 ```Subj:	Product form for the Lorentz factor
Date:	7/27/99
To:	ksbrown@seanet.com

Dear Kevin,
I have derived what I think is a new infinite product for the Lorentz
factor.  A proof of the identity (Theorem, VI) starts on page 4 of the
report at:
http://members.tripod.com/~EshlemanW/
Please reply as I think that it is a significant contribution to your
interests.
Sincerely,
Bill Eshleman

Subj:	 RE: Product form for the Lorentz factor
Date:	7/29/99 12:07:33 AM Eastern Daylight Time
From:	ksbrown@seanet.com (Kevin Brown)
To:	WDEshleman@aol.com ('WDEshleman@aol.com')

Hello,

The infinite product

oo
1/(1-x)  =  PROD (1 + x^(2^n))
n=0

is sort of a standard example of an infinite product.  It's
even given as the example under the definition of "infinite
product" in "The Harper Collins Dictionary of Mathematics" by
Borowski and Borwein.  (However, I notice they have a typo,
because they list the exponent on x as 2n instead of 2^n.)

Of course, Euclid showed that 1/(1-x) equals the geometric
series,  1 + x + x^2 + ..., so there are many different
infinite products, corresponding to the possible complete
numeration systems.  The one noted above corresponds to
the binary number system, whereas the base 3 numbers suggest
the infinite product

oo
1/(1-x)  =  PROD [1 + x^(3^n) + x^(2*3^n)]
n=0

=  (1 + x + x^2)(1 + x^3 + x^6)(1 + x^9 + x^18)...

and similarly for any other base.  Likewise, the factorial
number system suggests the product

oo
1/(1-x) = PROD [ 1 + x^(1*(n!)) + x^(2*(n!)) + ... + x^(n*(n!)) ]
n=1

=  (1 + x) (1 + x^2 + x^4) (1 + x^6 + x^12 + x^18)

(1 + x^24 + x^48 + x^72 + x^96)...

Since the nth partial sum of the geometric series is (1-x^n)/(1-x),
these products all just correspond to cyclotomic factorizations,
although not necessarily primitive factorizations.  In general
it is only necessary to define an infinite set of sets of integers
S1,S2,... such that every sum consisting of a single number from
each set is a unique non-negative integer, and every non-negative
integer is such a sum.  These are the same requirements for a number
representation, which explains why there is such a product for
each system of expressing numbers.

Hope this helps,

Kevin Brown

Subj:	Re: Product form for the Lorentz factor
Date:	7/29/99
To:	ksbrown@seanet.com

Thanks...I'm getting closer to my goal now that I know that my
Lemma II is in a book.  But how about Theorem VI?  I'm looking up
"cyclotomic factorization" and I've found much on the Internet, the
library of real books is next.
If you are interested, I can show you a quick way to go from
Lemma II to Theorem VI.  Thanks again.
Sincerely,
Bill Eshleman
http://members.tripod.com/~EshlemanW/

Subj:	 RE: Product form for the Lorentz factor
Date:	7/29/99 3:49:22 AM Eastern Daylight Time
From:	ksbrown@seanet.com (Kevin Brown)
To:	WDEshleman@aol.com ('WDEshleman@aol.com')

Your Lemma IV is in Euclid (IX,35), written circa 300 BC. Your
Theorem VI is just a particular algebraic identity, one of
infinitely many that can be derived based on the elementary
expressions for the finite and infinite geometric series.

Regards,
Kevin

Subj:	Re: Product form for the Lorentz factor
Date:	7/29/99
To:	ksbrown@seanet.com

on my webpage?  If you think of anything else of importance, please
let me know.
Sincerely,
Bill Eshleman
http://members.tripod.com/~EshlemanW/

Subj:	 RE: Product form for the Lorentz factor
Date:	7/30/99 1:14:56 AM Eastern Daylight Time
From:	ksbrown@seanet.com (Kevin Brown)
To:	WDEshleman@aol.com ('WDEshleman@aol.com')

Sure.  By the way, you might also be interested in a few ways of
seeing the identity you call "Theorem VI" directly.  For example,
bringing the factor of (1+x) over to the left side, the identity
is (use Courier font to read this formula)

/1 + x^2\1/2  /1 + x^4\1/4  /1 + x^8\1/8
1 = (1 - x^2)( ------- )   ( ------- )   ( ------- )  ...
