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תודות תודהלמנחהפרופסוריעקבפרידמןעלההכוונהלאורךכלהדרך,ועזרתוהרבהבמהלךהפרוייקט. תודהלסביוסבתיאליעזרולאהבר-עדעלהעזרהבהגהההלשונית. תודהלאשתיהיקרהאלישבעשבלעדיהפרוייקטזהלאהיהמסתייםכלל. The pre-potential of the EM field David Hai Gootvilig November 6, 2014 Abstract This report is continuation of previous work [ 8]-[12] on the prepotential of the electro- magnetic field. We define the Hodge dual operator on the standard representation of the Lie algebra of the Lorentz group on Minkowski space, denoted by πin this report. This operator defines a decomposition of the complexification of πinto eigen-subspaces o±, correspoding to to the eigenvalues ±ı. The elements of o±are called σ-dual, where σtake a value ±ı. We define a representation of the Lorentz group ˜π±on each of the subspaces o±. For any EM field tensor Fwe associate a σ-dual tensor F±. We show that outside of the sources of the field a σ-dual 4-potential can be defined. For any observer we find a constant ζ±, depending only on the null-displacement between the oserver and the source of the field which is invariant under ˜π±. The invariant ζ±is used to define the prepotential S±of the EM field. We have shown that the prepotential of a point charged particle in arbitrary motion produces the correct field. It is shown that outside of the sources the prepotential is always σ-dual. Maxwell’s equations for the prepotential are derived. We find conditions on the cur- rent density for the existence of a prepotential. The gauge freedom of the prepotential is calculated. Examples for the relation between the prepotential, the 4-current and the field are presented for: infinite charged rod, infinite charged plate and current through infinite solenoid. 3 Contents 1 Introduction 6 1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 A very short introduction to tensors 6 2.1 An intuitive approach to tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Numbers, Scalars, Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 Beyond vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Mathematical approach to tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Definition of tensors by their transformation law . . . . . . . . . . . . . . 9 2.2.2 Metric tensor, inner procduct and Index Lowering and Raising . . . . . . 10 2.2.3 Tensor addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.4 Tensor product, contraction and transpose . . . . . . . . . . . . . . . . . 10 2.2.5 Symmetric and antisymmetric tensors . . . . . . . . . . . . . . . . . . . . 11 2.2.6 Levi-Civita symbol and the Determinant . . . . . . . . . . . . . . . . . . . 12 2.2.7 Tensor Density and the Levi-Civita tensor . . . . . . . . . . . . . . . . . . 14 2.2.8 Hodge dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Tensor formulation of classical EM 16 3.1 Formulation of EM in the 4D Minkowski space . . . . . . . . . . . . . . . . . . . 18 3.1.1 The Minkowski space M. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 EM in Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 A scalar invariant under the Lorentz Group 19 4.1 Short introduction to Matrix Lie groups . . . . . . . . . . . . . . . . . . . . . . . 19 4.1.1 Group, subgroup, group action and group homomorphism . . . . . . . . . 19 4.1.2 The general matrix group GLn(F). . . . . . . . . . . . . . . . . . . . . . 21 4.1.3 Group representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.4GLn(F)and manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.5 Matrix Lie groups and Matrix Lie algebras . . . . . . . . . . . . . . . . . . 25 4.2 The Lorentz group, it’s Lie algebra and their representations . . . . . . . . . . . . 26 4.2.1 The Orthogonal group O(p,q)and it’s Lie algebra o(p,q). . . . . . . . . . 26 4.2.2 One dimensional Lorentz transform . . . . . . . . . . . . . . . . . . . . . 27 4.2.3 Representation π. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2.4 Complexified Minkowski space and the σ-dual subspaces o±ofo(1,3). . 30 4.2.5 Representations ˜π+and ˜π−. . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2.6 The projections of the Faraday tensor . . . . . . . . . . . . . . . . . . . . 34 4.2.7 Lorentz covariance relation of πand ˜π±. . . . . . . . . . . . . . . . . . . 34 4.2.8 Complexification of the 4-potential . . . . . . . . . . . . . . . . . . . . . . 35 4.2.9 Prepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 The ˜π±invariantζ±-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.1 The eigenvectors of ˜B± i. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3.2 The null basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.3 The matrix associated with 4-vector . . . . . . . . . . . . . . . . . . . . . 38 4.3.4 Decomposition of null-vectors in the null-basis . . . . . . . . . . . . . . . 41 4.3.5 Local orthonormal basis, and it’s null-vector decomposition . . . . . . . . 41 4 4.3.6 The invariant ζ±-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5 The prepotential of the EM field 43 5.1 The prepotential of a point charge . