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Electromagnetic Particles Guy Michel Stephan Electromagnetic particles Guy Michel Stephan To cite this version: Guy Michel Stephan. Electromagnetic particles. 2017. hal-01446417 HAL Id: hal-01446417 https://hal.archives-ouvertes.fr/hal-01446417 Preprint submitted on 2 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Electromagnetic particles. G.M. St´ephan 26, Chemin de Quo Vadis 22730, Tregastel email : [email protected] 25 janvier 2017 R´esum´e The covariant derivative of the 4-components electromagnetic potential in a flat Minkowski space- time is split into its antisymmetric and symmetric parts. While the former is well known to describe the electromagnetic field, we show that the latter describes the associated particles. When symmetry principles are applied to the invariants in operations of the Poincar´egroup, one finds equations which describe the structure of the particles. Both parts of the tensor unify the concept of matter-wave duality. Charge and mass are shown to be associated to the potential. 1 Introduction. When Born and Infeld published their work[1] on the foundations of non linear electromagnetism in the year 1934, their aim was to describe the electron and more generally the physical world from a purely electromagnetic theory. At that time, quantum mechanics was already developed but these authors wrote in the introduction of their article that it was in opposition to the ideas sustaining their work. The study which is developed here was motivated by the fact that both theories are not opposed, in fact they are com- plementary, simply because quantum mechanics is linear in the Hilbert space. Nonlinear electromagnetism has never been abandoned since[2] and led to interesting applications in cosmology [3]. The following work presents a broader view of the subject : instead of starting from the electromagnetic field, the corresponding potential in a flat Minkowski space is considered. The fundamental object is the covariant derivative of this potential. The antisymmetric part is the well known electromagnetic tensor. The symmetric part can be diagonalized and when constraints resulting from symmetry operations are applied to its invariants, one finds an equation for the potentials. Its solutions reveal a concentration of energy around the origin which corresponds to the structure of particles. An essential result is the explanation of the wave-matter duality which naturally arises from the intermingled parts of the tensor. We describe these calculations below and give examples of solutions and applications. 2 Frame of the theory. The theory is based on few ingredients : 1 i 0 1- A point M is defined in the flat Minkowski spacetime by its 4 coordinates−−! x where x = ct and 1;2;3 i x = x; y; z in the cartesian frame with an origin O. The associated vector OM = x ~ei is defined with respect to the orthonormalized basis ~ei . We choose the metric η = (+; −; −; −), the common dimension is length. 2- To each point is associated an electromagnetic potential Ai where A0 = ϕ/c, is the usual scalar potential and A1;2;3 = Ax;Ay;Az are the components of the vector potential. 2 3- The fundamental tensor is taken to be the covariant derivative (the gradient tensor) of A~i : @A~ a = i − ΓmA~ : (1) ij @xj ij m m It is simpler to use the cartesian basis first where Christoffel's symbols Γij cancel. When this tensor is split into its symmetric and antisymmetric parts, one is led to study the role of both. The latter part is the usual 1. Einstein summation convention is used. Upper and lower indices respectively refer to contravariant and covariant com- ponents. 2. aij is the covariant form 1 field tensor on which classical electromagnetism is based. We name the former the particle tensor and the following development is essentially devoted to the study of some of its properties. 4- We divide the potential into two parts : the first describes an isolated particle while the second describes the space in which it is embedded. This second part originates from the fundamental noise : this field has random properties, it is also isotropic and it can be represented by a diagonal tensor the modulus of which can be used as a reference for the amplitude of Ai. The following study will essentially be concerned by the isolated particle. 