\1 - x^2/     \1 - x^4/     \1 - x^8/

which we can instantly see is true, because the factors "telescope"
into each other, i.e., if we combine the first two terms on the
right side we get the equivalent relation

1/2  /1 + x^4\1/4  /1 + x^8\1/8
1   =        (1 - x^4)    ( ------- )   ( ------- )  ...
\1 - x^4/     \1 - x^8/

Then we combine the first two terms on the right side of THIS
expression to give the equivalent relation

1/4  /1 + x^8\1/8
1   =              (1 - x^8)    ( ------- )  ...
\1 - x^8/

and so on.  Thus it's clear that if |x| is less than 1 the right
hand infinite product equals 1.

Another way of seeing it is to write the original relation as

1         /1 + x^2\1/2  /1 + x^4\1/4  /1 + x^8\1/8
--------- =  ( ------- )   ( ------- )   ( ------- )  ...
(1 - x^2)     \1 - x^2/     \1 - x^4/     \1 - x^8/

and take the natural log of both sides, which gives the infinite
sum

1   /1 + x^2\    1   /1 + x^4\    1   /1 + x^8\
-ln(1 - x^2) = -ln( ------- ) + -ln( ------- ) + -ln( ------- ) + ...
2   \1 - x^2/    4   \1 - x^4/    8   \1 - x^8/

As you know, the two most common power series expansions involving the
natural log function are

u^2   u^3   u^4
-ln(1 - u) = u + --- + --- + --- + ...
2     3     4

and
1   /1+u \        u^3   u^5   u^7
-ln( ---- ) = u + --- + --- + --- + ...
2   \1-u /         3     5     7

so the logs on the right side of the preceding equation give the
overall sum

x^6           u^10           u^14
x^2    +     ---     +     ----     +     ----     + ...
3              5              7

x^4                         x^12
---            +            ----          +      ...
2                            6

x^8
---                     +            ...
4

which of course is the expansion of -ln(1 - x^2).

You could also express the above relations between logs in terms of
the respective continued fractions, noting that

u
ln(1 + u)  =   ----------
u
1 + ----------
u
2 + ----------
u
3 + ---------
...
and

1   /1+u\          u
-ln( --- ) =  -----------
2   \1-u/            u^2
1 - ----------
4u^2
3 - ---------
9u^2
5 - ----------
16u^2
7 - ----------
...

Obviously it's possible to construct infinitely many convoluted
identities in this way, such as an infinite sum of continued fractions,
and if we keep making more substitutions and inversions, etc., we
can create identities that would take quite a bit of labor to unravel,
though at base they are all just more or less disguised versions of the
simple geometric series.

Regards,
Kevin

Subj:	Product form for the Lorentz transform
Date:	7/27/99
To:	Paul.Marmet@Ottawa.com

Dear Paul,
I have derived what I think is a new infinite product for the Lorentz
factor.  A proof of the identity (Theorem, VI) starts on page 4 of the
report at:
http://members.tripod.com/~EshlemanW/
Please reply as I think that it is a significant contribution to your
interests.
Sincerely,
Bill Eshleman

Subj:	 Re: Product form for the Lorentz transform
Date:	7/30/99 2:13:51 PM Eastern Daylight Time
From:	Paul.Marmet@Ottawa.com (Paul Marmet)
To:	WDEshleman@aol.com

Dear Bill,
I have seen your paper transforming the Lorentz factor into an infinite
product.  You give the interpretation that this leads to Many Worlds
Interpretation.  Of course, mathematically, it is possible to transform
functions into a series expansion or infinite products.  Even constant
numbers can be represented by a series expansion.  The Lorentz
factor possesses a physical interpretation in a space of three
dimensions.  I believe that physics makes sense only in space of
three dimensions. Four dimensions can be used to shorten
calculations or make them easier, but there are only three dimentions
in PHYSICAL space.   The height, the width
and the depth. Everything can be located in that volume which exists
as a function of time.
I do not believe in many words interpretation.  I like it for fun, only
when I look at Star Trek or Star War.
This is a mathematical curiosity, but it is not the way Nature is done.
These mathematics are useful to study physics only when the provide
a shorter and better way to get the answer.  However, they do not
represents any PHYSICAL reality that needs only three dimensions.
Your mathematical demonstration is certainly interesting.
Thank you for letting me know.