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 The gradient of the prepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2.1 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2.2 Complex 4-potential of a point charge . . . . . . . . . . . . . . . . . . . . 45 5.2.3 EM Field of a point charge . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Prepotential of charge distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4 Riemann–Silberstein vector F−by the prepotential . . . . . . . . . . . . . . . . . 47 5.5 Maxwell’s equation, 4-current condition and gauge conditions . . . . . . . . . . 49 5.5.1 Maxwell’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5.2 Current condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5.3 Prepotential Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.6 Examples of calculations involving the prepotential . . . . . . . . . . . . . . . . 51 5.6.1 Infinitely long charged rod . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.6.2 Infinite charged plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.6.3 Infinitely long solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6 Discussion 52 6.1 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Advances with respect to previous works on the subject . . . . . . . . . . . . . . 52 תקציר 55 5 1 Introduction 1.1 Outline In section 2we introduce briefly the language of tensors, and in section 3we apply it to the classical Electromagnetic theory. In section 4we introduce the matrix Lie groups and their Lie algebras and the most necessary ideas of representation theory, we show that the canonical Lorentz representation πis a product of two commuting spin-half representations ˜π+and ˜π−. We investigate the σ-dual tensor field arising for the half-spin representations and find that outside of the sources a σ-dual complex 4-potential can be defined, and show that if the pre- potential satisfies the wave equation if and only if it is σ-dual. Considering only one of the half-spin representation alone on a complexified Minkowski space we give raise to non-trivial scalar invariant made out of a null displacement vector. In section 5we define the prepotential of a charged point particle using the invariant scalar derived in section 4. The relation between the prepotential and the complex 4-potential is defined, and the complex 4-potential induced by this prepotential is calculated. It’s real part is the Liénard–Wiechert potentials. The fields induces by this prepotential are calculated. And we show that the prepotetial is σ-dual everywhere outside of the source. The prepotential of a charge distribution is defined through super-position principle. The field tensor is formulated as a function of the prepotential and the gauge freedom of the prepo- tential is calculated. Maxwell’s equations in the prepotential formalism are presented. 1.2 Motivation • Fewer degrees of freedom in the description of the theory, the Electro-Magnetic field is represented by the Faraday tensor Fµν∼(E,B)which has 6 degrees of freedom. Using the Electro-Magnetic 4-potential Aµwe get down to 4 degrees of freedom, when using a gauge choice we reduce them to 3 degrees of freedom. The prepotential Sis a complex scalar and has 2 real degrees of freedom. • Locality. The field strength Tensor Fµνdepends on the position, velocity and acceleration of the sources, the 4-potential Aµdepends on the position and velocity of the source, while the prepotential Sdepends only on the position of the source. • Aharonov-Bohm effect indicates that there is a multiple-valued pre-potential of the 4- potential of an EM field. • The pre-potential is Conformally invariant. 2 A very short introduction to tensors 2.1 An intuitive approach to tensors In this section we will try to explain what are tensors in an intuitive way, as a naturally required generalization and specialization of scalars and vectors. 2.1.1 Numbers, Scalars, Vectors Number or ‘bare’ number is not a physical quantity but is a satisfactory answer when the in- formation we seek is a magnitude. For example: ”How many books are on your shelf?”, ”Ten” 6 might be a possible answer. Scalar is a physical quantity, it can be viewed as a number with a name - which gives us the unitwe use to express the quantity. For example: ”How much time does it takes for the world’s champion to solve a Rubik’s cube?”, ”Ten seconds” is a possible answer. Examples for scalars: the mass, charge, time, speed, temperature or electric potential at a point inside a medium. Vectors are abstract objects, which not only posses magnitude but also direction. The mag- nitude or ”length” of a physical vector must have physical units, for example, distance and displacement are measured in meters, velocity is measured in meters/second, force is measured in Newtons, electric field intensity is measured in Volts/meter, and so on. The direction is measured relatively to some possibly arbitrarily chosen reference direction. As stated vectors are abstract but their components can be manipulated when expressing them in some basis. For example in Newtonian Dynamics, the position or displacement of a point mass is expressed as a vector r(t)from the origin to the object, which depends on time. Using a coordinate system with a basis ei, we can write this displacement as r(t) =3∑ i=1ri(t)ei=∑ iri(t)ei=ri(t)ei=rj(t)ej where in the last two hands Einstein’s summation convention is used as defined in 2.1below. Definition 2.1. Einstein’s summation convention is a notational convention that implies sum- mation over a set of indexed terms in a formula, thus achieving notational brevity. We will define it with the following rules: 1. An index variable can appear up to twice in a single term, one as upper index and one as lower index. 2. When an index variable appears as both upper and lower index in a single term, it implies summation of the term over all the values of the index. 3. An index that is summed over is a summation index , and also called a dummy index since any symbol can replace it without changing the meaning of the expression, provided that it does not collide with another index symbols on that term. 4. An index that is not summed over is a free index and must be found in each term of the expression or equation. 5. Unless stated otherwise: • Small Latin alphabet ( i,j,k, etc.) indexes will have the set {1,2,3} • Greek alphabet ( µ,ν, etc.) indexes will have the set {0,1,2,3} • Small bold font Latin alphabet ( i,j,k, etc.) will be used as index arrays. in most cases the limits of implied summation is understood from context, an explicit summation is used where there is a possible confusion. Remark. We will use this convention extensively throughout this report. 2.1.2 Inner product Definition 2.2. The inner product of two vectors x=xieiandy=yjejis a scalar, expressed as x◦y= (ei◦ej)xiyj=δijxiyj whereδijis Kronecker’s delta. The argument (ei◦ej)here equals δijsince we assumed an Eu- clidean space with orthonormal basis, but later we will see this doesn’t have to hold in general. 7 Note. The definition of inner product given in 2.2will not be used in the rest of the report, and a better definition which generalize 2.2will be introduced in 2.9. Definition 2.3. A co-vector vjis an object that can be combined with a vector ujto create a scalars=vjuj. Even though a co-vector has similarities to a vector, they transform in a different way under a basis change, co-vectors change in the same way that the basis does and are called co-variant vectors or co-vectors for short, while vectors or contra-variant vectors transform in opposite manner so scalar combined from a co-vector and a vector stays invariant under a basis change. Example 2.4. The differential work dWis a scalar combined from the co-vector force Fjand the differential displacement drjasdW =Fjdrj. Example 2.5. The gradient operator ∂j=∂ ∂xjis a co-variant operator, when applied to scalar it creates a co-vector, the potential energy in classical mechanics Uis a scalar, and the force Fj is related to it by Fj=−∂jU=−∂U ∂xj. 2.1.3 Beyond vectors There are abstract geometric objects which are more sophisticated then vectors and require more information to be expressed, intuitively these objects have a magnitude and more then one direction. With these objects we can express relations which cannot be formulated when using only scalars and vectors. For example the magnetic flux density Bin volt*second/m2, and the magnetization Hin Amp/m, are related by the magnetic permeability µin Henry/m by the expression B=µ·HorBi=µijHj(2.1.3.1) thusBandHcan differ in magnitude and direction as happens in some materials. The classical example of the use of tensors in physics is related to stress in material object. Stress σhas units of N/m2so it is clear that (stress) ×(area) should equal (force), which will be the forces that creates the stress on this area. A differential area can be represented by a vector dSwhich is normal to the area pointing outward from the convex side, and has magnitude of the of the surface area dS. Since there are two types of stresses: the tensile stress (normal force) and sheer stress (tan- gential force) the stress cannot be a scalar, so the force dFdue to the stress σacting on the differential surface element dSis given by dF=σ·dSordFi=σijdSj(2.1.3.2) The stress tensor σwas the first tensor to be described and used by scientists and engineers. The word tensor derives from the Latin tensus meaning stress or tension. 2.1.4 Tensors The defining properties of tensors is that the abstract object they represent remains unchanged under a basis change, and are expressed by a single basis. It follows that not all scalars and vectors and other entities are tensors, only those that depend on single basis and transform in the correct way. For example the basis transformation matrix is not a tensor since it depends on two bases. 8 Assume in an Ndimensional space we have a basis {e1,e2,···,eN}, therefor for any vector vis represented as v=vµeµ. Now consider an invertible linear transformation that transforms the basiseµinto a basis eµ′as eµ=Lµ′ µeµ′=∂xµ′ ∂xµeµ′ (2.1.4.1) where∂xµ′ ∂xµis the inverse of the Jacobian matrix representing the first partial derivatives of the coordinate transformation xµ′(xµ). A scalarSis a tensor which is invariant under basis transform, thus S′=S. Vectors are also tensors, which represent the same object in any reference frame thus x=x′. From this requirement we get: x=xµeµ=xµLµ′ µeµ′=xµ′eµ′=x′⇒xµ′=Lµ′ µxµ⇒xµ= (L−1)µ µ′xµ′. (2.1.4.2) We can see from ( 2.1.4.2 ) that vectors components are transformed inversely to the basis transform. This is why they are called contra-variant, in contrast to vectors - co-vectors are transformed in the same way as the basis does. Also we see that the transformation matrices L arenottensors since they depend on two bases. Note. Now we can understand why in ( 2.1.4.1 ) we substituted∂xµ′ ∂xµforLµ′ µ. By differentiating the middle equation in ( 2.1.4.2 ) byxµassumingLis constant we get: ∂xµ′ ∂xµ=Lµ′ µ. The notion of the partial derivatives will be used in the rest of the report. Moreover it is generalizable to the more general settings of curvilinear coordinates and tangent spaces to differentiable manifolds which will not be discussed in this report. 2.2 Mathematical approach to tensors Tensors are geometric objects which encapsulate the meaning of the same thing, with respect to an arbitrary basis transformation, in physics the different bases are called a different reference frames. 2.2.1 Definition of tensors by their transformation law The following definition of tensors that we will use in this report defines tensors by their trans- formation laws with respect to a basis transformation. This is not the most general and abstract way but will suffice well for our needs. Definition 2.6. A tensorT(or more precisely tensor field) of type- (n,m )on a finite vector space Vwith dimension Nover the field F(for our cases Fis either RorC), is a multi-dimensional array with indexes i= (i1,i2,...,in);j= (j1,j2,...,jm)which transforms by the following rule with respect to a basis eµ→e′ µand coordinates xµ→xµ′transformation as: Ti′ j′=∂xi′ ∂xi∂xj ∂xj′Ti j whereTon the right hand is a tensor field with index arrays i′= (i′ 1,i′ 2,...,i′ n);j′= (j′ 1,j′ 2,...,j′ m), and∂xi′ ∂xi=∂xi′ 1 ∂xi1...∂xi′ n ∂xin, where∂xi′ ∂xiis given by ( 2.1.4.1 ). Definition 2.7. A tensor rank (also degree ororder ) is the dimensionality of the array needed to represent it, or the sum of it’s type indexes thus a tensor of type- (n,m )is of rankn+m. 9 2.2.2 Metric tensor, inner procduct and Index Lowering and Raising Definition 2.8. The inner product in general is defined by the metric tensor gµνassociated with the space, which is a type-(0, 2) tensor which obeys gµν=gνµ(more precisely it is non- degenerate symmetric form). Definition 2.9. The inner product x◦yof vectorsxandyis x◦y=gµνxµyν=gµνxνyµ Remark. We can now see that eµ◦eνfrom 2.2in the general case actually equals to gµν. Definition 2.10. The inverse metric tensor gµνis related to the metric by δµ ν=gµλgλν Definition 2.11. In metric space the metric and the inverse metric are used in order to lower and raise indexes. A vector vjis related to a co-vector vibyvi=gijvjor equivalently vj=gijvi, this can be done to any index on any tensor. Example 2.12. The mixed Faraday tensor Fµ νwhich is used in formulation of the Lorentz force dpµ dτ=qFµ νuν, is related by a raised index operation to the co-variant form FαβasFµ ν=gµαFαν. 2.2.3 Tensor addition and subtraction Definition 2.13. Tensors of the same rank can be added or subtracted, this is a component-wise operation, in detail the tensor addition (subtraction) of two tensors AandBof type (n,m ), is a tensorCof type (n,m )where the following holds component-wise: Ci j=Ai j±Bi j where a plus sign is used for addition and minus sign for subtraction, iandjare index arrays withnandmindexes respectively. 2.2.4 Tensor product, contraction and transpose Definition 2.14. The tensor product of two tensors AandBof types (n1,m1)and (n2,m2) respectively, is a tensor Cof type (n1+n2,m1+m2)where the following holds component- wise: Ci1i2 j1j2=Ai1 j1Bi2 j2 where in the left hand side i1,i2,j1andj2are index arrays of n1,n2,m1andm2indexes respectively, and in the right hand side the expressions i1i2andj1j2are index arrays, consisting of the indexes in the first index array followed by the indexes from the second index array. Example 2.15. The angular momentum Lijin classical mechanics is a subtraction of tensor products of the lowered index displacement xiand the momentum pi, as Lij=xipj−xjpi= x1p1x1p2x1p3 x2p1x2p2x2p3 x3p1x3p2x3p3 − x1p1x2p1x3p1 x1p2x2p2x3p2 x1p3x2p3x3p3  10 Example 2.16. A multiplication by a scalar λof a type (n,m )tensorTis defined through the tensor product 2.14, since the scalar is a type (0,0)tensor the result is a type (n,m )tensor and each of it’s components is the respective component of Tmultiplied by λthus recapturing the natural meaning of a scalar multiple. Definition 2.17. Tensor contraction is an operation that reduces a tensor Aof type (n,m )with index arrays i= (i,i′)andj= (j,j′), into a tensor Bof type (n−1,m−1)with index arrays i′ andj′withn−1andm−1indexes respectively, Bi′ j′=Aki′ kj′ whereNis the dimension of the underlying vector