain that everyone will say this, but will not be
able to extend their thoughts beyond that. I’ve studied and read as
much as I could in many different subjects and read books on the same
topic written by different authors. The basics are there but are seldom
expressed in a straightforward way.
Indeed we know what the number line is, but don’t
realize that it’s placement is relative. Also numbers aren’t connected to
anything physical and they can’t be created or destroyed.
The Number Line Based on Physical Reality
This work began with a discussion of the number line
and although the line night be thought of as something tangible it isn’t
it’s an abstraction. If the line’s existence requires that a ruler must be
used to establish its existence then a relation of Heisenberg’s
Uncertainty Principle will interfere with our intentions. However, for
different reasons.
We speak of a specific ‘length” as exactly # centimeters,
it’s just our concept of something that’s exactly # .It’s not possible to
prove that our idea of a length is anything more than just an
idealization. Any attempt to measure it will include errors.
There’s even something more unusual about the number
line; a physical representation of the number line could be bent and
joined to its other end, in other end, in other words a negative end
doesn’t exist, and that’s why a circle does. Our comprehension of the
universe is best understood by the number line concept.
Fractals forced these issues out into the open. So far I
haven’t seen any well founded ideas that are concerned with making
observations over the scale of magnification that is a fact. of life that’s
present all around us. Somehow, deep theoretical, mathematically
constructed arguments might be meaningful to some but they must
emerge as something that’s practical. To continue using the term “Self
Similarity” without considering its meaning in a broader sense just isn’t
a very wise thing to do.
It’s very difficult to stay in touch with the proper Mathematics
that’s connected to a multitude of different spaces .There are an
uncountable number of ways to become lost. I’ve tried to organize some
of this so that I wouldn’t be searching in the entirely wrong places. The
number line was basic to all of them so I started with this.
Space- time is something that is, by definition, a
necessity to establish the validity of a mathematical system.
Coordinates are created and become part of the reality. However,
specific coordinates of individual particles are never part of the
theoretical atomic framework. That’s because there’s always been an
interest large number of individual particles ( of any kind). Which
subsequently require identification or association with microsystems.
Thereby helping to define degrees of freedom necessary to establish the
dimensionality of the states of a system that are yet to evolve. The
validity of using probability in place of coordinate specificity is the major
purpose behind theoretical developments. Entropy is concerned with
the probability of observing a single microstate in a microcanonical
ensemble.
Iit’s apparent that magnification is more relevant to us if it’s
used having a different base. Something as simple as changing a base
can bring order into someone’s life. I use our monetary system as an
example.
There’s a good way to bring some understanding into this. If
either the base 10nor the “base” Fourier is expanded in a power series
\
These next two equations are series expansions for 10x
the term in Fourier’s equation. It’s easy to see that the exponent
in the base 10 equation can be used for magnification.
The expansion of either function ( if the unimportant garbage is
ignored) shows that they are similar since there’s a base to a power
that’s linear. In fact, we coincidently use magnification in terms of the
power of the base 10
DISCONTINUITY VS CONTINUITY
The concepts have been debated since the very beginning of
formalized mathematics ( I’m not certain that. anyone has reached solid
conclusions that have remained in place for any extended length of
time) then quantum mechanics was recognized and everything became
more complicated. Understanding a differential was always in
contention among the greatest mathematicians over subsequent time
periods: Newton, Leibnitz, Euler they all had different opinions. Then
Cantor changed everything and there was a realization that the
meaning of infinity had changed.
Probability and the a possibility that the number line might have
structure line has never come up.
But by what we understand today there’s a fundamental reason
why a number line shouldn’t be continuous
But if probability was available I’d expect that it might
stretched across a gap and reduce the possibility of not being able to
render an evaluation .
No one has said anything about the physical features of the
line. A form of noise that has a stable average might define a set of
conditions would predict the conditions that would have to be present
so that the convergence of a series might be possible at a certain
confidence level.
Thoughts about how a particle be influenced might behave is
not out of bounds in this thinking. If the number line is discontinuous will
it be necessary to redefine the Uncertainty Principle?. How would this
characteristic be manifested in a measurement.? If the positions of the
discontinuities were dynamic f (time) and ~ the size of a particle, would
it be possible to determine this? Any answers that I might provide are
not forthcoming. However, I have a suspicion that the only way to fix
this problem is to recast it in terms that include probability. Also,
whoever said that number line was quiescent?
I’m not merely suggesting because I consider this to be factual.
Everyone’s measurements depend on their measuring device. By
changing the nature of numbers themselves there’s no way to guarantee
that that the particle size hasn’t changed and the Uncertainty Principle
only ends up asserting that the uncertainty principle doesn’t apply to
that problem.
This is a pictorial representation of what I discussed above.
