omtrc algra quts gomtrc voluton and all that Alxandr M. OUNE opyrght 5 Astract: Th approach ntald n [] [] s usd for dscrpton and analyss of quts gomtrc phas paramtrs thngs crtcal n th ara of topologcal quantum computng [] [4]. Only physcal ojcts wth stats that can dntfd y lmnts of [5] [6] ar consdrd hr. Th usd tool omtrc lfford Algra [5] [6] s th most convnnt formalsm for that cas. nralatons of formal complx plan to an artrary varal plan n D and of usual opf fraton : g to th map g g g gnratd y an artrary unt valu lmnt g ar rsultng n mor profound dscrpton of quts compard to quantum mchancal lrt spac formalsm.. ntroducton Th workng nvronmnt wll vn sualgra of lmnts of gomtrc algra ovr Eucldan spac E. Algra has canoncal ass whr ar orthonormal ass vctors n aras vctors spannd y and ordrd dgs and. ualgra E j ar orntd mutually orthogonal unt valu j as dgs and. Varals and n s unt valu orntd volum spannd y s spannd y a scalar and ass vctors: ar ral scalars s a unt s orntd ara vctor lfthandd or rghthandd n an artrary gvn plan E. s lnar comnaton of ass vctors j. t was xpland n [7] [8] that lmnts only dffr from what s tradtonally calld complx numrs y th fact that E s an artrary plan. Puttng t smply ar complx Ths sum ars th sns mthng and mthng. t s not a sum of smlar lmnts gvng anothr lmnt of th sam knd s [] []. calars should always ral. omplx scalars ar not scalars.
numrs dpndng on E mddd nto E. Tradtonal magnary unt s gomtrcally m whn t s not ncssary to spcfy th contanng plan vrythng s gong on n on fxd plan not n D world. Usually s consdrd just as a numr wth addtonal algrac proprty. That may a urc of amguts as t happns n quantum mchancs.. Rotatons wth lmnts of Lt s tak a unt valu lmnt of : and gnrald opf fraton. wll us notaton unt radus sphr paramtrd y and f s a vctor for such lmnts. Thy can consdrd as ponts on caus. thn th map rot s rotaton of gvng nw vctor rot. Paramtr s cosn of half of angl of rotaton s unt valu vctor n plan orthogonal to axs of rotaton. Ths rotaton : gnrats map rot. f for xampl thn: o gnrats classcal opf fraton : whr. Al. shows that magnary numr n ths opf fraton wrttn n tradtonal trms should gomtrcally unt valu orntd ara orthogonal to th ass vctor. Orntatons of and must n agrmnt wth orntaton. opf fraton can gnrald assumng that t s gnratd y any vctor n D: : rot Axs of rotaton togthr wth th vctor orntaton should compatl wth orntaton of trvctor.
t s convnnt to hav rot xplctly wrttn n th ass vctor componnts. W know rsult for. For two othr ass vctors and w hav: rot. rot Thn:. rot.4 uppos vctor s xpandd not n ut n any ass of vctors whr s artrary unt vctor n D and ar two unt vctors orthogonal to and to ach othr. Thr mutual orntaton s assumd to satsfy th sam multplcaton ruls as do:.5 o. Thn for xampl th rsult of rotaton of s:
4 xactly th sam as n th cas. Rotatons of and gv rsults dntcal to. and.. t follows that.4 rmans vald f s rplacd wth. Th opf fraton can furthr xtndd to artrary gnratng lmnts from :.6. Quantum mchancal quts and lmnts of A pur qut stat n trms of convntonal quantum mchancs s two dmnnal unt valu vctor wth complx valu componnts: k k k k W can al thnk aout such pur qut stats as ponts on : } ; {. Tld s complx conjugaton; gomtrcal sns of magnary unt s not spcfd. Lt s fnd xplct rlatons twn lmnts as two-componnt complx vctors and lmnts longng to.