space, and the contraction was chosen over the first contra-variant and co-variant indexes, but any pair of contra-variant and co-variant indexes can be contracted. Example 2.18. The inner product x◦ydefined in 2.9is a contraction of the tensor gklxiyj(more precisely two contractions iwithkandjwithl), which is a tensor product of two vectors ( xi, yj) and the metric tensor gkldefined in 2.8, as x◦y=gijxiyj Definition 2.19. The transpose of a rank 2 tensor Aab,AaborAa bas (Aab)t=Aba ( Aab)t=Aba (Aa b)t=Aa b=gacAd cgbd this definition correlates with the known transpose of a matrix. Example 2.20. The transpose of a matrix Ai j=(a b c d) in an Euclidean space is ( Ai j)t=gii′Aj′ i′gj′j=δii′Aj′ i′δj′j=Ai j=(a c b d) 2.2.5 Symmetric and antisymmetric tensors Definition 2.21. Symmetry and anti-symmetry in tensors is defined on pairs of indexes. An exchange of symmetric pair of indexes, is equal to the original tensor component-wise, while an exchange of anti-symmetric pair of indexes, is equal to minus the original tensor component- wise. When the index pair is not of the same type (contra-variant or co-variant), the symmetry or anti-symmetry is observed after lowering the contra-variant or raising the co-variant index by the use of the mertic (or the inverse-metric). Example 2.22. The angular momentum Lijis anti-symmetric Lij=−Lji, and the stress tensor σijis symmetric σij=σji. Definition 2.23. Anti-symmetrization notation T[a1a2...an]=1 n!∑ σ∈PSign(σ)Taσ1aσ2...aσn whereTis a tensor, !is the factorial, Pis the set of all permutations of the set {1,2,...,n }, the sign of the permutation Sign(σ)is1for even-permutations and −1for odd-permutations, and the indexes σjare the permutation indexes. 11 Example 2.24. Forn= 2,3we have: M[ab]=1 2!(Mab−Mba) T[abc]=1 3!(Tabc−Tacb+Tbca−Tbac+Tcab−Tcba) Definition 2.25. The generalized Kronecker delta is related to the anti-symmetrization notation as δi jTi= (n!)T[j] where i,jare index arrays of size n, and !is the factorial. Example 2.26. Forn= 2,3we have: δcd abMcd= (Mab−Mba) δdef abcTdef= (Tabc−Tacb+Tbca−Tbac+Tcab−Tcba) Definition 2.27. Acompletely anti-symmetric tensor is a tensor which is anti-symmetric in each pair of it’s indexes. Theorem 2.28. In anNdimensional space any completely anti-symmetric tensor 2.27 of rank greater then Nistrivial , meaning any of it’s components in anybasis is identically zero. Proof. LetTibe a completely anti-symmetric tensor of rank m > N , where the index array is i= (i1,...,im). By the pigeonhole principle for any component of the tensor there must be two indexes with the same value, since a change of this two indexes will not change the index array i′=ibut by the anti-symmetry an exchange flips the sign as Ti=−Ti′, we findTi=−Ti′=−Ti which leads to Ti≡0, as required. 2.2.6 Levi-Civita symbol and the Determinant Definition 2.29. The Levi-Civita symbol εis not a tensor, it is a completely anti-symmetric object. And in an Ndimensional space in any reference frame it’s components are given by εi= Sign (σ)εi0 where i0= (1,2,...,N )andi=σi0are index arrays of size N,σis a permutation of index arrays of size N, andεi0= 1. In the cases where iis not a permutation of i0thenεi= 0. Example 2.30. The Levi-Civita symbol in two dimensions. It’s components are ε11= 0ε12= 1ε21=−1ε22= 0 or in matrix form ε=(0 1 −1 0) Definition 2.31. The relation between the covariant and the contra-variant Levi-Civita symbol is defined numerically to be: εi≡εi this is the one and only exception to the otherwise universal rule that when we use the same symbol on an object with upstairs indexes and an object with downstairs indexes, the former is related to the latter by raising the indexes with the inverse metric. 12 Lemma 2.32. The contraction of the Levi-Civita symbol with itself satisfies εiεi=N!. Proof. By2.31εiandεiare numerically equivalent, and since by 2.29 in any reference frame the non-zero components are ±1thus their square is always 1. Summing on all possible index ar- rangements without duplicate indexes (which yield the 0 components) which in Ndimensional space there are N!arrangements, the result is indeed N!as required. Definition 2.33. The determinant of a matrix or type- (1,1)tensorAi jis defined as det( Ai j) =1 N!εiεjAi j whereεiandεjare the covariant and contra-variant Levi-Civita symbol defined in 2.29 and 2.31,iandjare index arrays of size N, andAi j=Ai1 j1...AiN jN. Example 2.34. The determinant of A=(a b c d) . Using 2.30 and2.33 we get: detA=1 2!εi1i2εj1j2Ai1 j1Ai2 j2=1 2(ε12ε12ad+ε21ε12bc+ε12ε21cb+ε21ε21da) = =1 2(2ad−2bc) =ad−bc which is the expected result. Definition 2.35. In a similar way to 2.33 we define the determinant of a covariant and contra- variant tensors of rank 2 or rather type- (2,0)tensorAijand type- (0,2)tensorAijas det (Aij) =1 N!εiεjAij det( Aij) =1 N!εiεjAij whereεiandεjare the covariant and contra-variant the Levi-Civita symbol defined in 2.29 and 2.31,iandjare index arrays of size N, andAij=Ai1j1...AiNjN. Lemma 2.36. For any rank 2 tensor Aij,AijorAi j, the co-variant εiand the contra-variant εj Levi-Civita symbols are related by: εiAij= det( Aαβ) εj εiAi j= det( Aα β) εj εjAi j= det( Aα β) εi εiAij= det (Aαβ)εj(2.2.6.1) where iandjare index arrays of size N, andAij=Ai1j1...AiNjN. Proof. Using 2.35 to find the determinant of Aαβwe get det( Aαβ) =1 N!