On the undulatory theory of positive and negative electrons

Transcription

1 Su la théoie ondulatoie de électon poitif and negatif J. Phy. et le Rad. 7 (1936) On the undulatoy theoy of poitive and negative electon By AL. PROCA Intitut Heni Poincaé Pai Tanlated by D. H. Delphenich Intoduction. The Diac theoy of hole i actually the only one that pemit one a glimpe into the behavio of poiton. The expeimental dicovey of the latte ha confimed the fundamental hypothei and ha hown that the popoed equation account fo the poitive electon a well a the negative one. Nevethele the difficultie that one encounte emain conideable. Without peaking of infinite pope enegie the tuctue itelf of the theoy in fact aie othe poblem the leat inconvenient of which i to ende the analyi of the implet cae quite pathological. One know what thee difficultie conit of: viz. to decibe a poiton. One need to potulate the exitence of a unifom but infinite denity of negative electon with no mutual inteaction. Only the ditance fom that unifom ditibution mut be conideed to be obevable; any unoccupied place on a negative enegy level o hole contitute a poiton. One often emphaize that fom a phyical point of view it i difficult to accept the hypothei of an infinite ditibution of electon. Fom the mathematical viewpoint one i contantly obliged to evaluate the diffeence between the quantitie that we know in advance to be infinite. Meanwhile the eult that have been obtained up to the peent eem to how quite well that the theoy coectly epeent eality. Indubitably the peceding difficultie ae due to the olution of the Diac equation that ae chaacteized by a negative kinetic enegy. Thee olution have no phyical ene. In ode to elate them to anything that ha phyical ene one mut make uppoition of a neceaily atificial chaacte uch a fo example the hypothei of an infinite denity of electon. It i vey pobable that if the equation fo the electon automatically exclude thi type of olution then any difficulty of thi type will vanih. We have begun to pove thi. Pauli and Weikopf in a fundamental memoi ( 1 ) have emaked that if the wave function of the electon i given by the elativitic Godon equation then none of the peceding difficultie will aie. Indeed fom the Godon theoy of enegy of the elementay paticle i alway poitive wheea it chage might alo be poitive o negative. The ame equation epeent the negaton and poiton at the ame time without any need fo ecoue to the hypothei of any infinite denity of electon. Moeove Pauli and Weikopf have hown that the eult that one obtain in the poblem of the poduction of pai fo lage ( 1 ) Helvetica Phyica Acta 1934 vol. 7 fac. 7 pp Alo ee a lectue of W. Pauli Annale de l Intitut Heni Poincaé (in pe).

2 Poca On the undulatoy theoy of poitive and negative electon enegie o in that of the polaization of the vacuum ae the ame whethe one tat with the Godon equation o the hole hypothei. On the face of the poblem of the poiton two attitude ae theefoe poible: That of Diac Heienbeg etc. which take the Diac equation a it point of depatue (enegy with two ign chage alway poitive) o that of Pauli and Weikopf which advocate the ue of anothe equation that lead to a poitive enegy and a chage with two ign. It i clea that the expeimental fact will be in favo of the econd way of looking at thing if one know a convenient equation to tat with that i equivalent to that of Diac in it conequence. The Godon equation that wa employed by Pauli and Weikopf i not convenient. It doe not account fo the pin of the electon and thi i pobably it gavet hotcoming. It ha the conequence that the teatment that wa decibed by thee autho will apply only when the paticle in quetion admit Boe-Eintein tatitic which i obviouly incoect. It i impoible (Pauli loc. cit.) to employ Femi-Diac tatitic with the Godon equation a one would like to do. If one would then like to poceed like thee autho then the fit thing to do would conit in finding anothe equation than that of Godon that enjoy the ame popetie a the latte and i moe adapted to the deciption of the electon. Thee i good eaon to ecall the agument that led Diac to etablih hi equation and to modify it in uch a fahion that one obtain anothe moe atifactoy one.. Condition that a fundamental equation mut atify. In the light of ecent expeimental dicoveie it eem a pioi impoible to obtain an equation of thi type. We then examine the pincipal condition that one will have to atify. We fit emak that the agument of Diac that led him to etablih hi equation in an ea when he did not know of the exitence and popetie of the poiton i moe convincing today now that one know of the phenomena of the poduction and annihilation of pai. The coeponding dicuion wa made by Pauli and Weikopf who emaked that: 1. Due to the poduction of pai it i no longe poible to limit quantum mechanic to the one-electon poblem. The known expeimental eult can coincide with the theoetical pediction that ae deduced fom the olution of a many-body poblem.. Thee i no longe any ene in peaking of a denity of paticle. By contat the chage denity a well a the chage itelf peeve a pecie phyical ene and ae obevable quantitie. The chage that i contained in a given volume when conideed a an opeato ha pope value that ae popotional to the poitive and negative intege numbe. It eult fom thee emak that thee i no longe any valid eaon to potulate the paticula fom: ψ ψ (1)

