From: uunet!world!sweetser@ncar.UCAR.EDU (Doug B Sweetser) Subject: Re: Deriving Schrodinger using Quaternions Date: Tue, 6 Jun 2000 23:01:44 GMT Newsgroups: sci.physics.research Summary: Box differential operator Hello: In this post I will examine a suggestion made by Cl.masse over a week ago: "A simpler method (and perhaps a more valid one) is to Lorentz transform A directly, since it is a 4-vector. This transformation can be performed by quaternion operators. Also Box transforms under Lorentz." The Lorentz group has members that are classified as discrete, others that are continuous. The Lorentz transformation of a quaternion potential using the binary operator I called "Mirror" is a discrete transformation. Below I will define how continuous members of the Lorentz group can be defined, but it requires a subtle change from directly applying Masse's recommendation. Let's begin by reviewing the standard approach for a boost along the X axis. Here is how the 4-vectors that are used to generate the field strength tensor F transform: Box -> Box' = (gamma d/dt - gamma beta . d/dx, +gammma beta d/dt - gamma d/dx, -d/dy, -d/dz) A -> A' = (gamma phi - gamma beta . Ax, +gamma beta phi - gamma Ax, -Ay, -Az) Form the anti-symmetric tensor product of these two 4-vectors: F' = Box' A' - A' Box' F'_uu = 0, F'_uv = -F'_vu F'_xt = E'x = Ex F'_yt = E'y = gamma Ey - gamma beta Bz F'_zt = E'z = gamma Ez + gamma beta By F'_yz = B'x = Bx F'_xz = B'y = gamma By + gamma beta Ez F'_yx = B'z = gamma Bz - gamma beta Ey This is all very well know (and I am just copying from Feynman II, 26-9). The nuts and bolts of the quaternion multiplication are distinct from the standard approach with vectors and tensors. If the quaternion operator and potential are transformed just like the corresponding 4-vectors, the result will not be the transformed E and B fields correctly. Instead, a different transformation of the differential operator is required: Box -> Box' = (gamma d/dt + gamma beta . d/dx, -gamma beta d/dt -gamma d/dx, -d/dy, -d/dz) A -> A' = as before The vector beta goes to -beta for the differential operator which involves division by dx (inverting x to 1/x involves a conjugate, so this is not that strange). I will leave it as a serious amount of exercise to show that E' and B' are just like that in F' above. Something interesting does change, namely the gauge. Here is a rather tall order: "In other word, you have chosen the first alternative, that is, you don't use the field strengths. It remain to show that all equations of electro-magnetism can be so written, or better, that all observable phenomena can be so described." I will not make such a broad boast, but instead provide an observation. The reason that the 4 Maxwell equations and the Lorentz force appear at the beginning of graduate level electromagnetism books such as Jackson's, is because the rest of such works are largely (but not entirely) an examination of the mathematical consequences of those equations. For the particular case involved in this post, it took a while to find a different algebraic way with quaternions to the Lorentz transformations of the fields, but I believe it must necessarily exist. doug http://quaternions.com From kramsay@aol.commangled Mon Sep 4 13:35:38 CDT 2000 Article: 334411 of sci.math Path: news.math.niu.edu!husk.cso.niu.edu!vixen.cso.uiuc.edu!howland.erols.net!portc.blue.aol.com.MISMATCH!portc01.blue.aol.com!audrey05.news.aol.com!not-for-mail From: kramsay@aol.commangled (Keith Ramsay) Newsgroups: sci.math Subject: Re: Sensible Godel Theorem Question Lines: 159 NNTP-Posting-Host: ladder06.news.aol.com X-Admin: news@aol.com Date: 03 Sep 2000 23:51:59 GMT References: <39ABF5CC.BC802079@hot.rr.com> Organization: AOL http://www.aol.com X-Newsreader: Session Scheduler Message-ID: <20000903195159.29239.00000344@nso-cm.aol.com> Xref: news.math.niu.edu sci.math:334411 In article <39ABF5CC.BC802079@hot.rr.com>, "Ross A. Finlayson" writes: |What about this statement then: | | G_0 -> []G_0 We have this notation []X, "box X", which generally is the result of some kind of "necessity" operator applied to X, and sometimes the way it's used here, to mean "X is provable". A lot of statements have this property for a system satisfying the conditions of Goedel's theorem. Any statement of the form, "there exists a natural number n such that P(n)" where P(n) is a primitive recursive predicate does, because if there is such an n, it can be used to produce a proof that there exists one (giving n as example). Consider the converse, though, []X->X. If the system is correct, this is always true, so nobody who's happy with the system in the first place will have trouble with this principle, even though it's not a part of the system. There's a cute result about statements for which the principle can be