The consequences of complex Lorentz force and violation of Lorenz gauge condition
aa r X i v : . [ phy s i c s . g e n - ph ] S e p epl draft Consequences of complex Lorentz force and viola-tion of Lorenz gauge condition
A. I. Arbab (a) a Institute for Condensed Matter Theory Department of Physics University of Illinois atUrbana-Champaign, 1110 West Green Street, IL 61801, USA, and b Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khar-toum 11115, Sudan
PACS – Classical electromagnetism, Maxwell’s equations
PACS – Phenomenological theories
PACS – Exact solutions
Abstract –The complex Lorentz force is introduced and extended to include magnetic scalar. Thisscalar is found to be associated with a prevailing magnetic field permeating the whole space. Italso introduce an extra force in Lorentz complex force. The magnetic scalar is associated with thevacuum energy. The Proca-Maxwell’s massive electrodynamics is derived from the extend current-density transformations. Proca-Maxwell’s theory is found to be invariant under the extended gaugetransformations (current-charge density). The Lorenz gauge condition is shown to express thephoton charge conservation. Any violation of Lorenz gauge (photon charge) or electronic chargeconservation would lead to spin zero scalar particles. This is manifested in superconductivity. Thetotal charge comprising the electron and photon is always conserved. Owing to superconductivity,the photon charge is related to electron charge by e p = q m p m e e . Photons inside superconductorsare shown to be massive. It is shown that Maxwell’s equations expressed in complex form are moreconvenient to study duality transformations. Introduction. –
Maxwell’s equations are the basic equations for studying all elec-tromagnetic interactions [1]. They are invariant under Lorentz and gauge transformations.They entail that the photon is massless. In de Broglie hypothesis any micro-particle is ac-companied with wave nature. The photon on the other hand exhibits particle nature. Hence,duality is the fundamental nature existing for all micro-particles. Accordingly, electrons andphotons have dual nature. (a)
Present Address. [email protected] p-1. I. Arbab Proca extended Maxwell’s theory to include massive photons [2]. The addition of themass term in the Maxwell’s Lagrangian would break the gauge invariance. The loss of thisinvariance would make the theory less predictive. However, Proca theory is Lorentz invari-ant but not gauge invariant. Gauge invariance is applied only to the electric and magneticfields but not to the current and charge densities that are main ingredient of Maxwell theory.Maxwell’s theory guarantees the conservation of charge. This is expressed by the continuityequation relating the spread of charge and current densities. Maxwell’s equations are some-times expressed in terms of vector and scalar potentials instead of electric and magneticfields. If we redefine these potentials, Maxwell’s equations are still unchanged. We saythat Maxwell’s equations are gauge invariant. The question that will arise whether thesepotentials are physical observables or mere mathematical constructs. It is demonstrated ex-perimentally by Bohm and Aharonov that these potentials are physical objects [3]. To solveMaxwell’s equations in terms of potentials the Lorenz gauge is generally adopted, thoughother gauges are allowed. Hence, Lorenz gauge must have some physical aspect. The unifica-tion of quantum mechanics and Maxwell’s equations resulted in quantum electrodynamics.Here the electron and photon must interact with each other. The gauge particle, i.e., thephoton, is responsible to mediate the interaction between electrons (or charged particles).It is called gauge boson. It has spin equals to 1. While the electron is charged, the photonis left uncharged (neutral). Thus, the photon is chargeless and massless in the standardformulation of quantum electrodynamics. In Proca-Maxwell’s theory the photon is mas-sive but no charge is associated to it. However, in Bardeen -Cooper- Schrieffer theory, thesuperconductivity is described as a microscopic effect caused by a condensation of Cooperpairs into a boson-like state [20]. In this theory two electrons bind together to form