Massive photons and tachyon monopoles

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<ul><li><p>LV, TTERE AL NUOVO CIMENTO VOL. 20, N. 18 31 Dicembre 1977 </p><p>Massive Photons and Tachyon Monopoles. </p><p>G. DATTOLI and M. ~V~ATTIOLI </p><p>[stituto Nazionale di Fisiea Nucleate Sezione di Frascati, Italia Laboratori Nazionali - Fraseati, Italia </p><p>R. MIGNANI </p><p>Istituto di Fisiea dell' Universith - L'Aquila, Italia Islituto Nazionale di Fisica Nucleate - Sezione di Roma, Italia </p><p>(ricevuto il 14 Novembre 1977) </p><p>The possibility of nonzero rest-mass for photons has been the object of many theoretical and experimental investigations (1). The field equations describing a massive photon are, as is well known, Proca's equations, which can be written in the Maxwell- like form (*) </p><p>V" E -= ~--m2~, </p><p>V-H=O, </p><p>~H (1 ) V = - - - - </p><p>r~t ' </p><p>~E V -~ + j - - re=A, </p><p>with self-clear meaning of symbols. I t was conjectured, some time ago, by Recami and one of the present authors (R.M.) </p><p>that faster-than-light electric charges behave as magnetic monopoles (2). Indeed, they introduced superluminal Lorentz transformations for the elastic and </p><p>magnetic fields, which allow us to generalize Maxwell's equations to the case when both subluminal and superluminal electric charges are present (2). The remarkable feature </p><p>(1) See, e.g.: A. S. GOLDHABER and M. M. NIETO: Rev. Mod. Phys., 43, 277 (1971); A. ]3ARNES and J . D. SCA~GLE: Phys. Rev. Left., 35 (1975); L. DAvis jr., A. S. GOLDIt~BER and 1K. M. NIETO: Phy8 Rev. Lett., 35, 1117 (1975); A. S. GOLDtIABER and M. M. NIETO: Sci. Amcr., 234, 66 (1976). (*) We use the metr i c ( --, --, --). Summat ion is understood over the repeated indices. Natura l units (]~ = c = 1) and Gaussian rationalized units are used. (2) E. RECA~I and R. ]EiIGN~NI: Riv. Nuovo Cimento, 4, 209, 398 (1974); Left. Nuovo Cimento, 9, 479 (1974); Phys. Lett., 62 B, 41 (1976); R. MIG~ANI and E. RECA~I: Left. Nuovo Cimento, 9, 367 (1974); 11, 417 (1974); Nuovo Cimento, 30A, 533 (1975). </p><p>686 </p></li><li><p>MASSIVE P]:IOTONS AND TACItYON ~ONOPOLES 687 </p><p>of the generalized 3[axwell equations in that superluminal electric currents play the role of (spacelike) magnetic currents (2). </p><p>The basic tool, in order to extending Maxwell 's equations to tachyon charges, consists in the introduction of a (2)(*) e.m. tensor TP,': </p><p>where F~ is the usual e.m. tensor and the dual F~, is defined as </p><p>(3 ) =* 1 . ~e . I~l~ v ~ ~vt~vO.aJ-, 9 </p><p>In terms of T~ v, the Maxwell equations val id for both superluminal and subluminal charges read (2) </p><p>(4) ~ T~ = j . ( s ) - - i j~(S) , </p><p>where s = subhmina l and ,q - superluminal. Let us apply a similar procedure to Proca's equations. They can be generalized </p><p>as follows: </p><p>(5) ~, T~,, = [j~(s) - -m2Ag(s ) - - i( j~,(S) - - m'~A,(S) ) ] . </p><p>I f we confine ourselves to requir ing eovariance under the proper, orthochronous Lorentz group, eqs. (5) are equivalent to the fol lowing ones: </p><p>V. E= e(s)--m~(s), </p><p>V" H= - - e(S) - -m'2~(S) , </p><p>~H (6) V x E j (S ) + m'2 A(S) , </p><p>~ct </p><p>~E V "_&lt; H= c--- [ + j ( s ) - -m2A(s ) , </p><p>with clear meaning of the notation. As one easily realizes, eqs. (6) describe a massive eleetrodynamics with (spacelike) </p><p>magnetic charges--a very appealing (and expected) result. However, eqs. (6) suffer with some troubles, which are absent in the case of generalized Maxwell 's equations with zero-mass photon. The more serious one is that now two photons, one subluminM and the other superluminal, are put in the theory. Indeed, as it is easy to realize (either by the very (, dual i ty principle ))(2), or by applying to eqs. (6) the standard procedure to derive wave equations for E and H) , m' is the mass of the (faster than light) particle corresponding to the (superlumina.1) massive photon, and it is: m2=- -m '2 (as it must be). Obviously, the existence of two massive photons raises serious doubts about the possibil ity of electromagnetic interactions between subtuminal and superluminal charges. Therefore, we are led to conclude that there is a basic incompa- t ibi l i ty between the two theoretical ideas of tachyon monopoles, on one side, and massive photons, on the other side. </p><p>I t is only a matter of experiment to decide whether Nature made the one or the other choice--or, perhaps, none of them. </p><p>(*) In the sense of complex-manifold theory, i.e.: T ~ = iT *uv. See ref. (8). (3) E. J . FLAHERTY: Hermitian and Kdihlerian Geometry in Relativity (Berlin, 1976), p. 208. </p></li></ul>