I’d suggest that such things might have a direct influence of
some fundamental assumptions used in mathematics for example, the
Dirac Delta function is one that might be victimized P. Dirac was the
person who invented the Dirac Delta function. There is some graphical
information below that represents the present understanding of it.
Dirac was concerned with the relationship between x- location
and momentum and was thinking in terms of the Uncertainty Principle
as well as the wave particle theory. He created a rectangle of x and p
with p along the y axis using a geometric model. He, of all people, would
be aware of the issues that are involved with this problem. However, he
had taken derivatives out of the picture and had changed its nature .
But he had to face a new problem. His construction would
always be a rectangle with an area, in fact it would have an infinite
area. Which means as mentioned before that since infinity is number he
could cut off as much of it as he wished but it would never get any
smaller. I’m sure his solution was to turn it into a mathematical
equation, define some limits to make it useful and then adjust them a
little from time to time when it is necessary. Not only did he solve his
problem he created an extremely useful mathematical tool.
OBSERVATIONS MADE IN REAL SPACE
AND MOMENTUM SPACE
Thoughts about reconciling observations in real space and momentum
space are best done with mathematics.
The Fourier transform is a mathematical method exchanging a
continuous set of coordinates with a discrete set called wave numbers.
Probability amplitudes are a determinate factor in being able to observe
the result of an interaction.
QFT often choses to work in one space or the other, with good
reason. Sometimes it’s easier to make such a transition. To my great
surprise I encountered a very understandable reason for doing this it’s
due to the conservation of momentum . The law creates something that
I found surprising: momentum space is translationally invariant.
I consider that to be something very important that I’m just
beginning to understand the what some of consequences might be.
Such a thought is encouraging in light of the problem that’s
under discussion. Planck’s constant quantizes energy and brings noncommutativity and quantization into the picture. It’s also possible to
show in a more basic way that the two vectors ,x and p lie in a plane
their magnitudes can be related to energy as in QFT. However, if
quantization has to be included their inner product has to be included.
Algebraic Geometry has solved a basic problem that’s at the heart of
this problem: terms that have different dimensions aren’t to be used as
terms in the same equation .Algebraic Geometry uses the geometric
product in order to circumvent this problem.
x.p = x× p+ x Ù p
Once again, in order to develop more understanding about this
problem dimensions have to be considered to be a stumbling block in
resolving this problem.
Beside the problem of not toeing the line and choosing a
geometric direction in order to understand as much as possible (I found
that it was impossible to accommodate Fractals and QFT if I became
too involved with the details of either area.
I was inspired by them many directions that QFT had taken for
its development. In particular, I thought that by using creation/
annihilation (particles /operators) could be very helpful in a more general
context. Their quantizing came about du e to the impossibility of dealing
with n Schroedinger equations for an n-body problem.
By maintaining a canonical quantum basis, and by means of
second quantization, they transformed the many-body problem into an
a problem having infinite degrees of freedom. The -field particle
relationship was defined and a collection of particles could develop a
“collective state” known as the “vacuum state”. The vacuum energy
remained in equilibrium and defined a reference energy. The in certain
applications this is called “The Fermi Liquid” that has a Fermi surface
where all the interactions can be found.
The mathematical operations that were needed are never done
in the newly developed space but are always transferred into the space
that was required to support the mathematics. The answers were then
transcribed into the correct language.
This is just a beginning, an exploration into the same world with
energy matter always in its proper place but one that uses probability in
a different way. The famous ‘Wave function “ owes its existence to the
quantum scale. However the quantum scale supports the macroscale
and is part (as contained within) of it. However the manifestations of
quantum behavior are seldom directly observed on the macro scale.
MEASUREMENT OF LENGTH AND AREA
I didn’t want to leave my conjecture about the Pythagorean
theorem before I expanded my horizons a bit more. I had a strong feeling
about using something complicated but familiar, to start things going.
I made the choice and decided to use Schroedinger’s equation. I
knew why -because as a beginner I invented my own ways to
understand and to find solutions to the great equations. Then later I
switched over to the conventional methods. Therefore, I knew that the
harmonic oscillator is based on the Pythagorean Equation.
I investigated the possibility of deriving an equation that was
somewhat like the Schroedinger equation. I wanted to explore some of
the ideas that I mentioned before. Why not?
If something is consistent with what we think is “reality” then
such thoughts are worth considering. After all that’s what happened
when quantum mechanics came into being. I think that it’s reasonable
to assume that almost everyone considered it to be “dead in the water” ,
and somewhat outrageous.
I’m not suggesting that there’s anything like this in the works
but keeping an open mind is just a necessity. Also, try things for yourself.
THE PURPOSE OF CHALLENGES
I had no prior set of equations that I could use but I’d have to
pass judgment on whether it could offer a different viewpoint. Also it
would have to stand on logical grounds. As I discussed previously .
