Ingeniare. Revista Chilena de Ingeniería ISSN: Universidad de Tarapacá Chile

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1 Ingenare. Revsta Chlena de Ingenería ISSN: 78-9 Unversdad de Tarapacá Chle Torres-Slva, éctor IRAC MATRICES IN CIRAL REPRESENTATION AN TE CONNECTION WIT TE ELECTRIC FIEL PARALLEL TO TE MAGNETIC FIEL Ingenare. Revsta Chlena de Ingenería, vol. 6, arzo, 8, pp Unversdad de Tarapacá Arca, Chle Avalable n: ow to cte Coplete ssue More nforaton about ths artcle Journal's hoepage n redalyc.org Scentfc Inforaton Syste Network of Scentfc Journals fro Latn Aerca, the Carbbean, Span and Portugal Non-proft acadec project, developed under the open access ntatve

2 Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8, pp IRAC MATRICES IN CIRAL REPRESENTATION AN TE CONNECTION WIT TE ELECTRIC FIEL PARALLEL TO TE MAGNETIC FIEL MATRICES E IRAC EN REPRESENTACIÓN QUIRAL Y LA CONEXIÓN CON EL CAMPO ELÉCTRICO PARALELO AL CAMPO MAGNÉTICO. Torres-Slva Recbdo el 5 de septebre de 7, aceptado el 5 de dcebre de 7 Receved: Septeber 5, 7 Accepted: eceber 5, 7 RESUMEN En este trabajo se presenta una expresón de la transforacón general de Foldy-Wouthuysen a la representacón qural de las atrces de rac nteractuando con un capo de ferón. La hpótess es que a través de la ultplcacón de la atrz de Paul por las ecuacones qurales de Maxwell en el caso de E, se obtene la ecuacón qural de rac. Esta es la prueba del teorea de que la ecánca de ondas de partícula cuántca representa una electrodnáca especalzada. Palabras clave: Transforacón de Foldy-Wouthuysen, ecuacón qural de rac, electrodnáca. ABSTRACT In ths paper we offer an expresson of the general Foldy-Wouthuysen transforaton n the chral representaton of rac atrces nteractng wth feron feld. Our hypothess s that through the ultplcaton of the Paul atrx and Maxwell s chral equatons n the case of E, one obtans the rac s chral equaton. Ths s the proof of the theore that the wave echancs of quantu partcles represent a specalzed electrodynac. Keywords: Foldy-Wouthuysen transforaton, chral rac equaton, electrodynacs. CIRAL IRAC MATRICES The paper offers an expresson of the general Foldy- Wouthuysen transforaton n the chral representaton of rac atrces nteractng wth feron feld x, t. The paper [, ] dscuss the theory of nteractng quantu felds n the Foldy-Wouthuysen representaton []. These papers offer, n partcular, the relatvstc nonlocal altonan n the for of a seres n ters of powers of charge e. Quantu electrodynacs n the Foldy-Wouthuysen () representaton has been forulated usng altonan and soe quantu electrodynacs processes have been calculated wthn the lowest-order perturbaton theory. As a result, the concluson has been ade that the representaton descrbes soe quas-classc states n the quantu feld theores. Both partcles and antpartcles are avalable n these states. Partcles, as well as antpartcles, nteract wth each other. owever, there s no nteracton of real partcles wth antpartcles such nteracton s possble only n nteredate (vrtual) states. The representaton odfcaton s requred to take nto a account real partcle/antpartcle nteractons. In the papers [, ] such odfcaton has been ade usng the syetry dentcal to the sotropc spn syetry owng to nvarance of fnal physcal results under change of sngs n the ass ters of rac altonan and altonan. In the odfed Foldy-Wouthuysen representaton, real ferons and antferons can be n two states characterzed by the values of the thrd coponent of the sotropc spn S f ; real ferons and antferons nteractng wth each other ust have opposte sgns of S f. Quantu electrodynacs n the odfed representaton s nvarant under P, C, T transforatons. Volatons of the ntroduced Insttuto de Alta Investgacón. Unversdad de Tarapacá. Antofagasta Nº 5. Arca, Chle. E-al: htorres@uta.cl 48 Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8

3 . Torres-Slva: rac atrces n chral representaton and the connecton wth the electrc syetry of the sotropc spn lead to the correspondng volaton of CP nvarance. The standard odel n the odfed representaton was forulated n the papers [, 4]. It has been shown that forulaton of the theory n the odfed representaton doesn t requre that ggs bosons should oblgatory nteract wth ferons to preserve the SU () nvarance, whereas all the rest theoretcal and experental plcatons of the Standard odel obtaned n the rac representaton are preserved. In such a case, ggs bosons are responsble only for the gauge nvarance of the boson sector of the theory and nteract only wth gauge bosons W, Z, gluons and photons. In the papers entoned above, the energy representaton of rac atrces derved by rac hself s used: I I,, 5 I I, () ere we propose to change the Foldy-Wouthuysen transforaton for by usng the chral representaton of rac atrces. x p x p ; x ; x p x p x x ; p x p x x p p; p p ; Wth representaton (), relaton () looks lke R x p x p ; x ; L x p R x p R L x ; p L x p L x R x p p L x R x p p R x; p p R x L x p p L x; Relatons (4) use the operador equalty: () (4) c I I c c, I, 5 c c c I, () p E p The chral representaton () s coonly used n the odern gauge feld theores and n the Standard Model, n partcular. Frst consder the structure of equatons descrbng the coponents of the wave functons (x) for the two representaton of rac atrces consdered n the paper. In relatons (), () and below the syste of unts wth c s used; x, p, are 4-vectors; the nner product s taken as xy = x µ y µ = x y x k y k k µ =,,,, k =,,; p ; x are Paul atrces;, ; (x) s the, k, k,, four-coponent wave functon, x, x, x, x are the two-coponent wave functons. The followng operator relatons are vald for the free rac equaton wth representaton (): R L Coparson between relatons () and (4) shows that wth the substtuton below, p, 5 (5) These relatons transfor nto each other. The Foldy-Wouthuysen transforatons for the energy and chral representatons of rac atrces also transfor nto each other f the substtuton (5) s ade. Thus, the general Foldy-Wouthuysen transforaton wth rac atrces n the chral representaton U chr chr U..., as well chr chr chr as the feron altonan n the Foldy-Wouthuysen representaton chr chr chr chr 5 E qk q K q K... can be obtaned. Fro the correspondng expressons for en en U, wth rac atrces n the energy representaton (see [, ]) wth substtuton p, 5, we have Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8 49

4 Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8 the relatons p R x p R L x p L x p L x R x (6) ˆ ˆ A dva ˆ rota () n equaton () together wth the two dv equatons (9), transfor that syste () n to CIRAL APPROAC OF MAXWELL S EQUATIONS c ˆ ( ) ( ) c E E () Also the relatons (6) can be obtaned under the chral approach of Maxwell s Equatons where the electrc feld E s parallel to the agnetc feld [5], that s E, where /. If one wanted to descrbe the hydrogen gas by eans of electrodynacs one should start fro the frly establshed experence that the hydrogen gas ay absorb and reet electroagnetc energy, and that wthout external nterventon there s no ndcaton that the gas to contan electrc charges [6-]. Thus we consder the hypothess wtch vsualzes the gas as charge free electroagnetc feld as the startng pont wth the lest nuber assuptons; and so we try characterze the feld by the covarant chral Maxwell syste [5] rote T c t dv T e E ( ), ( ) (7) rot T c t E e dv T ( ), ( ) (8) ere, T s the chral scalar factor wth dvte, E( ) Te, dv E( ), and the condton of charge-free by eans of dve besdes dv (9) Solvng the wave equaton for E wth T T T / c, and by consderng we have e rote T E c ( ) ( ) E( ) () where E ( ) ˆ E( ˆ ). Below the syste of unts wth c equaton () s exactly equal to the chral rac equaton (6), f ( ) ( ). E R L To probe ths close connecton we can obtan the well known noral rac equaton, we get for (7, 8) the equatons rote T c t E ( ), dv () rot Te c t E dv ( ), (4) wth E grad, grad. Equatons () and (4) can be transfored as: ( T ) (, c ), ( T ) (, c ). So, scalar ultplcaton of the rot equatons n (, ) by the Paul-vector, and usng the algebrac relaton [] ˆ ˆ A dva ˆ rota we have E c t E c t E grad, grad (5) Equaton (5) can be expressed n ters of n atrz notaton ths reads Our hypothess s that through the ultplcaton of the atrx Paul for chral Maxwell s equatons wth E, one obtans the chral rac equaton (6). Usng the algebrac relaton [] E c t E grad, grad (6) 5 Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8