Sincerely,
Paul Marmet
E-Mail:   Paul.Marmet@Ottawa.com

Subj:	Re: Product form for the Lorentz transform
Date:	7/30/99
To:	Paul.Marmet@Ottawa.com

In a message dated 7/30/99 2:13:51 PM Eastern Daylight Time, Paul.Marmet@Ottawa.com writes:

> I do not believe in many words interpretation.  I like it for fun, only
>  when I look at Star Trek or Star War.

I also dismiss the parallel-worlds conclusions that some authors find
so fascinating.  On the other hand we must not "throw out the baby
with the wash water."  I don't want to believe that copies of people
exist in many-worlds, but I do see a beauty in having copies of
subatomic particles existing in many-worlds.  I see no need for time
to be a dimension, and see sqrt(x^2 + y^2 + z^2 - (ct)^2)) as only a
shortcut to the "real" maths (possibly my maths).  Thanks for your time.
my webpage?
Sincerely,
Bill Eshleman
http://members.tripod.com/~EshlemanW/

Subj:	 Re: Product form for the Lorentz transform
Date:	7/30/99 5:21:34 PM Eastern Daylight Time
From:	Paul.Marmet@Ottawa.com (Paul Marmet)
To:	WDEshleman@aol.com

Thank you Bill,
I did not know that we were sharing the same opinion.
It is strange.  The very same day, somebody else writes that he will
publish an article against what I wrote,  you kindly agree with me.
You can use my comments and refer to my web page, if you wish.
Sincerely,
Paul
E-Mail:   Paul.Marmet@Ottawa.com
http://www.physics.uottawa.ca/profs/marmet/

[HK]
If you set like this, this x is equal to

x = - i Deltat H.

In this setting, we have an exact identity

Psi(t+Deltat) = exp(-i Deltat H) Psi(t) = exp (x) Psi(t)

according to the Schroedinger equation. This equals

Psi(t+Deltat) = (1+ x + x^2/2! + x^3/3! + x^4/4! + ...) Psi(t),

which seems different from your calculation:

Psi(t+Deltat)/Psi(t) = [ 1/(1 - x) ],

i.e.

Psi(t+Deltat)=(1+x+x^2+x^3+x^4+...)Psi(t)

Do you mean to imply what we actually observe is different from the exact
physical process to this amount? If so, then why/how?

Best wishes,
Hitoshi

[WDE]
Hitoshi, Matti, and Stephen,

I wish I had said that. We are discussing some competing
notions of change.  Hitoshi's result for Schroedinger case,

Psi(t+Deltat) = exp(x) * Psi(t)
= (1+ x + x^2/2! + x^3/3! + x^4/4! + ...) Psi(t),        (A)

is partitioned between the extremes,

Psi(t+Deltat) = (1 + x) * Psi(t)                (B)
and,
Psi(t+Deltat) = Psi(t) / (1 - x)                  (C)

And A is very close to the average of B and C, below x = 0.1 .

B implies that the future is entirely determined by full knowledge
of the present.  Or, FUTURE = (1 + x) * PRESENT.

C implies that the present is determined by knowledge that
will only be complete upon arriving at the present. Or,
NOW = PAST + x * NOW => NOW = PAST/(1 - x).

A implies that the future is entirely determined by knowledge
of the present and additional knowledge of the past (or at
least past knowledge of the properties of exp(x) ).

Given a choice, I choose C because it is suggested
by Relativity. Eg., M^2 = (M_0)^2 + (v^2/c^2) * M^2
=> M^2 = (M_0)^2 / (1 - v^2/c^2).  Because it
seems to be a reason for believing that it is the
possibilities of the future that attract the present
to it.  And because I some interesting notions
and additional identities concerning 1/(1 - x).

Now, if Relativity turned out to be, as in A,
M^2 = exp(v^2/c^2) * (M_0)^2,
I could see a unification by  the similarity of their
"first principle of change."  Since this does not appear to be
true for Relativity, I am then prone to at least question
and speculate whether we ought to consider wave equations that
do follow C's notion of change?  If you reply with a wave
equation for the notion of C, I will appreciate it alot.

[WDE]
Why/how? Because I am at a stage where consistency is far
more important than being correct.

[HK]
I do not think these notions of change competing. Your claim for C is correct
in observation, while A is also correct inside an LS with respect to its own
time. These two notions of change are consistent, whose proof I refer to the
reference

[WDE]
I find my intuitions evaporate and my notions crushed, but
when I read your work I find that you agree that relativity alters
the subjective experience of the observer, but to say that the
Schrodinger perspective is the objective perspective for local
systems?  I will accept that.  It is interesting to note that a
"factorial operator" will transform

1/(1-x) = (1+x+x^2+x^3+ ...)    to
exp(x) = (1+x+x^2/2!+x^3/3!+...).

