QUATERNION ANALYTICITY OF HARMONIC OSCILLATOR
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1 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios QUATRIO AALYTICITY O ARMOIC OSCILLATOR Sm Rwt Dprtmt of Physis Zkir ussi Collg Jwhr hru Mrg w Dlhi Idi. mil: rwtsm@rdiffmil.om ASTRACT Thory of utrioi hrmoi osilltor hs lyzd for th systm of osos d frmios rsptivly i trms of ommuttio d tiommuttio rltios. Corrspodig igvlus of prtil miltoi d umr oprtors r lultd y imposig th rstritio o th ompot of utrio vrils. It hs show tht th suprhrg ommuts with osoi d frmioi umr oprtors whil th miltoi ostrutd from tiommuttio of suprhrgs. I SUSY lgr it hs show tht th suprhrgs ommut with th suprsymmtri miltoi.. Itrodutio: Qutrios wr th vry first xmpl of hypr omplx umrs hvig th sigifit impts o mthmtis d physis [-]. I utrioi frmwork turl lws [4] d si utios hv formultd y iklsti t l [5-7]. iklistit.l. [] first itrodud utrioi utum osilltor. Du. [8] usd utrios to ostrut osoi osilltor. Si utrios r th xtsio of omplx umrs ut thy diffrs i th wy tht thy do ot ommut mog thmslvs. This puts rstritios i th formultios of osoi utrio osilltor frmio osilltor d th suprsymmtri hrmoi osilltor. Aftr th ivtio of spiig prtils ttmpts wr md to itrodu utrios i trms of Puli mtris d ordigly th isomorphism tw mtris d utrios ws dvlopd y Jolly [9] &Silvri []. Suprsymmtry ws first pprd i fild thoris i trms of osoi d frmioi filds d th possiility ws rly osrvd tht it ould ommodt Grd Uifid Thory GUT for four si itrtios of tur strog wk ltromgti d grvittiol. Glfd d Liktm [] did th first work o supr lgr i sp-tim withi th frmwork of Poir lgr. Th SUSY lgr i utum mhis ws iititd withi th work of ioli []. Th osoi dgrs of frdom r hrtrizd y osoi rtio d ihiltio oprtors oyig th ommuttio rltios. Similrly frmioi dgrs of frdom r dsrid y frmioirtio d ihiltio oprtors d oy tiommuttio rltios. ioli s SUSY lgr ws dsrid s SUSY lgr hs xtdd y Witt []. Th xt suprsymmtry dsris symmtry tw osoi d frmioi dgr of frdom d is sstil igrdit i grd uifid thory. Th strutur of Li lgr tht
2 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios iorports ommuttio d tiommuttio rltios hrtrizs w typ of symmtry dymil symmtry whih is suprsymmtry i.. symmtry tht ovrts osoi prt ito frmioi prt d frmioi prt ito osoi prt. miltoi is o of th grtor of this supr lgr rmis ivrit udr suh trsformtios. So tht trmdous physil otts r iludd i it s it ots diffrt utum systms. I utum mhil systm SUSY hs foud to vry usful [4]. xt SUSY implis xt frmio oso msss whih hs ot osrvd so fr. Kpig i viw th ov poits d strtig from th most grl osidrtio tht osoi oprtor r utrio vlud d usig th fudmtl ommuttio rltio tht osoi oprtors stisfy w hv otid th osoi hrmoi osilltor i trms of miltoi with som rstritios i trms of utrio offiits. Qutrio ihiltio d rtio oprtor hs otid y rplig th imgiry umr i y utrio uit j. It hs show tht th umr oprtor ommuts with miltoi d h is ostt of motio. Th ommuttio rltio tw umr oprtor d rtio oprtor s wll s ihiltio oprtor r otid osisttly. Aordigly usig th igvlus of rtio ihiltio d umr oprtors th omplt igvlu sptrum of osoi hrmoi osilltor hs otid. It hs show tht th groud stt rgy of hrmoi osilltor oms out to first xitd stt oms whil tht of d so o d this givs th usul rsults of lgr usd for th systm of osos. It hs show tht thos prtils whih do ot follow th os-isti sttistis xtly d show dformtio from this sttistis r