εiεjAij. multiplying by N!, and using 2.32 we get εj[ εjdet( Aαβ) −εiAij] = 0 sinceεjis not zero everywhere we can omit it and find εiAij= det( Aαβ) εj as required. All other three relations are proved similarly using 2.35 or2.33 and2.32. 13 Note. Now it is clear why we said in the definition of the contra-variant Levi-Civita symbol 2.31 that the covariant and the contra-variant Levi-Civita symbols are not related by index raising and lowering, since using 2.36 where the rank 2 tensor is the metric gµνwe get εigij=gεj (2.2.6.2) whereg= det (gαβ). This is contrary to the result εigij=εjwe should have got if the rules of tensor index lowering and raising was applicable to the Levi-Civita symbols. 2.2.7 Tensor Density and the Levi-Civita tensor Definition 2.37. A tensor density Ti jof weightWand type- (n,m )is a quantity that under a basis transformation, transforms into tensor density Ti′ j′of the same weight and type as Ti′ j′=( det[ ∂xk′ ∂xk])W ∂xi′ ∂xi∂xj ∂xj′Ti j where i,i′,j,j′are pairs of index arrays of sizes n,manddet[ ∂xk′ ∂xk] is the Jacobian determinant. A tensor is just a tensor density of weight 0. The tensors operations: addition, subtraction, contraction, index raising and lowering, and the notions of symmetry and anti-symmetry are extended natively to tensor densities. Theorem 2.38. The Levi-Civita symbol is a co-variant εitensor density of weight +1or a contra- variantεitensor density of weight −1. Proof. Using 2.36 with the inverse Jacobian matrix∂xk ∂xk′we get: εi′det(∂xk ∂xk′) =∂xi ∂xi′εi multiplying by the Jacobian determinant det( ∂xk′ ∂xk) we find εi′= det( ∂xk′ ∂xk) ∂xi ∂xi′εi thus the transformation of the co-variant Levi-Civita symbol is indeed governed by the trans- formation of tensor density of weight +1as defined in 2.37, the proof for the contra-variant Levi-Civita symbol follows in a similar way. Remark. The Levi-Civita symbol has been defined in 2.29 so it will be an invariant tensor density and hence will have the same value in any reference frame. Definition 2.39. The tensor density product of a tensor density Aof type (n1,m1)and weight W1and a tensor density Bof type (n2,m2)and weight W2is a tensor density Cof type (n1+ n2,m1+m2)where the component-wise product holds as in 2.14, and the weight of CisW1+W2, Lemma 2.40. The determinant of a rank 2 tensor Lαβ,LαβandLα βis a scalar density of weight +2,−2and0respectively. 14 Proof. The lemma 2.40 follows immediately from 2.39,2.38,2.33 and2.35. Following 2.39 we can produce a tensor out of the Levi-Civita symbol which is a tensor density of weight ±1(plus for co-variant, minus for contra-variant), in order to do that we need ascalar density of weight ∓1, preferably one which is ”natural” and have a geometric meaning. Lemma 2.41. Given a tensor density Tof weightWthen|g|W 2Tis a pure tensor, where gis the determinant of the metric tensor. Proof. Since following 2.40gis scalar density of weight −2, therefor√ |g|is a scalar density of weight −1, hence the total weight in 2.41 is 0 as required. Note. The absolute value in 2.41 is necessary when dealing with a space with non-positive definite inner-product, such as the Minkowski space in Special and General relativity. Definition 2.42. In any metric space the Levi-Civita tensorϵis defined as a completely anti- symmetric co-variant or contra-variant tensor, which is the tensor density product of the Levi- Civita symbol ε2.29 and square root of the determinant of the metric√ |g|2.41, thus it is given by ϵi=√ |g|εi ϵi=(−1)s √ |g|εi where iis a vector index of size Nthe dimension of the underlying space, and sis the signature of the metric. Remark. In2.42 the definition of both the co-variant and the contra-variant Levi-Civita tensor is redundant since they are related by index lowering and raising. Lemma 2.43. The contraction of Levi-Civita tensor with itself obeys the following relation: ϵijϵi′j= (N−k)!(−1)sδi i′ where i,i′,jare vector indexes of sizes k,kandN−krespectively, δi i′is the generalized Kronecker delta defined in 2.25 andsis the signature of the metric. Proof. Using 2.42,2.31, and 2.25 we find ϵijϵi′j= (−1)sεijεi′j= (N−k)!(−1)sδi i′ The term (N−k)!arise since the contraction of completely anti-symmetric indexes occurs as all the possible arrangements of different indexes hence the factorial. 2.2.8 Hodge dual Since we established in 2.28 that the completely anti-symmetric tensors are ”bounded” in rank by the space dimension N, we check how much freedom there is in any rank 0≤k≤N, since a permutation of indexes gives the same object up to a sign flip, thus for tensor rank kthere is(N k) =N! k!(N−k)!distinct indexes choices. Since(N k) =(N N−k) it is suggestive to define an isomorphism between the kandN−k completely anti-symmetric tensors. While there are many possible isomorphisms the Hodge duality defines a unique one using the inner-product and orientation of the space. 