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Spinor and Hertzian Differential Forms in Electromagnetism

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Spinor and Hertzian Differential Forms in Electromagnetism Progress In Electromagnetics Research M, Vol. 16, 197{211, 2011 SPINOR AND HERTZIAN DIFFERENTIAL FORMS IN ELECTROMAGNETISM P. Hillion Institut Henri Poincar¶e 86 Bis Route de Croissy, Le V¶esinet78110, France Abstract|The purpose of this paper is to extend to spinor electromagnetism the di®erential forms, based on the Cartan exterior derivative and originally developed for tensor ¯elds, in a very compact way. To this end, di®erential electromagnetic forms are ¯rst compared to conventional tensors. Then, using the local isomorphism between the O(3,C) and SL(2,C) groups supplying the well known connection between complex vectors and traceless second rank spinors, they are generalized to spinor electromagnetism and to Proca ¯elds. These di®erential forms are ¯nally expressed in terms of Hertz potentials. 1. INTRODUCTION Di®erential forms pioneered by Cartan [1] (a di®erential p-form is a covariant skew-symmetric tensor ¯eld of degree p) and introduced some years ago in electromagnetism, may appear as a challenge to the conventional formalism described by scalars, vectors, tensors. All these entities, leaving aside General Relativity, are de¯ned either in the Einstein 4-world with the Minkowski distance ds2 = c2dt2 ¡ dx2 ¡ dy2 ¡ dz2 or in the Newton (3 + 1)-world with the space Euclidean distance ds2 = dx2 + dy2 + dz2 and the time t apart. Then, the electromagnetic ¯eld in the Minkowski space-time is ¹º de¯ned by the skew tensors F¹º(E; B);G (D; H) solutions of the Maxwell equations [2, 3] covariant under the SO(3,1) group locally isomorphic to the Lorentz group SL(2,C) @σF¹º + @¹Fºσ + @ºFσ¹ = 0 (1a) ¹º ¹ @ºG = J (1b) Received 15 November 2010, Accepted 6 January 2011, Scheduled 23 January 2011 Corresponding author: Pierre Hillion ([email protected]). 198 Hillion in which the 4-vector J ¹ is the electric current, the Greek indices take 1 2 3 4 j the values 1; 2; 3; 4 (with x = x, x = y, x = z, x = ct, @j = @=@x , @4=1=c@=@t) and the summation convention is used. But, this tensor formalism is not well suited to manage electromagnetic processes of practical importance with as consequence to promote electromagnetism in the Newton (3 + 1)-world. Then the components E (electric ¯eld), B (magnetic induction) of ¹º F¹º and H (magnetic ¯eld), D (dielectric displacement) of G become 3-vectors while the current J ¹ splits into a 3-vector j and a scalar ½ so that the Maxwell equations have the Gibbs representation r ¢ B = 0; r ^ E + 1=c@tB = 0 (2a) r ¢ D = ½; r ^ H ¡ 1=c@tD = j (2b) in which ^ is the wedge product symbol. Remark: For a Riemannian space-time with the metric ds2 = ¹ º g¹ºdx dx , Eq. (1a) is unchanged [4, 5] while Eq. (1b) becomes 1=2 ¹º ¹ @ºjgj F = J in which jgj is the determinant of g¹º. In a isotropic, homogeneous medium D = "E, B = ¹H with permittivityp "pand permeabilityp ¹, we introduce the complex vector ¤ = " E + i ¹H, i = ¡ 1, the ¤j's are in fact the components of ®¯ the self-dual tensor F¹º + i=2"¹º®¯F in which "¹º®¯ is the Levi- Civita tensor. Then, the Maxwell equations have the self-adjoint representation p p 1=2 r ^ ¤ ¡ i(n=c)@t¤ = ij ¹; nr ¢ ¤ = ½ ¹ ; n = ("¹) (2c) and, these equations are covariant under the 3D-complex rotation group O(3,C) isomorphic to the SL(2,C) group. In this work, the di®erential forms of Eqs. (2a), (2b) are ¯rst discussed and then, a generalization of the complex vector ¤ is used to de¯ne spinor electromagnetic di®erential forms. Finally, the electromagnetic ¯eld is expressed in terms of Hertz di®erential forms. 