dsrid i trms of -dformd lgr. w hv otid th utrioi -dformd osilltors d it hs show tht i th limitig s -dformtio rsults ovrlp with th tul rsults of systm of osos. Similrly w hv dsrid th utrioifrmioi oprtor whih stisfy tiommuttio rltio. Th frmioi hrmoi osilltor hs otid i trms of miltoi d igvlu sptrum. It hs show tht th igvlu utio of frmioi umr oprtor dos ot hg th stt d th frmio hrmoi osilltor osists oly two rgy vlus &. So it is possil to rt o frmioi prtil i o stt d thus oy Puli xlusio priipl. Lik osos w hv ostrutd - dformd frmio hrmoi osilltor for ths prtils whih xtly do ot follow th.d. sttistis d h tiommuttio rltios i trms of dformtio prmtr r otid osisttly d orrspodig igvlu sptrum is dsrid. Suprhrgs th grtors of suprsymmtri trsformtios r ostrutd from osoi d frmioi rtio d ihiltio oprtors. It hs show tht th suprhrg ommuts with osoi d frmioi umr oprtors whil th miltoi ostrutd from tiommuttio of suprhrgs. I SUSY lgr it hs show tht th suprhrgs ommut with th suprsymmtri miltoi. Aordigly w hv ostrutd rgy
3 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios 4 igvlu of miltoi y summig th osoi d frmioi rgy igvlus. SUSY miltoi hs rprstd i digol mtrix form d th splittd ito two supr prtrs miltoi +& - to oti th rgy igvlu sptrum of utrio SUSY osilltor.. Qutrioi osoi rmoi Osilltor: Lt us dfi osoi utrio osilltor s th xtsio of omplx osilltor hvig th dompositio whr & r oprtors d & r utrio uits.its ojugt is dfid s Lik othr osilltor lt us strt with th followig fudmtl oso ommuttio rltio i.. & Th oditio lds to whih givs 4 This is possil oly wh If w impos followig oditios & 5 whr w ssumd tht & ommut with osoi ompots & Or i grl A A A. Lt us dsri th miltoi for osoi hrmoi osilltor P 7 whih writt i trms of & s / 8 whr is lld osoi umr oprtor. I trms of osoi ompots giv s 9 W form th miltoi y p & p Th to rovr th ordiry ommuttio rltio P W th gt th vlu of P & oordit y P
4 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios I th ordiry ssustitutig ommuttio rltio tw P i. Th osoi umr oprtor with miltoi i.. [ ] whih stisfis th followig rltios i th & P rdus to - - [ ] [ ] [ ] [ ommuts ] [ ] [ ] 4 ilrt sp of th osilltor is dfid y â d â my rgrdd s rtio oprtor d ihiltio oprtor rsptivly.lt with 5 Th from utios 4 is igvtor of with ig vlu providd oly tht. Th surd orm of this vtor is â 7 rom utio 4 w gt 8 Th surd orm of this vtor is 9 Whih vr vishs us. Thus â is igvtor of with igvlu. y rptdly pplyig oprtor â w ostrut st of igvtors hvig th igvlus... Th su will gi with igvlu. So th orthoorml igvtors of dotd s will Si is proportiol to. So w my writ C whr C is th ostt of proportiolity otid from th orm of this vtor. C C rom this rsult w hv -! whih dsris th followig xprssio for th mtrix lmt ssoitd with th rtio d ihiltio oprtors o imposig th orthoormlity oditio to utio Shows tht mtrix hs its o-zro lmt o sp low th digol d o sp ov th digol. rom utios d w gt followig rsults 5