As you say in your paper, "The quantum phenomena occurring in a local
system follow non-relativistic quantum mechanics, but the observed
values of quantum mechanical quantities should be corrected according
to the classical relativity so that the corrected values equal the values
predicted by the (non-relativistic) quantum mechanics."

Would not the "factorial operator" qualify as a corrector?

[HK]
Yes, if you mean by the factorial operator the one that transforms n to n!,
your statement is right and justifies the transformation from QM to
Relativity and vice versa, on the level of calculus/mathematical rules. I postulated
this relation between QM and relativity as a mathematical framework
and proved its consistency as a mathematical theorem. We have justifications
on the same level: I think you can assure the consistency of the two views
related by the transformation by the factorial operator with some additional words.

As a corrector, the factorial operator transformation might be useful in
applications and would make the understanding of the consistent unification
of the two seemingly contradictory views easier.

[WDE]
Hitoshi,

You say in your paper that "The quantum mechanical
phenomena between two local systems appear only
when they are combined as a single local system.  In
the local system the interaction and forces propagate
with infinite velocity or in other words, they are
unobservable."

In my analysis of infinite products equal to 1/(1-x) there
is a reason to infer that black holes, atoms, and the
universe as a whole all have event horizons inside of
which we cannot observe.  That is, black holes and atoms
have event horizons at 1/0.7035 * GM/c^2 = 1.4 * GM/c^2
and the universe has an event horizon at,
0.7035 * c/2 * sqrt(3/pi/G/rho), where rho is the density
of the universe.  Interactions inside or beyond the
event horizons are unobservable, but I have reservations
as to whether Faster Than Light propagations occur in
these regions, or whether they are necessary at all.
Here is my reasoning:

1/(1-x) = prod{ [1+x^(2^n)]^(1/2^n) : n=0,infinity }
* prod{ 1/[1-x^(2^n)]^(1/2^n) : n=1,infinity }
or,
1/(1-x) = A * B

I am almost forced to admit that A is the objective part
and B is the subjective part.  Therefore to correct the
observation we must simply remove the relativistic part
to reveal what really happened.  Now we have another
candidate for the QM principle of objective change.
Here are the candidates:

1) Psi(t+dt) = (1+x) * Psi(t)
2) Psi(t+dt) = exp(x) * Psi(t)
3) Psi(t+dt) = prod{ [1+x^(2^n)]^(1/2^n) : n=0,infinity } * Psi(t),
and the mixture of objective and subjective change,
4) Psi(t+dt) = Psi(t) / (1-x)

If we accept  eq. 3 as a candidate for objective change,
we notice first that it is the closest yet to eq. 2.  Second,
eq. 3 does not go to infinity when x = 1; eq. 3 evaluates
to the value of 4 (not eq. 4) at x=1.  That is,
4 = 2 * 2^(1/2) * 2^(1/4) * 2^(1/8) * 2^(1/16) * * *.  While
eq. 2 is 2.718... at x=1.  Now, and here is the problem,
eq. 3 does not converge for x > 1.  I must conclude that
a) either the propagation inside the event horizon is at the
speed of light or b) that the speed of light inside the event
horizon is actually zero and that communication between
points is FTL due to the direct contact between
incompressible matter points.  I prefer b), but cannot
exclude a).  This may seem so academic and so
hypothetical as to be ignored, but at this time my main
effort is for consistency not believability.  The properties
of my infinite products are so beautiful that I can't put them
aside because of the concern that I may be correct. :-(

Your positive feedback so far is greately appreciated,
but this is where I tend to loose people, because, if I am
wrong, there no reason to keep "kicking a dead horse."
So, be critical, you may save me 20 years of work, after
which I would only be in possession of a pure mathematical
object having nothing to do with reality.  Come to think of
it, that might not be so bad after all...

In my analysis of infinite products equal to 1/(1-x) there
is a reason to infer that black holes, atoms, and the
universe as a whole all have event horizons inside of
which we cannot observe.

[HK]
You seem to think LS as the region beyond the event horizon.

[WDE]
Yes, but there are also local systems of the type you suggest;
that is, local systems inside the local system we observe from.
