Publiaions vailabl Onlin J Si Rs 8 3 49-58 06 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: 070-037 Pin; 070-045 Onlin ll ighs svd doi: hp://ddoiog/0339/jsv8i37033 J Si Rs 8 3 49-58 06 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
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
R Baizid al J Si Rs 8 3 49-58 06 5 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
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
R Baizid al J Si Rs 8 3 49-58 06 53 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
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 0 05000 v v0 030 500 w w 0 04 0 00 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: 8678 3095 8609 0754533 Fom quaion and w hav 0549697 Fom quaion 3 and 4 w g know ha B B os B o os B wh 0754533ˆ i 0549697 ˆj 0kˆ 0 764665 533796 B 0 sa sa 0754533 0549697 933534 and B 0764665ˆ i 0533796 ˆj 0kˆ B 87035903 0 Hn gain l ign os 99979 5 B 0764665 0533796 93566
R Baizid al J Si Rs 8 3 49-58 06 55 u u 0 05 0 00 v v 0 0 04 00 w w 0 05 03 00 u u 0 04 03 00 v v 0 0 05 00 w w 0 06 03 00 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 0755839 079643 06865 and 054758 0459595 0668553 0 0 0 069576 spivl 06539 0 sing simila poss as pvious on w hav h ign oaions in hs ass os 0994644 594 pp and os 0999778 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 0 05 0 00 v v0 03 0 5 00 w w0 04 0 00 u u 0 05 0 00 v v0 0 04 00 w w0 05 03 00 u u 0 04 03 00 v v 0 0 05 00 w w 0 06 03 00 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 094 088 0785 0785 07089 0754 spivl 064 06 05 05 06847 05988 058i 055i 0409i 0409i 0404i 0 08535i 7
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 0 05 0 00 v v0 03 0 5 00 w w0 04 0 00 u u 0 05 0 00 v v0 0 04 00 w w0 05 03 00 u u 0 04 03 00 v v 0 0 05 00 w w 0 06 03 00 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 366 366 3678 36586 5999 5999 spivl 393 393 367 39 5558 5558 04 04 0388 0395 094 0 94 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
R Baizid al J Si Rs 8 3 49-58 06 57 and Spifiall o illusa ign oaions fo GPLTvloi vos a dfind as u u 0 05000 v v 0 030 500 w w 0 04000 u u 0 05000 v v 0 0 0400 w w 0 0503 00 u u 0 0403 00 v v 0 0 05 00 w w 0 0603 00 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 0738 07497 07636 07497073 070896 06754 06 05998 0604678 05998 06847 0697834 00409 0040936 0040936 0040936 008743 0 087433 spivl sing simila poss as MGLT w hav h ign oaions of h abov ass a ign os 99997 074 ign os 9959 os 099945 9 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 0 5 0 v v 0 3 5 0 w w 0 4 0 3 ign Roaion u u 0 5 0 v v 0 4 0 w w 0 53 0 ign Roaion u u 0 43 0 v v 0 5 0 w w 0 63 0 SLT No appliabl No appliabl No appliabl MGLT 5 594 MNLT 0 0 0 QLT 0 0 0 GPLT 074 9 4 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
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 40 49 939 hp://ddoiog/0307/96855 H Kom m J Phs 7 5 004 3 J D Jakson Classial Elodnamis 3d diion il Nw Yok 999 4 N J Papasamaiou H Masumoo and H mzawa Pog Tho Phs 695 647 983 5 S Panhaanam Gnalizd Tho of Infn and is ppliaion - Po Indian ad Si 44 47 956 6 M B J Mod Op 34 40 987 hp://ddoiog/0080/0950034874553 7 R Y Chiao and T F Jodan Phs L 3 77 988 hp://ddoiog/006/0375-960889055- 8 H P Yun Phs Rv 3 6 976 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 56 6065 997 hp://ddoiog/003/phsrve566065 D Han Y S Kim and M E Noz Phs Rv E 6 5907 000 hp://ddoiog/003/phsrve65907 E C G Sudashan N Mukunda and R Simon Opia a 3 855 985 hp://ddoiog/0080/738807 3 S Baskal and Y S Kim J Op So m 69 049 009 4 S Baskal and Y S Kim Phs Rv E 66 06604 00 hp://ddoiog/003/phsrve6606604 5 J J Monzon and L L Sanhz-Soo m J Phs 64 56 996 hp://ddoiog/09/835 6 R Rsnik Inoduion o Spial Rlaivi il Easn limid 994 7 C Moll Th Tho of Rlaivi Ofod nivsi pss London 97 8 M S lam Sud of Mid Numb - Po Pak ad Si 37 9 000 9 M S lam J Thois 34 00 0 M S lam Ind J Phs 77 47 003 MS lam and K Bgum Jahanginaga Phs Sudis 5 89 009 Kala Thoial Phsis B Saunds Compan Philadlphia London 967 3 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 6-3 003 7 B K Daa D Sabbaa and L Ronhi Il Nuovo Cimno 3B 7 998 8 B K Daa and R Daa Found Phs L 998 9 M S lam and M D Chowdhu J Na Si Found Si Lanka 343 43 006