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 Thanks again, Kevin. May I use your comments, name, and address 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. May I use your comments, name, and address (or just comments) on 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. (You forgot minus sign in the above). 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 http://www.kitada.com/time_I.tex [WDE] I've now read your paper on local times. Usually when I read 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 meaning is that of many-worlds made compatible with your local 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 about FTL. Is your event horizon transparent for the observer? [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?

Back to Page 1: