Hoshi R

Wakasa A

Hoshi Reiun

Wakasa Akira

若狭 昭

Kanazawa University Repository for Academic Resources

1 The ScienじeReports of th'e K口nazawaUniversity， Vol. VII， No. 1， pp. 1， June司 1960Dirac Particleswith Anomalous班a伊leticMoment in Arbitrary External Electromag'netic Fields By Reiun HOSHザ andAkira W AKASA制CReceived Octob巴r1， 1%9) 1. Introduction The suggestion of Lee and Yang (1) that the neutrino is to be desc1'ibed by a two-component theo1'Y indicates that the巴lectronsemitted in f3 decay and the muons emittecl inπ-μdecay will be longitudinally pola1'izecl. As pointed out by Ca回(~)、 it is very inte1'est to see to what extent it is possible to give a rigorous clescription of the behaviour of such pa1'ticles in an exte1'nal electromagnetic field. The results obtained up to the present day would certainly be exp巴cted(3ョ4)on a nonτelativistic model fo1' spinning particle. Recently， Case (2) and Tolhoek心1'oot仁5)have shown Independently that the same conclusions can be ca1'1'ied ove1' to 1'elativistic energies for Di1'ac pa1'ticles. 1n this pape1' we t1'昨日tgene1'ally with Dirac particles taking into account an anomalous 一歩 一与magnetic moment. Here， we introduce phenomenologically te1'ms pgd.μσ.H in the Hamiltonian where d.μ 1'epresents aIi anomaldus magnetic niom巴nt.2. Calcula tion Th巴 tr巴atmentis simplest in the Heisenberg repres巴ntation. We use the Heisenbe1'g equation of motion for an :;trbitrary operator 0. d0/dt = (d0/ dt)+(i/元)[3C，0J and we take the Hamiltonian in the form 3C = cp10'oπ+ ps111C2 十 elf十 p3ð.ρσ • H.(1) (2) → Here， the last t邑rmp3ð.f1σ • H is the interaction between an external magnetic五eldH ancl the anomalous magnetic moment D，.fi， σ;'s are (4 x 4) Pauli's spin matrices 10 1 0 0¥ 10 -i 0 0¥ 1 0 0 0¥ σ_11 0 0 0¥ _ _Ii 0 0 01σ=12-~ ? 21 1=¥0 0 0 11'σ2=¥0 0 o-il' 0'3=¥0 0 1 01; '0 0 1 0' ¥0 0 QI 、o0 0-1' (3) ? ? ?， ，? ? ? ?? 、 ? 、? ?? ??? ?? ，? ? ? ??????????????』?????、??????、。??? ?? ? ???? ????日)向=(i:-1!) (4) 梧 Presentaddress is Usetsu High School， Ishikawa prefecture. 骨持 Department of Physics， Faculty of Science， Kanazawa University. 2 R. HOSHI and A.羽TAKASAand 71=ρ-(e/c)A and A， !fare vεctor and scalar potentials for the external五eldφIn the following calculation， we make us巴 ofthe commutation relations for (l/s and .0/8: (J"σE十σj(l，，= (k， l=l， 2， 3)， PkP!+PIPk=2d1c! (k， 1=1， 2， 3)， (JkP!-P!σ，=0 (k，l=lラ 2，3); (5) σ"O'! =U7m， [σkヲσ!]= 2iO"踊 iρ"PI =ip町 [Pk，Pi]=2ip踊[(k， 1， m) is a cyclic permutation of (1， 2.， 3)J. ( 1 ) Longitudinal spin 0 = (Jふ.→→i If in Eq. (1) w記 put0=σ@π， we have 〆f →→日→→ a →→ E → dA →→ すく門)=すt(刊 +(i/約四， (1'71J =ー ヲ一σ at-e，+ .~炉 斗 .0 》ー → 砂ー →ト 〉ー+Paóp[-iσ，(マメ H)--ir~c UJ.(HxA-A xH)J -+ -} -? -} -+ -iD ...f-""'" がー + =ed.E-it:.仰伊[(マ×H〉-jL(H×A-AXH〉]+ (II) ~omentum 母 =π.→ L抗告=πinEq. (1). We obtain d マ a→→→ ( e -，今 ie2 . -~ -: _ ) πatπ+(ijli )[瓜 π]28E+CP470XH-JFE川 ，AJ} -名[ヂ，A一同ポ〔Lb(III) Spin (9 = a . → TVVhen we take 0 = 0， itleads to すよ=JJL十(i/日札 2]=一!?PIJd-23巴paJJIn deriving these equations we mak巴 useof the relations → 式 1 aA → b=一一一一一一一-'i7!f!，c at 一歩 → -> H=i7xA， → ー捗'i7.H=O; and the useful formulas summarized in appendix. 