5 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios & 4 illy th igvlus d igvtors of osoi hrmoi osilltor r dsrid s 4 whr. I othr words w gt followig ig-vlu sptrum Th stt vtors s th sp th ilrt sp! Whr is osidrd s groud or vuum stt whih must ormlizd 7 Th givs th fmilir rsults + +/ whr ow o ostrut th igvlu sptrum s +/ 9 This is th groud stt rgy of osoi hrmoi osilltor ow th rgy of first xitd stt is giv y 9 Similrly. 5 9 d so o. It shows tht ig vlu is groud stt. W my ow xprss th miltoi for i th rprsttio i whih is digol w lult th igvlus of. Th igfutio x x tk s xpsio offiit of strt igvtors of d i trms of igvtors of.w my ow lult th mtrix lmt spd y th sis of lmts formd y th igvtors of d of â i th followig mtrix form usig utio for projtio oprtor d utio i... -Dformd osoi Osilltor: o ` Somtims prtils do ot follow os isti s sttistis ompltly. Thy show dformtio from it. I tht s thy r dfid y itroduig th dformtio prmtr. Suppos it is dotd y. Th th osos r supposd to follow th followig rltios [ ]
6 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios 7 whr is th osoi umr oprtor. I symplti rprsttio dfid s So utio writt s Th ottio i trms of dformtio prmtr dfid s x x x 4 5 I I I 5 Th th rltio 9 d i trms of dformtio prmtr writt s. I I P 7 d th umr oprtor is giv y utio utio 4 d 5 writt s 8 P 9 Tkig th rgy of -dformd osilltor is giv s follows 4 As suh w my hv th followig st of utio of rgy sptrum i.. for groud stt rgy or 4 Corrspodig to th usul rsults of utum mhis. i or 4 ii or 4 d so o. I th limitig s w gt th fmilir rsults ut wh th rgy stts r ot ully spd lthough w gt wll kow rsults tht rgy is / for groud stt still. 4. Qutrioi rmioi rmoi Osilltor: Similrly w writ th followig ti ommuttio rltio for frmioi hrmoi osilltor
7 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios whr is utrio d my domposd s 45 whr d. ow for to th ompot of frmioi hrmoi osilltor th ssry d suffiit oditio tht ths must hv to stisfy ll th rltios of utio 44. So w my pply ths rltios i followig stps. i Th oditio or 4 iith oditio i iii Th oditio i y omiig utio w gt followig oditios i i j i j i 49 whr th pitl rkt dfis th ommuttor d th urly rkt dfis th tiommuttor. Lt us ow writ th frmioi miltoi s Sustitutig th vlu of i utio 5 from utio W my xprss th frmios i trms of Puli mtris s & I 5 y xpdig th ommuttig d tiommuttig rkt w gt & - 5 & 5 r d formd from utrio mtris s follows
8 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios i 5 i 5 Lt us oti th ig vlu sptrum of frmio osilltor for whih th ilrt sp with igvtor ostrutd s ; ; & 54 So tht th ig vlu utio 5 is giv s - - or Dformd rmioi Osilltor: W ow stlish th dformto for th systm of frmios o th sm sis s w hv disussd rlir for th systm of osos. Th umr oprtor frmios stisfy th rltio tks vlus d C 5 istd of tiomuttio rltio giv y utio 44. Thus w gt th followig rltio for frmio oprtors 5 d th frmioi miltoi is dsrid y utio 5. Th sptr of whih hv oly two lvls & 57 Th rgis r ow -idpdt d groud stt rgy is thus gtiv. or 55 lso shows th ormlizd stt. w gt th igstt & i this osilltor.th groud stt rgy of this osilltor is. Si. So it is possil to rt o frmioi prtil i o stt. Thus oy Puli xlusio priipl.. Suprsymmtri rmoi Osilltor: Lt us ow ostrut simpl suprsymmtri utum mhil systm tht ilud th osoi osilltor dgr of frdom d frmioi spi dgrs of frdom.w ll it s suprsymmtri hrmoi osilltor.th suprsymmtry is otid y ihiltig simultously o osoi utum d rtig o frmioi utum or vi vrs. W illustrt th f f ihiltig suprsymmtri hrgs 9