2. ELECTROMAGNETIC DIFFERENTIAL FORMS Following De Rham [6] fsee [7] for the physical applicationsg and Meetz-Engl [8], Deschamps [9] introduced in electromagnetism the notion of exterior di®erential forms, gene-rating numerous works [10{ 16] on this subject. Exterior means that the algebra of di®erential forms is endowed with the Cartan exterior derivative d assigning to a p-form w (in this Note, di®erential forms are represented by underlined expressions) a (p + 1)-form dw such that d is linear and p d(w1 ^ w2) = dw1 ^ w2 + (¡1) w1 ^ dw2; ddw = 0 (3) Progress In Electromagnetics Research M, Vol. 16, 2011 199 In the (3+1)-world, the Maxwell equations with d = dx@x+dy@y +dz@z have the di®erential form representation d ^ E + (1=c)@tB = 0; d ^ B = 0 (4a) d ^ H ¡ (1=c)@tD = j; d ^ D = ½ (4b) E, H are the 1-form E = Exdx + Eydy + Ezdz; H = Hxdx + Hydy + Hzdz (5a) and B, D the 2-forms B = B dy ^ dz + B dz ^ dx + B dx ^ dy; x y z (5b) D = Dxdy ^ dz + Dydz ^ dx + Dzdx ^ dy while J is a 2-form and ½ a 3-form J = jxdy ^ dz + jydz ^ dx + jzdx ^ dy; ½ = ½dx ^ dy ^ dz (5c) In the Minkowski 4-world, the following 2-forms are de¯ned [13] with ¹º the F¹º, G tensor ¯elds ¹ º F = 1=2F¹ºdx ^ dx ; (6a) ¹ º G = 1=2G¹ºdx ^ dx (6b) ®¯ with in (6b) G¹º = "¹º®¯ G where "¹º®¯ is the antisymmetric Levi- Civita tensor. ¹ Let d = dx @¹ be the Cartan exterior derivative. Applying d to (6a), (6b) gives the 3-form ½ ¹ º dF = 1=2@½F¹ºdx ^ dx ^ dx ; (7a) ½ ¹ º dG = 1=2@½G¹ºdx ^ dx ^ dx (7b) and, according to the property of the Cartan exterior derivative, the electromagnetic ¯eld is solution of the 3-form equations dF = 0; (8a) dG = J; (8b) in which J is the 3-form ¹ º ½ σ J = 1=3!j "¹ºρσdx ^ dx ^ dx (9) The constitutive relations between the components (E, B) and (D, H) of the electromagnetic ¯eld in the Minkowski space-time have according to Post [17] the covariant expression ¹º ¹º®¯ G = 1=2 F®¯ (10) the tensor ¹º®¯ has symmetries reducing to twenty the number of its independent components. Then, the 2-form G is changed into ¤F (a notation introduced by analogy with the Hodge star operator [8{16]) ¤ ¹º ½ σ F = 1=2Âρσ F¹ºdx ^ dx (11) 200 Hillion so that ¤dF has to be substituted to G in (8). p To thep self-dual ¯eld ¤ is associated the di®erential form L = "F + i ¹G with according to (6), (9), "ijk being the Levi-Civita tensor ³ ´ ¡ j ¢ k l F = Ej dx ^ cdt + 1=2"jklBj dx ^ dx (12a) ³ ´ ¡ j ¢ k l G = Hj dx ^ cdt ¡ 1=2"jklDj dx ^ dx (12b) ³ ´ k l J = 1=2"jklJj dx ^ dx ^ cdt + ½ (dx ^ dy ^ dz) (12c) so that the di®erential form L becomes ³ ´ ¡ j ¢ k l L = ¤j dx ^ cdt + in=2"jkl¤j dx ^ dx (13) p and, the di®erential form Eq. (8) reduces to dL = J ¹. So, with di®erential forms, electromagnetism is written very compactly. 