3 Poca On the undulatoy theoy of poitive and negative electon 3 fo the chage denity a Diac did in ode to etablih hi equation. An immediate conequence of thi i that the fundamental equation mut not neceaily be linea in / t; the Godon equation cetainly fall into thi categoy. In the following paagaph we will how the manne in which one may utilize the Diac agument in ode to etablih the new equation. Be that a it may the new equation mut atify thee condition: 1. It mut peent elativitic and electomagnetic (i.e. gauge) invaiance.. The wave function mut have fou component. Indeed in the Diac theoy fou component uffice to account fo the known phenomena and thee i no eaon to exceed that numbe. In addition it eem eay to etablih a theoy with a lage numbe of component but it i clea that a imila attempt will have no inteet in the context of expeiment o we will not impoe that equiement. 3. The paage to the cae in which an extenal field i peent mut be done a h alway by adding the opeato Φ viz. the potential of the field to the opeato. i x Thee i no eaon to enounce that condition which will be impoed fo eaon of coepondence. 4. One mut be able to fom a wold vecto that epeent the cuent and chage denity. 5. Thi cuent mut atify a conevation law by vitue of the fundamental equation a well a in the peence of an extenal field. 6. The tempoal component of thi vecto viz. the chage denity mut be capable of being poitive o negative. Thi condition i eential. 7. One mut be able to fom a ymmetic teno of econd ank that epeent the teno denity of enegy-quantity of motion. 8. Thi teno mut atify a conevation law by vitue of the fundamental equation Tk Tk that ha the fom = 0 in vacuo o = j k x x F whee j i the cuent and F k i the extenal field.

4 Poca On the undulatoy theoy of poitive and negative electon 4 9. The enegy denity mut be capable of being poitive o zeo o it expeion mut be a poitive definite fom which mut be tue in the peence o abence of an extenal field. Thi condition i eential. 10. One mut be able to exhibit the exitence of a pin and a magnetic moment. 3. Indication egading the choice of a fundamental equation. The agument by which one aive at the fundamental equation might not have an abolute chaacte; in fact it eve only to guide ou choice. Once thi choice ha been made one potulate the equation thu found and then jutifie thi potulate by the eult obtained. Thi i the cae in the Diac theoy moeove and ome of the eaoning that wa employed on that occaion when conveniently modified may eve again. Ou point of depatue i condition 9 accoding to which the enegy denity mut be epeented by a poitive-definite fom: ΦΦ whee the Φ may be eithe the component of a function that call the wave function o moe geneally function of it. The conevation of enegy will be expeed by: d dt Φ Φ Φ + Φ dv = 0 t t t ΦΦ = the integal being taken ove a convenient volume o even all of pace. One may deduce fom thi a in the Diac theoy ( 1 ) that at a well-defined intant one may imultaneouly aign abitay value to: Φ / t Φ / t and Φ Φ. Theefoe the Φ atify equation that linea equation in / t and by ymmety alo linea in / x. On the othe hand the fundamental elation: lead to the econd-ode equation: W c = p1 + p + p3 + m c ( 1 ) PAULI Handbuch de Phyik vol. 19 pp. 17.

