Wigner Rotations of Different Types of Lorentz Transformations

Transcription

1 Publiaions vailabl Onlin J Si Rs JORNL OF SCIENTIFIC RESERCH wwwbanglajolinfo/indphp/jsr ign Roaions of Diffn Tps of Lonz Tansfomaions R Baizid * M S lam Dpamn of Businss dminisaion Lading nivsi Slh Bangladsh Dpamn of Phsis Shahjalal nivsi of Sin and Thnolog Slh Bangladsh Rivd 4 Mah 06 apd in final visd fom pil 06 bsa hav sudid ign oaions of diffn ps of Lonz Tansfomaions aoding o h nau of movmn of on inial fam laiv o h oh inial fam hn h moion is along an abia diion hn w an find h fomula fo ign oaions using h vloi addiion fomula fo mos gnal mid numb quanion and gomi podu Lonz ansfomaions Finall w hav usd simulad daa fo appling h ign oaion fomula in pion da hain and onludd h sul Kwods: Spial Lonz ansfomaion; Mos gnal Lonz ansfomaion; Mid numb Lonz ansfomaion; Quanion Lonz ansfomaion; Gomi podu Lonz ansfomaion; ign Roaion 06 JSR Publiaions ISSN: Pin; Onlin ll ighs svd doi: hp://ddoiog/0339/jsv8i37033 J Si Rs Inoduion Th omposiion of wo Lonz booss whn h a no ollina suls a Lonz ansfomaion [LT] ha is no a pu boos bu is h omposiion of a boos and a oaion This oaion is alld Thomas oaion Thomas ign oaion o ign oaion Th oaion was disovd b Thomas in 96 and divd b ign in 939 [] If a squn of non-ollina Lonz ansfomaions uns an obj o is iniial vloi hn h squn of ign oaions an ombin o podu a n oaion alld h Thomas pssion [] Th Thomas ff in nula sposop is mniond in Jakson s book on lodnamis [3] In fa h ign oaion is h k issu in man banhs of phsis involving LTs [4] Th ign oaion appas in phsial posss whos undling mahmaial languag inluds h Lonz goup; B s phas is an ampl of i [5 6] This banh of phsis dals wih a phsial ssm whih gains a phas angl af oming * Cosponding auho: susaik@gmailom

2 50 ign Roaions bak o h oiginal sa a h nd of a sis of ansfomaions If h ansfomaions inlud hos of a goup isomophi o h Lonz goup h ign oaion plas a vial ol in ha as [7] In n ims h Lonz goup has bom an impoan sinifi languag in boh quanum and lassial opis Th ho of squzd sas is a psnaion of h Lonz goup [8 9] Opial insumns a vwh in modn phsis basd on lassial a opis I is njoabl o obsv ha h Lonz goup is h ssnial sinifi languag fo a opis inluding polaizaion opis [0] infoms [] lns opis [3] las aviis [4] and muli-la opis [5] I is possibl o pfom mahmaial opaions of h Lonz goup b aanging opial insumns Fo insan h goup onaion is on of h mos sophisiad opaions in h Lonz goup Sin h a man mahmaial opaions in Quanum fild ho and opial sins osponding o LTs h ign oaion boms on of h impoan issus in lassial and quanum opis Th a diffn ps of LTs fis w hav disussd hs LTs Spial Lonz ansfomaion L us onsid wo inial fams of fn S and S wh h fam S is a s and h fam S is moving along h X-ais wih vloi wih sp o h S fam Th spa and im o-odinas of S and S a z and z spivl Th laion bwn h o-odinas of S and S is alld h spial Lonz ansfomaion SLT an b win as [6] Y Y Z S S X Z X Fig Spial Lonz ansfomaion z z and h invs SLT an b win as z z

3 R Baizid al J Si Rs Mos gnal Lonz ansfomaion hn h moion of h moving fam is along an abia diion insad of X- ais i h vloi has h omponns and z hn h laion bwn h spa and im o-odinas of S and Sʹ is alld h mos gnal Lonz ansfomaionmglt an b win as [7] Y Y S X S X Z Z Fig Mos gnal Lonz ansfomaion 3 and h invs MGLT an b win as 4 wh z k j i k z j i 3 Mid numb Lonz ansfomaion Consid sam as as MGLT hn using h mid podu [8-0] B i B B h mid numb Lonz ansfomaion [] MNLT an b win as 5 i and h invs MNLT an b win as 6 i

4 5 ign Roaions 4 Quanion Lonz ansfomaion gain onsid sam as as MGLT hn using h quanion podu [-4] B B B h quanion Lonz ansfomaion QLT [5] an b win as 7 and h invs QLT an b win as 8 5 Gomi Podu Lonz ansfomaion gain onsid sam as as MGLT hn using h gomi podu of wo vos [6-8] B B B h gomi podu Lonz ansfomaion [] GPLT an b win as 9 and h invs GPLT an b win as 0 ign Roaion Consid h pion da hain wh pion π is moving wih vloi wih sp o lab fam S muon μ is moving wih vloi wih sp o π lon is moving wih vloi wih sp o μ hn w wan o find h vloi of lon wih sp o lab fam in diffn ps of LTs Th a wo was o g h vloi of lon wih sp o lab fam and Th angl bwn hs wo vloi vos is alld ign oaion wh dnos h Lonz sum

5 R Baizid al J Si Rs S S Fig 3 ign oaion of Lonz ansfomaions ign Roaion fo spial Lonz ansfomaion SLT is on dimnsional Th vloi of h moving fam is along -ais So h is no ign oaion fo SLT ign Roaion fo mos gnal Lonz ansfomaion If b h vloi of muon wih sp o lab fam hn aoding o h vloi addiion fomula fo h MGLT [] w an wi Now if muon movs wih vloi wih sp o lab fam and lon movs wih vloi sp o muon hn aoding o h vloi addiion fomula fo MGLT [] w an wi o gain fom muon and lon aoding o h vloi addiion fomula fo MGLT [] h sulan vloi of and an b win as 3

6 54 ign Roaions Finall if lon movs wih vloi wih sp o pion hn h sulan vloi of lon wih sp o lab fam an b win as o 4 Spifiall o illusa ign oaion w hav usd simulad daa fo vloi vos in uni of dfind as u u v v w w vloi of pion laiv o lab fam; vloi of muon laiv o pion ; vloi of lon laiv o muon Th osponding γ faos a as follows: Fom quaion and w hav Fom quaion 3 and 4 w g know ha B B os B o os B wh ˆ i ˆj 0kˆ B 0 sa sa and B ˆ i ˆj 0kˆ B Hn gain l ign os B

7 R Baizid al J Si Rs u u v v w w u u v v w w b wo ss of vloi vos of pion da hain as Fig 3 hn using quaions 3 and 4 w hav h vloi vos of lon laiv o lab fam a and and spivl sing simila poss as pvious on w hav h ign oaions in hs ass os pp and os pp ign spivl ign 3 ign Roaion of mid numb Lonz ansfomaion If b h vloi of muon wih sp o lab fam hn aoding o h vloi addiion fomula fo h MNLT [] w an wi i 5 Now using simila poss as MGLT h vloi vos of lon laiv o lab fam fo MNLT as Fig 3 w hav 6 and i i Spifiall o illusa ign oaion fo MNLT vloi vos a dfind as u u v v w w u u v v w w u u v v w w b h ss of vloi vos of pion da hain as Fig 3 hn using quaions 5 6 and 7 w hav h vloi vos of lon laiv o lab fam a a and spivl i 055i 0409i 0409i 0404i i 7