5- The next ingredient is the Poincar´egroup of coordinates transformations. We will associate the transfor- mations of the Poincar´egroup to the idealized isolated single entity (field/particle), or cluster of entities, which can be characterized by similar properties, such as a given velocity, a symmetry of rotation or trans- lation. The field tensor and the particle tensor do not mix in a coordinate change and their invariants are the fundamental quantities which will provide the description of the system. We describe below how a fundamental equation can be deduced from these ingredients. The solutions are the components of the 4-potential. These components are mixed together in a non-linear way to form invariants of the system. One of this invariant is related to the density of energy. We give illustrations for a series of spherically symmetric solutions. An application of the theory to make the link with Maxwell's equations is finally given with use of the principle of least action. Here, the elementary electric charge is shown to be a characteristic of only a single type of solution. 3 The fundamental tensor. 3.1 Notation. The electromagnetic potential (4-vector) Ai is defined by its contravariant components in the direct space. We will write the covector A~ in the reciprocal space. Its components are : A~0;1;2;3 = ϕ/c, −A~x; −A~y; −A~z . The dimension of A~ and A is that of an ordinary electric potential (MLT −1Q−1). The scalar product is a Lorentz invariant : j j2 i ~ km ~ ~ m k 2 − ~2 − ~2 − ~2 2 − x2 − y2 − z2 A = A Ai = η AmAk = ηkm A A = (ϕ/c) Ax Ay Az = (ϕ/c) A A A : (2) i The components of the potential vector [A ] (tensor type (0,1)) and the covector [A~i] (type (1,0)) have different signs. Our fundamental object is the covariant derivative written in a cartesian frame : 2 3 ϕ;t ϕ;x ϕ;y ϕ;z 6 − ~ − ~ − ~ − ~ 7 ~ 6 Ax;t Ax;x Ax;y Ax;z 7 [aij] = D[A] = 4 5 : (3) −Ay~ ;t −Ay~ ;x −Ay~ ;y −Ay~ ;z −Az~ ;t −Az~ ;x −Az~ ;y −Az~ ;z j This is the standard matrix representation of D[A~] where the first index in aij = @A~i=@x is the line index and the second the column index. We have used the compressed notation for the partial derivatives in the cartesian frame : @(ϕ/c) @ϕ/c @ϕ/c @ϕ/c ϕ ≡ ; ϕ ≡ ; ϕ ≡ ; ϕ ≡ ;t c@t ;x @x ;y @y ;z @z @A~ @A~ @A~ @A~ Ax~ ≡ x ; Ax~ ≡ x ; Ax~ ≡ x ; Ax~ ≡ x (4) ;t c@t ;x @x ;y @y ;z @z and the same for the derivatives of A~y and A~z. 3.2 The field and the particle tensors. 3 [aij] can be split into its symmetric and antisymmetric parts : [aij] = [s(ij)] + [f[ij]] (5) 3. This splitting makes sense for covariant or contravariant tensors because their symmetry remains invariant in a coordinate change. This property does not apply to mixed tensors. 2 with : 1 1 s = (a + a ) and : f = (a − a ) (6) ij 2 ij ji (ij) 2 ij ji The antisymmetric part is explicitly written : 2 3 0 ϕ;x + Ax~ ;t ϕ;y + Ay~ ;t ϕ;z + Az~ ;t 6 ~ ~ ~ ~ ~ 7 6 −Ax;t − ϕ;x 0 −Ax;y + Ay;x −Ax;z + Az;x 7 [fij] = 0:5 4 5 : (7) −Ay~ ;t − ϕ;y −Ay~ ;x + Ax~ ;y 0 −Ay~ ;z + Az~ ;y −Az~ ;t − ϕ;z −Az~ ;x + Az~ ;x −Az~ ;y + Ay~ ;z 0 apart the factor 0.5, [fij] is the transpose of the standard electromagnetic field tensor : 2 3 0 Ex=c Ey=c Ez=c 6 7 6 −Ex=c 0 −Bz By 7 [Fij] = 4 5 : (8) −Ey=c Bz 0 −Bx −Ez=c −By Bx 0 The electric (Ex;Ey;Ez) and magnetic (Bx;By;Bz) fields are defined from the derivatives of the 4-potential covector components : @A~ @ϕ/c @A~ @ϕ/c @A~ @ϕ/c E =c = − x − ;E =c = − y − ;E =c = − z − : (9) x c@t @x y c@t @y z c@t @z and : @A~ @A~ @A~ @A~ @A~ @A~ B = z − y ;B = x − z ;B = y − x : (10) x @y @z y @z @x z @x @y k l m The identity : @flm=@x + @fmk=@x + @fkl=@x = 0 , when developed, leads to the first set of Maxwell equations. The symmetric part of [aij] is : 2 3 2ϕ;t ϕ;x − Ax~ ;t ϕ;y − Ay~ ;t ϕ;z − Az~ ;t 6 ~ ~ ~ ~ ~ ~ 7 6 −Ax;t + ϕ;x −2Ax;x −Ax;y − Ay;x −Ax;z − Az;x 7 [sij] = 0:5 4 5 : (11) −Ay~ ;t + ϕ;y −Ay~ ;x − Ax~ ;y −2Ay~ ;y −Ay~ ;z − Az~ ;y −Az~ ;t + ϕ;z −Az~ ;x − Az~ ;x −Az~ ;y − Ay~ ;z 2Az~ ;z We will name [sij] the "particle tensor" to balance [fij] which is the field tensor.
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