5 . Torres-Slva: rac atrces n chral representaton and the connecton wth the electrc enotng the quantty on wtch the dfferental operators act by, that s E E E wth X X X connecton ' E ' ' ' 4 4 (7) and consderng the well-known (8) between the Paul and rac atrces, we get for () syste c t E grad, grad (9) ere one has to bear n nd that each of both coluns atrx (4) that s E E E and E E E () Independently represent a syste of functons solvng (6). Fro ths, a separaton of the te dependence accordng to t fnally yelds the apltude equaton e () c () If we use equaton () n (), t s agreeent wth the rac apltude equaton s coplete. Now noralzng eq. () wth, c, p, we can wrte as p x p ; x ( ) And the transforaton to a chral rac equaton s trval by usng relaton (5). The equatons () as well as () or (), show n addton that the electrodnacal and the wave echancal feld coponent are connected by sple lnear relaton, the sae holdng true for the refracton (, µ) n relaton to the scalar T. Ths soorphs can be checked easly and drectly because the eght Eq. (, ) ay be cobned nto two systes of four equatons each, n the followng way: rot c E dv rot c E rot c E (4) dve rote c rote c rote c Insertng here the frst or second wave functon of () nto the frst syste (upper sgns) or the second one (lower sgns), respectvely, the wave functons of () ends up edately, n both cases and we are back to rac agan CONCLUSION Usng a chral representaton of the Foldy-Wouthuysen transforaton for the rac equaton we show that the sae result can be obtaned wth a chral electrodynacs usng the atrx Paul. Wth ths we proof the theore that waves echanc of quantu partcle represents a specalzed electrodynacs. The result sees unabguous and ncopatble wth the current doctrne whch rest on a partcle nterpretaton. REFERENCES [] V.P. Neznaov. Physcs of Eleentary Partcles and Atoc Nucle (EPAN). Vol. 7 Nº. 6. c c c () [] V.P. Neznaov. Voprosy Atono Nauk I Tekhnk. Ser: Teoretcheskaya I Prkladnaya Fzka. Issues -, p. 4, hep-th/ Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8 5

6 Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8 [] L.L. Foldy and S. A. Wouthuysen, Phys. Tev 78, [4] V.P. Neznaov. ep-th/ [5]. Torres-Slva and M. Zaorano Lucero. Chral Electrodynac. URLs: [6] J.R. Oppenheer. Phys. Rev. Vol. 8, p [7].E. Moses. Sup. Nuovo Cento. Sere X. Vol. 7. Nº [8] T. Ohura Prog. Theor. Phys. Vol. 6, p [9] S.N. Gupta. Theory of longtudnal photons n quantu electrodynacs. Proc. Phys. Soc. Vol. 6, pp [] F. Renes and W.. Sobel, Test of the Paul Excluson Prncple for Atoc Electrons, Phys. Rev. Lett. Vol., pp [] W. etler Quantu Theory of Radaton, nd Ed., Oxford Unversty Press, Oxford, p []. Sallhofer. Maxwell rac soorphs. Z. Naturforsch. Vol. 4 a, p Ingenare. Revsta chlena de ngenería, vol. 6 núero especal, 8