Volume 1 PROGRESS IN PHYSICS January, 2009 A Note of Extended Proca Equations and Superconductivity Vic Christianto, Florentin Smarandachey, and Frank Lichtenbergz Sciprint.org — a Free Scientific Electronic Preprint Server, http://www.sciprint.org E-mail: [email protected] yChair of the Dept. of Mathematics, University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] zBleigaesschen 4, D-86150 Augsburg, Germany E-mail: [email protected] It has been known for quite long time that the electrodynamics of Maxwell equations can be extended and generalized further into Proca equations. The implications of in- troducing Proca equations include an alternative description of superconductivity, via extending London equations. In the light of another paper suggesting that Maxwell equations can be written using quaternion numbers, then we discuss a plausible exten- sion of Proca equation using biquaternion number. Further implications and experi- ments are recommended. 1 Introduction The background argument of Proca equations can be sum- marized as follows [6]. It was based on known definition of It has been known for quite long time that the electrody- derivatives [6, p. 3]: namics of Maxwell equations can be extended and general- 9 @ @ @ @ @ > ized further into Proca equations, to become electrodynam- @ = = ; ; ; = @0; r => @x @t @x @y @z ics with finite photon mass [11]. The implications of in- ; (3) @ > troducing Proca equations include description of supercon- @ = = @0; r ; ductivity, by extending London equations [20]. In the light @x of another paper suggesting that Maxwell equations can be @a0 @ a = + r~a; generalized using quaternion numbers [3, 7], then we discuss @t (4) a plausible extension of Proca equations using biquaternion @2 @2 @2 @2 number. It seems interesting to remark here that the proposed @ @ = = @2 r2 = @@ ; 2 2 2 2 0 (5) extension of Proca equations by including quaternion differ- @t @x @y @z 2 ential operator is merely the next logical step considering where r is Laplacian and @@ is d’Alembertian operator. already published suggestion concerning the use of quater- For a massive vector boson (spin-1) field, the Proca equation nion differential operator in electromagnetic scattering prob- can be written in the above notation [6, p. 7]: lems [7, 13, 14]. This is called Moisil-Theodoresco operator 2 @@ A @ (@A ) + m A = j : (6) (see also Appendix A). Interestingly, there is also a neat link between Maxwell 2 Maxwell equations and Proca equations equations and quaternion numbers, in particular via the Moisil-Theodoresco D operator [7, p. 570] and [13]: In a series of papers, Lehnert argued that the Maxwell pic- @ @ @ ture of electrodynamics shall be extended further to include a D = i1 + i2 + i3 : (7) @x1 @x2 @x3 more “realistic” model of the non-empty vacuum. In the pres- ence of electric space charges, he suggests a general form of There are also known links between Maxwell equations the Proca-type equation [11]: and Einstein-Mayer equations [8]. Therefore, it seems plau- 1 @ sible to extend further the Maxwell-Proca equations to bi- r2 A = J ; = 1; 2; 3; 4: quaternion form too; see also [9, 10] for links between Proca c2 @t2 0 (1) equation and Klein-Gordon equation. For further theoretical Here A = (A; i=c), where A and are the magnetic description on the links between biquaternion numbers, Max- vector potential and the electrostatic potential in three-space, well equations, and unified wave equation, see Appendix A. and: J = (j; ic) : (2) 3 Proca equations and superconductivity However, in Lehnert [11], the right-hand terms of equa- tions (1) and (2) are now given a new interpretation, where In this regards, it has been shown by Sternberg [20], that the is the nonzero electric charge density in the vacuum, and j classical London equations for superconductors can be writ- stands for an associated three-space current-density. ten in differential form notation and in relativistic form, where 40 V.Christianto, F. Smarandache, F. Lichtenberg. A note of Extended Proca Equations and Superconductivity January, 2009 PROGRESS IN PHYSICS Volume 1 they yield the Proca equations. In particular, the field itself Another way to generalize Proca equations is by using acts as its own charge carrier [20]. its standard expression. From d’Alembert wave equation we Similarly in this regards, in a recent paper Tajmar has get [6]: shown that superconductor equations can be rewritten in 1 @ 2 terms of Proca equations [21]. The basic idea of Tajmar ap- r A = 0J ; = 1; 2; 3; 4; c2 @t2 (12) pears similar to Lehnert’s extended Maxwell theory, i.e. to include finite photon mass in order to explain superconduc- where the solution is Liennard-Wiechert potential. Then the tivity phenomena. As Tajmar puts forth [21]: Proca equations are [6]: 2 “In quantum field theory, superconductivity is explain- 1 @ 2 mpc r + A = 0 ; = 1; 2; 3; 4; (13) ed by a massive photon, which acquired mass due to c2 @t2 ~ gauge symmetry breaking and the Higgs mechanism. The wavelength of the photon is interpreted as the Lon- where m is the photon mass, c is the speed of light, and ~ is don penetration depth. With a nonzero photon mass, the reduced Planck constant. Equation (13) and (12) imply the usual Maxwell equations transform into the so- that photon mass can be understood as charge density: 2 called Proca equations which will form the basis for 1 mpc J = : (14) our assessment in superconductors and are only valid 0 ~ for the superconducting electrons.” Therefore the “biquaternionic” extended Proca equations Therefore the basic Proca equations for superconductor (13) become: will be [21, p. 3]: 2 @B mpc r E = ; }} + A = 0 ; = 1; 2; 3; 4: (15) @t (8) ~ and 1 @E 1 The solution of equations (10) and (12) can be found us- r B = j + A: ing the same computational method as described in [2]. 0 c2 @t 2 (9) Similarly, the generalized structure of the wave equation The Meissner effect is obtained by taking curl of equation in electrodynamics — without neglecting the finite photon (9). For non-stationary superconductors, the same equation mass (Lehnert-Vigier) — can be written as follows (instead (9) above will yield second term, called London moment. of eq. 7.24 in [6]): Another effects are recognized from the finite Photon 2 mpc a a mass, i.e. the photon wavelength is then interpreted as the }} + A = RA ; = 1; 2; 3; 4: (16) ~ London penetration depth and leads to a photon mass about 1/1000 of the electron mass. This furthermore yields the It seems worth to remark here that the method as de- Meissner-Ochsenfeld effect (shielding of electromagnetic scribed in equation (15)-(16) or ref. [14] is not the only pos- fields entering the superconductor) [22]. sible way towards generalizing Maxwell equations. Other Nonetheless, the use of Proca equations have some known methods are available in literature, for instance by using topo- problems, i.e. it predicts that a charge density rotating at an- logical geometrical approach [16, 17]. gular velocity should produce huge magnetic fields, which is Nonetheless further experiments are recommended in or- not observed [22]. One solution of this problem is to recog- der to verify this proposition [25,26]. One particular implica- nize that the value of photon mass containing charge density tion resulted from the introduction of biquaternion differential is different from the one in free space. operator into the Proca equations, is that it may be related to the notion of “active time” introduced by Paine & Pensinger 4 Biquaternion extension of Proca equations sometime ago [15]; the only difference here is that now the time-evolution becomes nonlinear because of the use of 8- Using the method we introduced for Klein-Gordon equation dimensional differential operator. [2], then it is possible to generalize further Proca equations (1) using biquaternion differential equations, as follows: 5 Plausible new gravitomagnetic effects from extended Proca equations (}})A 0J = 0 ; = 1; 2; 3; 4; (10) where (see also Appendix A): While from Proca equations one can expect to observe gravi- tational London moment [4,24] or other peculiar gravitational @ @ @ @ } = rq + irq = i + e + e + e + shielding effect unable to predict from the framework of Gen- @t 1 @x 2 @y 3 @z eral Relativity [5, 18, 24], one can expect to derive new grav- @ @ @ @ itomagnetic effects from the proposed extended Proca equa- + i i + e + e + e : @T 1 @X 2 @Y 3 @Z (11) tions using the biquaternion number as described above. V.Christianto, F. Smarandache, F. Lichtenberg. A Note of Extended Proca Equations and Superconductivity 41 Volume 1 PROGRESS IN PHYSICS January, 2009 Furthermore, another recent paper [1] has shown that 7 Concluding remarks given the finite photon mass, it would imply that if m is due to a Higgs effect, then the Universe is effectively simi- In this paper we argue that it is possible to extend further lar to a Superconductor. This may support De Matos’s idea Proca equations for electrodynamics of superconductivity to of dark energy arising from superconductor, in particular via biquaternion form. It has been known for quite long time that Einstein-Proca description [1, 5, 18]. the electrodynamics of Maxwell equations can be extended It is perhaps worth to mention here that there are some and generalized further into Proca equations, to become elec- indirect observations [1] relying on the effect of Proca energy trodynamics with finite photon mass. The implications of in- (assumed) on the galactic plasma, which implies the limit: troducing Proca equations include description of supercon- ductivity, by extending London equations. Nonetheless, fur- 27 mA = 310 eV: (17) ther experiments are recommended in order to verify or refute this proposition.