5 Poca On the undulatoy theoy of poitive and negative electon 5 m c Φ Φ = 0 () h and it i natual to potulate a in the Diac theoy that thi i atified in the cae whee a field i abent. We mut then lineaize () i.e. we mut find a ytem of equation that have () a thei conequence. Now only the Diac olution lead to a tue lineaization and that i not convenient in ou poblem. One deduce fom thi that the Φ ae not wave function o athe that the expeion fo enegy doe not uniquely contain tem that depend diectly upon the wave function ψ. Thee will alo be tem that ae neceaily combination of ψ and / x and they collectively pemit a ot of lineaization; fo example it will make it eaie to undetand what it conit of. We take the lineaized equation to be: The ytem: Φ 1 = Φ k Φ = 0 ( = 0 1 3). () x Φ = y Φ 3 = z Φ 0 = 1 c t (3) Φ Φ Φ + + x y z c t 1 Φ = k ψ (4) whee ψ i a cala ha the elation () a a conequence; it uffice to ubtitute (3) in (4) and diffeentiate. Meanwhile it i clea that the fundamental equation emain the econd-ode one that define ψ: ψ = k ψ ; (5) the Φ ae deduced fom it by mean of fit-ode equation. Be that a it may if one i given the complex cala ψ which atifie (5) then one may deduce the Φ fom it by mean of (3) and conequently wite a econd-ank teno whoe tempoal component: = 1 Φ Φ + Φ Φ + k ψ ψ (6) will be a poitive-definite fom. Thi how that one may etablih a theoy of a paticle with poitive enegy in that fahion. The fundamental equation (5) i the Godon equation and the coeponding theoy wa developed by Pauli and Weikopf. It indeed atifie the condition of the peceding paagaph except and 10. Upon genealizing the peceding poce we find anothe ytem that alo atifie the lat two condition. In the peceding example the wave function ψ i a cala. Now we have need fo a magnitude with fou complex component; one may take two pino ψ χ. One may take a vecto complex vecto that wa fomed by combining the fou component of the

6 Poca On the undulatoy theoy of poitive and negative electon 6 two vecto a b. One may alo take the = 4 complex component of a pino of econd ank g and top with the implet poibilitie ( 1 ). The olution by two pino i the Diac olution which i not convenient. A a eult being given uch a g amount to being given an anti-ymmetic teno of econd ank with eal component ( ) and two invaiant. Finally the latte olution which take the fom of a wave vecto i the implet; thi i why we hall adopt it. The peceding example i completely chaacteized when one i given the coeponding Lagange function which i following Godon: L = h c The enegy denity teno: + m c ψ ψ x x (µ = 1 4). 4 4 T µν = µ = 1 µ µ h c + + L δµ xµ xν xν x µ the field equation etc. ae deduced fom L immediately. We look fo the poible L in the cae of a wave function that i the wold vecto ψ. L i an invaiant; it appea additively in the expeion T 44 and conequently will be compied like enegy of a um of tem of the fom A * A. Among them one will have: 1. Tem that contain the component of the function ψ explicitly. The implet invaiant combination that one may fom fom it i: ψ ψ and:. Tem that ae fomed by combining / x and ψ. The implet combination will be the divegence A = / x ; one confim that it doe not wok. The econd degee of complexity will be A = a econd-ank teno of type / x and moe paticulaly the one that ha the minimum numbe of component namely the otation of the vecto ψ : A = x x. The implet Lagangian will then be of the fom: ( 1 ) One may nonethele take magnitude with eveal indice by impoing cetain ymmety condition on them; fo example g t when it i anti-ymmetic in and t ha only fou non-zeo component. ( ) Cf. LAPORTE and UHLENBECK Phyical Review (1931) pp. 155.