¶2y é
éx, y é
é
a
4
a
¶
=y
¶x
¶x2
+ x2y [x, y] = i
m ¶y [x, y
r ¶y
é
¶ éé
¶é
2
2
x
i
a
x
+
i
a
=
x
iy
x
+
iy
=
x
+
y
é
¶x é
¶x é
é
éé
é
é
(
(xy - yx) = [x, y] = ia
)(
)
This is how the wave function looks. When I have more time
later I’ll finish the first part of the story.
I’m somewhat encouraged by the fact that the
Pythagorean theorem has been in existence and has been so useful over
such a period of history that there’s a good reason for it. Simply because
it correctly accounts for the obvious and it is constantly used. Meaning
that’s what people commonly observe and involves something that
everyone observes on a daily basis. It’s certainly the basis for the
foundation of so many things that it’s overwhelming
There’s something that needs to be stated in regard to this
 The Theorem as taught to every student in our school
system is incorrect, not by a little bit but totally incorrect
 I feel a need to mention this because it has a bearing on
what I’m about to propose.
This is taught at the present time
This is correct and should be taught
That just means that the Pythagorean Theorem isn’t a
fundamental issue in the problem.
Self similarity can also be implicated in the problem.is
This is the structural form of the Dumbell Ruler
.
This is the dumbbell ruler. It’s a physical example of the
measurement of length and the errors that occur from the measurement
process. The ends are not only enlarged but they have a different
dimension than the ruler does and that happens to be the source of the
errors, having to determine where the end of the ruler is where there’s
an abrupt change in dimensionality. This way of making measurements
is a case of having to combine one and two dimensions to get a
measurement.
The ruler can only measure an invariant length along with some
electronic clicks , the errors. I used a Log[#] to record the errors but I
could have used
Log[Errors] =
instead.
Dn
N
As it stands it is a scalar measurement of length with
area being recorded as an error.
It looks a great deal like some very familiar thermodynamic
equations.
𝐸𝑛𝑒𝑟𝑔𝑦
AN EQUATION IN TERMS OF
𝑉𝑜𝑙𝑢𝑚𝑒
I needed to calculate the energy density found in a reliable
equation that’s used in macroscopic applications. I also decided not to
1
consider using anything unless it had a
dependence.
𝐿𝑒𝑛𝑔𝑡ℎ
1
hc
compare it to E =
x
l
I was looking to compare the energy density in Fermi energy
dependent applications with others. I didn’t expect to find anything that
would be significant, and I’d probably not be able to determine whether
a correspondence would be direct.
EÞ
My inverse length requirement was made for practical reasons.
Everyone is familiar with certain length scales , but usually they haven’t
related it to an energy scale or a mechanism that goes with it.
This is very important equation that has length in - the grain
size of a microstructure. There’s no mystery about this but there should
be some more definitive scientifically based reasons to eliminate a sense
of mystery
My choice was based on the conservation of energy. There’s a
consistency that’s present that stabilizes the different worlds: atomic->
nano-> micro. I think that it’s (once again), the conservation of matter
and energy.
In most of the situations that we conjure, our thinking is usually
directed to ask a question about something that’s divorced from
everything and is converted into an object or thing. This just makes a
conversation more efficient.
In discussions of scaling between the atomic and larger scales “
how? and how long? Should be included because I don’t expect that
there would be an immediate change
The other piece of advice is
“Keep your eye on the transfer of energy.
An exchange of energy would be required in instance of
wavelength to grain size transformation. In this case there’s no problem
because the grain is so overweight. That without a serious effort it’ll
never be able make such a change even with a weight loss program
lasting many years.
EÞ
1
hc
compare it to E =
x
l
This calculation was based on the most recent form of the HallPetch equation. Grainboundaries are commonly found in materials.
DEFORMATION OF METALS
The Hall Petch Equation is associated with a size dependent
property the yield stress , which is the grain size. In topological terms
they define a structurer surrounding randomly distributed points in 3D
space. having structure which can be described as topological objects
that are determined by all of the other topological members of a set,.The
k
s =so +
d
stresses are typically 100-200 MPa
s =so +
e = eo +
e n->¥ =
k ' Ln[d]
d
kDd
d2
(
(x - xo )3
xo 3x2 - 3xxo + 3xo2
)
Below is the calculation showing how the energy changes during
deformation.
The values indicate that the energies are in the range of
EW » 1- 2 ×1010
ergs
cm3
Fracture Energy/Volume
There was a set of fracture experiments where I
was able to determine a value for a fracture energy directly from the
fracture surface .It was
Dislocation Storage 10%
Dissipation Energy 90 %
ESTIMATION OF THE FERMI ENERGY DENSITY
I found answers that came from an expertLandau who else ?
(
)
h2
EF =
3p 2 n
2m
3
2
n » 1023 / cm3
EFermi » 1 ev or 10-11
erg
cm3
.