Cooperpairs with spin equals to zero. London related the supercurrent to photon field (the vectorpotential) [5]. In this sense, the gauge transformations would alter this supercurrent. Weshould then think to restore the original picture. To this end we should assume that gaugetransformations include current density as well as vector potential. Thus, the charge-currentdensities and scalar-vector potentials should be equally treated.In this paper we study the consequences of employing the extended complex Maxwellequations. We introduced in section 2 the generalized Maxwell’s equations. We proposed amagnetic scalar field that can give rise to longitudinal waves besides the ordinary electromag-netic transverse waves. This scalar is found to be connected with the charge conservation.p-2onsequences of complex Lorentz force and violation of Lorenz gauge conditionIn section 3 we introduced an extended Lorentz force employing quaternions. In thissection we obtained an extended Lorentz force embodying the new magnetic scalar. Withthis representation we obtain the complex Lorentz force relating the force on magnetic andelectric charges placed in an electromagnetic field. Besides, we obtained the power associatedwith these charges. We further show that the extended Lorentz force is duality invariant.In section 4 we showed that when no electromagnetic field is present the magnetic scalarinduces a force on the charges present. We associate the energy with this state to that of thevacuum. This energy is reflected in the amount of vacuum energy found today as prescribedby the cosmological constant riddle.In section 5 we study a complex magnetic scalar and its effect on duality transformationof Maxwell’s equation and Lorentz force. In section 6 we relate the Lorenz gauge conditionto conservation of photon charge. We showed here that the violation of Lorenz gauge orcharge conservation leads immediately to massive photons. We compare the formulation ofProca with that of the the extended complex Maxwell’s formulation. We employ here theLondon’s ansatz for the superconductivity and relate the photon mass with photon charge.This relation agrees with the recent proposition by Chu that gravity (mass) resulted fromthe non-neutrality of photons [6]. The charge and mass of the photon are exceedingly small,but non-zero.
The generalized Maxwell’s equations. –
We have shown recently that one cangeneralize Maxwell’s equations to include scalar (longitudinal) wave [7]. This is achieved byexpressing Maxwell’s equations in quaternionic form employing an electromagnetic vector ~F = ~Ec + i ~B [7]. This yields e ∇ ∗ e F = µ e J , (1)where e F = (Λ , i ~F ) , e J = ( iρ c , ~J ) , e ∇ ∗ = (cid:18) ic ∂∂t , − ~ ∇ (cid:19) , (2)and the scalar Λ defines some scalar ’magnetic’ function representing the fourth component ofthe electromagnetic 4-vector. Using the quaternionic multiplication rule for two quaternions,˜ A = ( a , ~a ) and ˜ B = ( b ,~b ), one finds, ˜ A ˜ B = (cid:16) a b − ~a · ~b , a ~b + ~a b + ~a × ~b (cid:17) . Hence, theexpansion of eq.(1), using eq.(2), yields1 c ∂ Λ ∂t + ~ ∇ · ~F = µ cρ , (3)p-3. I. Arbab and − ~ ∇ Λ − c ∂ ~F∂t − i~ ∇ × ~F = µ ~J , (4)where µ and ε are the permeability and permitivity of the free space, respectively, and c − = ε µ . Equations (3) and (4) generalize Maxwell’s equations to incorporate scalarwave, Λ. Taking the divergence of eq.(4), differentiating eq.(3) partially with respect totime, and adding the two resulting equations, one finds1 c ∂ Λ ∂t − ∇ Λ = µ (cid:18) ~ ∇ · ~J + ∂ρ∂t (cid:19) . (5)It is interesting to see that Λ doesn’t influence the electromagnetic wave nature. The extended Lorentz force. –