5 Though magnary unt n quantum mchancs dos not hav xplct gomtrcal sns ovous tact assumpton should that and n. hav th sam complx plan othrws complx conjugaton s not wll dfnd [7]. Tak an artrary. f vctor s chosn as th on dfnng complx plan w hav: o w gt corrspondnc: For anothr ass vctor w gt: And for : o w hav at last thr dffrnt maps j j dfnd y complx plans j. Actually th numr of such maps s nfnt caus complx plan can an artrary plan n D. nstad of th ass of thr vctors w can tak artrary mutually orthogonal unt vctors satsfyng multplcaton ruls.5. f s xpandd n ass : thn for xampl takng as complx plan w gt:
6 and th corrspondnc s: - artrary vctor n D. ummarng f w hav thr artrary mutually orthogonal plans n D and unt orntd vctors longng to thm and satsfyng multplcaton ruls.5 thn for any w hav th followng mappngs: forgottn forgottn forgottn ; ; ;. All that mans that to rcovr or stalsh whch n D s ascatd wth t s ncssary frstly to dfn whch plan n D should takn as complx plan and thn to choos anothr plan orthogonal to th frst on. Th thrd orthogonal plan s thn dfnd y th frst two up to orntaton that formally mans. wll call lmnts pur qut stats or smply g -quts. 4. Quantum mchancal osrvals and pur qut stats n ths scton w wll consdr m rlatons of pur qut stats wth Paul s matrcs spcal cas of osrvals n quantum mchancs trmnology. Thr xsts on-to-on corrspondnc twn rmtan matrcs and lmnts of. f a matrx s: a d c d c a
thn ts countrpart s: a a a a c d or nvrsly f n trms thn th corrspondng rmtan matrx s: Th corrspondncs follow from th Paul s matrcs 4 dfnton: ˆ ˆ ˆ and thr on-to-on corrspondnc wth ass vctors. Namly all gomtrc algra quatons can xprssd n trms of ˆ k usng rlatons and ˆ whr ˆ ˆ ˆ [8]. ˆ ˆ Lt complx plan s spannd y } thn usual quantum mchancal pur qut stat { corrspondng to wth s: Matrx-vctor calculatons wth Paul s matrcs gv th followng xpctaton valus of th osrvals: ˆ ˆ ˆ k ˆ whr s lmnt conjugatd to on row matrx: Each of th products ˆ gvs xactly on -th componnt of usual opf fraton though from. w gt vctor sum of all thr componnts n on gomtrc algra opraton 4 mad m modfcatons of th matrcs to follow th handdnss and corrspondnc to th ass. 7
8. Th dffrnc follows from quantum mchancal matrx formalsm ovr wth gomtrcally unspcfd magnary numr whr osrvals ar rmtan matrcs. n spt of th corrspondncs 4. and algrac concdnc of multplcaton ruls for Paul matrcs and ass vctors j furthr algrac opratons ar not quvalnt spcally caus 5 4. s not on-to-on. To rtrv corrspondng to w nd n addton to dntfyng of wth to rcall th xstnc of and. Thn and Du to th fact that s not gomtrcally spcfd a quantum mchancal stat s consdrd as formally quvalnt to n th sns that and ar undstngushal that s oth hav dntcal xpctaton valus. For xampl: ˆ ˆ Th products ˆ ar scalar products of two on column matrcs not lk th map rot transformng a vctor to vctor. Transformaton xprssd n trms dos not gnrally rturn dntcal stat. 5 al low c.5 how n th map nformaton s lost.
9 5. Nw dfntons asd on th aov rlatons and dffrncs twn usual quantum mchancal qut stats and th pur qut stats shall gv nw dfntons of stats osrvals masurmnts and osrval valus n th cas of D ojcts dntfal as lmnts: Dfnton.. tats of a qut and osrvals of a qut ar unt valu lmnts of. Th notatons wll usd n th followng manly as: tat - Osrval - Dfnton.. Masurmnt of osrval masurd n stat s a gnrald opf fraton gnratd y : : whch s a nw lmnt from xplctly gvn n ass componnts y.4. Dfnton.. Valu of osrval masurd n stat s a map of th masurmnt of th osrval to a st of masurmnt valus.
Th frst two dfntons dntfy as th structur ntgratng stats osrvals and osrval masurmnts. Dfnton. s th ass for proalty calculatons usng gomtrc masurs on actually on. Th aov dfntons lookng may a t strang from th frst sght ar n full agrmnt wth rgorous dstngushng twn oprators and oprands vn n th smplst physcal stuatons. Lt s tak lmntary physcal procss as dsplacmnts of a pont on a straght ln. ts stat s vctor of dsplacmnt appld to osrval ntal poston. Th rsult of ths s masurmnt: osrval stat masurmnt of osrval n gvn stat f th sam s gong on n D ut w rstrct ourslvs to D smlar to th cas of unspcfd complx plan th pctur coms: nfntly many stats dash rd gv th sam masurmnt of osrval n D That s qut smlar to what was shown n.. Exampl. ood xampl of usng th aov dfntons s th tossd con xprmnt []. t was shown thr that th xprmnt dos not formally dffr from quantum mchancal dscrpton whn formulatd n gomtrc algra trms. tat s lmnt wth th vctor part qual to th plan of con rotaton. Th stat scalar part s a half of nstant angl of con rotaton. Osrval s vctor dfnng opf fraton of th stat n that cas ntal con poston vctor. Masurmnt s nstant vctor of th opf fraton. Th spcfc valu of osrval may
tradtonally th rsult whch on of th two dstngushal valus whch sd of th masurmnt vctor s osrvd from gvn drcton. t was shown that f th stat s unspcfd varal unformly dstrutd ovr thn th had/tal rsults hav qual proalts / that was provd y drct calculatons. Formally ths valu of osrval s th sgn of cosn of angl twn th osrvr vctor and masurmnt vctor. As was mntond aov quantum mchancal stat transformaton dos not hav osrval rsult. f w do th sam n trms that s multply y m on th lft osrval and hnc ts masurmnts wll not chang only f for xampl plan s qual to th plan of osrval. Ths s gomtrcally ovous n ths xampl of tossd con: nothng can chang n th rsult of xprmnt f vctor s rotatd n ts own plan formal proof n [] []: t s not gnrally tru for an artrary plan. Th gomtrc algra formalsm s much mor nformatv du to dfnng qut stats as lmnts compard to th two-dmnnal complx spac formalsm. n th two-way corrspondnc th to map: gvs complx numrs and wrttn wth formal magnary unt. f to furthr work n th formalsm th plan of s not ncssary at all may fully gnord forgottn forvr to say nothng of two othr orthogonal plans s.. And that s th cas of xstng quantum mchancs. n th oppost way map f w tak a quantum mchancal stat thn to rtrv g -qut on can us any plan and artrary choos two othr plans orthogonal to and ach othr. Ths s xplanaton how nformaton s lost n th qut stat formalsm compard to gomtrc algra formalsm whr qut stat s. Not surprsng that t was possl to formally prov [9] that hddn varals do not xst n th quantum mchancs lrt spac formalsm.
6. Qut stats n corrspondng to quantum mchancal ass stats Quantum mchancal pur qut stat s and. n lnar comnaton of two ass stats trms ths two stats ar as follows from. th classs of quvalnc 6 : lmnts: - tat corrsponds to any on of th followng sts of f th complx plan s slctd as or: f th complx plan s slctd as or: 4. f th complx plan s slctd as. lmnts: - tat corrsponds to any on th followng sts of f th complx plan s slctd as or: f th complx plan s slctd as or: 4. f th complx plan s slctd as. For any of th stats corrspondng to th valu of osrval s: 4. For any of th stats corrspondng to wth th mod mod mod mod agrmnt mod= snc ndx dos not hr xs th valu of osrval s: mod mod mod mod mod mod mod mod mod mod 4.4 Ths s th actual manng of quantum mchancal ass stats: Valu of osrval tslf: s for any pur qut stat from th st of all th vctor 6 Rcall that s an artrary vctor plan n D and ar two vctors orthogonal to and to ach othr wth mutual orntaton satsfyng handdnss:
Valu of osrval : s for any pur qut stat from th st of all flppd vctor Lts tak artrary vctor osrval xpandd n ass. Wthout losng of gnralty w can thnk that complx plan s dfnd y vctor. Thn and gnrald opf fratons masurmnts of th osrval for th stats ar corrspondngly: cos sn sn cos through paramtraton cos sn and: through paramtraton cos sn sn cos cos sn. Ths s a dpr rsult compard wth usual quantum mchancs whr. y usng corrspondng lmnts and w gt th followng: Masurmnt of osrval n pur qut stat and s vctor wth th componnt qual to unchangd valu. Th and masurmnt componnts ar qual to and componnts of rotatd y angl dfnd y cos and sn whr plan of rotaton s. Masurmnt of osrval n pur qut stat s vctor wth th componnt qual to flppd valu qut flps n plan. Th and masurmnt componnts ar qual to and componnts of rotatd y angl dfnd y cos sn whr plan of rotaton s. Th alut valu of angl of rotaton s th sam as for ut th rotaton drcton s oppost to th cas of.
