Physics Notes. Note 22. The Electron and the Ilectron

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1 Physis Ntes Nte 9 Deeber 6 The Eletrn and the Iletrn I.L. Galln 4 St. Katherine s Aenue, Bridprt, Drset, DT6 DE, UK D. V. Giri Pr-Teh, -C Orhard Curt, Ala, CA 9457 USA, Dept. f ECE, Uniersity f New Mexi, Albuuerue, NM 877 USA and C. E. Bau (Psthuusly based n ur nersatins with hi befre he left us) Abstrat The eletrn was the first eleentary partile t be disered. We knw its bsered ass, its harge, its angular entu and we assue that it has infinite lifetie. Fr re than a entury n nsensus has been reahed er the euatin f tin fr a radiating eletrn and the standard del wuld hae us beliee that the eletrn is a pint partile. We briefly nsider the histry f the euatin f tin f the eletrn and pint t the reasn fr the failure f what was nsidered t be the rret result f lassial physis. Furtherre, a gien ass has an eent hrizn, and any ass apprahing a pint wuld lie within and s be a blak hle, whih wuld rapidly disappear ia Hawking radiatin. Beause a pint partile with ass annt spin, it has been delared that the eletrn angular entu is a purely uantu ehanial effet. The preditin f zitterbewgung (trebling tin in Geran) by Shrödinger ffered a partial slutin, in that it suggested that the angular entu was due t irular tin arund a pint, but still left the prble f a assie partile (the eletrn) traelling at, the speed f light. We hae the rules fr aelerating eletrns t exist in statinary states, but n explanatin as t why they d nt radiate. It was knwn fr the beginning that the eletrn ust hae ehanial ass. Unable t separate this ass it was assued t be 'pushed int a nn-bserable real'. Despite these prbles and lak f explanatins, Quantu Eletr Dynais (QED) ntinues t prdue the st aurate preditins eer prdued by thery. We als nsider a plex binatin f harge and ass whih leads t the nsideratin f pssible 5th fre and a new partile, whih we all the iletrn. The iletrn has deried prperties that ake it a ntender as a WIMP.

2 . Intrdutin The eletrn and in partiular its tin has a lng histry. The Abraha-Lrentz euatin f tin is F () Jaksn [] writes '..an be ritiised n the grunds that is send rder in tie...' and '...anifests itself iediately in the s-alled runaway slutins.' The deriatin is uh the sae in Heitler [], Jienez and Caps [], Erber [4] and Dira [5]. The prble is aused by linearizing a nn-linear euatin [6]. The eletrn is a negatiely harged assie partile with the attributes listed in Table. Attribute Fundaental Partile the eletrn rest ass kg harge (- ) C lassial radius r e uantu ehanial radius angular entu Ω = / kg /s agneti ent.5965 B B is the Bhr Magnetn, = J/T Table. Attributes f the eletrn All these uantities exept fr r e hae been deterined by easureent. The radius is deterined by euating the integral f the eletri field energy t the kineti energy ( ) as fllws. 4 re r dr 4 r e () The result is r e () 4 with = perittiity f free spae = F/. This is t be understd as eaning, if anything, that this is the rder f agnitude f the size f the eletrn. Quantu physiists nw regard the eletrn bth as a pint partile and a wae, but a pint partile is in nflit with relatiity. If the eletrn had zer radius, the integral in euatin () wuld dierge and nseuently a free eletrn wuld hae infinite eletrstati energy. Classially, the eletrn has t hae a nn zer radius, and this iplies a struture. T fr a del it is neessary t inlude nt nly field energy, but als a ehanial ass (pages - f []).

3 Measureents suggested (page 84 f []), and slutins t Dira's euatin (page 6 f [7]) nfired that the eletrn has angular entu. Let us nsider a spinning spherial ass, radius r spinning abut its entre, as shwn in Figure. Suh a slid bdy has a ent f inertia I gien by 5 I r () Figure. A slid bdy f ass and radius r its kineti energy is E 5 5 I r r (4) where is the angular elity. The angular entu is I r (5) 5 If we redue the radius t zer, the angular entu ges t zer and a pint partile is nt aeptable lassially. This is why the eletrn spin is desribed as a purely uantu ehanial effet. An bius interpretatin f the slutin t Dira's euatin fr a statinary eletrn [] is that it is ing in a irle f radius / where is the redued Cptn waelength. Using a lassially deried euatin f tin that intrdues the lassial statinary state [6] prdues a radius se 5% greater, and a elity se 5% lwer, that is nsistent with relatiity. It is this rtatinal tin that was tered zitterbewgung by Shrödinger, the uantu ehanial result being that the elity was. Befre the deelpent f uantu ehanis the struture f ats had been deterined by experient and the results were in nflit with lassial physis. The appliatin f ideas ntributed by Plank, de Brglie, Einstein, Shrödinger and Heisenberg, ulinating in Dira's euatin, laid dwn

