CHIRAL FIELD IDEAS FOR A THEORY OF MATTER IDEAS DE CAMPO QUIRAL PARA UNA TEORÍA DE LA MATERIA

Transcription

1 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8, pp CHIRAL FILD IDAS FOR A THORY OF MATTR IDAS D CAMPO QUIRAL PARA UNA TORÍA D LA MATRIA H. Toes-Silva 1 Reibido el 5 de septiebe de 7, aeptado el 1 de diiebe de 7 Reeived: Septebe 5, 7 Aepted: Deebe 1, 7 RSUMN n este tabajo, paa el desaollo de una teoía unifiada de apos eletoagnétios y gavitaionales se usa un étodo quial. Los fotones que satisfaen las euaiones de Maxwell, paa una onda eletoagnétia se onsidean oo oponentes físios básios. l objetivo de esta teoía es unifia el fenóeno de la invaianza elativístia, eánia de onda y la eaión del pa eletón positón, on las euaiones de Maxwell, paa obtene una teoía de la ateia totalente eletoagnétia. Consideando esta teoía se disuten algunos aspetos de los sisteas GPS (Global Positioning Systes). Palabas lave: Potenial quial, teoía de la ateia, onda-patíula. ABSTRACT In this pape, a hial appoah is used fo developing a unified theoy of eletoagneti and gavity fields. The photons whih satisfy Maxwell s equations fo an eletoagneti wave ae taken as the basi physial oponents. The goal of the theoy is to unify the phenoena of elativisti invaiane, wave ehanis and pai eation with Maxwell s equation to obtain an eletoagneti field theoy of atte. With this theoy soe aspets of GPS (Global Positioning Systes) systes ae disussed. Keywods: Chial potential, atte theoy, wave-patile. INTRODUCTION A hial appoah is suggested fo developing a unified theoy of eletoagneti and gavity fields. Photons whih satisfy Maxwell s equations fo an eletoagneti wave ae taken as the basi physial oponent. The extent of the photon in its dietion of tavel peits a pat of the photon to odify the geodeti of anothe pat. A photon with a self distubed obit, fo whih a entoid an be defined, has the key popety by whih atte diffes fo light. Matte has a speed whih is less than that of light. The entoid of the obit has a speed whih is less than the speed of the photon whih tavels with the speed of light. We efe to this hial appoah as the eletoagneti field theoy of atte. Chial appoah eans that ou Univese is obsevable aea of basi spae-tie whee tepoal oodinate is positive and all patiles bea positive asses (enegies). The io Univese is an aea of the basi spae-tie, whee fo viewpoint of egula obseve tepoal oodinate is negative and all patiles bea negative asses. Also, fo viewpoint of ou-wold obseve the io Univese is a wold with evese flow of tie, whee patiles tavel fo futue into past in espet to us. The two wolds ae sepaated with the ebane - an aea of spae-tie inhabited by light-like patiles that tavel along light-like ight o left-handed (isotopi-hial) spials. The goal of the theoy is to unify the phenoena of elativisti invaiane, wave ehanis and pai eation with Maxwell s equation fo eletoagneti waves. Setion enueates advantages of an eletoagneti field theoy of atte. Setion 3 onsides how the de Boglie elation and the Shodinge equation ight be deived fo Maxwell s wave equation. Setion 4 teats the elation between eletoagneti and inetial enegy. Setion 5 oents on paity failue in weak inteations. Appendix A deives an appliation on GPS satellites using hial potential. 1 Instituto de Alta Investigaión. Univesidad de Taapaá. Antofagasta Nº 15. Aia, Chile. -ail: htoes@uta.l 36 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8

2 H. Toes-Silva: Chial field ideas fo a theoy of atte ADVANTAGS OF AN LCTROMAGNTIC FILD THORY OF MATTR An eletoagneti field theoy should not be onfused with an eleti hage theoy of atte whih is not elativistially invaiant [1]. An eletoagneti field, howeve, is elativistially invaiant fo the stat. One sipliity is that speial elativity is not a sepaate hypothesis. The Loentz ontation of eletoagneti fields was ealised befoe speial elativity. If atte is oposed only of eletoagneti fields then atte is autoatially Loentz invaiant. In patiula, atte annot exeed the speed of light. Anothe aea of sipliity is pai eation whee two eletoagneti fields (photons) podue an eleton and a positon. If the patiles ae eletoagneti fields, then pai eation is like the tansfoation of eletoagneti field fo one state of otion to anothe. We suggest sine asses ae unique that this should be thought of as the onstution of a quantized state of the M wave (i.e. standing wave a de Boglie type phase elation) []. The distintion between atte and antiatte would then be sought as a natual law of onsevation of a popety inheent in the sepaate