3. SPINOR ELECTROMAGNETIC DIFFERENTIAL FORMS 3.1. Spinors in Electromagnetism Maxwell's equations, in absence of charge and current, (see Section 4.2 when charge and cur-rent exist) have the Gibbs representation r ^ E + (1 + c)@tB = 0; r ^ H ¡ (1 + c)@tD = 0 (14a) r ¢ D = 0; r ¢ B = 0 (14b) and, introducing the complex vectors ¤ = E + iH; ­ = B + iD (15) they become r ^ ¤ ¡ (i=c)@t­ = 0; (16a) r ¢ ­ = 0 (16b) In a homogeneous isotropic medium with permittivity " and permeability ¹, B = ¹H, D = "E, but more generally ", ¹ are some tensor with components "ij , ¹ij . Now, ¯rst rank spinors [18] and fin [19, 20] the spinors of higher ranks are discussedg are geometrical objects ÃA de¯ned on a two- 0 dimensional complex space satisfying the transformation law ÃA = B B tAÃB, A; B = 1; 2 with the complex 4 transformation matrix jjtAjj and Progress In Electromagnetics Research M, Vol. 16, 2011 201 a well known connection exists [21, 22] between complex vectors and s s traceless second rank spinors Ãr, Ár, r; s = 1; 2: 2 1 1 2 ¤x + i¤y = Ã1; ¤x ¡ i¤y = Ã2; ¤z = Ã1 = ¡Ã2 (17a) 2 1 1 2 ­x + i­y = Á1; ­x ¡ i­y = Á2; ­z = Á1 = ¡Á2 (17b) So, D, Dt being the 2£2 matrix derivative operators with components z D11 = ¡D22 = @ ;D12 = @x + i@y;D21 = @x ¡ i@y t t t t (18) D11 = D22 = 1=c@t;D12 = D21 = 0 the Maxwell equations have the Proca representation [19] Dª ¡ Dt© = 0 (19) In which ª, © are the matrices with components 1 ¡ 2 2 1 1 2 2 1 Ã1 = Ã2;Ã1;Ã2; Á1 = ¡Á2;Á1;Á2 (19a) Taking into account (17(a), (17b), it is proved in Appendix A that Eqs. (19) imply Eq. (14). Now, let ¤ be the 1-form and ­ the 2-form ¤ = ¤xdx + ¤ydy + ¤zdz; (20a) ­ = ­x(dy ^ dz) + ­y(dz ^ dx) + ­z(dx ^ dy) (20b) then, using the exterior derivative operator d = dx@x + dy@y + dz@z, we get d¤ =(@y¤z ¡ @z¤y)(dy ^ dz) + (@z¤x ¡ @x¤z)(dz ^ dx) + (@x¤y ¡ @y¤x)(dx ^ dy) (21) d­ =(@x­x + @y­y + @z­z)(dx ^ dy ^ dz) so that the Maxwell equations have the di®erential form representation in terms of Proca ¯eld d¤ ¡ (i=c)@ ­ = 0 t (22) d­ = 0 it is su±cient to observe that making null the coe±cients of the di®erentials dxj ^ dxk gives the Maxwell Eq. (14) which may be considered as strong solutions of Eq. (22). 3.2. Spinor Di®erential Forms Let us introduce the complex coordinates » = x + iy, ´ = x ¡ iy so that @x = @» +@´; i@y = @» ¡@´; dx = 1=2(d»+d´); idy = 1=2(d»¡d´) (23) 202 Hillion Then, substituting (23) into (20a), (20b) and taking into account (17a), (17b), we get 2 1 1 ¤ = 1=2Ã1d» + 1=2Ã2d´ + Ã1dz (24a) 1 2 1 i­ = 1=2Á2(d» ^ dz)+1=2Á1(dz ^ d´)+1=2Á2(d» ^ dz) 1 +1=2Á1(d´ ^ d») (24b) while the exterior derivative operator becomes d = d»@» + d´@´ + dz@z and a simple calculation gives ¡ 1 1¢ ¡ 2 1¢ d¤ = @»Ã1 ¡ 1=2@zÃ2 (d» ^ dz) + 1=2@zÃ1 ¡ @´Ã1 (dz ^ d´) ¡ 1 2¢ + 1=2@´Ã2 ¡ 1=2@»Ã1 (d´ ^ d») (25) Substituting (24) and (25) into (22) gives the spinor di®erential form equation d = 0 (26) 1 2 1 d = Â2(d» ^ dz) + Â1(dz ^ d´) + Â1(d´ ^ d») (26a) in which with @¿ = 1=c@t, the components of the spinor di®erential form are 1 1 1 1 Â2 = @»Ã1 ¡ 1=2@zÃ2 + 1=2@¿ Á2 2 2 1 2 Â1 = 1=2@zÃ1 ¡ @´Ã1 + 1=2@¿ Á1 (27) 1 1 2 1 Â1 = 1=2@´Ã2 ¡ 1=2@»Ã1 ¡ 1=2@¿ Á1 and ¡ 2 1 1¢ @»Ã1 ¡ @´Ã2 ¡ @zÃ1 (d» ^ d´ ^ dz) = 0 (26b) s The strong solutions Âr = 0 of (26a) are, taking into account (17), the Maxwell Eq.
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