Why Baryons May Be Yang-Mills Magnetic Monopoles

Transcription

1 Why aryons May e Yan-Mlls Manetc Monooles ay. Yablon Schenectady ew Yor Abstract: We deonstrate that Yan-Mlls Manetc Monooles naturally confne ther aue felds naturally contan three colored ferons n a color snlet and that esons also n color snlets are the only artcles they are allowed to et or absorb. hs aes the worthy of serous consderaton as baryons. Introducton and Suary he thess of ths aer s sle: anetc onoole denstes whch coe nto exstence n non-abelan Yan-Mlls aue theory are synonyous wth baryon denstes. aryons are Yan-Mlls anetc onooles! We exane three llars of suort for ths: a Yan-Mlls anetc onooles naturally confne ther aue felds secton ; b they naturally contan exactly three ferons whch we dentfy wth colored quars n a color-neutral snlet sectons and ; and c the only artcles crossn ther surface or observed as decay roducts are esons also n colorneutral snlets secton 4. Secton 5 aes bref concludn rears about chral roertes of these roosed onoole baryons for further develoent whch ay rovde touchstones for exerental valdaton.. aue eld onfneent rst we deonstrate how Yan-Mlls anetc onooles naturally confne ther aue felds. We use the lanuae of dfferental fors and assue the reader has suffcent falarty so no tutoral exlanatons are requred. In an Abelan coutn feld aue theory such as QD the feld strenth tensor s secfed n relaton to the vector otental aue feld e.. hoton A accordn to da. he anetc onoole source densty s then 4 secfed classcally for hh-acton S ϕ d xl ϕ >> h where the uler Larane equaton ay be aled by the feld equaton d dda 0. hs aes use of the eoetrc law that the exteror dervatve of an exteror dervatve s zero dd 0. In nteral for ths becoes d dd da 0. All of the foreon zeros are what tell us that there are no anetc onooles n an Abelan aue theory such as QD. hs absence of anetc onoole chares at all attanable exerental eneres s well borne out n the 40 or so years snce aes ler Maxwell ublshed hs 87 A reatse on lectrcty and Manets. In a non-abelan non-coutn feld Yan-Mlls aue theory such as QD the fundaental dfference s that the feld strenth tensor s now secfed n relaton to the vector otental aue feld e.. luon n QD accordn to d. In ths relatonsh [ ] dx dx exresses the non-coutn nature of the aue felds and the non-lnearty of Yan-Mlls aue theory. herefore althouh dd 0 as always because of the

2 d d d d exteror eoetry the classcal hh-acton anetc onoole densty becoes whch s non-zero. In nteral for usn auss Stoes law ths becoes: d d d d d and fro the last two ters n the above we ay also derve the coanon equaton:. d 0.. Of course. albet wth the dfferent feld nae s ust the relatonsh da 0 whch tells us that there are no anetc onooles n Abelan aue theory. ut n lht of. whch rovdes us wth a non-zero anetc onoole 0 what can we learn fro. whch s the Yan-Mlls analoue to the Abelan no anetc onoole relatonsh da 0? If we erfor a local transforaton d on the feld strenth whch n exanded for s wrtten as ' [ ] then we fnd fro. as a drect and edate result of the Abelan no onoole relatonsh d 0 n. that: d.. hs eans that the flow of the feld strenth across a two densonal surface s nvarant under the local aue-le transforaton ' [ ]. We now n QD that nvarance under the slar transforaton A A ' A Λ eans the aue araeter Λ s not a hyscal observable. We now n { } ravtatonal theory that nvarance under ' Λ lewse eans the aue vector Λ s not a hyscal observable. In ths case the nvarance of under the transforaton [ ] ' tells us the aue feld s not an observable over the surface throuh whch the feld s flown. ut s sly the aue feld whch n QD s the luon feld. So sly ut: the Yan-Mlls aue felds ncludn luons n SU are not observables across any closed surface surroundn a anetc onoole densty. Whatever oes on nsde the volue reresented by the aue felds rean confned. an ths a ste further we see that the orns of ths aue feld confneent rest n the 40-year old ystery as to why there are no anetc onooles n Abelan aue theory. In dfferental fors the stateent of ths s dd 0. In nteral for ths becoes d 0 equaton.. Yet t s recsely ths sae zero whch renders nvarant under [ ] ' n.. So the hyscal observaton that there are no anetc onooles n Abelan aue theory translates nto a syetry condton n non-abelan aue