initial photon and divided between the patiles. Suh a diffeene is inheent the photon polaization. Fo exaple, the photons an have ightand left-handedness. It should be futhe noted that a fast eleton is like an M wave having a tansvese M field with equal eleti agneti field enegies. Also the oentu of fast patiles, like M waves, is the enegy divided by. The unetainty piniple of quantu ehanis would also be oe onsistent with an eletoagneti theoy of atte. That is, patiles whih have an inheent wave natue would be expeted and not a supise. Also, quantized absoption of M enegy would not be viewed as a hage aeleated by an eleti field but a eging of two M waves. The eged M fields would have the equied fequeny and wavelength to be the quantized wave of the eleton in the final state. The appeaane of the fine stutue onstant, = e /h, in the atio of the asses of fundaental patiles would be expeted and not a oinidene as in the atio of the ass of the eson and the eleton. The developent of an M field theoy of atte equies the aoplishent of at least two objetives, naely (i) pedit the Coulob foe between eleton and positon (et.), and (ii) deive the Shodinge equation fo Maxwell s equation fo an eletoagneti wave. We suggest appoahes to these objetives in the next setions. DRIVATION OF TH PARTICL WAVLNGTH FROM CHIRAL POTNTIAL WAVS We stat with the potential veto equation. Assuing e jot tie dependene, Maxwell s tie-haoni equations [] fo isotopi, hoogeneous, linea edia (without hages) ae j B (1) H j D () B (3) D (4) Chiality is intodued into the theoy by defining the following onstitutive elations to desibe the isotopi hial ediu [3] D T (5) B H H (6) Whee the hiality fato indiates the degee of hiality of the ediu, and the y µ ae peittivity and peeability of the hial ediu, espetively. Sine D and ae pola vetos and B and H ae axial vetos, it follows that and µ ae tue salas and T is a pseudo sala fato. This eans that when the axes of a ighthanded Catesian oodinate syste ae evesed to fo a left-handed Catesian oodinate syste, T hanges in sign wheeas and µ eain unhanged. Sine B = always, this onditions will hold identially if B is expessed as the ul of a veto potential A sine the divegene of the ul of a veto is identially zeo. Thus B A (7) And A ust be pependiula to both and B and lie in and xb plane. Howeve, A is not unique sine only its oponents pependiula to ontibute to the oss podut. Theefoe, A, the oponent of Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8 37

3 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8 A paallel to, ust be speified. The ul equation fo, as in quation, and quation give x ( + ja) = whee the quantity in paentheses should be paallel to and the ul of the gadient of a sala funtion is identially zeo; so the geneal integal of the above equation is Let xˆ yˆ zˆ A ik x ikz Ax Ay Az j A (8) Substituting quation into quation we obtain xˆ( ik os A ) y yˆ( ik sin A ik os A ) zˆ( ik sin A ) so equation (1) is expessed as z x y (13) A H T H=B (9) Substituting quation and quation into quation gives T T A j j B, 1k T 1k T 1k T o o o with k T = T/. Plaing the value of x fo equation into the above equation, using the veto identity x xa = ( A A and equation (8) enables us to wite the above equation as k A A 1 kt (1) T T A A j 1 k T 1 k T Hee Ais abitay, so in ode to speify A, fo unique A, we ay define a hial Loentz gauge A j 1 k T (11) And eliinate the te in paentheses. Then quation will be siplified to k T A A ( A) (1) 1 k T 1 k T The solution of the potential veto equation an be solved as follows: o k ( 1 kt ) k ik kt os A x ik kt os k ( 1 kt ) k ik kt os Ay ik kt sin k ( 1 k T k ) A The solution of the deteinant is k ( 1 k T ) k / k k / 1 k T (14) fo the longitudinal field, and k ( 1 kt ) k ) 4kT (sin os ) k k / ( 1 k T) (15) fo the tansvese fields. Ou appoah to deiving the Shodinge equation fo Maxwell s equation stats with the assuption that an eleton is an eletoagneti wae tavelling in a iula obit in the obseve s est fae. We suggest that the obit is a geodeti in a spae-tie uved by the photon s own eletoagneti enegy. We note that the odel of the self- tapped wave ust look like an eleton to obseves in all inetial faes. The obseve at est only sees the stati Coulob field. The oving obseve, with speed u, sees (i) soe agneti field fo the uent assoiated with the haged patile, and (ii ) (the wave-otion of the patile with a wavelength = h/u. Fo the obseve whose speed elative to the iulating photon appoahes the veloity of light, the photon appeas as a photon. This is appopiate sine the eleti and agneti fields of a fast eleton beoe tansvese as its speed appoahes. Fo equation (14), whee k k / 1k T, if we ake k = /, k T = T/ u/ then patile oentu is onsistent with the photon odel. The obseve with z 38 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8

4 H. Toes-Silva: Chial field ideas fo a theoy of atte speed u in the usual theoy of speial elativity notes that the eleton has enegy [3]. This esult is the sae as obtained fo the theoy of the eleton fo v - using quation (17) 1 (16) h h p h () whee = u/ = T/, and oentu p u 1 (17) Moe geneally, fo any value of u fo equations (19) and (1) h u p (3) Fo equations (16) and (17) as v appoahes and the est ass beoes a sall pat of the enegy p / (18) In speial elativity the fequeny tansfos as the enegy and this is the oet expession fo the oentu of a photon. The oentu of a photon in the est fae is effetively zeo beause the geodeti loses on itself. We will ty to illustate how the photon fequeny an lead to the appopiate patile wavelength fo all inetial obseves. Fo the obseve at est with espet to the iulating photon. The field appeas stati, the peiod is infinite, the fequeny is zeo, and the wavelength infinite. At high speed we will show that the photon s wavelength in the obseve s fae is onsistent with the quantu ehanial expession fo the oesponding patile. We elate the photon enegy hv to the est ass of the eleton using the speial elativity elation =, by. h (19) and inveting equation (19) h / / v () In the liit of u the atte wavelength is the wavelength assoiated with the photon edued by the usual speial elativity Loentz ontation fato 1. Then using equations (), (19) and (16) in ode we obtain h h 1 / 1 (1) whee h / 1. Defining a wavelength elated to the patile h 1 (4) p u u u In tes of fequenies fo equation (4) u (5) We intepet this esult as follows. The eleton has intinsially the fequeny of its paent photon. The obseve going by at the speed of light sees the iulating wave stethed out to its liit and assoiates the full fequeny, v, with the patile fequeny. The obseve oving oe slowly passes the nodes in the M wave oe slowly and intepets this as a lowe fequeny, v <v. The obseve at est sees no hange and onludes v =. These fequenies ae onsistent with the de Boglie wavelength fo atte. The Shodinge equation desibes the wave otion of the entoid of a photon whih is a solution of Maxwell s wave equations in a distoted spae tie. A wod is in ode about the poble of intefeene pattens of satteed eletons suh as podued by the diffation by two slits in a baie. Magenau [4]onsides the diffiulties of this poble. He onludes that thee ae no known inteations that an explain how an eleton an go though one slit and be appopiately satteed by the othe. An eletoagneti field whih ats like a patile, ay possible avoid this dilea by eithe (i) the eleti and agneti field goings patly though eah slit, o (ii) having a satteing whih diffes fo known inteations beause of the flutuating M field popeties of the Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8 39

5 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8 eleton suh as in ou photon odel. We think that the seond possibility is the oet explanation. RLATION BTWN LCTROMAGNTIC AND INRTIAL NRGY In this setion we elate the photon eletoagneti and the eleentay hage eleti field enegies with the inetial and gavitational enegies. When an eleton positon pai is eated we distinguish two hanges in enegy: (i) the eletoagneti field enegy of the photons is tansfoed into the eleti field of the pai, and (ii) the tansfoed photon whih we eognise as a patile has inetia with espet to the osology. By speial elativity the inetial enegy pe unit ass is. The inetial ass elates to the eletial enegy of the hage by = e /. Now a oe detailed disussion of the effet of the osology on inetia and the gavitational ed shift ae equied to laify the distintions between the fou types of enegy. We assue, following Mah, that the inetia depends on the osology. That is, we take inetial enegy equal to the negative of the osologial gavitational potential. Fo geneal elativity, then, we assue MG u k T (6) R whee M is the total ass of the univese (~1 56 g), R is the adius of the univese (~1 8 ) and G is the gavitational onstant (,67x1-8 dyne /g ) [5-7]. Consequently, in pai eation, the inetial enegy gained is anelled exatly by the loss of gavitational potential enegy. We note that equation (11) is independent of. We have assued to be invaiant and hene the gavitational potential enegy pe unit ass ust also be independent of loation in