proven, i.e. X satisfying []([]X->X). One obvious possibility is []X: if []X then since X implies []X->X in an elementary way, []([]X->X) for reasonable formal provability notions []. It turns out that for the sort of formal system which is strong enough for Goedel's theorem to apply, this is the only way: []([]X->X)->[]X. I was thinking about the proof of this result. It has a strange air of hocus-pocus. It occurred to me that although it was constructive, and presumably then presents a way of constructing a proof of X in the system given a proof of []X->X, it isn't constructive in a transparent way. On the bus to work one day I reworked it a bit in my head. The brief sketch I came up with goes like this: We apply Goedel's diagonal trick to get a sentence Y which is equivalent to [](Y->X) and provably so in the system. Goedel got an unprovability statement, but for the constructive approach it often is better to use positive statements; unprovability is negative. Inside the system: Assume Y. Then [](Y->X). Since Y is equivalent to a provability, Y->[]Y. So []Y. Combining [](Y->X) and []Y we get []X. But []X implies X. Thus Y->X. Outside: But if the system proves Y->X, we have [](Y->X). That's what Y is. Since Y is equivalent to a provability, Y->[]Y. So []Y. Combining []Y and [](Y->X) we get []X. Notice how similar the argument being illustrated inside the system is to the argument "outside". I asked myself what does the proof of X which this proof constructs look like. If you think about it, it looks an awful lot like the proof itself! We take this embedded proof of Y->X which is carried out within the system, and convert it into a proof of [](Y->X) within the system-- by showing in the system that the sequence of steps follows the rules of the system. But then we finish up by inferring Y in the system, and applying the same internal subproof again to deduce X from it. I thought that was pretty cute, and illustrates why the proof seems so peculiar. It's often been pointed out that Goedel's second incompleteness theorem is the special case of []([]X->X) -> []X where we substitute "false" for X. The provability of "false" is by definition the inconsistency of the system. [](false)->false is equivalent to ~[](false) which is the consistency of the system. So a system with enough infrastructure which proves its own consistency is inconsistent. I tried filling in steps here, and the result came out somewhat messy. I don't feel like trying to clean it up any further right now, but perhaps someone will enjoy reading it as it is. -------- We assume that the system has certain deductive powers: A. For any sentence P, []P -> [][]P. A proof of P can be used to demonstrate the P is provable, simply by exhibiting it and showing that each step satisfies the rules of the system. For any sentences P, Q, and R: B. If [](P<->Q) then [](P->Q) and [](Q->P). C. If []P and [](P->Q), then []Q. D. If [](P->Q) and [](Q->R) then [](P->R) E. [](P & Q->R) -> [](P->(Q->R)) F. If [](P->Q) and [](P->R) then [](P->Q&R). We assume also that the system is "aware" of the fact that it has deductive rules A-F, although I'm pretty sure we don't need all these: A'. []([]P->[][]P) B'. []( [](P<->Q) -> [](P->Q) ); []( [](P<->Q) -> [](Q->P) ) C'. []([]P & [](P->Q) -> []Q) D'. []([](P->Q) & [](Q->R) -> [](P->R)) E'. []([](P&Q->R) -> [](P->(Q->R))) F'. []([](P->Q) & [](P->R) -> [](P->Q&R)) We are given 0. []([]X->X) Goedel's trick gives us a statement Y satisfying 1. Y <-> [](Y->X). 2. [](Y<->[](Y->X)). That is, Y is equivalent to [](Y->X) in such an elementary way that it can be proven so in the system itself. Goedel's sentence is more analogous to the negation of Y-- asserting an unprovability-- but for constructive purposes it's good to let Y be a positive statement. Since Y is a statement of provability, it should entail it's own provability, and that implication should be provable within the system itself. It got messier than I thought to fill in the details here 3. Aim to show that [](Y->[]Y) a. [](Y->[](Y->X)) from (2) and (B). b. []([](Y->X)->[][](Y->X)) from (A'). c. [](Y->[][](Y->X)) from a, b, and (D). d. []([](Y->X)->Y) from (2) and (B). e. [][]([](Y->X)->Y) from d and (A). f. []([]([](Y->X)->Y) & [][](Y->X) -> []Y) from (C'). g. []( []([](Y->X)->Y) -> {[][](Y->X)->[]Y}) from f and (E). h. []([][](Y->X)->[]Y) by e, g, and (C). i. [](Y->[]Y) by c, h, and (D). 4. [](Y -> []Y & [](Y->X)) by 3a, 3i, and (G). 5. []([]Y & [](Y->X) -> []X) by (D'). 6. [](Y->X) by (E). I guess that deducing what the system is capable of deducing, while it seems transparent enough on the surface putting yourself in the shoes of the system, actually involves a certain amount of tedium due to its indirectness. At this point we realize that Y actually is true, by 1. Moreover, it can be proven in the system as well: 7. [][](Y->X) by (6) and (A). 