IL NUOVO CIMENTO VOL. 112 B, N. 6

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1 IL NUOVO CIMENTO VOL. 112 B, N. 6 Giugno 1997 Comment on Describing Weyl neutrinos by a set of Maxwell-like equations by S. Bruce (*) V. V. DVOEGLAZOV (**) Escuela de Física, Universidad Autónoma de Zacatecas Antonio Dovalí Jaime s/n, Zacatecas 9868, ZAC, México (ricevuto il 2 Settembre 1996; arovato il 1 Dicembre 1996) Summary. Results of the work of S. Bruce (Nuovo Cimento B, 11 (1995) 115) are comared with those of the recent aers of D. V. Ahluwalia and myself, devoted to describing neutral articles of sin j41/2 and j41. PACS.5 Classical field theory. PACS.65.Pm Relativistic wave equations. PACS 11..C Lorentz and Poincaré invariance. PACS 11..Er Charge conjugation, arity, time reversal, and other discrete symmetries. The main result of ref. [1] is the roof of the ossibility of deriving the generalized Maxwell s equations (eqs. (24) of the cited aer) from a set of the j41/2 equations for Weyl sinors. Another imortant oint discussed there is the connection between arity-even and arity-odd arts of the CP conjugate states. The consideration is restricted by the massless case. On the other hand, in recent aers of Ahluwalia and Dvoeglazov [2] on the basis of ideas of Majorana [] and McLennan and Case [4] the construct for self/anti-self charge conjugate states has been resented. It ermits one to take into account ossible effects of neutrino mass and to exlain origins of the question of missing right-handed neutrino [5]. Tye-II self/anti-self charge conjugate sinors are defined in the momentum reresentation from the beginning, in the following way (ref. [2a], eq. (6)): (1) l( m ) fu (z l U [ j ] ) f* L ( m ) f f L v, r( m ) fu R ( ( m ) m ) (z r U [ j] )* f* R ( m ) v. (*) The author of this aer has agreed to not receive the roofs for correction. (**) valerihcantera.reduaz.mx 847

2 848 V. V. DVOEGLAZOV z l and z r are the hase factors that are fixed by the conditions of self/anti-self conjugacy, U [ j ] is the Wigner time-reversal oerator for sin j. They satisfy the equations ( 1 ) (2a) (2b) (2c) ig m m l S (x)2mr A (x) 4, ig m m r A (x)2ml S (x) 4, ig m m l A (x)1mr S (x) 4, (2d) ig m m r S (x)1ml A (x) 4. It was shown there (cf. ref. [2b], eqs. (22) or [2c], eqs. (67)-(7)) that tye-ii sinors are connected with the Dirac sinors in the Weyl reresentation u s ( m ) and v s ( m ) as follows: (a) l S H( m ) (u 11/2( m )1iu /2 ( m )2v 11/2 ( m )1iv /2 ( m )), (b) l S I( m ) (2iu 11/2( m )1u /2 ( m )2iv 11/2 ( m )2v /2 ( m )), (c) l A H ( m ) (u 11/2( m )2iu /2 ( m )2v 11/2 ( m )2iv /2 ( m )), (d) l A I ( m ) (iu 11/2( m )1u /2 ( m )1iv 11/2 ( m )2v /2 ( m )). and (ref. [2a], eqs. (48)) (4a) r S H( m ) 42il A I (m ), r A H (m )41il S I( m ), (4b) r S I( m ) 41il A H (m ), r A I (m )42il S H( m ). We assumed that f 11/2 L, R ( im )4column (1 ), f /2 L, R ( im )4column ( 1); in the oosite case we have to include additional hase factors in the mass terms of eqs. (2a)-(2d). They can be fixed if the theory is imlied invariant with resect to the intrinsic arity. Using identities (4a), (4b) and rewriting eqs. (2a)-(2d) in the momentum reresentation with taking into account the chiral helicity quantum number [2] we are able to obtain the following equations in the two-comonent form (hase factors are restored): (5a) (5b) (5c) (5d) [ 1sQ] f H L ( m )2me 1ix Uf I L *( m )4, [ 2sQ] Uf H L *( m )1me 2ix f I L ( m ) 4, [ 1sQ] f I L ( m )1me 1ix Uf H L *( m )4, [ 2sQ] Uf I L *( m )2me 2ix f H L ( m ) 4, ( 1 ) As we got knowing recently, this set of equations has been roosed in ref. [6] for the first time.