9 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios grtors s Q Q whr Q & Q stisfis for SUSY osilltor y lttig Q Q Q Q Q Q d Q if log with if Q if if 59 Thus rgy stts d r dgrt i rgy with th stt. So tht miltoi oms Q Q 59 d os spi oso rmio Q Q 59d rgy igvlu is &.... I trms of utrio ompot th totl osilltor is writt s whih visulizd logously to th followig xprssio of hrmoi osilltor i o dimsiol form disussd rlir [44] i.. d x 4 whr d x 4 &. r th trm giv i utio rmovs th zro poit rgy. ut o-lir.i tur h i th grl s w writ susy miltoi i th form d w w' I ordr to writ th xpliit form of grl suprsymmtri hrmoi osilltor miltoi i thr dimsiol rprsttio d ordigly to visuliz th prst thory of utrioi hrmoi osilltor i thr dimsio w my sustitut th followig rltios tw th oprtors; d W d dy d dz W W x Wy y 4 4
10 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios As suh th SUSY rmoi osilltor oms d d dy W d dz W x Wy Wz 5 This is thr-dimsiol rprsttio of utrioi hrmoi osilltor. d rdud furthr to thr s of o dimsio oly y sttig Y Z i..o-dimsiol hrmoi osilltor is d w W x d w W x d w W x d w W whr d x 7 r dotd s utrioisuprprtr miltois i.. AA 8 A A whr A is grlizd omid osoi d frmioi i.. suprsymmtri ihiltio oprtor d A is grlizd rtio oprtor d giv y d W x 9 A d W x 7 A Th suprhrgs giv y utio 58 my thus rprstd y d 58 rsptivly for osoi d frmioi oprtors log with th xprssio usd y utio for positio d momtum oprtors d ordigly w my thus oti th suprsymmtri miltoi oprtor giv y utio. Rturig to utio th ig stt is dsrid s d groud stt s so tht ; or 7 W lso hv Q + Q + 7 Ths suprhrgs rprsts ovrsio of frmioi stt to osoi stt d osoi stt to frmioi stt Q oso frmio Q frmio oso 7 utio 59 is th dirt logous of followig utios of supr symmtry { Q 74 Q } P 75 Q or d. Suprhrgs lwys ommut with usul miltoi. Thus th tiommutig hrgs i utrio formlism omi to form th grtors of tim trsltio mly th miltoi.th groud stt of this systm is th stt or whr oth os spi oso frmio osoi d frmioi dgrs of frdom r i th lowst rgy stt. This stt is uiu o d stifis 4
11 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios Q Q 7 As suh w my lult th rgy of suprsymmtri hrmoi osilltor from utio.59 i.. 77 whih shows tht SUSY dos ot rk for groud stt d w hv th highr rgy stts i th followig mr d ordigly 78 Illustrtio s follows rgy oso Stt 4 4 rmio 78d d so o. It shows tht th rgy stts d r dgrt stts. Similrly d d r lso dgrt. As suh th xitd stts form towr of dgrt lvls tl with rgy whr th sig of th sod trm is dtrmid y whthr th spi is plus or mius. stt Th towr of stts dsris th oso frmio dgry for xt suprsymmtry. Th osoi stt lld osoi i fild thory logy us thy oti o frmios hv th sm rgy s thir frmio prtr i. Thus th utrio rformtio of supr symmtry givs ris to simpl rprsttio of supr symmtry i utum mhis. It is howvr trivil si it dsris o-itrtig oso osilltor d frmios spi / prtils. 4
12 Itrtiol Jourl of Pur d Applid Physis. ISS Volum umr 7 pp. -4 Rsrh Idi Pulitios 7. Rfrs [] W. R. milto; lmts of utrios Vol.I II &III 899. [] W. R. milto; Pro. Roy.Irish Ad [] W. R. milto; Trs Roy.Irish Ad [4] K. Imd; Rport PL--8- Okoym Uiv. of si Jp.98 [5] iklsti D Juh J.M. Shimiovihs S. Spisr D.; J. Mth. Phys;vol. o.9 7 [] iklsti D Juh J.M.Shimiovihs S. Spisr D; J. Mth. Phys; 4 9 [7] iklsti D. Juh J.M. Shimiovihs S. Spisr D; J. Mth. Phys; [8] Vo. D. Du. Ad V. Thhuog; ommuitios i phys vol8 [9] D.C. Jolly; Ltt. uovoim.; [] A.D. Silvri; Ltt.uovoim [] Y.A. Gl fd d.p. Likhtm; JTP Ltt;97. []. ioli ;Suprsymmtry d spi systm J.Phys.A [].Witt; ul. Phys [4] G.Jukr; suprsymmtri mthods i utum mhis d sttistil physissprigr rli. 4
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terms of discrete sequences can only take values that are discrete as opposed to
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