8. []Y by (3d), (7) and (D). But that's all we need to get X in the system too: 9. []X by (6), (8), and (D). Keith Ramsay From torkel@sm.luth.se Mon Sep 4 13:35:55 CDT 2000 Article: 334453 of sci.math Path: news.math.niu.edu!husk.cso.niu.edu!vixen.cso.uiuc.edu!howland.erols.net!news.algonet.se!algonet!feed2.news.luth.se!news.luth.se!luth.se!not-for-mail From: Torkel Franzen Newsgroups: sci.math Subject: Re: Sensible Godel Theorem Question Date: 04 Sep 2000 07:08:26 +0200 Organization: University of Lulea, Sweden Lines: 22 Sender: torkel@beta13.sm.luth.se Message-ID: References: <39ABF5CC.BC802079@hot.rr.com> <20000903195159.29239.00000344@nso-cm.aol.com> X-Trace: news.luth.se 968044139 11440 130.240.3.1 (4 Sep 2000 05:08:59 GMT) X-Complaints-To: abuse@luth.se X-Newsreader: Gnus v5.5/Emacs 20.3 Xref: news.math.niu.edu sci.math:334453 kramsay@aol.commangled (Keith Ramsay) writes: > We apply Goedel's diagonal trick to get a sentence Y which is > equivalent to [](Y->X) and provably so in the system. This is a variant - the sentence used in the orignal proof of Lob's theorem was a sentence Y equivalent in T to []Y->X. Your variant yields a "Ramsay sentence" R for T: "This sentence is refutable in T". If R is true, it is refutable in T, but also provable in T So for consistent T, R is false, but not refutable in T. > I tried filling in steps here, and the result came out somewhat > messy. The properties of [] needed are the deducibility conditions of Hilbert and Bernays, which can be formulated (1) For any sentence P, if P is provable in T, so is []P. (2) For any sentence P, []P->[][]P is provable in T. (3) For all sentences P,Q, if P->Q is provable in T, so is []P->[]Q. From pkuzmic@biokin.com Sun Nov 12 00:17:45 CST 2000 Article: 2247 of sci.math.num-analysis Path: news1!news.math.niu.edu!husk.cso.niu.edu!vixen.cso.uiuc.edu!uchinews!newsfeed.cs.wisc.edu!loops.cs.wisc.edu!newsspool.doit.wisc.edu!news.chorus.net!not-for-mail From: Petr Kuzmic Newsgroups: sci.math.num-analysis Subject: Re: George Box said: Date: Fri, 10 Nov 2000 10:50:58 -0800 Organization: BioKin Ltd. Lines: 65 Sender: reguser@204.227.174.103 Message-ID: <3A0C4392.8ED1B138@biokin.com> References: <3A0BD216.CE23D24F@civil.ist.utl.pt> NNTP-Posting-Host: 204.227.174.103 Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: news.chorus.net 973882182 13491 204.227.174.103 (10 Nov 2000 18:49:42 GMT) X-Complaints-To: abuse@chorus.net NNTP-Posting-Date: 10 Nov 2000 18:49:42 GMT X-Mailer: Mozilla 4.72 [en] (WinNT; U) X-Accept-Language: en Xref: news1 sci.math.num-analysis:2247 Manuel Ritto Correa wrote: > > "All models are wrong. Some models are useful." > > I like that quote! Can you tell me something more about who is George > Box? George Box is Professor Emeritus of statistics at the University of Wisconsin in Madison. As far as I know he started in the chemical industry and never quite got out of that mode. This makes his writings very interesting for all experimentalists and amateur statisticians [like me] trying to investigate the mechanisms of chemical or biochemical reactions via statistical analysis of experimental data. Website: -------- http://www.engr.wisc.edu/ie/faculty/box_george.html. A popular book: --------------- G.E.P. Box, W.G. Hunter, and J.S. Hunter (1978) "Statistics for Experimenters" Wiley, New York Two volumes of collected journal articles (< 1985): --------------------------------------------------------- "The Collected Works of George E. P. Box" - Volumes I, II George C. Tiao - Editor Wadsworth, Belmont, 1985 ISBN 0-534-03307-5 (vol I) ISBN 0-534-03308-3 (vol II) My favored paper by Box ----------------------- "Science and Statistics" J. Amer. Statist. Assoc. 71, 791-799 (1976) This is the paper where he used the charming terms 'cookbokery' and 'mathematistry', which denote two different but equally wrong way of doing statistics: one is stuck in the experiment and the other in theory. Another interesting tidbit in that paper is where Box reports how he looked some old handwritten family records belonging to Fisher, and found out where Figure 1 in Fisher's "Statistical Methods for Research Workers" comes from. Fisher says in that book: 'Figure 1 represents the growth of a baby weighed to the nearest ounce...'. Box found out that that was actually Fisher's second baby, Harry Leonard Fisher, born in 1923 shortly before the first edition of "Statistical Methods". One final factoid: I know people who remember R. A. Fisher visiting in Madison with Box, but I don't remember which year that was supposed to be. > Thanks in advance You're welcome. Happy reading! -- Petr _____________________________________________________________________ P e t r K u z m i c, Ph.D. mailto:pkuzmic@biokin.com BioKin Ltd. * Software and Consulting http://www.biokin.com