3 theory that aue boson flow s not an observable over the surface of a anetc chare. Aan: In Abelan aue theory there are no anetc onooles. In non-abelan theory ths absence of Abelan anetc onooles translates nto there ben no flow of aue bosons e.. luons across any closed surface surroundn a Yan-Mlls anetc onoole. onsequently the absence of luon flux hence color across surfaces surroundn non-abelan chrooanetc onooles s fundaentally equvalent to the absence of anetc onooles n Abelan aue theory. And because ths s turn ornates n dd 0 we see that ths confneent s eoetrcally andated osed by sacete. he very sae zero whch n Abelan aue theory says that there are no anetc onooles n non-abelan aue theory says that there s no observable flux of Yan-Mlls aue felds across a closed surface surroundn a Yan-Mlls anetc onoole. We do not fnd a free luon n Yan-Mlls aue theory any ore than we fnd an Abelan anetc onoole n electrodynacs for dentcal eoetrc reasons.. atural hree-eron Syste: art I Whle color confneent s necessary rerequste for Yan-Mlls anetc onooles to be consdered as baryon canddates t s not suffcent. At nu we ust also show that these onooles are caable of naturally contann three ferons n sutable color eenstates because we now that baryons contan three colored quars. or ths urose we eloy the classcal feld equatons D : D [ ] D D D D.. toether wth the Yan-Mlls feld strenth tensor: [ ] [ ] D D D. where the rou enerators are related by the rou structure f [ ] and where and are x atrces for any ven SU. Above. and. resectvely are ust exanded restateents of the classcal feld relatonshs d and d whch we used n.. As soon as one substtutes the non-abelan. nto Maxwell s equaton. whle the ters based on contnue to zero out by dentty n the usual way va 0 aue felds one nonetheless arrves at a resdual non-zero anetc chare: dd whch as shown n secton confnes the [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] hs s a lonhand verson of d d used n.. Let s now study ths o ben we ae use of the coutator relatonsh [ ] closely...4 to relace the varous n.4. xandn aears throuhout so these ters dro out. e-consoldatn yelds:

4 [ ] ] [ ] ] [ ] ]..5 ow we see an nverse relaton I to relace each above wth a whch can then be used to ntroduce feron wavefunctons va. Aan usn [ ] nverse I s secfed n ters of a syetrzed confuraton sace oerator { } [ ] [ ] D D n. wth a hand-added roca ass by: I..6 We also use a syetrzed I A [ ] to calculate I. In don so we ee n nd that the { } s an x atrx for the Yan-Mlls aue rou SU so anyte aears n a denonator we ust actually for a Yan-Mlls atrx nverse. So that exressons we develo have a slar loo to falar exressons fro QD we wll use a quoted denonator notaton M M to desnate a Yan-Mlls atrx nverse. hus etc. hs nverse s calculated to be: I [ { }] [ ].7 [ ] and can only be fored f we sultaneously ose the covarant aue condton n confuraton sace: { } 0 {..8 } ote that the often-eloyed [ ] 0 s not a aue condton here; ths s relaced by.8. ow nverse.7 has any nterestn roertes whch we shall not tae the te to exlore here. Secal ; 0 cases of nterest nclude [ ] 0 ; both 0 and 0 ; and on shell 0 for 0 or 0 for 0. We wll also note that when worn towards a quantu ath nteral forulaton [ ] n.7 s relace by a aue-nvarant erturbaton oent s n the low-erturbaton lt [ ] 0 [ ] 0. ut our nterest at the. hus usn.7 n nverse relaton I wth all the quoted denonators becoe ordnary denonators and we obtan:..9 We have reduced ths usn the fact that n oentu sace current conservaton x 0 0 becoes see [] after I.54. he above s ust le the exressons we encounter for nverses wth a roca ass n QD. It says not unexectedly that n the low-erturbaton lt QD loos le QD. he ont of develon ths nverse s to be able to use.9 n.5 and then deloy feron wavefunctons va. ecause.5 contans sx dfferent aearances of there are sx ndeendent substtutons of 4