8 56 ign Roaions sing simila poss as MGLT w hav h ign oaions of ah as os 0 ign 4 ign Roaion of Quanion Lonz ansfomaion If b h vloi of muon wih sp o lab fam hn aoding o h vloi addiion fomula fo QLT [9] w an wi 8 Now using simila poss as MGLT h vloi vos of lon laiv o lab fam S fo QLTas Fig 3 w hav 9 and Spifiall o illusa ign oaion fo QLT vloi vos a dfind as 0 u u v v w w u u v v w w u u v v w w b h ss of vloi vos of pion da hain as Fig 3 hn using quaions 8 9 and 0 w hav h vloi vos of lon laiv o lab fam a ; and spivl sing simila poss as MGLT w hav h ign oaions of ah as os 0 ign 5 ign Roaion of Gomi podu Lonz ansfomaion If b h vloi of muon wih sp o lab fam hn aoding o h vloi addiion fomula fo GPLT [] w an wi Now using simila poss h vloi vos of lon laiv o lab fam fo GPLTas Fig 3 w hav

9 R Baizid al J Si Rs and Spifiall o illusa ign oaions fo GPLTvloi vos a dfind as u u v v w w u u v v w w u u v v w w b h ss of vloi vos of pion da hain as Fig 3 hn using quaions and 3 w hav h vloi vos of lon laiv o lab fam a ; and spivl sing simila poss as MGLT w hav h ign oaions of h abov ass a ign os ign os 9959 os spivl ign 3 Compaison of h Sud 3 Compaison of ign Roaions of spial mos gnal mid numb quanion and gomi podu Lonz ansfomaions Nams of Lonz ansfomaions ign Roaion u u v v w w ign Roaion u u v v w w ign Roaion u u v v w w SLT No appliabl No appliabl No appliabl MGLT MNLT QLT GPLT Conlusion hav disussd h ign oaions fo diffn ps of LTs In h as of MGLT and GPLT w hav found ign oaions bu h valus a diffn in ah

10 58 ign Roaions as In h as of MNLT and QLT h is no ign Roaion an appl ou suls in h sud of hiddn magni fos manifs in som poblms of Elomagnism Spin-obi inaion of lon wih nulus in an aom in Quanum Mhanis sud of h quanizd lomagni fild in phas spa and of h inaion bwn aoms and phoons in aviis in quanum opis Rfns E P ign nn Mah hp://ddoiog/0307/96855 H Kom m J Phs J D Jakson Classial Elodnamis 3d diion il Nw Yok N J Papasamaiou H Masumoo and H mzawa Pog Tho Phs S Panhaanam Gnalizd Tho of Infn and is ppliaion - Po Indian ad Si M B J Mod Op hp://ddoiog/0080/ R Y Chiao and T F Jodan Phs L hp://ddoiog/006/ H P Yun Phs Rv hp://ddoiog/003/phsrv36 9 Y S Kim and M E Noz Two Diffn Squz Tansfomaions old Sinifi Singapo 99 0 D Han Y S Kim and M E Noz Phs Rv E hp://ddoiog/003/phsrve D Han Y S Kim and M E Noz Phs Rv E hp://ddoiog/003/phsrve65907 E C G Sudashan N Mukunda and R Simon Opia a hp://ddoiog/0080/ S Baskal and Y S Kim J Op So m S Baskal and Y S Kim Phs Rv E hp://ddoiog/003/phsrve J J Monzon and L L Sanhz-Soo m J Phs hp://ddoiog/09/835 6 R Rsnik Inoduion o Spial Rlaivi il Easn limid C Moll Th Tho of Rlaivi Ofod nivsi pss London 97 8 M S lam Sud of Mid Numb - Po Pak ad Si M S lam J Thois M S lam Ind J Phs MS lam and K Bgum Jahanginaga Phs Sudis Kala Thoial Phsis B Saunds Compan Philadlphia London Quanion 06 hp://mahwoldwolfamom/quanionhml 4 D Saah's Quanion Dmo 06 hp://wwwsappsadu/~sjg/lass/30/mahfsalg000/quanionshml 5 M S lam and S Bauk Quanion Lonz Tansfomaion Phsis Essas mian Insiu of Phsis Canada 0 6 M S lam Nws Bull Cal Mah So B K Daa D Sabbaa and L Ronhi Il Nuovo Cimno 3B B K Daa and R Daa Found Phs L M S lam and M D Chowdhu J Na Si Found Si Lanka