Lorentz Transformation Properties of Currents for the Particle-Antiparticle Pair Wave Functions

Transcription

1 Open Aess Library Journal 17, Volume 4, e373 ISSN Online: ISSN Prin: Lorenz Transformaion Properies of Currens for he Parile-Aniparile Pair Wave Funions Raja Roy Deparmen of Eleronis and Elerial Communiaion Engineering, Indian Insiue of Tehnology, Kharagpur, India How o ie his paper: Roy, R. (17) Lorenz Transformaion Properies of Currens for he Parile-Aniparile Pair Wave Funions. Open Aess Library Journal, 4: e373. hps://doi.org/1.436/oalib Reeived: June 9, 17 Aeped: June, 17 Published: June 3, 17 Absra The Lorenz ransformaion properies of harge urren four veor for Dira spinor pariles are examined one more espeially for he zierbewegung erms whih are inegral pars of his heory. Subje Areas Quanum Mehanis Copyrigh 17 by auhor and Open Aess Library In. This work is liensed under he Creaive Commons Aribuion Inernaional Liense (CC BY 4.). hp://reaiveommons.org/lienses/by/4./ Open Aess Keywords Dira Equaion, Zierbewegung, Lorenz Transformaion 1. Inroduion Four veors like eleri or harge parile urrens ransform under Lorenz boos obeying erain ransformaion laws in lassial physis [1]. I is expeed ha he quanum mehanial harge urrens for elemenary pariles (and/or anipariles) as derived from he probabiliy urrens [], by muliplying he pariles harge wih i, follow hese ransformaion laws if he orrespondene priniple [3] relaing quanum and lassial physis is o remain valid. As is saed in referene [] he Dira equaion desribing elemenary pariles like he eleron give rise o erms in he probabiliy urren four veor whih have rapid osillaion or zierbewegung in ases where saes are formed by ombining parile and aniparile wave funions. In his presen leer we show ha hese zierbewegung erms do no follow he Lorenz ransformaion law. I is also demonsraed ha he wave funion for a salar parile (along wih is aniparile) whih follows he Klein Gordon equaion [4] gives rise o no suh zierbewegung erms a leas in he zeroh omponen of harge urrens whih vi- DOI: 1.436/oalib June 3, 17

2 olae is Lorenz ransformaion properies.. Lorenz Transformaion Properies of Classial and Quanum Mehanial Currens Any lassial four veor J µ ransforms from a referene frame S o anoher S whih is moving wih respe o S wih a onsan veloiy v along he x direion aording o (we are wriing he zeroh ha is he µ = omponen only) 1 v 1 1 J = J J J osh J sinh = ω ω v 1 as given for example by Jakson [1] (see p. 56 of his referene). If in his frame 1 he value of J (he x omponen of urren) is zero hen he raio of harge densiy ρ = J in S o is value ρ in S is given by ρ = oshω (1) ρ On he oher hand he Dira parile-aniparile pair, like he eleron and posiron an be represened in frame S in whih i is assumed o be a res by he wave funion as given for example by Bjorken and Drell [] (see p. 1 of his book) o be 1 im im ψ ( x, ) Ae = + Be () 1 Here he omplex oeffiiens A and B represen he proporion in whih he parile and aniparile omponens are presen in he ombined sae. There should no be any objeion o forming suh ombined saes wih he help of Dira spinors where boh parile and aniparile wave funions are involved as suh saes are inluded in forming wave pakes as given for example by Equaion (3.3) of referene []. Also we need no normalize he wave funion o sudy he Lorenz ransformaion properies of he harge urren four veor (ha is he eleroni harge e muliplied by he probabiliy urren four veor). This quaniy an be expressed as + ( x, ) ψ ( x, ) γγψ ( x, ) J µ = e µ (3) following he presripions of pages 3 and 9 of referene []. Here ψ + ( x,) is he omplex onjugae of he ranspose of ψ ( x,). The zeroh omponen of his veor J ( x, ) is he harge densiy ρ ( x, ) and an be evaluaed in boh S and S frames o see if hey follow Equaion (1) above provided ψ ( x,) in he frame S is given by Equaion () implying ha parile and aniparile are boh a res in his frame. Thus in he S frame ρ( x, ) = J ( x, ) = eψ + ( x, ) ψ ( x, ) (4) /4