Th aov rsults ar gomtrcally prtty clar. Th two stats and rprsntd n corrspondngly y and only dffr y addtonal factor n. That mans that masurmnts of vctor osrval n stats ar quvalnt up to addtonal wrappr : 4.5 That smply mans that th masurmnt on th lft sd s rcvd from th and masurmnt just y flppng of th lattr rlatv to th plan. All that al maks mannglss standard quantum mchancal statmnts lk that non-commutatvty of th oprators Paul matrcs n our smpl cas s rsponsl for classcally paradoxcal proprty of quantum osrvals: on can fnd stats wth wll-dfnd masurd valu of on osrval ut such stats wll not hav a wll-dfnd masurd valu othr non-commutng osrvals. All that s fault of th usd mathmatcal formalsm vctor/matrx algra on complx spacs wth gomtrcally unspcfd magnary unt. 7. rry paramtrs for th qut stats Quantum mchancal transformatons consdrd as transformatons on calld lfford translatons. Lt s look at lfford translatons n trms of our changng n tm. ar oftn cas wth th angl lfford translaton of stat wth xplctly dfnd complx plan l should wrttn as l aout lfford translatons tak plac:. Only n th cas whn l concds wth th wll known facts - lfford translaton satsfs ts gnrc proprts: t dos not chang norms of and dstancs twn lmnts lmnts that s why t s calld translaton. For xampl: cos sn cos sn cos sn - Th vlocty of lfford translaton n such a cas s: 4
d dt d t cos sn dt sn cos Ths lmnt s orthogonal n th sns of scalar part of vctor produc to th lfford translaton ort t and spans on-dmnnal vrtcal suspac of th lfford ort tangnt spac on. Orthogonalty can asly vrfd y calculatng th scalar part of th product: cos sn cos sn cos sn cos sn cos sn sn cos cos sn sn cos - Two mor lmnts and whr and ar two plans n D orthogonal to and to ach othr ar orthogonal to and to ach othr. Thy span two-dmnnal horontal suspac of th lfford ort tangnt spac on o w hav th followng: - th vrtcal tangnt of th lfford translaton spd along fr t s ;. - th two othr tangnts spannng th horontal suspac of th tangnt spac ar and Th conclun s that f a. qut pur stat rotats whl movng along lfford ort wth m angular spd n tangnt to ort drcton thn t al rotats wth th sam angular spd n plan orthogonal to that drcton. n th cas of artrary plan l not concdng wth all that coms dffrnt. Evrythng low ncludng scton 8 ar xampls of gnralaton of lfford translatons to artrary and varal complx plans and magnary unts. Lt s tak transformaton: t t 7. whr s constant valu gnrc amltonan of th systm vctor of concdng wth. larly satsfs chrödngr quaton: wth th plan not 5
d dt wth unt vctor n D rplacng magnary unt of th convntonal quantum mchancs cas. Ths s mportant thng: n trms of th chrödngr quaton s quaton for th rsult of t transformaton of a qut stat gnratd y amltonan whch s gnrc amltonan of th systm. l n trms of transformaton actually changs vctor qut stat. f th vctor part of an osrval dos not hav th plan dntcal to th plan of l th masurmnt l l s dffrnt from s al aov xampl of a tossd con. nrald lfford translaton l wll rturn undstngushal quantum stat f only th plan of t s qual to th plan of osrval. n convntonal quantum mchancs an mportant thng s lvng gnstat/valu prolm E. For th gnstats th corrspondng gnvalus E ar valus of osrval n convntonal trmnology n th stat : E E. All that gnvctors gnvalus tc. - dos not mak a lot of sns n th for xampl whr approach. f w tak s unt vctor of th amltonan plan thn w know s 4. 4.4 that th stats whch only multply th osrval n a masurmnt actually y plus or mnus ar: and and - any two scalars satsfyng rsultng n s 4.5: - any unt vctor orthogonal to and 6