4 the priniples f uantu ehanis. Belw we nsider the speifi failure that spurred the deelpent f uantu ehanis fllwed by a brief lk at a lassial del f zitterbewgung. This is fllwed by a disussin f results generated by a plex transfratin f the eletrn leading t speulatin abut a new partile, its prperties and ipliatins.. Classial Apprah t the Hydrgen At Cnsider the Hydrgen at illustrated in Figure. Figure. The Hydrgen at with a single rbiting eletrn The eletrn experienes a entripetal fre and a entripetal aeleratin. Fr any bjet t stay in a irular tin, there is a entripetal (enter- seeking) fre, Hweer, as per Newtn s laws, there is a reatin fre that is entrifugal. This pseud-fre that is entrifugal is balaned by the Culb fre s that the eletrn an stay in its irular rbit. r Z (6) 4 r where is the speed f the eletrn, r the rbit radius, and Z = ati nuber (= fr Hydrgen) and = perittiity f free spae = F/. Euatin (6) an als be written s as t relate the speed and rbit radius as, r r 4 r 4 (7). The result f the rbit radius is nsistent with euatin () fr energy nsideratins. Hweer, lassially, an aelerating eletrn radiates energy (E) at a rate gien by the Larr frula [4] de dt ( J / s) (8) (s) (9) 6 4

5 The elity is ~ where is the fine struture nstant gien by and the radius is knwn as the Bhr radius and is gien by x 4 7 () a. () () Writing E k fr the kineti energy, the lifetie f the rbiting eletrn is gien by E k de dt a 4 = 4.67 (s) () r whih is hardly lng enugh t build a unierse! This prble was ere by wae ehanis by speifying rules fr the tin f eletrns in ats. Dira s deelpent f his relatiisti euatin inrprated these rules, but surprisingly prdued the result that eletrns nt ated n by any fre, e abut at the elity f light.. Classial Explanatin f Zitterbewgung An euatin f tin deelped in (euatin 7. f [6]) fr a radiating eletrn, dified by the Statinary State Hypthesis and applied t an eletrn with n applied fres is where ( / ) and the radiatin nstant i is d i. i 4 () dt i (4) 6 i This euatin has the expeted slutin f zer aeleratin, but it als has a slutin t exp i exp i t (5) τi and the bsered rest ass bees r i (6) 5

6 ~.65 i (7) where i is the intrinsi ass, whih inludes bth ehanial ass and eletragneti ass. Euating the angular entu t the knwn alue f the eletrn spin i Sling this fr i τ i (8) 4 α η e (9) α and the angular elity is fund t be 4α 4 ω e () τ The uantu ehanial result fr the freueny f the eletrn Zitterbewgung is ω 4α τ () a nseuene f Dira's failure t inlude the ehanial ass f the eletrn. The rtatin radius fr the eletrn is gien by r s τ = 4 α τ () 4 αη e Intrduing the Cptn Waelength ia the relatin the radius is gien by r s ().5557 (4) ηe This result is nsistent with the unertainty priniple, the unertainty f the entu being and the unertainty in psitin r giing r r x r r ω h/( ΔpΔx ) (5) 6