GETTING VEXED BY STATISTICAL MECHANICS
I perused Statistical Mechanics and ended up asking some
obvious questions having figured out that there was no way to answer
the most important one. Because the justification for using the theory
guarantees that a level of indistinguishability must accompany it..
 At this time I’m certain that I’m the only person who has
known anything about this. It’s very important because
I’ve found that it’s impossible to ignore the Binomial
Probability distribution and use a Gaussian Distribution in
its place,. The Binomial Distribution is a discrete
distribution
 I’d consider using the equation for Entropy in a different
but equivalent form, that more amenable to analytical
revisions
I soon realization that both statistical mechanics and QFT both
assumed that the number of particles were unlimited i.e
I didn’t have to look fpr this because it has bothered me for a
long time. I was concerned with some other things such as not being
able to describe how the evolution of a microstate can influence a
macrostate.
But now this opens the door for accepting fractals as being
fundamental part of the entire process. I hope that everyone in our
project realizes that since Fractals and probability are inseparable
partners and also that Fractals are quantized gives the impression that
quantum mechanics share some similarities with all of this.
But there’s even more to consider: because fractals are selfsimilar there’s no to way to ignore the possibility that Fractals presence
is operational everywhere..
SETTING UP AN ANALYTICAL METHOD
This section is presented so that I can demonstrate and
familiarize you with how we might use or invent a special method based
on what we understand about fundamental material science Obviously
our methods will consist of detailed methods that are relevant. But
experience has taugh us that’s probably too much to expect because
discovering the nature of a problem is where most of your time is spent.
The demonstration is nothing more than thatI I used grain
boundaries to start and then to followed an analytical path that is
analogous to some aspects of QFT. I hope that It’s possible, by means of
this presentation, that it’s possible to find some similarity existing
between QFT and this.
I start by defining a refence state – a definition of the vacuum
state. This is a universal geometric “state” that provides states -higher
order harmonics that support higher energy states based on the
fundamental idea of a circle.
To do this it’s necessary to define a “fiducial” circle. That’s done
by counting the number of points attached to the grains in the area
that’s to be analyzed.
This is the space occupied by the grains. Boundaries iesthat
arepresent occur can be thought of as the set of plane that pass
through the midpoints of the lines connecting the points. This approach
resembles a Vornoi construction. However, such constructions are still
idealizations and are simply are different ways to speak about these
things.
There are some useful concepts that enter the picture after
trying to grasp such simple/complicated subject. Topology makes it
possible to speak of many things that are not obvious but are present. It
doesn’t produce measurements but can provide concepts that will help
to develop principles about how to measure something.
Starting at low magnification, count and determine the
number of (grains -or something else that you know is relevant to your
problem) this is just a matter of needing some type of coordinate system
that can be used as a basis set (orthogonal) that you understand and
know what the relations can be used to “span” the set. Should you need
to switch to another coordinate system such as “k” space then you must
give up something in order to get something.
This figure shows the placement of a reference circle on a
boundary and that the fine scale structures are distributed around as
deviations. The picture below shows the structures that can be found in
the separations.
From left right: the first is an idealization, a crystal a reference
state. The second shows the quite periodic features that are man made.
I’ve always found that in making measurements on something
that was important it might be possible to get some wonderful results on
what you thought was important.
The problem on hand is seldom solved because we’re never
certain that our knowledge base is complete. The reason that I prefer to
think in terms of spectra is due to the fact that it’s easier to account for
what’s missing. Furthermore, there’s so much more information that’s ‘
available and is expressed in spectral terms that interpretations
improve and develop greater relevance
FRACTALS AND SPECTRA
I started using spectral methods for analyzing fractures in 1974. I
came to the conclusion that since I had found that there so few
instances that I had found that an “average” something measured on a
fracture surface could be determined, that something new had to be
considered. A spectral analysis offered so many opportunities that I
ended up building my own apparatus and developed the methods that I
needed to get good results.
My encounter with fractals came about from some facture
experiments that I had done that showed that there was something
unusual in the fracture spectra. Every spectra had a
.
structure, not just one line that wasn’t related to the type of
material.
The most intriguing new aspects came from different directions.
The frequencies had been created by the separation of matter ( certainly
solids). So there was no doubt that they were derived from matter.
The second part of the intrigue that I experienced, occurred
when I found a relationship between the fractal dimension and the
fracture energy. I don’t think that anything had stirred my curiosity to
such a heighten level before.
I soon found something even more interesting, no one noticed!
They weren’t paying attention. I was fired up because the relationship
could not be understood by using today’s physics.
The spectrum above doesn’t suggest, it exposes an important
question. Does fracturing starts on the atomic level?
Since the body ends up it two or more parts there’s no other way that
something like that could happen. Then does fracture and the
separation of the bodies just depend on atomic structures. Absolutely
not! I’ve presented some images below that are similar to ones that can
be seen in an observation/.