7 Poca On the undulatoy theoy of poitive and negative electon 7 L = A A + k ψ ψ whee k i a contant that we take to be popotional to the pope ma fo eaon that will become appaent in the equel. By hypothei fo the cae of no extenal field we explicitly wite: h c 4 L = + m c ψ ψ x x x x with the uual ummation convention. The Lagangian fo the cae of an extenal field will be deduced fom the peceding one by the known ule (condition 3). * * * 4. Notation. Lagange function. Fo eaon that will become obviou hotly we do not adopt notation that ae analogou to the one that wee employed in the Diac theoy; thi hould not intoduce any difficulty but will offe no immediate advantage eithe. Theefoe conide a vecto ψ = a + ib and it complex conjugate ψ * = a ib. We may develop the theoy in a univee that i defined by x 1 = x x = y x 3 = z x 0 = ct. Fo the ymmety of the fomula it i meanwhile advantageou to allow a fouth imaginay coodinate. Nevethele in that cae one mut take cetain pecaution in ode to obtain the coect eality condition. Fo that it will uffice to wite the coodinate in the fom: x 1 = x x = y x 3 = z x 4 = ε ct (7) ε being a complex unit uch that ε = 1 it commute with the othe imaginay unit i that appea in the expeion fo the ψ. It will then be incoect to wite ε i = 1 but one will have ε i = i ε. The ateik that indicate the complex conjugate will change the ign of i but not that of ε. The patial component of ψ will be a + ib a b eal ( = 1 3) and the tempoal component will be ψ 4 = εa 4 + iεb 4 with a 4 b 4 eal. The complication that i intoduced by the ue of two complex unit i compenated by the advantage of ymmety. Moeove ε alway diappea when one pae to the vaiable t. In addition fo geate claity we avoid the ue of the ateik fo function of ψ * ; we ue diffeent lette to indicate that function and it complex conjugate. Finally et: = x ( = 1 3 4). Having done thi conide the cae of no extenal field. We et: F = ψ G = ψ ψ. (9)

8 Poca On the undulatoy theoy of poitive and negative electon 8 F and G ae complex conjugate; they contain an ε a a facto wheneve one of the indice o i equal to 4. By hypothei the Lagangian of the poblem will be: L = h c F 4 G m c ψ ψ + (10) in thi cae h being Planck contant divided by π. One then pae to the cae whee a field exit that i defined by a potential: by the ubtitution ( 1 ): Φ 1 Φ Φ 3 Φ 4 = ε Φ 0 (Φ 0 Φ 3 eal) (11) x x ie Φ ψ hc x ie + Φψ. (1) x hc To implify the notation we et: We futhe et: A = e hc Φ (13) k = mc h. (14) F = ( + ia ) ψ ( + ia ) ψ G = ( ia ) ψ ( ia ) ψ ; F G ae anti-ymmetic teno of econd ode. The Lagange function in the cae of an extenal field i the witten with the uual ummation convention: (15) h c 4 L = F G + m c ψ ψ. (16) 5. Fundamental equation. The momenta ae: One ha in tun: = h c F ( ψ ) = h c G ( ψ. ) ( 1 ) We adopt the convention of Pauli and Weikopf fo the ign of e; cf. loc. cit. pp. 7.

9 Poca On the undulatoy theoy of poitive and negative electon 9 4 = m c ψ 4 ih c A F = m c ψ + ih c A G = ih c ( ψ F ψ G ). A The fundamental equation: take the fom: ( ) = ( ψ ) = ψ (0) ( + ia ) F = k ψ ( ia ) G = k ψ (1) to which one agee to add equation (14) which define the intemediay magnitude F and G. Thee have the fom of Maxwell equation (fo an imaginay potential ) completed by element that epeent the influence of an extenal field. One may then deduce a cetain numbe of conequence fom thi fact that we will pa ove fo the moment. They indeed peent elativitic and electomagnetic invaiance; the fit thee condition of paagaph ae thu atified: Eliminate F and G ; one ha in geneal: Set: ( ia )( ia ) ( ia )( ia ) = i( A A ) ( + ia )( + ia ) ( + ia )( + ia ) = + i( A A ). = ( + ) = ( ) µ iaµ µ iaµ µ = 1 µ = ( µ µ ) ψ µ ( µ µ ) ψ µ µ = 1 µ = 1 J = + ia I = ia () (3) The fundamental equation become: H = hc e ( A A ). (4) ie ψ ( + ia ) J = k δ + H ψ hc ie ψ ( ia ) I = k δ H ψ. + hc + (5)

10 Poca On the undulatoy theoy of poitive and negative electon 10 In the cae of no field and povided that k and theefoe the ma of the paticle i non-zeo which i an eential condition one deduce that: Indeed in thi cae (5) become: ψ = J I = J = 0. (6) = k ψ k ψ ψ I and upon applying the opeato one deduce the tated eult (6). Theefoe in the abence of a field the equation become equation of the Godon type: ψ = ψ = 0 ψ = k ψ with mc (7) ψ = k ψ k =. h In the geneal cae we apply the opeato ie hc F H = k J to (1); one will have: ie hc G H = k I. Theefoe if k = 0 and only in thi cae then: ieh 1 J = hc mc F ieh 1 H I = hc mc G H. (8) We immediately point out ome othe inteeting elation. By vitue of (17) the ae anti-ymmetic teno; one thu ha: ( ψ ) ( ) = ( ψ ) = 0. The fundamental equation pemit one to deduce: = ψ = 0. (9) 6. Exitence and conevation of cuent. We define the cuent a Godon did following Mie by mean of the deivative of the Lagange function with epect to the potential and with the ign changed:

11 Poca On the undulatoy theoy of poitive and negative electon 11 o by vitue of (19): j = Φ = e (30) hc A j = i ehc ( ψ G ψ F ). (31) It i clea that (30) i a wold-vecto. It alway atifie a conevation law j = 0. Indeed one ha: j = i ehc ( ψ G G ψ F ψ F ). By vitue of the fundamental equation thi i: = i ehc [G ( + ia ) ψ F ( ia )ψ ] and by vitue of the anti-ymmetic chaacte of F G thi i: = i ehc G F F G 0. Thee thu exit a vecto j that atifie a conevation law. In ode fo u to be able to confim that it indeed epeent a cuent it i futhe neceay that when it i combined with the electomagnetic field it indeed funihe the expeion fo the foce that eult fom the divegence of the enegy teno; we how that thi i indeed the cae. In any cae the tempoal component i: it can take on value that ae eithe poitive o negative. j 4 = i ehc ( ψ G 4 ψ F 4 ); (3) 7. Enegy-quantity of motion teno and conevation law. In ode to etablih both the expeion fo thi teno and it conevation law we poceed a Schödinge did. Diffeentiate the Lagange function with epect to x abitay; one will have: L = ψ + ψ + ψ + ψ + ( ) ( ) ψ ψ ψ ψ A A o by vitue of the fundamental equation: L = (L δ ) = ψ + ψ + ( ) ( ) A A. Let H be the extenal electomagnetic field and let j be the cuent:

12 Poca On the undulatoy theoy of poitive and negative electon 1 j = e hc A. (30) We have poved that j = 0; theefoe: H j = A + A. (33) A A Add (3) and (33); upon taking into account the fundamental equation: H j = ψ + ψ + A δ L (34) ( ) ( ) A one find that: H j = {h c F ( ia )ψ + h c G ( + ia ) ψ δ L}. (35) Now one eaily ee that one ha the identity: 0 ψ + ψ ψ ψ ; (36) ( ) ( ) it uffice to take (9) into account along with the fact that the ae antiymmetic. ( ) Upon making (36) explicit with the aid of the fundamental fomula (17) and (18) one ha: 0 {h c F ( ia )ψ + h c G ( + ia ) ψ m c 4 ( ψ ψ ψ ) }. (37) Subtact (37) fom (35); by vitue of (15) one ha: namely: H j = {h c (F G + G F ) + m c 4 ( ψ ψ ψ ) δ L}. (38) T = H j (39) whee: T = h c ( F G + F G ) + m c ( ψ ψ ψ ) δ L. (40) 4

13 Poca On the undulatoy theoy of poitive and negative electon 13 Thi T i a ymmetic teno; (39) how that it divegence i equal to the Loentz foce. It may thu be choen to be the enegy-quantity of motion teno. The enegy i contantly poitive. The enegy denity i defined by E = T 44 and one ha upon witing out L explicitly: E = F G h c F4 G4 + m c ( ψ ψ ψ 4ψ 4) (41) whee vaie fom 1 to 4 and vay fom 1 to 3; thi i again by witing it out explicitly: E = h c (F 3 G 3 + F 31 G 31 + F 1 G 1 F 14 G 14 F 4 G 4 F 34 G 34 ) + m c 4 ( ψ 1ψ 1 ψ 3ψ 3 ψ 4ψ 4) (4) a quantity that i eentially poitive ince G i the complex conjugate of F. 8. Electomagnetic moment of the electon. Let the cuent be: j = i ehc ( ψ G ψ F ). (31) One may egad it a a um of two tem of a paticula fom. Indeed upon witing it out explicitly one ha the expeion: j = i ehc[ψ ( + i A ) ψ ψ ( i A )ψ I ψ J] + ( ψ ψ ψ ψ ). (43) In the cae of a zeo extenal field I = J = 0 and the cuent educe to: 0 j = i ehc [ψ ψ ψ ψ + ( ψ ψ ψ ψ ) ]. (44) One may thu epaate the cuent denity into two pat j = j + j jut a in the Diac theoy. The econd one: j = i ehc ( ψ ψ ψ ψ ) = m peent itelf a the cuent denity that i due to an electic o magnetic moment teno m (which doe not depend upon the extenal field explicitly). We may thu aume that the electon poee an electomagnetic moment that i given by: The emaining one: m = i ehc( ψ ψ ψ ψ ). (45) j = i ehc[ψ ( + i A ) ψ ψ ( i A )ψ I ψ J] (46)