15 Definition 2.44. TheHodge dual operator maps completely anti-symmetric tensors of rank k into completely anti-symmetric tensors of rank N−k. The Hodge dual of an arbitrary completely antisymmetric tensor ηof rankkis a completely antisymmetric tensor ⋆ηof rankN−kas (⋆η)i=1 k!ϵj iηj where iandjare index arrays of size N−kandkrespectively and ϵis the Levi-Civita tensor. Theorem 2.45. The square of the Hodge dual is the identity operator, up to sign. The Hodge dual square of a completely anti-symmetric tensor ηof rankkin anNdimensional space is given by (⋆⋆η )i= (−1)k(N−k)+sηi where iis an index array of size kandsis the signature of the metric. Proof. Using 2.43 we find: (⋆⋆η )i=1 k!ϵj i(⋆η)j=1 k!ϵj i1 (N−k)!ϵi′ jηi′ =1 k!(N−k)!ϵijϵji′ηi′=(−1)k(N−k) k!(N−k)!ϵijϵi′jηi′ =(−1)k(N−k) k!(N−k)!(N−k)!(−1)sδi′ iηi′=(−1)k(N−k)+s k!k!ηi = (−1)k(N−k)+sηi the term (−1)k(N−k)appears when rearranging the indexes i′andjon a completely anti- symmetric tensor we get k(N−k)index pair exchanges. Example 2.46. In 4-dimensional space with metric signature s= 1, the Hodge dual of a com- pletely anti-symmetric tensor of rank 2 Fαβ, is(⋆⋆F )αβ=−Fαβ. On the other hand in an Euclidean space ( s= 0) we would have (⋆⋆F )αβ=Fαβ. 3 Tensor formulation of classical EM The classical four Maxwell’s equations (in flat vacuum) in differential formulation, can be di- vided into two source independent equations ∇ ·B= 0 (3.0.8.1) ∇ ×E+∂B ∂t= 0 (3.0.8.2) called respectively Gauss’s law for magnetism, and the Maxwell-Faraday equation (also called Faraday’s law of induction), and two source dependent equations ∇ ·E=1 ϵ0ρ (3.0.8.3) ∇ ×B−µ0ϵ0∂E ∂t=µ0J (3.0.8.4) called respectively Gauss’s law and Ampere’s circuital law (with Maxwell’s addition). 16 Whereϵ0andµ0are the electric permittivity and the magnetic permeability of the vacuum, EandBare the electric and magnetic field components, they do not form vectors or co-vectors, their true nature is as components of unified electromagnetic rank-2 tensor to be introduced below in ( 3.1.2.2 ). In this section we will introduce the naive 3D approach. Using tensorial formulation the source independent equations are Bi ,i= 0 (3.0.8.5) εijkEi,j+˙Bk= 0 (3.0.8.6) where the dot over means differentiation by time ˙Bk=∂ ∂tBk, and for the source dependent equations are Ei ,i=1 ϵ0ρ (3.0.8.7) εijkBi,j−µ0ϵ0˙Ek=µ0Jk(3.0.8.8) Since the magnetic field is divergence free ∇ ·B= 0we can define it as a curl of a potential co-vector A Bi=εijkAj,k orB=∇ ×A (3.0.8.9) substituting ( 3.0.8.9 ) into ( 3.0.8.2 ) or ( 3.0.8.6 ) we find that the curl of the electric field Eequals to the curl of the time derivative of the vector potential −∂A ∂t, hence they differ by an arbitrary gradient term ∇ϕin the following way Ei=−ϕ,i−˙AiorE=−∇ϕ−∂A ∂t(3.0.8.10) we can verify that ϕis the known electrostatic potential by substituting ( 3.0.8.10 ) into ( 3.0.8.3 ) or (3.0.8.7 ) and setting A= 0to find the Poisson’s equation relating the electric potential to a charge distribution ϕ,i ,i=−ρ ε0or∇2ϕ=−ρ ε0(3.0.8.11) in the general case the two source dependent equations takes the form ϕ,i ,i+˙A,i i=−ρ ε0(3.0.8.12) µ0ε0¨Ai−Ai,j ,j+∂i(µ0ε0˙ϕ+Aj ,j) =µ0Ji(3.0.8.13) or ∇2ϕ+∇ ·∂A ∂t=−ρ ε0(3.0.8.14) µ0ε0∂2A ∂t2− ∇2A+∇(µ0ε0∂ϕ ∂t+∇ ·A) =µ0J (3.0.8.15) There is a gauge freedom in the potential formulation, if we transform the potentials by the following transformation ϕ′=ϕ+∂ξ ∂torϕ′=ϕ+˙ξ A′=A− ∇ξorA′i=Ai−ξ,i (3.0.8.16) 17 whereξis arbitrary scalar field, the fields they represented by this potential doesn’t change. There are several important gauge choices, but we will focus only on the Lorenz gauge since it is the covariant gauge choice. The Lorenz gauge condition is µ0ε0˙ϕ+Aj ,j= 0 orµ0ε0∂ϕ ∂t+∇ ·A= 0 (3.0.8.17) using this gauge Maxwell’s equations in the potential formalism is a wave equation µ0ε0¨ϕ−ϕ,i ,i=ρ ε0or (µ0ε0∂2 ∂t2− ∇2)ϕ=ρ ε0(3.0.8.18) µ0ε0¨Ai−Ai,j ,j=µ0Jior (µ0ε0∂2 ∂t2− ∇2)A=µ0J (3.0.8.19) 3.1 Formulation of EM in the 4D Minkowski space 3.1.1 The Minkowski space M Definition 3.1. In special-relativity time is one of the space-time dimensions, and the space is called the Minkowski spacetime, a displacement of an event is xµ= (ct,x,y,z ). The metric of the (flat) Minkowski space is ηµν= 1 0 0 0 0−1 0 0 0 0 −1 0 0 0 0 −1  this metric express the idea of relative motion in it’s formulation which is in the heart of special relativity theory. The metric ηαβin3.1give rise to the following classification of 4-vectors in Minkowski space, letxbe a 4-vector then:   xlike time is ⇐⇒x·x> 0 xlike light is ⇐⇒x·x= 0 xlike space is ⇐⇒x·x< 0(3.1.1.1) 3.1.2 EM in Minkowski space Define the 4-current Jµis a tensor density of weight −1which is written in 1+3D decomposition as Jµ= (cρ,J) (3.1.2.1) and define a covariant rank-2 anti-symmetric Faraday tensor field Fµνas F0j=1 cEj Fij=εijkBk (3.1.2.2) or in a matrix form as Fµν= 01 cE11 cE21 cE3 −1 cE1 0B3−B2 −1 cE2−B3 0B1 −1 cE3B2−B1 0 ;Fµ ν= 01 cE11 cE21 cE3 1 cE1 0−B3B2 1 cE2B3 0−B1 1 cE3−B2B1 0 (3.1.2.3) 18 Using ( 3.1.2.1 ), (3.1.2.2 ) we can express Maxwell’s equations in flat vacuum using two equations: ∂βFαβ=µ0√−gJα F[αβ,λ ]= 0 or∂β(⋆F)αβ= 