the osology. This is onsistent with the assuption that evey loation in the osology senses the expansion of the osology in the sae way and that thee is no distinguished loation in the osology. The next ipotant hypothesis is that the photon enegy does not hange as it oves though the osology. This onept ust be distinguished fo the dependene of fequeny on the loal gavitational potential. We use a subsipt i to oespond to initial value, i.e. hv i is the photon enegy when it is eitted by an ato o podued as a esult of pai annihilation. Now hv; depends on the gavitational potential at the loation whee it is eitted sine fo atoi adiation hv i e 4 i / h and fo pai adiation hv i = i. In this way we see that hv i i, and both depend in the sae way on the gavitational potential. This is onsistent with geneal elativity [7]. In patiula it agees with the gavitational ed shift. Fo exaple, a photon adiated fo the sun has enegy hv s s, and the oesponding ato on the eath has the tansition enegy hv e e ; v s is to the ed of v e as given by geneal elativity beause the asses ae elated by e s 1 (7) whee is the diffeene in the gavitational potential enegy pe unit ass. Thus the photon enegy hv s has not hanged enegy duing its tavel fo sun to eath. This is the justifiation fo the assuption that the photon enegy does not hange as it oves though the osology. We now etun to the elation between the eletial enegy and inetial enegy and opae both with the photon enegy. The subsipts elating to loation in the gavity field ae etained h e / (8) i i i and we ust subsipt the patile adius fo onsisteny. Thus photons whih ae podued by annihilation an only eeate the oet aount of enegy whee the gavitational potential is the sae as at point of oigin. The poble an be solved by speial elativity. Thee ae fou enegies with signifiant diffeenes. A photon an be tansfoed into an eleton and both photon and eleton have equal and positive enegy. The photon enegy does not hange as it taveses the osology. The patile enegy depends on the loal osologial eti. No net gavitational enegy is podued by the eation of the patile sine the gain of inetial enegy is just anelled by the loss in gavitational enegy. As a patile taveses spae the inetial enegy is always equal to the eletial enegy of the patile 4 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8

6 H. Toes-Silva: Chial field ideas fo a theoy of atte and the inetial enegy is always anelled by the gavity enegy. PARITY The photon odel of the eleton has a natual explanation of the failue of efletion syety (paity) in weal inteations. Wigne [8] has given a phenoenologial disussion of the ipliation of the C deay expeient in whih the spin of the is deteined to be opposite to its oentu. He points that the io iage of an eleton is a positon. We invoke the polaization of the photons the eleton positon pai. We assue that one patile has ighthanded polaization photon and the othe a left- handed one. Thus, the patiles ae io iages as equied. In fat, it is the diffeene in handedness whih distinguishes two hage states. APPNDIX A: RLATIVISTIC FFCTS ON CLOCKS ABOARD GPS SATLLITS Conside a lok aboad a satellite obiting the ath, suh as a Global Positioning Syste (GPS) tansitte. Thee ae two ajo elativisti influenes upon its ate of tiekeeping: a speial elativisti oetion fo its obital speed and a geneal elativisti oetion fo its obital altitude. Both of these effets an be teated at an intodutoy level, aking fo an appealing appliation of elativity to eveyday life. Fist, as obseved by an eathbound eeive, the tansitting lok is subjet to tie dilation due to its obital speed. Fo ou esults of hial potential, (14), a lok aboad a spaeship taveling at speed u uns slow (opaed to a stationay lok) by a fato of [9] 1 u u (A1) povided u <<, as would be the ase fo a satellite. Thus when one seond of pope tie elapses, the oving lok loses u / = K/ seonds, whee K and ae the kineti and est enegies of the lok, espetively. Seond, a lok at the highe gavitational potential of obit uns faste than a sufae lok. The gavitational potential enegy of a body of ass in ath s gavity is U = V, whee V = G / is ath s gavitational potential (at distane fo the ente of the ath of ass ). In the ase of a photon, we eplae by /, whee = hf is the photon s enegy. If the photon tavels downwad in ath s gavitational field, it theefoe loses potential enegy of (hf/ )V and gains an equal aount of kineti enegy hf. We theeby dedue that the falling photon is gavitationally blue-shifted by f f f (A) (This expession an also be staightfowadly dedued [1] using the equivalene piniple to teat ath s downwad gavitational field as an upwad aeleating fae, and then alulating the Dopple shift in the light between eission high up and obsevation low down in this oving fae.) If the lok s tiking is synhonized to a light wave, the obiting lok will be obseved at ath s sufae to be tiking faste due to this gavitational fequeny shift. Theefoe, when one seond of ath tie elapses, the lok at high altitude gains V/ = U/ seonds, whee U is the gavitational potential enegy of the lok. The su of the two elativisti effets an be opatly expessed as t K U (A3) whee t is the tie lost by the obiting lok when a tie inteval elapses on the sufae-bound lok. Hee K U is the Lagangian of the obiting lok whee the efeene level fo the gavitational potential enegy is hosen to lie at ath s sufae. As a onete exaple, let s alulate the size of these two effets fo a GPS satellite, loated at an altitude of = 6,58 k, about fou ties ath s adius of = 638 k. Fo Newton s seond law, we have F v G g a v (A4) whee ath s sufae gavitational fieldis g G / = 9.8 /s. Hene the fational tie loss due to the satellite s obital speed is g / pe seond, o 7. µs/day. Meanwhile, the geneal elativisti fational tie gain due to the satellite s altitude is Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8 41

7 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8 V 1 G G g ( 1 ) (A5) whih woks out to be µs/day. Notie that the gavitational effet is oe than six ties lage than the speed effet: the doinant GPS oetion is geneal, not speial elativisti! If we instead onside satellites in pogessively lowe altitude obits, thei speeds will inease aoding to q. (4), while the gavitational potential diffeene in q. (5) will deease. ventually we will eah an altitude at whih the two oetions exatly anel, so that the satellite s lok will un synhonously with an eathbound lok [1, 11]. This ous when u V g g ( 1 ) 1. 5 (A6) i.e., at an altitude of half an ath adius. SUMMARY An outline has been pesented of an eletoagneti field theoy fo atte. The advantages of the theoy ae given in setion. Thei seeingly distint aeas of physis see unified with Maxwell s equation fo M waves. They ae elativisti invaiane, pai eation, and wave ehanis. Light is elativistially invaiant, hene, patiles ade out of photon ae elativistially invaiant. If atte is a fo of eletoagneti enegy, then pai eation is a tansfoation fo enegy. If patiles ae ade out of photons they have an intinsi wave natue and thei wave otion is expeted. In setion 3 we elaboated on how the wave natue of photon, whih fos an eleton, leads to wave ehanis of the patile. The full fequeny is appoahed as the veloity of the patile, elative to the obseve, appoahes the veloity of light. In setion 4, we gave the elation between the eletoagneti and inetial enegy. No enegy is added in pai eation: the eletoagneti enegy is tansfoed between photon and patile states. The inetial enegy is a popety of the patile in the osology. The osologial gavitational potential is the negative of the inetial enegy so that these utually anel. Theefoe, no net inetial plus gavitational enegy is equied fo pai eation. Setion 5 pointed out that the io iage popety of the positon and eleton equied by the failue of paity onsevation in weal inteations an be attibuted to the handedness of the photon whih ae tansfoed into the eleton positon pai. RFRNCS [1] H.A. Loentz. The Theoy of letons. B.G. Teubne, Leipzig [] H. Toes-Silva. New intepetation of the atoi speta of the hydogen ato: a ixed ehanis of lassial LC iuits and quantu wave-patile duality. Ingeniae. Rev. hil. ing. Vol. 16 Nº 1, pp [3] H. Toes-Silva. letodináia quial: eslabón paa la unifiaión del eletoagnetiso y la gavitaión. Ingeniae. Rev. hil. ing. Vol. 16 Nº 1, pp [4] H Magenau. Open Vistas. Yale Univ. Pess, New Haven [5] N. Salingaos. Invaiants of the eletoagneti field and eletoagneti waves. A. J. Phys. Vol. 53, pp [6] C.W. Allen. Astophysial Quantities. 3d ed. Athlone, London [7] A. instein. The Meaning of Relativity. Pineton Univ. Pess. Pineton, New Jesey [8] M. Gogbeashvili, Otonioni vesion of Dia equations, Intenational Jounal of Moden Physis A. Vol. 1 Nº 17, pp [9] D.C. Gianoli. Physis fo Sientists and nginees. 3d ed. Chap. 37. Pentie Hall, Uppe Saddle Rive, NJ.. [1] N. Ashby. Relativity and the Global Positioning Syste. Phys. Today. Vol. 55, pp May,. [11] S.P. Dake. The equivalene piniple as a stepping stone fo speial to geneal elativity. A. J. Phys. Vol. 74, pp. -5. Januay 6. 4 Ingeniae. Revista hilena de ingenieía, vol. 16 núeo espeial, 8