7 Furtherre this del [6] allws the alulatin f the fine struture nstant t the sae auray as the QED alulatin [7]. With this lassial relatiisti del f the eletrn euatin f tin tgether with a del f the hydrgen at, it was pssible t alulate FSC by first deterining an apprxiatin fr a nstant using the then aailable alue f FSC and n ther nstant. The nstant was apprxiately Fr the thery the nstant had t be prie, and 4 is prie. Further deelpent gae Then inerting prble and alulating FSC assuing 4, gies a alue within abut f the Gabrielse result and is nly.x - fr the latest ean CODATA result. 4. The Cplex Eletrn It is ften adantageus t fr a plex ariable fr tw real ariables, as in plex analysis, by diiding by suitably defined diensinal nstants s as the transfred ariables hae the sae diensin. We an d this fr the harge and ass f eleentary partiles, as fllws. Cnsidering the eletrstati and the graitatinal fre between tw eletrns, we hae f K 4 r r (6) f G 4 gr r (7) where, in the usual ntatin 4 K 9 x 9 ( N / C ) (8) G 6.67 x 4π g ( N /Kg ) (9) In the graitatinal fre abe, the nentin is a negatie fre fr attratin and a psitie fre fr repulsin. We nw define tw nstants g d. 86x ( C / Kg ) and d. 6 x ( Kg / C) g () We bsere that a diensinless nstant fr the eletrn is gien by d 9 g. 6 x C d d. 75x d. x 9. x Kg = F F e G () This apprah uld be extended t all the eleentary partiles, in partiular the prtn and the neutrn. It is nted that eery fundaental partile that has nn-zer ass will hae an assiated nstant d. This nstant wuld anish fr a neutral partile ( = ) suh as a neutrn. We an nw express ass and harge with the sae diensin by fring a plex Mass (M) r a plex harge (Q). There are arius ways t fr these binatins, as fllws. 7

8 M g M i d i a and Q i d i (b) g i and Q i d g g Fr exaple, if we use the binatin f euatin ( ) and nsider plex r bined fres, between tw plex asses M i and M i () g g we hae the plex fre gien by Expanding this expressin using euatins ( ), F M MM (4) 4 r F M i 4 g r 4 r 4 g r f i f i (5) Nte that the real part gies the nentinal graitatinal and Culb fres. If we nw hse M t be a neutral partile by setting F M f i 4 gr (6) Setting i (7) g we btain F M f i f i i where fi (8) 4 g r The strength f a fre is gien by Matt Strassler [5] r f S (9) Cparing the strength f this iaginary fre t the eletri fre, and assuing the neutral partile is a neutrn n 9 S / ~ 9.x (4) i e g The strengths f the fur standard fres tgether with this new fre are gien belw in Table. 8

9 Strng Weak Eletr Magneti....7 (=α) Iaginary Graity Eletr- Mass Table. Strengths f the fie fres These strengths are alulated in the regie where they are effetie, the strng fre within the nuleus, and the weak fre within a nulen while the eletragneti fre perates utside the eletrn. A fre is nsidered weak if F r ( ) and is nsidered strng if F r. In partile physis the eletr-ass fre wuld be undetetable, but if the neutral partile is a neutrn star, this fre wuld be the dinant fre ating n eletrns. A typial neutrn star ntains ~5 56 neutrns. 5. The Iletrn Regnising that iaginary ass an be interpreted as harge, and iaginary harge as ass, we an nsider the iaginary eletrn, where we interhange the ass and harge, but this transfratin is nt erely an algebrai transfr. We ust transfr the intrinsi ass f the eletrn t btain the new harge, the harge transfring as befre. Ealuating the pnents g i i i i ii, g, i, i, i (4) g 9 ii e. 86 kg and i i =.7 x -4 C (4) g. 65 g In euatin (4), ii is the intrinsi ass euialent f the eletrn s harge and i is the harge euialent f eletrn s intrinsi ass. We are alling this new partile with its ass and harge gien by euatin (4) as the iletrn r an iaginary eletrn. 6. Iletrn Spin and Magneti Ment If we assue the spin f / is transferred t the new ass we an deterine the agneti ent. The fine struture nstant is gien by (4) 4 Replaing by i i 8 i i (44) Fr euatin (9) replaing by i, the rtatinal elity is αi i i ηe. 89 α i (45) 9

10 Fr euatin (6) where r ii / (46) i i i i ii 6 (47) i ii and is the redued Cptn radius. The radius f rtatin is then 4 i r 6. / (48) The agnitude f the agneti ent is then gien by ir 6 4 B 48.. B (49) 9. 7 The abe ideas hae been ritiised as 'ere speulatin', but where wuld siene be if se f us did nt asinally break away fr what is knwn. 7. Eletrns and Blak Hles Briefly, it is wrthwhile t lk at se analgus relatinship between the eletrn and a blak hle. The lassial eletrn radius is gien by the Lrentz frula [6] (als euatins () and (7) abe),.87 x 4 (5) r e 5 Replaing e / by / g in the abe, the radius f the eletrn is gien by r 4 g (5) A blak hle is a regin f spae and tie with a ery strng graitatinal pull s that n partile r EM radiatin an esape fr it. The bundary fr whih n esape is pssible is alled the eent hrizn. The abe frula fr the radius f an eletrn is siilar t the frula fr the radius f a Shwarzshild blak hle r S g G (5) Exaples f Shwarzshild radii f se n planets are:

11 sun ( k), earth (8.7 ), Mn (. ) and Jupiter (.). If the Shwarzshild radius exeeds the physial radius, the bjet is a blak hle. Hene these planets are nt blak hles. Alternatiely we uld estiate the Shwarzshild radius fr an eletrn as e 58 res g (5) whereas the lassial radius f the eletrn is.8 x -5. With the lassial radius f an eletrn being uh larger than its Shwarzshild radius, it is nt a blak hle. 8. Suarizing Rearks By fring a plex eletrn we hae raised the pssibility f a new fre any rders weaker than the eletragneti fre, yet still furteen rders greater than graity. Cntinuing with this idea we nstruted a hypthetial iaginary eletrn whih turns ut t hae a substantial ass f ~.9-9 kg and effetiely n harge r agneti ent. The iletrn wuld appear t be a gd ntender fr a WIMP (a weakly interating assie partile), a s far undeteted hypthetial partile in ne explanatin f dark atter. If the iletrn exists, presuably it has an anti-partile and an iletrn eeting an anti-iletrn wuld result in a pair f high energy gaa phtns. Assuing a siilar ibalane in the relatie nubers as is fund fr ther partile pairs, suh interatins wuld be rare. The detetin f these gaa rays wuld in all prbability reuire the design f new detetrs. The iletrn has n ipat n psitrn. The anti-partile t the iletrn is the ipsitrn. The ass f the iletrn is.86 x -9 kg whih nerts t ~ MeV = 5 TeV The axiu energy f the LHC in CERN is TeV. The LHC wuld hae t be upgraded by a fatr f arund 8x t prdue an iletrn! At this tie the nly pssibility f nfiratin is if the iletrn eets its anti-partile, the ipsitrn and the resulting gaas pass thrugh ur detetrs. Gaas abe TeV are lassified as ultra-high energy, and s far nne hae been deteted. If they are the elusie WIMP they wuld hae t pride a ass f ~9% f the Milky Way. The ass f the sun is ~x kg and s the nuber f iletrns reuired is N = [ x kg /.86 x -9 ] ~ 9 (54) With the lue f the Milky Way being ~ 48 the reuired density is 8 N (55) Detetin f gaas f this energy wuld nt nly supprt their existene but wuld als enable estiates f the density f iletrns. Sattering f these gaas with eletrns ay als ntribute t an explanatin f gaa ray bursts. Fr lassial thery, the eletrn has a radius whih is larger than its Shwarzshild radius, and hene, is nt a blak hle. This bseratin f the siilarity f the radius f the eletrn [euatin (5)] and the radius f a blak hle [euatin (5)] raises an interesting uestin if the harges f partiles are

12 uantized, are the asses f blak hles uantized? That is a uestin fr Astr-Physiists and ay hae been already answered. Referenes: [] Jaksn, Classial Eletrdynais, Jhn Wiley & Sn. [] W. Heitler, The Quantu Thery f Radiatin, Clarendn Press, 954, Chapter, p 5. [] Jienez and Caps, A ritial exainatin f the Abraha-Lrentz euatin fr a radiating harged partile, A J Phys 55() N 987 [4] Erber, The Classial Theries f Radiatin Reatin, Frtshritte der Physik 9, 4-9 (96) [5] Dira, Classial thery f radiating eletrns, Preedings f the Ryal Siety, 98 [6] I. L. Galln, Extending Classial Physis int Quantu Dain, Physis Nte, June. [7] J MCnnell, Quantu Partile Dynais, Nrth Hlland Publishing Cpany, 958. [8] J. Larr, On a dynaial thery f the eletri and luiniferus ediu, Philsphial Transatins f the Ryal Siety 9, (897) pp. 5 (Third and last in a series f papers with the sae nae) [9] [] The Feynan Letures n Physis, l I.