The extended Lorentz complex force can be writtenas ˜ f = q ˜ V ˜ F ∗ , ˜ V = ( ic , ~v ) . (6)Using the quaternion multiplication rule, eq.(2) and eq.(6) yield˜ f = (cid:16) iq ( c Λ + ~v · ~F ) , q Λ ~v + qc ~F − iq~v × ~F (cid:17) . (7)The vector part of the quaternion in eq.(7) is the extended Lorentz complex force. Hence, ~f Λ = q Λ ~v + qc ~F − iq~v × ~F . (8)This force reduces to the ordinary complex Lorentz force for magnetic and electric charges ~f L = qc ~F − iq~v × ~F . (9)It is important to point that we have derived the complex Lorentz force in eq.(9) fromMaxwell’s equations [7]. Now, eq.(8) generalizes eq.(9) to include magnetic scalar, Λ. Thisscalar affects both the field and force equations. It is interesting to see whether eq.(8) can beobtained from eqs.(3) and (4) following the same steps that led to eq.(9) [7]. The generalizedLorentz force is associated with the force on electric and magnetic charges. It is generallyassociated with the symmetrised Maxwell’s equations. But in the present formulation wedidn’t presume a priori the presence of magnetic charges. It seems that the magnetic chargeis associated naturally with the electric charge. This is evident if we express the total chargeof a particle as, q = q e − i q m µ c . Equation (8) is thus the extended complex Lorentz force thatis associated with the complex Maxwell’s equations above.p-4onsequences of complex Lorentz force and violation of Lorenz gauge conditionThe scalar part of eq.(7) represents the power dissipated by moving charges in the electricand magnetic fields, i.e. , P = iqc Λ + iqc~v · ~F = − qc~v · ~B + iq ( c Λ + ~v · ~E ) . (10)Equations (8) and (10) can be written as ~f Λ = q (Λ ~v + ~E + ~v × ~B ) + iqc ( ~B − ~vc × ~E ) . (11)and P Λ = − qc~v · ~B + i ( qc Λ + q~v · ~E ) . (12)Equation (9) can be written as ~f L = q ( ~E + ~v × ~B ) + iqc ( ~B − ~vc × ~E ) . (13)We have recently derived from Maxwell’s equations the equation [7] q d ~Fdt = ~ ∇ × ~f D , (14)where ~f D = − q~v × ~F − iqc ~F (15)Under duality transformations, ~F → − i ~F , so that eq.(15) transforms to eq.(9). Thus, ~f D is the dual of ~f L . Furthermore, the invariance of eq.(8) under the duality transformationsdictates that a real Λ must vanish. Now the invariance of the complex Lorentz force ( ~f L = m~a ), eq.(9), under the duality transformation ( ~F → − i ~F ) requires that eithercase (a): m → − i m and q → q , orcase (b): q → i q and m → m . Case (a) suggests that one can write the mass as a complex quantity, consisting of the massof an electric charge ( m e ) and the mass of a magnetic charge ( m m ). That means, m = m e + i m m , so that under duality transformation, m e → m m and m m → − m e . Case (b) suggests also, q = q e − i q m µ c , so that under duality transformation, µ cq e → − q m and q m → µ cq e . If wep-5. I. Arbab consider case (a), then Lorentz force is invariant under duality transformation. Similarly,Lorentz force is invariant under the duality transformation suggested in case (b). Case (a)conforms with Maxwell’s field equations where, ρ → − iρ , i.e. , µ cρ e → ρ m and ρ m →− µ cρ e . However, case (b) needs some attention. It will be effective for the quantumelectrodynamics theory where the particle and the fields are involved. It is interestingto remark that we have dealt with case (b) where we propose the invariance of a theoryunder the complex space-time transformation of massive particles [10], generalizing t’Hooft-Nobenhuis transformation [11].Now eqs.(11) and (12) can be expressed as ~f Λ = q e (Λ ~v + ~E + ~v × ~B ) + iq m ( ~H − ~v × ~D ) . (16)and P Λ = − q m ~v · ~H + iq e ( c Λ + ~v · ~E ) , (17)where q = q e and q m = q e cµ are the electric and magnetic charges, ~B = µ ~H , ~D = ε ~E .It is pertinent to mention that the first derivation of the Lorentz force was due to OliverHeaviside in 1889, but Lorentz derived it afterwards. However, the Lorentz force in eq.(13)describes the force not only on electric charge but on magnetic charge too. This is so despitethe fact that we didn’t employ the symmetrised Maxwell equation in our exposition. Theelectric charge gains energy from the electric field, i.e. , q e ~v · ~E , abd the magnetic scalarpresent in space. Moreover, we observe that Λ influences the force on the particle and thepower associated with it. The vacuum contribution. –