Each of the many local systems that are inside our observation
local system is either a large collection of matter points (fermions)
or a small collection of matter points.  The largest and densest
collection of matter points is the black hole that, although it is in
our local system (the galaxy or universe), it is on the other side of
an event horizon and is in a way unobservable to us. In the same
way, atomic nuclei are each on the other side of their own event
horizon.  Between these event horizons are local systems of grains
of dust on up to local systems of stars and galaxies; i.e., the local
system of our universe.  The event horizons are constructions of
subjective observations extrapolated to locations we will never get
to observe directly.  In the same sense, the universe must itself
be confined to a subjective event horizon so that there must exist
other local systems (universes, galaxies, black holes, stars,
grains of sand, molecules, atoms, etc.) that are really beyond the
event horizon of our universe local system.  I know that you will
agree that there are many local systems open to our observation
in our own universe, and I will argue that there are many local
systems in the objective sense of many-worlds.

That is, black holes and atoms
have event horizons at 1/0.7035 * GM/c^2 = 1.4 * GM/c^2
and the universe has an event horizon at,
0.7035 * c/2 * sqrt(3/pi/G/rho), where rho is the density
of the universe.  Interactions inside or beyond the
event horizons are unobservable, but I have reservations
as to whether Faster Than Light propagations occur in
these regions, or whether they are necessary at all.

[HK]
The FTL propagation inside an LS in my context seems to
have different meanings from yours.

[WDE]
My meaning is not mine at all.  If it was, I would be smart.  My
systems.  The event horizons are what I previously believed to be
objective structures generated by my mathematics, but I am
happy to see them as limitations of subjective observation
predicted by my mathematics.  FTL communication is not
necessary in many-worlds, but I'll consider subjective FTL though.
Is that what you mean?  Here is the Everett idea as explained
by M. C. Price:

"Many-worlds is local and deterministic. Local measurements
split local systems (including observers) in a subjectively random
fashion; distant systems are only split when the causally transmitted
effects of the local interactions reach them. We have not assumed any
non-local FTL effects, yet we have reproduced the standard predictions
of QM."

[WDE]
Here is my reasoning:

1/(1-x) = prod{ [1+x^(2^n)]^(1/2^n) : n=0,infinity }
* prod{ 1/[1-x^(2^n)]^(1/2^n) : n=1,infinity }
or,
1/(1-x) = A * B

I am almost forced to admit that A is the objective part
and B is the subjective part.  Therefore to correct the
observation we must simply remove the relativistic part
to reveal what really happened.  Now we have another
candidate for the QM principle of objective change.
Here are the candidates:

1) Psi(t+dt) = (1+x) * Psi(t)
2) Psi(t+dt) = exp(x) * Psi(t)
3) Psi(t+dt) = prod{ [1+x^(2^n)]^(1/2^n) : n=0,infinity } * Psi(t),
and the mixture of objective and subjective change,
4) Psi(t+dt) = Psi(t) / (1-x)

If we accept  eq. 3 as a candidate for objective change,
we notice first that it is the closest yet to eq. 2.  Second,
eq. 3 does not go to infinity when x = 1; eq. 3 evaluates
to the value of 4 (not eq. 4) at x=1.  That is,
4 = 2 * 2^(1/2) * 2^(1/4) * 2^(1/8) * 2^(1/16) * * *.  While
eq. 2 is 2.718... at x=1.  Now, and here is the problem,
eq. 3 does not converge for x > 1.  I must conclude that
a) either the propagation inside the event horizon is at the
speed of light or b) that the speed of light inside the event
horizon is actually zero and that communication between
points is FTL due to the direct contact between
incompressible matter points.  I prefer b), but cannot
exclude a).

[HK]
If the region inside the event horizon could be objective in
your sense and is observable, it might be meaningful to wonder

[WDE]
My objective event horizon has evaporated.  That is, if
I use either,

Psi(t+dt) = exp(x) * Psi(t),
or,
Psi(t+dt) = prod{ [1+x^(2^n)]^(1/2^n) : n=0,infinity } * Psi(t),

it is not there anymore.  The way I analyze 1/(1-x) depends
on the mathematical fact that it is defined for x only up to
x=0.7035.  Above x=0.7035 it cannot calculate orbital
motion due to the lack of an inverse procedure to give
position and velocity in the region up to what I called the
event horizon.  Anyway, if FTL is subjective, FTL might
as well be observed even if it is not happening. Is FTL
in your context, merely subjective?  Is the constant
c in a vacuum, subjective, and FTL objective?

```

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