But here’s something that’s never been discussed. I don’t think
that it will be easy to answer and that’s why I like it.
This figure shows irregular crack paths, and how structures
appear when looking through them, which is quite common. It’s
common to encounter an orderly discreteness meaning that there are
small regions of order held together by indistinguishable
“glassy”regions.
Below is an example of what needs to be done in order to begin
doing quantitative analysis. It’s necessary to estimate the volume of
material, to account and to identify what’s there. etc I’ve presented an
extremely simplified example which usually is a starting point for
forming a hypothesis.
I’m not surprised anymore when I’ve found something has been
neglected. Indeed something very important hasn’t been discussed in
fact, it hasn’t reached a level of perception that would indicate anything
will happen.
It’s come up before but in other contexts. The fracture spectra
show that there’s a relationship between amplitude and frequency when
matter “comes apart”
But there’s a famous story that involves a relationship between
amplitude and frequency, in fact it brought the house down some years
ago. It’s the same old conflict between classical and quantum physics,
one claims that by increasing the amplitude you should expect to have a
stronger effect. The other side has taken the opposite view, power is
only a small part of the story, it’s frequency that matters.
 Einstein’s interpretation of the photelectric effect gave me a
reason to interpret the fracture spectra in a different way; in
the case of photo emission the intensity of emission depends
(up to a limit) on the power input from the source (amplitude
dependence) but at the onset of photo emission the output
becomes frequency dependent.
MATHEMATICS, PROBABILITY AND STATISTICAL
MECHANICS
Conversation recently overheard between two enlightened
females
“Probability isn’t a thing!”
“It has no physical characteristics whatsoever! “
What about
… “collapse of the wave function”
But I heard you say ..”no a physical characteristics !”...
Well …those are just different they account for things
happening
All the others have no purpose whatsoever…
“Oh You know what I mean”
The role that (supposedly) probability plays isn’t limited to the
quantum scale. I ‘ve always considered it to be generic
Early on I constructed a mind map of probability and its
affiliations. It determined how and where I looked and helped to
establish boundaries. . It’s more or less a map of my interests an
encounters with different types of subjects that to have had caught my
interest
Gibbs
Boltzmann
N Particle sample
Equations for
Distribution Functions
Gibbs Ensemble
Based on Probability
Poisson
Deals with combinations
of incoherent wavefunctions
Density Matrix
Wave Function
Scaling
Gaussian
Normal
Partition function
can be used to calculate
thermodynamic properties
Fact: Scaling behavior is
commonly observed in
Nature
Fractals
Scaling a Part
of its formal structure
Mandelbrot
Classical
coordinates treated by integrating
over 6N coordinates of phase space
and quantum mechanically treated by summing over
6N discrete coordinates
Classical and quantum
viewpoints accommodated
Based on the probability of
phase space
Statistical Mechanics
Born's Viewpoint Established
a Probability Basis for
Quantum Mechanics
Delannoy
Quantum Mechanics
Binomial
Characteristics of
System Size
Squared Wave
Function is a
Probability
Density Function
Delannoy
Structured Scaling
Pascal
Probability
Gibbs
BEING VEXED BY STATISTICAL MECHANICS
I perused Statistical Mechanics and this time compared to earlier
years I developed a healthy skepticism because I had expanded my
understanding considerably and had garnered an enormous amount of
experience.
that fractals might be in a good position of working with
systems of the size that has been neglected, that is, before someone
uses Stirling’ approximation to replace the Binomial Distribution which
happens to be a key point where one of most important theoretical
developments begins.
Although the Laplace De Mioivre Theorem guarantees that
there’s nothing to be concerned about since for larger numbers of
particles the Gaussian distribution is equivalent to the Binomial
Distribution.
I didn’t have any comfort there because I wanted to view
everything as it actually is in its the initial. Stages. There are some
considerable differences:
 The Binomial Distribution is a discrete distribution
 I’d consider using the equation for Entropy in a different
but equivalent form, that more amenable to analytical
revisions
I soon realization that both statistical mechanics and QFT both
assumed that the number of particles were unlimited i.e
N ® N Av ® N¥
That was because NAv was so incredibly large. Furthermore, I’d
never have the ability to count so many particles.
Although the Laplace De Mioivre Theorem guarantees that
there’s nothing to be concerned about since for larger numbers of
particles the Gaussian distribution is equivalent to the Binomial
Distribution.
Every theoretical derivation that I’ve seen begins with the
Binomial Distribution. This helps to develop certain aspects of the
derivation but there’s a recognition that it isn’t suitable for dealing with
large numbers of particles and the factorial terms for large numbers are
difficult to work with are replaced by using Stirling’s approximation. The
Laplace D-Moiver Theorem is quoted as a justification for doing this
This is a form of Stirling’s approxinmation
There’s a major problem in doing this, the binomial distribution is
a discrete distribution and the Gaussian Distribution is not .