5 5.9 nto.5 and what we ust resue to be sx ndeendent roca asses. o trac ths we wll use the frst sx letters of the ree alhabet to carry out the nternal ndex suatons and to label each of these sx roca asses. hs substtuton yelds:..0 Here we see sx assve vector boson roaators each couled wth a current vector. We rase the ndexes on all the currents and absorb the. We use... to exlctly ntroduce the SU enerators. We factor out the resultn coutators [ ]. And fnally we eloy and the le to ntroduce feron wavefunctons. Wth ths and ovn all currents nto the sae nuerator.0 becoes: [ ].. he above now shows feron wavefunctons and s the startn ont for the next stae of develoent.. atural hree-eron Syste: art II ow we ae the follown sequence of substtutons for and the other two le ters n. above: uu.. Let us now exlan each ste n the sequence. In the frst ste we use the Drac snors u u and su over sn states. Often the sn su s wrtten as uu sns see [] secton 5.5. ut there s an led covarant noralzaton n ths exresson. o be exlct ths should really be wrtten [] roble soluton 5.9: uu sns. So n the second ste we aly. n..

6 6 ext we tae the affratve ste whch as we wll shortly dscuss requres soe accountn for derees of freedo that wll render the aue bosons assless of dentfyn the rest ass n the resultant wth the labeled ass n the denonator so we now set. hs of course started out n.0 as a aue boson ass n a aue boson roaator denonator but by ths ste we turn t nto a feron rest ass. And we sultaneously roote nto the oentu four-vector for ths feron wth ass. And we label and. nally we use the well-nown relatonsh:. but eloy an nverse n reconton of the fact that whenever an SU atrx ncludn the u u needs to o nto a denonator we ust for ts nverse. hus alyn. to. yelds: [ ]..4 ut there s one fnal ece of the uzzle that s requred to ae ths all wor roerly. We ust balance the derees of freedo used to turn. nto.4 and n artcular to turn boson rest asses nto feron rest asses. In.0 we started wth sx vector bosons wth resued roca asses. A assve vector boson has three derees of freedo so the sx bosons n.0 brouht x68 derees of freedo nto. ut then we too three of those boson asses and turned the nto feron asses. Massve ferons however have four derees of freedo not three. So to roote a assve boson ass nto a feron ass we ust transfer one deree of freedo over fro the boson to the feron. So these bosons ust dro down to two derees of freedo aece and thus becoe assless.e. that we ust now set these to zero 0. ow the 8 derees of freedo that ntally beloned three aece to sx assve vector bosons have been redstrbuted: of these now belon to the ferons and only 6 belon to the reann bosons. hs should see very falar as ths s the sae way n whch assless aue bosons frst becoe assve by swallown a deree of freedo fro a scalar feld va the oldstone echans. Here ferons swallow a deree of freedo fro bosons.