Note that in the absence of electric and magneticfields and their sources, there exists a non-zero force acting on the charge, viz. , ~f = q e Λ ~v ,where Λ is now constant, and the power delivered is, P = q e c Λ. Thus, the total forceacting on a neutral system is zero. Note that when Λ is constant, the extended Maxwell’sequations reduce to the ordinary Maxwell’s equations. The Lorentz force in this case lookslike a frictional (viscous drag) force that a particle will experience when moves in space.However, no force can act on the magnetic charge when no fields are present. This forcecan be attributed to the vacuum (ether) that acts like a fluid (magnetic) in which thecharge moves. Therefore, the speed of a negative charged particle (an electron), decreasesexponentially with time as v ∝ e qe Λ m t , and the distance traveled is x = mq e Λ v + x , where x is ρ = ρ e + i ρ m µ c . p-6onsequences of complex Lorentz force and violation of Lorenz gauge conditionsome constant. Since this force is proportional to Λ, which has a dimension of magnetic field,we can ascribe this force to a relic (background) magnetic field prevailing the whole spacethat is unidirectional. A moving charged particle will be separated away from its antiparticleby the ether. This may help account for the non-annihilation of all matter in during the earlycosmic evolution. To have an idea of this magnetic field one can define a characteristic time, τ Λ = mq e Λ , that reflects the properties of this field. If we equate for instance, τ Λ and m toPlanck’s time (10 − s) and mass (10 − kg), respectively, then Λ P amounts to, Λ P ∼ T .Similarly, for the present epoch (10 s), one has Λ ∼ − T . These two values are obtainedconsidering that the universe is a quantum system [12, 13]. This prevailing magnetic fieldmay arise from the charge the photon carries. Hence, the magnetic scalar Λ reflects the levelto which the electric charge is violated. This scalar field may also be associated with themagnetic field carried by the photons in the cosmic background radiation. This is possibleif we associate a charge with the photon. The smallness of Λ today signifies that even if thephoton does have a charge, its value is exceedingly small. At any rate, we should expect that f to be very small. When the universe was created f P ∼ N . This coincides with themaximal force existing in nature, i.e., c G , where G is the gravitational constant [12,13]. Themaximum present acceleration ( a = q e Λ cm ) that can act by the ether on a charged Planck’smass amounts to 10 − ms − . This coincides with the acceleration exhibited by Pioneer10/11 acceleration anomaly [14]. We therefore think of the space as a fluid endowed withintrinsic magnetic properties. The maximum energy embedded in the ether is E = q e Λ c ∆ t .Thus, the energy absorbed by the ether during the entire cosmic expansion is equal to theone absorbed during Planck’s time (10 J). The mass creation rate is defined by ˙ m = e Λ.Hence, the mass created since Planck’s time is M = e Λ ∆ t ∼ kg, where ∆ t ∼ sec,and Λ ∼ T. The corresponding power also coincides with the maximal power, i.e. , c G [13]. In this case one can write Λ = c q e G . This gives the magnetic field produced by acharged particle moving in gravitational field. Thus, a magnetic field can originate fromgravity. The energy associated with Λ will act as a vacuum energy in the universe. It seemsthat the magnetic field induced at the time of the universe creation (big bang) is still notyet fully extracted from the space. The prevalence of magnetic scalar in our universe mimicsthe phenomenon of hysteresis in magnetic material. This ushers that space has magneticnature retaining the time with it.Let us now calculate the magnetic flux produced in a system of area A . If this flux isp-7. I. Arbab quantized, one can write, A = τ Λ ¯ hm . This shows that the area containing the magnetic fluxis quantized. For instance for the whole universe today, A = 10 m , where ¯ h ∼ J s, m ∼ kg, and at Planck’s time, A P = 10 − m [12, 15]. These are of the same orderof magnitudes for these values. Furthermore, one finds the rate of decay of the magneticflux is constant. This means the induced electromotive force during the time it is developedis constant throughout cosmic expansion. The vacuum energy density associated with thismagnetic scalar is ρ Λ = Λ µ . This yields a value of 10 − Jm − for the present epoch. Hence,one can associate the cosmological constant (Λ v ) with this magnetic scalar [16]. This can bewritten as Λ v = Gc k Λ , where k is the Coulomb’s constant. It is interesting to see that theratio, Λ v Λ = Gc k remains constant. Thus, the equality between the two contributions, onedue to Λ v and the other due to Λ, which is negative, can lead to a zero total contributionsolving the cosmological constant dilemma. Furthermore, the above ratio gives the inducedmagnetic capacitance per mass. If we consider our universe to be spherical of radius R ,then this capacitance, C = 4 πε R . Hence, R = Gmc . This is of the same order as theSchwarzschild radius of a gravitating object.Because today monopoles (magnetic charges) are not abundantly observed, duality symme-try is a broken symmetry. Particle physicists can associate an energy scale at which dualitysymmetry is broken. Notice that a real magnetic scalar breaks the duality invariance, but acomplex one preserves the symmetry. Thus, the occurrence of Λ implies a duality breaking.The effective temperature associated with a uniformly accelerating observer ( a ) in vac-uum is given by [17] a = 2 πck B T ¯ h , where k B is the Boltzman constant. If we assume the origin of this acceleration to the onementioned above, then Λ = 2 πmk B Te ¯ h , where we set q e = e , representing the remnant (vacuum) magnetic field. The correspondingenergy density associated with this magnetic field is thus u Λ = (cid:18) π k B m µ e ¯ h (cid:19) T . This may provide the vacuum energy density associated with cosmological scale. Note thatthe energy density of an ordinary boson gas (photon) is related to the temperature by u ∝ T . It is interesting to note that Gasperini related the cosmological constant to T ,p-8onsequences of complex Lorentz force and violation of Lorenz gauge conditionthus the cosmological constant can be interpreted as a parameter measuring the intrinsictemperature of the empty space [18]. Thus, because of Λ space and time are tightenedtogether.The magnetic scalar can be written asΛ = 4 π λ (cid:18) ¯ he (cid:19) , λ = h √ πmk B T . where λ is the wavelength (thermal) of the system at which quantum effects are effective,and ¯ h/e is the quantum flux. Hence, Λ can be related to the magnetic field of the emittedblack body radiation. The mass m can be related to the mass of the photon.The force associated with this vacuum is thus F = 2 πmck B T ¯ h . If we equate this force to the maximal force in nature ( c / G ), we obtain the maximaltemperature T = c ¯ h πGmk B . This is exactly the Hawking temperature of a black hole radiating like a black body radiation[19]. Note that the vacuum in the uniformaly accelerated frame is supposed to be in thermalequilibrium with its own radiation. Therefore, the above Hawking relation is a manifestationof quantum gravity and thermodynamics. It is not necessarily connected with black holes.The maximal force occurs when the gravitational energy is converted into relativistic massenergy. In this case pair-production is created. While the mass m in our above relationis the mass of the newly created particles from the conversion of gravitational energy intomass energy, the mass in Hawking is the mass of the evaporated black hole. Thus, if thePlanck mass, 10 − kg, was converted into radiation, its temperature would be 10 K. Andif the whole present universe mass, 10 kg, is converted to radiation, its temperature willbe 10 − K. Therefore, the former temperature represents the temperature of the vacuumfilling the primeval universe (big bang), and the latter one is the temperature of the presentvacuum. It is shown that in a strong magnetic field the vacuum behaves as a superconductor,and similarly at very low magnetic field [20]. The magnetic field was so enormous at thetime of big bang, and exceedingly small at the present time.p-9. I. Arbab The complex magnetic scalar. –