3 COMMENT ON DESCRIBING WEYL NEUTRINOS BY A SET OF MAXWELL-LIKE EQUATIONS ETC. 849 which answer for the McLenann-Case-Ahluwalia construct. A remarkable feature of this set is that it is valid both for ositive- and negative-energy solutions of eqs. (2a)-(2d). The corresonding equations for f R ( m ) and Uf R *( m ) sinors follow after substitution K2 in the matrix structures of the equations (not in the sinors!). The hase factors x R, L fw 1 R, L 1w 2 R, L are defined by exlicit forms of the 2-sinors of different helicities ( 2 ) and can be regarded at this moment as arbitrary. Considering roerties of 4-sinors with resect to the K2 after Bruce we state ( ) (6) j L even4f L I 1Uf L H *fu B 1iB B 1 1iB 2v, j L odd4f I L 2Uf LH * fu 1v 2E 1iE. 2E 2 1iE The oosite helicity arts are connected by the Wigner time-reversal oerator: (7).` /` f L H 1Uf L I *fu Kj L odd4u E 2 1iE 1 2E 2iE v, f L H 2Uf L I *f2 U Kj L even4u B 1 2iB 2 2B 1iB v. Adding and subtracting relevant equations of the set (5a)-(5d) we obtain (8a) u vu E 2 1iE 1 2E 2iE v1u 1 1i 2 1 2i 2 2 vu B 1 2iB 2 2B 1iB v2 2 m cos xu E 2 1iE 1 2E 2iE v1im sin xu B 1 2iB 2 2B 1iB v 4, (8b) u vu B 1 2iB 2 2B 1iB v1u 1 1i 2 1 2i 2 2 vu E 2 1iE 1 2E 2iE v1 1m cos xu B 1 2iB 2 2B 1iB v2im sin xu E 2 1iE 1 2E 2iE v 4, (8c) u vu B 1iB B 1 1iB 2v1u 1 1i 2 1 2i vu 2 2E 1iE 2 2E 2 1iE 1v1 1m cos xu B 1iB B 1 1iB 2v2im sin xu 2E 1iE 2E 2 1iE 1v 4, ( 2 ) Of course, different choices of x R, L will influence eqs. (4a), (4b). ( ) We refer to use the conventional notation MKE, the olar vector, and NKB, the axial vector. We still leave room for different interretations of these vectors in relevant hysical cases.

4 85 V. V. DVOEGLAZOV (8d) u 2E vu 1v1u 1iE 2E 2 1iE 1 1i 2 1 2i 2 2 vu B 1iB B 1 1iB 2v2 2m cos xu 2E 1iE 2E 2 1iE 1v1im sin xu 2v B 1iB 4. B 1 1iB They are recast into the vector form (9a) (9b) (9c) (9d) E2 B1E 2mB cos x2me sin x4, B1 E1B 2mE cos x1mb sin x4, E 2 (QB)2mE cos x1mb sin x4, B 1 (QE)1mB cos x1me sin x4. For arity conservation of these vector equations we should assume that E would be a seudoscalar and B would be a scalar, furthermore, x4 or. In a matrix form with the Majorana-Oenheimer matrices (1a) a 41 44, 4u a 1 i 2i v, (1b) 4u a 2 2i 4u i a v, i 2i v, we can obtain the equations (a fa, a i f2 a i ) (11a) (11b) a m m C 2 ( m )2me 2ix C 1 ( m ) 4, a m m C 1 ( m )2me 1ix C 2 ( m ) 4, satisfied by the field functions (12a) C 1 ( m )42 CC 2 *( )4u2i(E 2iB ) E 1 2iB 1 m E 2 2iB 2 E 2iB v, C2( m )42 CC 1 *( )4u2i(E 1iB ) E 1 1iB 1 m v, E 2 1iB 2 E 1iB

5 COMMENT ON DESCRIBING WEYL NEUTRINOS BY A SET OF MAXWELL-LIKE EQUATIONS ETC. 851 where (1) C4 C 4u1 Ca v, m C 4 a m * in accordance with the definition of Dowker [7] in the orthogonal basis. Some remarks have already been done that these equations can be written in the same form for both ositive- and negative-frequency solutions. If we choose C 2 ( m ) as resenting these arts of the field oerator and x4, then we have in the coordinate reresentation: (14a) (14b) i[ u, v a m m C(x m )2m C c (x m ) 4, i[ u, v a m m C c (x m )2mC(x m ) 4, with [ u, v 461 deending on what solutions, of ositive or negative frequency, are considered (cf. [8]). Next, we would like to mention that similar formulations (but without a mass term) were met in literature [9-12]. Probably, alying them was caused by some shortcomings of eq. (1) of the aer [1], which the author of the commented work aid attention to. To his critical remarks we can add that it contains the acausal solution with E4, ref. [9, 1, 1]. Acausal solutions of similar nature aear for any sin (not only for sin-1 equations), ref. [1]. While Oenheimer roosed a hysical interretation of this solution as connected with electrostatic solution and recently another solution, the B( ) longitudinal field, to the Maxwell s equations was extensively discussed [14], the roblem did not yet find an adequate consideration. In the mean time, the equations for sin-1 massless bosons, resented by Majorana, Oenheimer, Giantetto, and the ones of this aer for massive sin-1 case, are free of any acausalities; they are of the first order in time derivatives and reresent the Lorentz-invariant theory ( 4 ). As a rice we have additional dislacement current and a ossible mass term. Other equations which can be considered as suitable candidates for describing sin-1 bosons are the second-order Weinberg equations [15, 1, 16]; they have only causal solutions E 46 in the massless limit. Moreover, their massless limit also can be reduced [16] to eqs. (24) of the commented aer. Finally, I would like to note that resented ideas deserve further rigorous elaboration, since we are still far from understanding the nature of electron, hoton and neutrino. Their secific features seem not to lie in some secific reresentation of the Poincaré grou but in the structure of our sace-time. Thus, the equations for the fields in the (1, )5 (, ) and the (, 1)5 (, ) reresentations, given above, could rovide additional information for our goals. ( 4 ) The question of the relativistic invariance of the equations (14a), (14b) is a tune oint. A searate aer will discuss the relativistic invariance of new equations. But one should note here that roviding new frameworks we are not going to disute results of the Dowker s consideration ((7a),. 18).