7 7 Loon closely at.4 we now also see a ath to choosn noralzatons for whch sultaneously are covarant retan the ornal ass densonalty of for u u and reatly slfy.4. Secfcally we now choose the covarant ass denson-reservn noralzatons: ; ;..5 Usn these toether wth 0 n.4 yelds the vastly slfed: [ ]..6 roceedn aace the coutator [ ] oerates to coute the vertces and n artcular the oeraton t erfors s [ ] [ ]. hs s the sae coutaton [ ] of free ndexes wth whch everythn started bac n.5 and even further bac n the underlyn feld densty [ ] of. whch s the heart of Yan-Mlls theory. So.6 now becoes: [ ] [ ] [ ]..7 All that now reans n.7 s the fnal coutator wth oentu ters such as. on bac to [ ] whch tells us that coutn a sacete feld wth s ust a clever way to tae ts dervatves we can slarly wrte [ ] M M for a second ran tensor feld x M. So f we also ae use of the second ran Drac covarant [ ] and also relabel wth slar labeln of the assocated wavefunctons.7 now becoes:..8 ow let s exlan what we have done. We deduce leadn to.8 that a Yan-Mlls anetc onoole densty naturally contans three feron wavefunctons and related roaators. ut for any SU these are - coonent colun vectors. So because there are three we ntroduce the SU aue rou of QD wth rou enerators 8 ;... noralzed to tr and assocate each of the three wth a quar n an eenstate of color. hus 0 ; ; 0 0 and 8 ; 0 0. hs sultaneously forces excluson so that no two quars n ths syste have the exact sae quantu nubers. If one then assocates each color eenstate wth the sacete ndex n the related oerator n.8.e. ~ ~ and ~ and ees n nd that s antsyetrc n all

8 ndexes then we ay exress ths antsyetry wth wede roducts as ~. So the natural antsyetry of the anetc onoole coare the to lne of.4 leads straht to the requred antsyetrc color snlet wavefuncton confuraton [ ] [ ] [ ] for a baryon see [] equaton [.70]. And because of what we dd to et to the feron asses n.4 we are requred to ee the SU aue bosons assless ust as s also requred n QD. he above.8 whch also confnes ts aue felds as develoed n secton we therefore nterret as a baryon. nally by contrastn.8 wth.4 we see that the feld coutator [ ] at the heart of Yan-Mlls theory n the feld strenth [ ] of.4 has now turned nto three Drac tensors: [ ] ; [ ] ; [ ] Mesons Another hallar of hadron nteracton are esons so now let s loo for those. A eson s a artcle antartcle conuate ar so let s start wth the well-nown Drac conuaton relatonshs and. One ay deduce fro these that:. 4. Secfcally sultaneously coutes ndexes and reverses sn. lus the well-nown dentty So usn [ ] fundaental Drac relatonshs [ ] and { } for the antsyetrc Drac tensors n.8 or.9 nto: derved fro cobnn the we ay decoose one of the nuerator ters. 4. ecause s a tensor and s a conuate tensor the addtve cobnaton ay s be understood as a tensor eson that s as a eson artcle for whch e.. n l n wth n whch the arallel ntrnsc sns totaln s are arallel to the orbtal exctaton l. Alternatvely ths ay be n l s n also wth n whch the s s antarallel to the stretched orbtal exctaton l. urther s a scalar and s a conuate scalar so the addtve cobnaton ay be understood s as a scalar eson for whch n l n 0 wth 0. So by vrtue of 4. and.9 we see that the 8

9 coutator [ ] actually contans a tensor eson lus a scalar eson! See [] [4] for a full exoston of exerentally-observed esons and ther sn classfcatons as scalars vectors tensors etc. and axal varants. In fact contrastn wth fro. and notn that dx dx dx let us ultly both sdes of.8 by the antcoutn volue eleent dx dx dx tae the trle nteral then aly auss Stoes law to the rht hand sde and renae ndexes. What we et s: dx dx dx dx dx. 4. ' We showed n. that nvarance of under a aue-le transforaton [ ] eans that there are no aue bosons allowed to flow across a closed surface surroundn a Yan-Mlls anetc onoole whch eans for SU ts luons are confned. So far so ood. ut that tells us what cannot flow. he above 4. tells us what can and does flow. What s allowed to flow across any boundary are sn tensors whch va 4. ay be decoosed nto tensor esons and scalar esons. Moreover the aussan nteraton has reoved the oerators and what reans by nsecton n 4. s the wavefuncton color confuraton whch s recsely the syetrc snlet color cobnaton requred for a eson! So 4. would see to say that only colorless tensor and scalar esons flow across a closed surface surroundn a anetc onoole densty. However contrastn 4. wth 4. we fnd that the scalars dro out dx dx 0 v because s a syetrc tensor whle { } 0 dx dx v are antcoutn so that! dx dx v 0 where we show the wede roduct to ae ths ont clear. So the eoetry tself acts as a flter ust as t does to confne luons! and shuts down the flow of scalar esons such as n S0 across the boundary and forces ther confneent as well. All that ay cross are sn tensor esons such as n or n both wth or any other esons that ay be constructed entrely out of quars and conuate quars e.. qq qq wth arallel sn alnents n 5 S wth. Of course after they have exted the closed surface these tensor esons ay thereafter decay nto other tensor or vector or scalar or axal eson by-roducts and so be observed as they are when studyn baryons and esecally nucleons. ut what 4. says s that to actually cross a closed surface surroundn a Yan-Mlls anetc onoole densty whatever s nsde the volue ust frst be excted nto a color-neutral sn tensor or axal tensor as we shall dscuss oentarly n order to cross throuh the surface va after whch the sn eson ay decay nto other observed esons of other sns. Sn eson s the assort 9