Furthermore, the Laplace De Moivre theorem isn’t a blessing, quite the
contrary, because it guarantees that it’s impossible to determine there’s
no influence from Binomial distribution in the theory of Statistical
Mechanics Formalism. there’s difference between them. So, once again
the Uncertainty Principle wins out since the Binomial Distribution is here
to stay.
I show how the two would appear if shown together.. A fractal
and Gaussian combination is shown below that.
Clustering is also part of the story and should no longer be
ignored.
There’s a very important fact to be remembered that concerns
fractals. All of the prime fractals are related in. one way or another to
the Binomial distribution furthermore since fractals are self - it’s quite
obvious that their many forms should be influential and extending
throughout most of the atomic and macroscopic worlds.
It seemed relevant to bring fractals into the foreground at this
point. They have some important characteristic features that are
discussed below. I till avoid any reference to Pascal’s triangle, simply
because I consider it to be a generating function, and nothing more than
that.
I’ve used another systematic method for subdividing space
that’s not fractal to show how a mathematical function can be used to
perform a task and that it converges to one value. But value of a fractal
doesn’t converge and it depends on the number of times that you reduce
it.
In this exercise there are no numbers only the systematic
subdivision of space. Anything whatsoever could be placed in any of the
subdivisions. The exercise ends up with a power series that replaces the
concept of a continuum.
I expect that Qantum Field Theory The binomial distribution
accounts for probability of the of n excitations that will occupy the
states that are available in the ground state.
Since Fractals are a quantized probability they provide a
means of selecting and defining a subset from the collection of particles
and identifying it with a number.Each subset has a different probability
distribution and are relationships that are known that can be used to
develop understanding how the relationships work
..
Two are of primary interest to QFT the Bose Einstein
distribution and the Fermi Dirac distribution. All of the particles are
quantized and their statistical behavior depends on their particle statecollective interactions. Their identity is the prime factor that influences
their behavior. The identification of every particle isn’t possible- rather-a
“particle” is used to describe an entire class of particles. Bosons are
indistinguishable meaning that they can change places and have no
consequences. It also means that an energy state can be occupied by a
Boson but since there’s no way to determine that there’s more one in
any state., many Bosons can occupy the same vacuum state. Electrons Fermions come in two types and can be distinguished by their spins.
The derivation is based on the Pauli Principle ( using it for
Fermions because there will be some other instances where this
can be applied)).
I’ve chosen to use the above figure for several reasons. It
represents something that I learned from R.Feynman it’s possible to
represent mathematical equations as cartoons. There’s something that
has emerged from this that has been very helpful
This shows the equations that are associated with the theory for
the quantum harmonic oscillator as developed in statistical mechanics.
This is becomes necessary in order to deal with large number of particles.
The assumption that everything starts from a state of equilibrium is
necessary in order to have a collection of stable states that define a
reference. The equations below describe that state of equilibrium.
Z=
-( E- m )
kT
e
- Ei
kT
åe
This is important, that by means the partition function every
thermodynamic function can be derived from.
Partition functions are multiplied because they are probabilities.
This equates the sum to the product.
Multiple sums become turned into sums in exponents simply
because the the sums are an equivalent mathematical form.
Once again, when multiple systems are to be accounted for
products of sums are used.
ENTER- CONTINUED FRACTIONS
An observation concerning Continued Fractions. I’ve mentioned
that the difference it has with some terms in statistical mechanics is
very interesting.In statistical mechanics the order that is used is related
to the standard treatment of independent probabilities. Below I’ve
shown one of Euler’s representations. Notice how the strings of integers
are associated. His diagrams are instructions related to the construction
of the continued fractions.
If the terms happen to be complex then this applies
I’ll discuss, in just a few pages, continued fractions. I’ve used
them in developing the idea “of a “Fractal in a Fractal”. I’ve already done
a calculation on this idea using a fractal in a matrix as the number 3. At
first it the whole method was so different and removed from the usual
way that this is treated that it was totally unfamiliar. I persevered
followed the rules that applied to matrices and produced an answer. I
haven’t spent any time, since then, in trying to understand the results. It
can’t be that difficult though because it has to be a yet undiscovered
number.
Fractals provide the means of being selective since it is a special
form of probability that’s quantized. It’s possible to find a specific set
say Mod[2] in a random distribution. This might be a very useful method
to place coded messages since there are an infinity of primes where they
could be hidden
FERMI DIRAC
There are three different distributions shown here the
Maxwell Boltzmann distribution- for distinguishable particles, the Fermi
Dirac distribution for Fermions distinguishable and Bose Einstein Bosons which are not distinguishable.
The Fermi Dirac distribution is of primary importance to the
entire semi- conductor industry. It’s use is found everywhere to diagnose
and to understand the electrical characteristics of devices.