If we now express Λ as a complex function, viz. ,Λ = Λ m + ic Λ e , [7], then eqs.(11) and (12) read ~f Λ = q e (cid:16) Λ m ~v + ~E + ~v × ~B (cid:17) + iq m (cid:16) ε Λ e ~v + ~H − ~v × ~D (cid:17) . (18)and P Λ = − q m (cid:18) Λ e µ + ~v · ~H (cid:19) + iq e (cid:16) c Λ m + ~v · ~E (cid:17) . (19)Hence, a complex Λ scalar affects the electric and magnetic charges. It is worth to mentionthat the magnetic charge is analogous to electric charge, but is accelerated by a magneticfield, and its path is bent by an electric field. With such behavior, magnetic charges can bedetected.Let us now consider the duality transformations, i.e. , ~E ( ~cB ) → c ~B ( − ~E ) , c Λ m (Λ e ) → Λ e ( − c Λ m ) , cµ q e ( q m ) → q m ( − cµ q e ) , (20)so that ~f Λ → ~f Λ , but P Λ → − P Λ . This resembles the transformation of force and powerunder time reversal. Does this mean the duality transformation is equivalent to time reversalelsewhere?Now for Maxwell’s equations, eq.(5), to be invariant under duality, ~J ( ρ ) → − i ~J ( − iρ ),and Λ → − i Λ . This implicitly implies that ~J and ρ are complex quantities too. This is in-deed the case if we define ~J = ~J e + i ~J m cµ and ρ = ρ e + i ρ m cµ . In this case, the ordinary, eqs.(5),and extended, eqs.(3) & (4), Maxwell’s equations are invariant under the duality transforma-tions. Therefore, it is more convenient to study duality transformations involving electricand magnetic charges and their belongings, if we express Maxwell’s equation in complexform rather than the ordinary real form. The energy conservation of the electromagneticfield can be written as ∂∂t µ ( ~F · ~F ∗ ) + ~ ∇ · ic µ ( ~F × ~F ∗ ) = − ~J · ~F ∗ . (21)This equation is invariant under the duality transformation. The charge-current densities transformations. –
Charge conservation dictatesthat the right hand-side of eq.(5) to vanish. Hence, the scalar function Λ satisfies the waveequation. Thus, besides the electromagnetic field which has a transverse nature, a scalarfield with a longitudinal character is predicated from the generalized Maxwell’s equations, This because Λ has a dimension of magnetic field. p-10onsequences of complex Lorentz force and violation of Lorenz gauge condition viz. , eqs.(3) and (4). These equations can be obtained from the ordinary Maxwell’s equationsif relax the Lorenz gauge condition to read [21], ~ ∇ · ~A + 1 c ∂ϕ∂t = − Λ . (22)Equations (3) and (4) can be expanded to yield ~ ∇ · ~E = ρε − ∂ Λ ∂t , ~ ∇ · ~B = 0 , (23)and ~ ∇ × ~E = − ∂ ~B∂t , ~ ∇ × ~B = µ ~J + 1 c ∂ ~E∂t + ~ ∇ Λ . (24)Notice that Λ is a real (physical) scalar wave since it has energy and momentum. Moreover,note that Λ has a dimension of magnetic field.Equations (23) and (24) mimic the vacuum solution if we let ρε = ∂ Λ ∂t , µ ~J = − ~ ∇ Λ (25)These two equations state that ∂ρ∂t + ~ ∇ · ~J = 1 c ∂ Λ ∂t − ∇ Λ = 0 , (26)and ~ ∇ ρ + 1 c ∂ ~J∂t = 0 , ~ ∇ × ~J = 0 . (27)Hence, charge is conserved as long as Λ satisfies the wave equation. It is interesting thatthis case corresponds to our generalized continuity equations [22]. Equations (26) and (27)are invariant under the following charge-current gauge transformations ~J ′ = ~J + ~ ∇ β , ρ ′ = ρ − c ∂β∂t , (28)where β is some scalar function satisfying the wave equation. Equations (23) and (24) can beobtained by applying the transformations in eq.(28) in Maxwell’s equations, where Λ = µ β .If we now allow charge non-conservation, in particular we consider ~ ∇ · ~J + ∂ρ∂t = − µ µ Λ , (29)where µ = mc ¯ h and m is a mass scale, then eq.(26) yields1 c ∂ Λ ∂t − ∇ Λ + µ Λ = 0 . (30)p-11. I. Arbab This is the Klein-Gordon equation of spin zero particles. Thus, the violation of chargeconservation or Lorenz gauge condition lead to an emergence of massive scalar particle. Ifthis scalar is associated with the photon, then the photon would have three polarizationsates instead of two. But if it is linked to a different particle then one can associate this toHiggs boson. Thus, photons inside superconductors can develop a non-zero mass. Equations(22) and (29) suggest the following relations ~J = − α ~A , ρ = − αc ϕ , α = µ µ . (31)Under gauge transformations (scalar-vector potentials) one has ~A ′ = ~A + ~ ∇ λ , ϕ ′ = ϕ − ∂λ∂t . (32)The definition in eq.(31) is the one considered by London in his theory of superconductivity,with α = ne /m [5]. Equation (31) suggests also the following transformations in Maxwell’stheory ~J → ~J − α ~A , ρ → ρ − αc ϕ . (33)that can lead to massive photon. This is indeed the case, since the application of eq.(33)in Maxwell’s equations yields the Proca-Maxwell’s electrodynamics for massive photon [2].These are ~ ∇ · ~E = ρε − µ ϕ , ~ ∇ · ~B = 0 . (34)and ~ ∇ × ~E = − ∂ ~B∂t , ~ ∇ × ~B = µ ~J + 1 c ∂ ~E∂t − µ ~A . (35)Equations (34) & (35) are equivalent to eqs.