1 Quaternion Analysis

Transcription

1 Quatrnion Analysis Complx numbrs ar a subfild of uatrnions. My hypothsis is that complx analysis should b slf-vidnt ithin th structur of uatrnion analysis. Th challng is to dfin th drivativ in a non-singular ay, so that a lft drivativ alays uals a right drivativ. If uatrnions ould only commut... Wll, th scalar part of a uatrnion dos commut. If, in th limit, th diffrntial lmnt convrgd to a scalar, thn it ould commut. This ida can b dfind prcisly. All that is ruird is that th magnitud of th vctor gos to zro fastr than th scalar. This might initially appars as an unrasonabl constraint. Hovr, thr is an important application in physics. Considr a st of uatrnions that rprsnt vnts in spactim. If th magnitud of th -spac vctor is lss than th tim scalar, vnts ar sparatd by a timlik intrval. It ruirs a spd lss than th spd of light to connct th vnts. This is tru no mattr hat coordinat systm is chosn. fining a Quatrnion A uatrnion has 4 dgrs of frdom, so it nds 4 ral-valud variabls to b dfind: a, a, a, a Imagin ant to do a simpl binary opration such as subtraction, ithout having to spcify th coordinat systm chosn. Subtraction ill only ork if th coordinat systms ar th sam, hthr it is Cartsian, sphrical or othris. Lt,,, and b th shard, but unspcifid, basis. No can dfin th diffrnc btn to uatrnion and that is indpndnt of th coordinat systm usd for th masurmnt. d a a, a a /, a a /, a a / What is unusual about this dfinition ar th factors of a third. Thy ill b ncssary latr in ordr to dfin a holonomic uation latr in this sction. Hamilton gav ach lmnt parity ith th othrs, a vry rasonabl approach. I hav found that it is important to giv th scalar and th sum of th -vctor parity. Without this scal factor on th -vctor, chang in th scalar is not givn its propr ight. If d is suard, th scalar part of th rsulting uatrnion forms a mtric. dˆ da da da 9 da 9 9, da da, da da, da da What should th connction b btn th suars of th basis vctors? Th amount of intrinsic curvatur should b ual, so that a transformation btn to basis -vctors dos not contain a hiddn bump. Should tim b tratd xactly lik spac? Th Scharzschild mtric of gnral rlativity suggsts othris. Lt,, and form an indpndnt, dimnsionlss, orthogonal basis for th -vctor such that: This unusual rlationship btn th basis vctors is consistnt ith Hamilton s choic of, i, j, k if ˆ that cas, calculat th suar of d: d da da da da 9 9 9, da da, da da, da da. For

2 Th scalar part is knon in physics as th Minkoski intrval btn to vnts in flat spactim. If ˆ dos not ual on, thn th mtric ould apply to a non-flat spactim. A mtric that has bn masurd xprimntally is th Scharzchild mtric of gnral rlativity. St ˆ ( - GM/cˆ R), and calculat th suar of d: d da G M da.da da c, da R G M 9, da da, da da c R This is th Scharzchild mtric of gnral rlativity. Notic that th -vctor is unchangd (this may b a dfining charactristic). Thr ar vry f opportunitis for frdom in basic mathmatical dfinitions. I hav chosn this unusual rlationships btn th suars of th basis vctors to mak a rsult from physics asy to xprss. Physics guids my choics in mathmatical dfinitions :-) An Automorphic Basis for Quatrnion Analysis A uatrnion has 4 dgrs of frdom. To compltly spcify a uatrnion function, it must also hav four dgrs of frdom. Thr othr linarly-indpndnt variabls involving can b dfind using conjugats combind ith rotations: a, a /, a /, a / a, a /, a /, a / a, a /, a /, a / Th conjugat as it is usually dfind (*) flips th sign of all but th scalar. Th * flips th signs of all but th trm, and * all but th trm. Th st, *, *, * form th basis for uatrnion analysis. Th conjugat of a conjugat should giv back th original uatrnion.,, Somthing subtl but prhaps dirctly rlatd to spin happns looking at ho th conjugats ffct products:, Th conjugat applid to a product brings th rsult dirctly back to th rvrs ordr of th lmnts. Th first and scond conjugats point things in xactly th opposit ay. Th proprty of going half ay around is rminiscnt of spin. A tightr link ill nd to b xamind. Futur Timlik rivativ Instad of th standard approach to uatrnion analysis hich focuss on lft vrsus right drivativs, I concntrat on th ratio of scalars to -vctors. This is natural hn thinking about th structur of Minkoski spactim, hr th ratio of th chang in tim to th chang in -spac dfins fiv sparat rgions: timlik past, timlik futur, lightlik past, lightlik futur, and spaclik. Thr ar no continuous Lorntz transformations to link ths rgions. Each rgion ill ruir a sparat dfinition of th drivativ, and thy ill ach hav distinct proprtis. I ill start ith th simplst cas, and look at a sris of xampls in dtail. finition: Th futur timlik drivativ: Considr a covariant uatrnion function f ith a domain of H and a rang of H. A futur timlik drivativ to b dfind, th -vctor must approach zro fastr than th positiv scalar. If this is not th cas, thn this dfinition cannot b usd. Implmnting ths ruirmnts involvs to limit procsss applid suntially to a diffrntial