Lt s now consdr th amltonan corrspondng to constant valu magntc fld slowly rotatng around axs orthogonal for xampl to plan axs along vctor. ntal plan of th fld s nclnd y m angl rlatv to vctor. t may ntald as rotaton of n plan : Th nclnd fld rotats around dpndng on angl : 7. 7 From that w wll al to calculat th rry conncton and rry phas. Lt s rduc vrythng to usual quantum mchancal trms. alculat 7. n two stps. Frst ncln : that gvs cos th rsult of rotaton of th nclnd vctor: sn. Thn a t tdous calculatons gv sn cos sn sn sn cos cos 7. Lt s now wrt usual quantum mchancal stat for th. n th currnt xampl t s natural to tak thrd opton: stat 7.. Thr ar dffrnt ways to map. Thn from th. cos cos sn sn sn cos cos sn cos sn sn sn Ths s on of th two gnvctors of. Forgt tmporarly aout. Known ruls of calculatng of th rry conncton s for xampl [] namly A u u A u u gv n th currnt cas whn and cos u : sn 7 mply addng xponntals dos not hlp too much snc plans of rotaton ar dffrnt 7
sn sn sn cos sn cos A cos cos A cos sn sn sn And thn th rsult for th rry curvatur s: F A A sn Lt s now calculat all that not n th rducd quantum mchancs trms rcvd y ut n full nstad of u w gt: U sn sn trms. Usng notaton U for corrspondng sn cos U cos cos sn sn cos ojct 8 thn A U U From anothr drvatv: U sn w gt: W s that full valus A A and U U sn sn cos A dffr from usual quantum mchancal cas y addtonal vctor trms. Th rry curvatur al has addtonal vctor trm: F A A sn cos Th addtonal vctor trms may an ndcator of a knd of torn causd y 7. g -qut transformaton though t nds furthr laoraton. 8 am al usng asly vrfd 8
8. omtrc voluton. Wth xplctly dfnd varal magnary unt many thngs com not just mor nformatv ut al much smplr. As an xampl tak th mchancal approach through holonomy s for xampl [4]. Transformaton 7. can straghtforwardly gnrald to: varant of gomtrc voluton drvd n usual quantum Th complx plan dfnd y unt vctor angl n ths plan. Th voluton quaton for th rsult of transformaton s: d dt t t thn coms varal and plays th rol of whch s a gnralaton of chrödngr quaton wth varal magnary unt. Lt s mak assumpton that U n U U just gnrats rotaton. f th ntal valu of amltonan al ls on d dt wth th luton: 8. coms: t t d dt t t 9. oncluns. Th da of usng varal plan n D for playng th rol of complx plan n comnaton wth algra rsultd n mor dtald dscrpton of quantum stats and osrvals usually formald n trms of two-dmnnal complx vctors and rmtan matrcs. Nw addtonal componnts n rry paramtrs may usd to analy possl anyonc stats n th topologcal computng rlatd structurs. 9
Works td [] A. ogun "A tossd con as quantum mchancal ojct" ptmr. [Onln]. Avalal: http://arxv.org/as/9.5. [] A. ogun "What quantum "stat" rally s?" Jun 4. [Onln]. Avalal: http://arxv.org/as/46.75. []. Nayak.. mon A. trn M. Frdman and. Das arma "Non-Alan Anyons and Topologcal Quantum omputaton" Rv. Mod. Phys. vol. 8 p. 8 8. [4] J. K. Pachos ntroducton to Topologcal Quantum omputaton amrdg: amrdg Unvrsty Prss. [5] D. stns Nw Foundatons of lasscal Mchancs Dordrcht/oston/London: Kluwr Acadmc Pulshrs 999. [6]. Doran and A. Lasny omtrc Algra for Physcsts amrdg: amrdg Unvrsty Prss. [7] A. ogun "omplx onjugaton - Raltv to What?" n lfford Algras wth Numrc and ymolc omputatons oston rkhausr 996 pp. 85-94. [8] A. ogun Vctor Algra n Appld Prolms Lnngrad: Naval Acadmy 99 n Russan. [9] J. v. Numann Mathmatcal Foundatons of Quantum Mchancs Prncton: Prcton Unvrsty Prss 955. [] D. Xao M.-. hang and Q. Nu "rry Phas Effcts on Elctronc Proprts" Rvws of Modrn Physcs p. 959 July. [] J. W. Arthur Undrstandng omtrc Algra for Elctromagntc Thory John Wly & ons.