3 ino whih subsiuion of he value of ψ ( x,) from Equaion () yields ( x, ) ρ = e A + B (5) Before applying Equaion (4) o he frame S we mus ransform he spinor ψ ( x,) o ψ ( x, ) (3.5) of referene [] ha is using he marix operaor whih appears in Equaion im oshω sinhω Ae oshω sinhω ψ ( x, ) = (6) sinhω oshω sinhω oshω im Be Thus from Equaions (3) and (6) we ge where + ( x, ) = ( x, ) ( x, ) ρ eψ ψ im im (7) * * e( A B ) oshω e = + A Be + AB e sinhω * A is he omplex onjugae of A. Thus he seond erm in Equaion (7) is he zierbewegung erm and if he equaion is divided by Equaion (5) we ge im im * * ρ A Be + AB e = oshω ρ A + B sinhω (8) Comparison of Equaions (1) and (8) shows ha lassial Lorenz ovariane is violaed by he zierbewegung erms whih are essenial feaures of he Dira desripion of a parile even hough hese may exis in a small proporion (see p 39 of referene []). Furhermore harge densiy is an observable in quanum mehanis and so his will lead o some onsequenes as far as experimenal resuls are onerned. 3. Salar Parile Charge Densiy and Is Lorenz Transformaion The quanum mehanial wave funion φ ( x,) of a salar parile follows he Klein-Gordon equaion and an be wrien following he guidelines of referene [4] as ( x, ) φ 1 φ m φ x, + x, = where we have reained he veloiy of ligh expliily. The harge densiy ρ x, an be expressed following Equaion (3.17) of his above referene as * * φ x, φ x, ρ( x, ) = ei φ( x, ) φ( x, ) We evaluae his in boh he S and S frames o obain from (9) (1) 3/4

4 im im φ x, = Ae + Be and im osh ω x sinh im osh x sinh ω ω ω + + φ x, = Ae + Be he expressions 3 m e ( x, ) ρ = A + B 3 m e ( x, ) ρ = A + B oshω The expressions for φ ( x, ) above are obained from ( x,) φ by using he x Lorenz ransformaion equaion for whih is = oshω+ sinhω. (11) (1) Here he omplex numbers A and B have he same meaning as in Seion. Now we are in a posiion o hek from Equaions (11) and (1) how he salar field harge densiy ransform. Indeed one obains he same relaion beween ρ and ρ as in Equaion (1) if we divide Equaion (1) by Equaion (11). 4. Conlusion and Response o he Consruive Criiism of he Referee We would like o hank he referee for his onsruive ommens espeially on he need o make a onrasive analysis of he resuls obained in seions and 3. The immediae poin ha is o be noed is he fa ha Equaions (5) and (7) in onras o Equaions (11) and (1) have he erms involving A and B wih he same signaure. Anipariles are supposed o be opposiely harged as ompared o pariles and his fa is made explii in Equaions (11) and (1). In fa on page 76 of referene [4] i is made lear ha he onep of harged urren for he parile-aniparile pair as given in he ase of he zeroh omponen by Equaion (1) above was inrodued for he firs ime by Pauli and Weisskopf in 1934 whih onsiss of erms of boh signaures. The Dira equaion fails o show suh onsiseny if we are o aep his definiion of he parile (and aniparile) urren and his as we know is a dire resul of he desire o keep probabiliy densiy posiive. However we would like o add ha he Dira equaion has explained he fine sruure of hydrogen aom along wih a hos of experimens in parile physis so i would be premaure o pu a quesion mark on his equaion. Referenes [1] Jakson, J.D. (1999) Classial Elerodynamis. 3rd Ediion, Indian Reprin. John-Wiley & Sons (Asia) Pv. Ld., Singapore. [] Bjorken, J.D. and Drell, S.D. (1964) Relaivisi Quanum Mehanis. MGraw-Hill New York. [3] Powell, J.L. and Crasemann, B. (1961) Quanum Mehanis: Addison-Wesley In., Indian Reprin by Narosa Publishing House, 1995, New Delhi. [4] Halzen, F. and Marin, A.D. (1984) Quarks and Lepons: An Inroduory Course in Modern Parile Physis. John-Wiley & Sons, New York. 4/4

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