3 uatrnion. First th limit of th thr vctor is takn as it gos to zro, ( - *)/ ->. Scond, th limit of th scalar is takn, ( *)/ -> (th plus zro indicats that it must b approachd ith a tim gratr than zro, in othr ords, from th futur). Th nt ffct of ths to limit procsss is that ->. f,,, limit as d, limit as d, > d, f d,,,, f,,, d, Th dfinition is invariant undr a passiv transformation of th basis. Th 4 ral variabls a, a, a, a can b rprsntd by functions using th conjugats as a basis. f,,, a f a / f a / / / f a / Bgin ith a simpl xampl: f,,, a a a a / lim lim d, d, a Th dfinition givs th xpctd rsult. A simpl approach to a trickir xampl: f a / a a a lim lim d, a / d, So far, th fancy doubl limit procss has bn irrlvant for ths idntity functions, bcaus th diffrntial lmnt has bn liminatd. That changs ith th folloing xampl, a tricky approach to th sam rsult. f,,, a / a a lim lim d, / d,

4 lim lim d, lim d, / d, / d, Bcaus th -vctor gos to zro fastr than th scalar for th diffrntial lmnt, aftr th first limit procss, th rmaining diffrntial is a scalar so it commuts ith any uatrnion. This is hat is ruird to danc around th and lad to th cancllation. Th initial hypothsis as that complx analysis should b a slf-vidnt subst of uatrnion analysis. So this uatrnion drivativ should match up ith th complx cas, hich is: z a b i, b Z Z /i b z i b z Ths ar th sam rsult up to to subdits. Quatrnions hav thr imaginary axs, hich crats th factor of thr. Th conjugat of a complx numbr is rally doing th ork of th first uatrnion conjugat * (hich uals -z*), bcaus z* flips th sign of th first -vctor componnt, but no othrs. Th drivativ of a uatrnion applis ually ll to polynomials. lt f f lim lim d, d, lim lim d, d, d, d, lim lim d, d, d, lim d, This is th xpctd rsult for this polynomial. It ould b straightforard to sho that all polynomials gav th xpctd rsults. Mathmaticians might b concrnd by this rsult, bcaus if th -vctor gos to - nothing ill chang about th uatrnion drivativ. This is actually consistnt ith principls of spcial rlativity. For timlik sparatd vnts, right and lft dpnd on th inrtial rfrnc fram, so a timlik drivativ should not dpnd on th dirction of th -vctor. Analytic Functions Thr ar 4 typs of uatrnion drivativs and 4 componnt functions. Th folloing tabl dscribs th 6 drivativs for this st a a a a / / / / / / / / This tabl ill b usd xtnsivly to valuat if a function is analytic using th chain rul. Lt s s if th idntity function is analytic. Lt a, a, a, a Us th chain rul to calculat th drivativ ill rspct to ach trm: 4