The graph shows the junction carrier concentration as a function
of temperature. The exponential term has the chemical potential which
𝜕𝐸
is responsible the energy concentration dependence,  =
this an
𝜕𝑛
energy / concentration term which is responsible for the junction’s
dependence on the current level.


The Fermi energy is shown to be =1 obviously a reduced
energy, and the concentration – 0.5.
+
There are three different representations of the
harmonic oscillator shown below the graph. They are different ways to
represent the HO. The first one shows the classical,<-> quantum
representation, the second one is a pictoral one somewhat like Feynman
would use sticking with a representation of a sum, and then using
Feynman diagrams at every point to work out the mathematics of the
interactions. The last one resemembles the Fermi-Dirac equation, except
for g, which is related to the occupation numbers.
I realized that the form of equation similar to the one that I had
found for Mod[3].
I asked myself the most obvious question:
What’s the best way to have a more comprehensive
mathematical understanding of the structure of the Fermi Surface? I
thought of several conventional ways.
What if I could analyze a fractal within a fractal?
Was the first question…. As I mentioned previously .that was an
idea that I had analyzed in terms of fractals some time ago. I used prime
fractals directly in algebraic equations. But in this application time I
thought of using in a similar way. There would be a small possibility that
the theories of continued fractions could be easily transferred into my
application. But using a fractal based approach might open new
avenues that I hadn’t thought of before.
The new ideas of using mathematical function in the form of a
continued fraction an the “Fractal within a Fractal” were too appealing
to put aside. Also, I’d just have to build up a store of knowledge before
I’d have any understanding of their usefulness.
To a statement such as “build a background
” I’d have to ask ..”about what”?
In this case that’s not difficult. But I knew that there would be so
many new ( presently unknown) discoveries that the question looked
unimportant. In fact, I have several examples that demonstrate this in
the next section.
However, the results of the following experiments turned my
attention to continued fractions leading to the exploration of some new
territory that included fractals.
The next figure shows the problem that mentioned by using
matrix-fractal in its geometric form it demonstrates how.a fractal within
a fractal appears, but used the fractal in matrix form ,geometric form
and followed the standard rules of matrix algebra. I was willing to try
this because the prime fractals followed the known mathematical rules
of addition etc..
I never had an answer that wasn’t symmetrical. But I had no
way to interpret the answer. However, one came to mind that was only
2
one that I had it’s , I simply did the algebra.
5
I haveencountered one of the most amazing and unexpected
result This has been a most fortunate occurance because I it helped me
to identify an area of mathematics based on number theory, where
everything that was encountered was related to the project’s goals.
I spent time doing calculations and found that although theories
of Continued Fractions were helpful in interpreting answers and looking
at functions as though they had properties that were similar to those of
numbers. This created different expectations and introduced different
goals.
For example, the figure below showns how integers appear in
continued fraction expansions. If the integers were experimental
probability, then this would be a way to analyze the probability
propagation which involves the clustering of independent probabilities.
.
Merely by chance I happened to notice that there’s a
difference between Euler’s continued fraction formulation and the
manner in which partition functions are combined to make a composite
system in many body theory. Euler’s equation for complex continued
fractions. Note -it’s a sum of products
Continued fractions and solving the equation shows that it
converges.
Defining the Partition function for a system in many body theory
The partition function of a state’s configurations is a product . The
identifcation of a ground state is based on:
This gives rise to the partrticle the the probability distributions
for Fermions and Bosons.
Shown below is the Fermi Dirac distribution . It’s of primary
importance to the entire semi- conductor industry. It’s use is found
everywhere to diagnose and to understand the electrical characteristics
of devices.
The graph below shows the junction carrier concentration as a
function of temperature. The exponential term has the chemical
potential which is responsible the energy concentration dependence,
𝜕𝐸
 = 𝜕𝑛 this an energy / concentration term which is responsible for the
junction’s dependence on the current level.


The Fermi energy is shown to be =1 obviously a reduced
energy, and the concentration – 0.5.
+
THE “FRACTAL IN A FRACTAL “ EXPERIMENT
I did several experiments :
1. Verify the first equation
2. Find what the relationship between the level of
reduction and z is
3. Find the spectrum of the curves.
4. Use the two new methods to study its structure
I used Z as the term to replace the energy ratio and always
solved for it throughout this work.
I haveencountered one of the most amazing and unexpected
result This has been a most fortunate occurance because I it helped me
to identify an area of mathematics based on number theory, where
everything that was encountered was related to the project’s goals.
I spent time doing calculations and found that although theories
of Continued Fractions were helpful in interpreting answers and looking
at functions as though they had properties that were similar to those of
numbers. This created different expectations and introduced different
goals.