10 n and out of a anetc onoole baryon; all other assae s forbdden. here s no couln to the eoetry n 4. that allows a sn eson to ass sn luons are not ertted to ass for the sae reason that there are no anetc onooles n Abelan aue theory and sn 0 esons are fltered by dx dx 0. v 5. oncluson: Hadronc hral Asyetry and xerental aldaton We conclude wth a bref coent about axal esons whch are also wdely observed n hadron hyscs ost notably the I π seudoscalar esons wth n S0 and All such axal obects nvolve oeratn on a wavefuncton to roduce A 5 where a vector wavefuncton s defned as a wavefuncton for whch the related current densty transfors as a Lorentz four-vector n sacete. 5 0 ased on cobnn the relatonsh wth dualty based on the wor of ench [5] later elaborated by Wheeler [6] whch uses the Lev-vta forals see [7] at aes t turns out that there s a whole syste of chral dualty that s an nteral albet aarently heretofore undeveloed feature of the Drac alebra. or 5 0 exale ven the dualty relatonsh * A one ay wrte n the alternatve for A! 5 *. hen one ay for * A by sandwchn between wavefunctons. urther t s also well nown because the second ran dualty oerator ** that one can for contnuous lobal rotatons usn e A * θ cosθ *snθ. or exale: cosθ snθ A. 5. snθ cosθ A Slar transforatons ay be develoed for frst thrd and even zeroth fourth ran dualty wth the result that 5 0 tensors x wth axal tensors vectors wth axal vectors and scalars wth seudoscalars. And when s aled to.8 as art of a ordon decooston really recooston of a vector current t turns out that baryon and eson hyscs s endecally orancally non-chral whch s consstent wth what s exerentally observed. he dualty anle θ coes to be assocated wth the strenth of the runnn stron couln S whch n turn bears wellstuded relatonshs [8] [9] to the exerental oentu transfer Q. So by fully develon the chral dualty of Drac s equaton and alyn ths to.8 t ay well becoe ossble to exerentally confr the thess that aryons are Yan-Mlls anetc onooles: sly robe nucleons at varyn eneres study the chral characterstcs of the debrs that eeres fro those robes and correlate those chral roertes to the robe eneres that were aled. 0

11 eferences [] Zee A. Quantu eld heory n a utshell rnceton 00. [] Halzen. and Martn A. D. Quars and Letons: An Introductory ourse n Modern artcle hyscs. Wley & Sons 984. []. erner et al.d D see secfcally hater 4. Quar Model htt:d.lbl.ov0revewsr0-rev-quar-odel.df. [4]. erner et al. artcle Data rou D htt:d.lbl.ov0tablesr0-suesons.df [5] ench.y. lectrodynacs n the eneral elatvty heory rans. A. Math. Soc. ol [6] Wheeler. A. eoetrodynacs Acadec ress [7] Msner. W. horne K. S. and Wheeler. A. ravtaton W. H. reean & o. 97. [8] K. Hawara et al. Quantu hroodynacs hyscal evew D htt:d.lbl.ov00qcdr.df. [9] S. ethe he 009 World Averae of S ur. hys htt:arxv.orabs [he-h].