5 a a a a / / / Us combinations of ths trms to calculat th four uatrnion drivativs using th chain rul. a a a a a a a a a a a a a a a This has th drivativs xpctd if is analytic in. Anothr tst involvs th Cauchy-Rimann uations. Th prsnc of th thr basis vctors changs things slightly. Lt u a,,,, V, a, a, a u V u a a, V u a a, V a a This also solvs a holonomic uation. u V V V Scalar,,, a a a a,,, Thr ar no off diagonal trms to compar. This xrcis can b rpatd for th othr idntity functions. On noticabl chang is that th rol that th conjugat must play. Considr th idntity function *. To sho that this is analytic in * ruirs that on alays orks ith basis vctors of th * varity. Lt u a,,,, V, a, a, a u V u a a, V u a a, V a a This also solvs a first conjugat holonomic uation. u V V V Scalar,,, a a a a,,, Por functions can b analyzd in xactly th sam ay: 5

6 Lt a a a 9 a 9 a a, a a, a a u a a a 9, a 9,,, 9 9 V, a a, a a, a a u a u a u a a a a V a V a V a This tim thr ar cross trms involvd. u a V a 9 a u a V a 9 a u a V a 9 a At first glanc, on might think ths ar incorrct, sinc th signs of th drivativs ar suppos to b opposit. Actually thy ar, but it is hiddn in an accounting trick :-) For xampl, th drivativ of u ith rspct to a has a factor of ˆ, hich maks it ngativ. Th drivativ of th first componnt of V ith rspct to a is positiv. Kping all th information about signs in th s maks things look non-standard, but thy ar not. Not that ths ar thr scalar ualitis. Th othr Cauchy-Rimann uations valuat to a singl -vctor uation. This rprsnts four constraints on th four dgrs of frdom found in uatrnions to find out if a function happns to b analytic. This also solvs a holonomic uation. u V V V Scalar,,, a a a a,,, a a a a Sinc por sris can b analytic, this should opn th door to all forms of analysis. (I hav don th cas for th cub of, and it too is analytic in ). 4 Othr rivativs So far, this ork has only involvd futur timlik drivativs. Thr ar fiv othr rgions of spactim to covr. Th simplst nxt cas is for past timlik drivativs. Th only chang is in th limit, hr th scalar approachs zro from blo. This ill mak many drivativs look tim symmtric, hich is th cas for most las of physics. A mor complicatd cas involvs spaclik drivativs. In th spaclik rgion, changs in tim go to zro fastr than th absolut valu of th -vctor. Thrfor th ordr of th limit procsss is rvrsd. This tim th scalar 6

7 approachs zro, thn th -vctor. This crats a problm, bcaus aftr th first limit procss, th diffrntial lmnt is (, ), hich ill not commut ith most uatrnions. That ill lad to th diffrntial lmnt not canclling. Th ay around this is to tak its norm, hich is a scalar. A spaclik diffrntial lmnt is dfind by taking th ratio of a diffrntial uatrnion lmnt to its -vctor, - *. Lt th norm of approach zro. To b dfind, th thr vctor must approach zro fastr than its corrsponding scalar. To mak th dfinition non-singular vryhr, multiply by th conjugat. In th limit */(( - *)( - *))* approachs (, ), a scalar. f,,, f,,, limit as, > limit as d, >, f d, f d,,,, f,,, d,,,, f,,, d, To mak this concrt, considr a simpl xampl, f ˆ. Apply th dfinition: Norm limit, > limit as d, >, a, B d, a, B d, a, B d, a, B d, lim a, B, a, B, norm,, a, B, a, B, norm,, Th scond and fifth trms ar unitary rotations of th -vctor B. Sinc th diffrntial lmnt could b pointd anyhr, this is an arbitrary rotation. fin: a, B, a, B, norm, Substitut, and continu: lim a, B a, B, lim 4aˆ B. B B. 4aˆ B. B B. B, B a, B a,. B. < B, B, Look at ho ondrfully strang this is! Th arbitrary rotation of th -vctor B mans that this drivativ is bound by an inuality. If is in dirction of B, thn it ill b an uality, but could also b in th opposit dirction, lading to a dstruction of a contribution from th -vctor. Th spaclik drivativ can thrfor intrfr ith itslf. This is uit a natural thing to do in uantum mchanics. Th spaclik drivativ is positiv dfinit, and could b usd to dfin a Banach spac. fining th lightlik drivativ, hr th chang in tim is ual to th chang in spac, ill ruir mor study. It may turn out that this drivativ is singular vryhr, but it ill ruir som skill to find a tchnically viabl compromis btn th spaclik and timlik drivativ to synthsis th lightlik drivativ. 7