For example, the figure below showns how integers appear in
continued fraction expansions. If the integers were experimental
probability, then this would be a way to analyze the probability
propagation which involves the clustering of independent probabilities.
.
Yes tive in a very wonderful and u were extroardinary and
unexpected. I’ll report the some of the results first because it will be
easier to identify later how they are related to the curves.
1. I started with the standard FD equation then
operated at several levels or. [[..Fractal..]]
2. At the second level the upper shelf dropped to ~
0.6 and the population at the Fermi Energy was
the inverse of the divine section.
3. The upper shelf also became equal to
 -1and everything soon stabilized,
The solutions were in a form that lent themselves to polar
plotting.. I haveencountered one of the most amazing and unexpected
result This has been a most fortunate occurance because I it helped me
to identify an area of mathematics based on number theory, where
everything that was encountered was related to the project’s goals
After finding the unexpected appearance of Phi, and having
written several papers on it, I recalled some of its characteristics from
Number Theory but initially I didn’t think that there could be any
connection between Phi and /or QFT or Fractals.
Phi and the nature of its CF characteristics ts Continued Fraction
expansion could very useful in identifying occupation numbers and
classifying particle types in QFT.
The fact that I’ve been dealing with the Fermi Level and the
carrier population brings that into focus. In an obvious way.
I continued, I wondered if there was a way to analyze a
fractal that some aspect of fractals might be hidden that I haven’t
realized. My approach was direct, to use what I’ve been using to analyze
a fractal. Using the fractal Mod[2] that I’ve That wouldn’t be too
difficult.
I deconstructed the Mod [2] fractal and compared it to Phi. I
knew tat Phi would consist of all ones, and a mMod [2] fractal wouldn’tt
They had some similarity but I couldn’t pass any positive
judgment. They couldn’t possibly be equivalent because phi’s matrix
would end up being a constant one. But learned that it’s possible to
deconstruct a fractal and turn it into a number in any base.
.
The solutions were in a form that lent themselves to
polar plotting. That caused me to look for other applications and later I
began to explore different representations for Fourier transforms.
The next figures show all of the results.
This is the original Fermi Dirac. These figures were found by
solving the Equations for z the Fermi energy.
Phi appearance was welcome, astonishing and completely
unexpected.
.
Phi and it’s inverse are the roots of this equation.
Phi is 1.61803… and its inverse is 0.61803 which classifies it as
unusual but there’s more: Phi’s representation as a Continued Fraction
the only number known to have {1:1,1,1,1,,} as a continued fraction
expansion It’s a member of the Metallic Mean Family which consist of a
special group of numbers having the same property: n:n,n,n,n…
Another member of that group is the silver mean
=1+√2=2.414..  2:2,2,2,2,2…
The numbers must be irrational. I’ve studied 1024 members of
the string of irrational numbers from several of them. They all qualified
as noise generated by a discrete distribution
Euler had a form that he used for continued fractions.
.
I had used the idea before so the “Fractal in a Fractal” approach
was not foreign to me .Although a fractal reveals smaller and smaller
geometric details, in my prior work I had used a fractal as a numerical
object. Geometric in geometric terms perhaps, but I had been able to
work with a fractal in a geometric form in algebraic equations. I was very
happy to find an opportunity to develop a broader understanding about
these new ideas.
The fact that I’ve been dealing with the Fermi Level and the
carrier population brings that into focus. In an obvious way.
Bosons here ={1}.1,1,1,1….
Fermions here =2],[2] [2] [2]2[2] 22… just another
way to express what we already do. The difference is that I’ve given the
state an identity.
I continued, I wondered if there was a way to analyze a
fractal that some aspect of fractals might be hidden that I haven’t
realized. My approach was direct, to use what I’ve been using to analyze
a fractal. Using the fractal Mod[2] that I’ve That wouldn’t be too
difficult.
A continued fraction expansion cannot be used directly as a
number because it must be converted back into a number. By using a
computer in is easy to do this.
I’ve found a method that has multiple uses once it is done. The
starting sequential entries in a matrix can be extracted and used as
continued fractions. In the case of fractals a matrix such as the Serpinski
Gasket has a well-defined configuration. sequential deconstruction gives
an order to the string of numbers that correspond to the matrix i,j
addresses. Once the string is converted into a number then it becomes
more than just a number. I’m working with the assumption that it might
be possible to find rules that determine how to change one fractal into
another
They had some similarity but I couldn’t pass any positive
judgment. They couldn’t possibly be equivalent because phi’s matrix
would end up being a constant one. But learned that it’s possible to
deconstruct a fractal and turn it into a number in any base.
,,
A Fractal Involvement ?
As I continued, I suspected that some aspect of
fractals might be hidden that I haven’t realized. My approach was
direct, to use what I’ve been using to analyze a fractal. Using the fractal
Mod[2] that I’ve That wouldn’t be too difficut.
,,