(23) & (24) if we let µ ϕ = ∂ Λ ∂t , − µ ~A = ~ ∇ Λ , (36)which satisfy the Lorenz gauge condition ~ ∇ · ~A + 1 c ∂ϕ∂t = 0 . (37)It is worth to mention that the Proca-Maxwell’s theory, that is not invariant under the nor-mal gauge transformations, is now invariant under the matter-field transformations, eq.(33),including both ~J ( ~A ) as well as ρ ( ϕ ) as dictated by eqs.(28) & (32), where β = α λ (orΛ = µ λ ). The mystery of the scalar Λ is still worth further consideration. At any rateone can associate Λ with the amount by which the charge is violated or to the mass of thephoton. It is also related to the gauge scalar, λ .p-12onsequences of complex Lorentz force and violation of Lorenz gauge condition Electron-photon interaction. –
In standard quantum electrodynamics the photonis considered to be chargeless and massless. In this section we would like to explore theconsequences that the photon has both mass and charge. In this sense the photon carriesenergy and charge. Adding eqs.(22) and (29) using eq.(33), one finds ~ ∇ · ~J T + ∂ρ T ∂t = 0 , (38)where ~J T = ~J + ~J p , ρ T = ρ + ρ p , (39)where ~J p = − α ~A and ρ p = − αϕ/c , are the current and charge densities of photons. Thisurges us to interpret the Lorenz gauge condition as a manifestation of the conservation ofphotonic charge, or possibly magnetic charge. If the photon had mass it would be possibleto have charge. Dirac however, associated a magnetic charge ( q m ) with the electric charge( q e ) quantization [23], q m q e = n ¯ h/ , (40)where n is an integer. This relation can be seen as expressing the amount of charge violating.This is associated with the Heisneberg’s uncertainty in determining the electric and magneticcharges simultaneously, viz. ∆ q m ∆ q e = ¯ h/ , (41)It is interesting to see that under duality invariance, the quantization condition in eq.(40)requires that n → − n . Equations (38) and (39) reveal that the total charge of the systemis conserved. We further assume that the constant α is the same for photon and electron,and that the photon number density ( n p ) is equal to the electron number density ( n e ). Thisimplies that the photon charge is e p = r m p m e e . (42)This equation agrees with the relation suggested recently by Chu assuming that gravity hasa purely electromagnetic origin embedded in the non-neutrality of photons. He proposedthat m = Cq , for a particle with mass m , charge q and the constant C is related to thegravitational constant and is equal to C = 3 . × kgC − [6]. He obtained a value of e p = 7 × − e . Equation (42) relates the photon charge to its mass. Using the black-bodyradiation distribution, we have recently deduced that the photon mass today is ∼ − kgp-13. I. Arbab [12, 13]. Hence, eq.(42) states that the photon charge would be e p ∼ − e . (43)This is in a remarkable agreement with Chu proposition. If we now connect the baryon-to-photon ratio (10 − ) to equality of electronic charge to photonic charge, i.e. , N p e p = N e e ,assuming charge neutrality, thus N e /N p = e p /e . Then, e p /e = 10 − so that eq.(42) yields m p ∼ − kg. The photon charge-mass relationship depends on what photon we arestudying. Thus, this ratio may differ from one case to another. An upper limit for thephoton charge is found recently by Semertzidis et al. from Laser light deflection in magneticfield to be 8 . × − e [24].If we now equate the ratio of the electric force to the gravitational force between twophotons to the ratio between the vacuum energy at Planck’s time to the one at the presenttime (i.e., ∼ ), then one finds e p /m p ∼ [12, 13]. This result is compatible witheq.(40). If we consider now that photons are accumulated inside superconductors with restenergy density equals to that of a black-body radiation, ρ γ = π
15 ( k B T ) ( c ¯ h ) = n p m p c , where k B is the Boltzman constant, then one finds m p ∼ − kg for a temperature of few Kelvins.For a comprehensive limit on the photon mass we suggest that the reader should refer toreference [25]. Concluding remarks. –
We have investigated the consequences of of using ComplexLorentz force and abandoning the Lorenz gauge condition in formulating electromagnetictheory. Several physical effects would arise as result of enlarging the Maxwell formulation.
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