3. Di呂cu富島ion自(6) (7) (8) (9) 一歩In Eq.'sく6)，(7)， and (8) th巴巴lectromagneticpotential Aμ=(A，ヂ)is the external field i. e. c-number quantity. Therefore commutators betwεen each other of these vanish and then Eq.'s (6)， (7)， and (8) are lead to the following A -令 日.， → 一歩 + 一歩すT(J'n=伊 E-iムf1Pa(J'(VxH)， (10) Dirac Particles with Anomalolts Magnetic Momenf in A rbitrary External Elecfromagnetic Fields 3 ↓ ?↓ ???? ? ?? ?。?，?， ，??↓??↓?? ????↓???? ?↓??? ? (11) and JJ=-jipμ-2jEpaJd (12) 1n these equations last terms depending onムμrepresentthe effect of the anomalous magnetic moment to b巴 contrastedthe results given by Case and others. (A) First let us consider a pure magn巴tic五eld.Then， バ→→→→→dTaon = -it.f1P3a.('V X H)， d 一歩 今 一歩 +→ー 夕立Jπ=叩1σXH-t.f1P3巾 .H)，and d → 2c →→ 2t.ρ →+ dt σT(llσ × π ~-hP3σ xH. (13) →》→→From Eq. (13)σ・π，πandσar巴 nolonger constants of motion. Then， for example， it is possible to chang巴 astate of longitudinal polarization to one of transverse polarization using purely magnetic五eldsand a longitudinally polarized beam will be depolarized on passing through purely magnetic fields. (B) Second we consider a pure electric五eld. 1n this case we have d →→→之一 σ・π=eσ・1エ，dt ~ -" -~~ ~ d →=ヒヨ子π=叫 pand d → 2c →→ dFσT(llσ×π固 (14) →→→→ Frol11 Eq. (14)σ・π，πandσarenot constants of l110tion. Then particles passing through a purely electric field are depolarized. These conclusions can be applied， of course， so long as the Hamiltonian is given by Eq. (2). Final1y we expect that for the baryons the巴任巴ctof anol11alous l11agnetic moment will be very important. From the degree of the depolarization of particles passing through a external field， we can巴stimatethe order of the magnitude of their anomalous magnetic moment. 1n conclusion， the authors wish to express their sincere gratitude to Prof. T. Tati for his encouragement， and wish to thank Dr. S. Hori for his suggestion and valuable discussion. Appendix 1n this appendix， we sUl11marize the useful forl11ulas of which we make use →→ →→ 一歩今 日歩 -'1・ → (σ.B)(σ・C)=(B.C)+iσ・(BxC)， (A1) (A2) (A3) ベシ →シ → → → → → → → σ〔σ・B)C=(B・C)-iσ・(BxC)，一歩 → → → → → → → → B(σ。C)σ=(B・C)-ia-(BxC)，4 今+R. HOSHI and A. W AKASA .; .;.→炉-> -+ σ.(σxB)=2iσ.B， ←ヤ→→→->(σxB).ぴ=--2ia・→ →〉 ヤ → 一歩 → (j" (σ.B)=B-iσxB， ベヤ ヤー 令ー + ゅ →p 〔σ.B)σ=B+iaxB，(A5) (A6) (A7) wherε B and C ar巴 arbitraryoperators which don't commute each other but commute wIth spin matrIces 0" /s References [1] T. D. Lee and C N. Yang， Phys. Rev. 105，. 1671 (1957). See ulso S. Hori， H. Segawa and A. Wakasa， Prog. Theor. Phys. lSL 249 (1958)固[ 2 ] K. M. Case， Phys. Rev. 1自S，173 (1957). [3 J Garwinラ Ledermun，ancl Wεinrich， Phys. Rev. 105， 1415 (1957)切[4] Louisell， Picld， and Crane， Phys. Rev. 94， 7 (1954) [ 5 ] H. A. Tolhoek and S. R. de Grootラ Physicu17， 17 (1951)， H. A. Tolhoek， Rev. Mode工nPhys固28， 277 (1956). 

Dirac particles with Anomalous Magnetic Moment in Arbitrary External Electromagneic Fields

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Dirac particles with Anomalous Magnetic Moment in Arbitrary External Electromagneic Fields

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