6 852 V. V. DVOEGLAZOV *** I areciate encouragements and discussions with Profs. D. V. AHLUWALIA, M. W. EVANS, I. G. KAPLAN and A. F. PASHKOV. Many internet communications from colleagues are acknowledged. I am grateful to Zacatecas University for a rofessorshi. This work has been artially suorted by the Mexican Sistema Nacional de Investigadores, the Programa de Aoyo a la Carrera Docente and by the CONACyT under the research roject 27P-E. R E F E R E N C E S [1] BRUCE S., Nuovo Cimento B, 11 (1995) 115. [2] AHLUWALIA D. V., Int. J. Mod. Phys. A, 11 (1996) 1855; DVOEGLAZOV V. V., Rev. Mex. Fis. Sul., 41 (1995) 159; Int. J. Theor. Phys., 4 (1995) 2467; Nuovo Cimento A, 18 (1995) [] MAJORANA E., Nuovo Cimento, 14 (197) 171 (English translation: Tech. Trans., TT-542, Nat. Res. Council of Canada). [4] MCLENNAN J. A., Phys. Rev., 16 (1957) 8; CASE K. M., Phys. Rev., 17 (1957) 7. [5] BARUT A. O. and ZIINO G., Mod. Phys. Lett. A, 8 (199) 111; ZIINO G., Int. J. Mod. Phys. A, 11 (1996) 281. [6] MARKOV M. A., Ž. Ėks. Teor. Fiz., 7 (197) 6 (see also ref. there: GEHENIAN, Comt. Rend., 198 (194) No. 8). [7] DOWKER J. S. and DOWKER Y. P., Proc. R. Soc. London, Ser. A, 294 (1966) 175; DOWKER J. S., Proc. R. Soc. London, Ser. A, 297 (1967) 51. [8] AHLUWALIA D. V., JOHNSON M. B. and T. GOLDMAN, Phys. Lett. B, 16 (199) 12. This aer resents an exlicit examle of the quantum field theory of the Bargmann-Wightman- Wigner-tye (WIGNER E. P., in Grou Theoretical Concets and Methods in Elementary Particle Physics, Lectures of the Istanbul Summer School of Theoretical Physics, 1962, edited by F. GÜRSEY (Gordon & Breach) 1965,. 7). [9] MAJORANA E., Scientific Manuscrits ( ), edited by R. MIGNANI, E. RECAMI and M. BALDO, Lett. Nuovo Cimento, 11 (1974) 568; see also GIANETTO E., Lett. Nuovo Cimento, 44 (1985) 14. [1] OPPENHEIMER J. R., Phys. Rev., 8 (191) 725. [11] IMAEDA K., Prog. Theor. Phys., 5 (195) 1. [12] OHMURA T. (KIKUTA), Prog. Theor. Phys., 16 (1956) 684, 685. [1] AHLUWALIA D. V. and ERNST D. J., Mod. Phys. Lett. A, 7 (1992) [14] EVANS M. W. and VIGIER J.-P., Enigmatic Photon, Vol. 1 and 2 (Kluwer Academic Publ., Dordrecht) ; see also DVOEGLAZOV V. V., TYUKHTYAEV YU. N. and KHUDYAKOV S. V., Russ. Phys. J., 7 (1994) 898. [15] WEINBERG S., Phys. Rev. B, 1 (1964) 118. [16] DVOEGLAZOV V. V., Can the 2( 2j11) comonent Weinberg-Tucker-Hammer equations describe the electromagnetic field?, Prerint he-th/941174, Zacatecas, October 1994.