14 Poca On the undulatoy theoy of poitive and negative electon 14 will be the conduction cuent. A in the Diac theoy each of thee patial cuent atifie an equation of continuity: = 0 and j = 0. j 9. Spin. Stating with the enegy-quantity of motion teno T one obtain in geneal the value of the kinetic moment by foming the integal: in which: 1 P = ( x T4 x T4 ) dv εc ( = 1 3) (47) 4 T4 = h c ( F4 G + F G4 ) + m c ( ψ 4ψ ψ 4) 4 (48) T4 = h c ( F4 G + FG 4 ) + m c ( ψ 4ψ ψ 4). Conide an electon in the abence of a field; one will have in thi cae: P = = 1 { 4 4 ( ) h c F xg xg + m c ψ 4 ( xψ xψ ) + conjugate} dv εc 1 { h c [ F ] 4 x ψ F4 x ψ F4 x ψ + F4 x ψ εc + m c 4 ψ 4 (x ψ x ψ ) + conjugate} dv. Fo = the component P ae zeo; fo one ha: and: One may thu wite: x ψ = (x ψ ) x ψ = (x ψ ) (49) F4 x ψ = F4 ( xψ ) ψ F4 (50) F4 x ψ = F4 ( xψ ) ψ F4. P 4 = 1 { h c [ F ( ) ( ) ( ) ( )] 4 xψ F4 xψ F4 xψ + F4 xψ εc + h c (ψ F 4 ψ F 4 ) + m c 4 (x ψ x ψ ) + conjugate} dv. Upon integating by pat and auming a one uually doe that the ψ ae annulled on the bounday one will have: 1 P 4 = h c { x ( ψ F4 G4 ) x ( ψ F4 G4 )} dv εc 1 + h c ( ψ F4 ψ F4 G4 ψ G4 ) dv εc. (51)

15 Poca On the undulatoy theoy of poitive and negative electon 15 One may egad the fit integal which ha the fom (x A x A ) dv a equivalent to an obital momentum and the econd one whee the x x no longe appea explicitly a the pope momentum of the paticle. The pin denity of the latte will thu be: M = h c {ψ F 4 F 4 G 4 G 4 ). (5) It i obviou that thi decompoition i abitay which coepond to the natue of thing moeove. We emak that the elativitic vaiance of the pin i that of the total momentum (47) that we tated with; it cannot be othewie. Thi pin diffe fom the one that one encounte in the Diac theoy. Indeed it i cetainly compied of the thee tempoal component of a teno of the fom P l but it i not completely anti-ymmetic a in the cae of the Diac equation. It i meanwhile eay to modify the decompoition (51) o a to epaate the total momentum into an obital momentum and a pin that i a completely anti-ymmetic teno of ank thee. Indeed one ha fo = 1 3 4: o: m c 4 x ψ 4 ψ = h c ψ 4 (x F ) h c ψ 4 F m c 4 x ψ 4 ψ = h c ψ 4 (x F ) h c ψ 4 F 4 m c ψ 4( xψ xψ ) h c ψ 4 ( xf xf ) = h c ψ 4 F 4 m c ψ 4( xψ xψ ) h c ψ 4 ( xg xg ) = h c ψ 4 G. (53) One then deduce a epaation of the total momentum P into an obital moment and a pin : 1 h c ( ψ F4 F 4 4F + conjugate) dv εc. (54) In thi fom the pin denity will be compoed of the tempoal component of a completely anti-ymmetic teno of thid ank exactly a it i in the Diac theoy. Manucipt eceived on 8 May 1936.