0(3.1.2.4) whereµ0is the magnetic permeability of the vacuum, and Fαβis the Faraday tensor with raised indexes. In order to reduce the number of the equations we will define a potential which will always satisfy the second equation. We define the 4-potential Aµ, so that Fαβ=∂αAβ−∂βAα= 2∂[αAβ]= 2A[β,α] (3.1.2.5) which satisfies the second equation because the partial derivatives commute, and under an anti-symmetrization they cancel out. In 1+3D decomposition the 4-potential reads Aµ= (ϕ c;−A) (3.1.2.6) the gauge freedom takes a simple form as A′ µ=Aµ+χ,µ (3.1.2.7) Substituting ( 3.1.2.5 ) into the source depended equation we get □Aβ−Aα,β ,α=−µ0Jβ(3.1.2.8) by a gauge choice we can remove one degree of freedom out of the 4-potential. For example the covariant gauge is the Lorenz gauge which states that the divergence of the 4-potential vanishes Aα ,α=A,α α= 0 (3.1.2.9) so Maxwell’s equation for the 4-potential in the Lorenz gauge is □A=−µ0J (3.1.2.10) which is the inhomogeneous wave equation. 4 A scalar invariant under the Lorentz Group 4.1 Short introduction to Matrix Lie groups In this section we will introduce essential aspects of group theory. After that we will define the concept of Lie group and explore matrix Lie groups through construction of the general matrix Lie groupGLn(F). Finally we will develop the matrix Lie algebra as a tool to deal with matrix Lie groups in a linear manner. 4.1.1 Group, subgroup, group action and group homomorphism Definition 4.1. A group is a set, G, equipped with an operation •(called the group operation ofG), that combines any two elements aandbto form another element, denoted a•bor as we prefer by the juxtaposition ab. Any group (G,•)or simplyG, has to satisfy four requirements called the group axioms : 19 Closure For alla,b∈G, the result of the operation ab, is also inG. Associativity For alla,b,c∈G, the following relation holds (ab)c=a(bc). Identity element There exists an element e∈G, so for any a∈G, the relation ea=ae=a holds. Such an element is unique, hence it’s called theidentity element. Inverse element For eacha∈G, there exists b∈Gsuch thatab=ba=e, whereeis the identity element. Example 4.2. The group that consists of the set of integers Zwith the integer addition ”+” as group operation, is a group. We get closure since the addition of any two integers is an integer, associativity since integer addition is associative, the identity element is 0since for any integer zthe sumz+ 0 = 0 +zequalsz, inverse element is exists for any zthere exists −zinZsuch that (z) + (−z) = 0 . This group is also commutative since for any two integers z1,z2the sum is commutative z1+z2=z2+z1. Definition 4.3. A subgroup Sof a groupG, is a group consisting of a set which is subset of the set ofG, and having the same group operation. Example 4.4. The group of all even integers with integer addition as group operation is a subgroup of the group defined in 4.2. Remark. Every group Ghas two trivial subgroups. The subgroup that it’s associated set contains only the group’s identity element e, and the subgroup which is the group Gitself. The idea of group by itself might have useful outcomes but for our purpose as we want to model symmetries and transformations of space, we will define next the concept of group action which explain how a group ”acts” on a set which could be a model of physical space of our interest. Definition 4.5. A group action ϕof a groupGon a setXis a function ϕ:G×X→X that satisfies the following two axioms Compatibility For allg,h∈Gand allx∈X,ϕ(gh,x ) =ϕ(g,ϕ(h,x)), whereghis the result of the group operation on gandh. Identity For allx∈X,ϕ(e,x) =x, hereeis the identity element of G. Example 4.6. The circle group U(1)consists of all complex numbers z∈Chaving unit modulus ∥z∥= 1where the group operation is multiplication of complex numbers. We define an action ϕofu∈U(1)on a point in the Euclidean plane r∈R2as ϕ(u,r) = (xℜu−yℑu,xℑu+yℜu) wherer= (x,y)while ℜuandℑuare the real part and the imaginary part of urespectively. Since for some real θwe can express uasu=eıθ= cosθ+ısinθwe get ϕ(u,r) = (xcosθ−ysinθ,xsinθ+ycosθ) which is the known formula for rotating R2by angleθcounterclockwise around the origin. Definition 4.7. Let two groups (G,⋆),(H,·), a group homomorphism from (G,⋆)to(H,·), is a functionh:G→H, such that for all g1,g2∈G h(g1⋆g2) =h(g1)·h(g2) which is the way of saying that his ”compatible” with the group structure. 20 Example 4.8. Homomorphism from U(1)defined in 4.6toSO(2)defined in 4.25 which is the matrix group of rotations of the Euclidean plane R2around the origin. Let hbe the homomor- phism (which is actually an isomorphism) between U(1)toSO(2)defined by: h(uθ) =(cosθ−sinθ sinθ cosθ) =Rθ whereuθ=eiθ. This is expected indeed since by rearranging the result of 4.6we get: r′=ϕ(uθ,r) =(cosθ−sinθ sinθ cosθ)(x y) =h(uθ)r=Rθr 4.1.2 The general matrix group GLn(F) Definition 4.9. We will note the associative algebra consisting of square matrices of size n×n over the field F(whereFisRorC) byMn(F). Example 4.10. M2(R)the algebra of 2×2real matrices. It is the set M2(R) ={(a b c d) :a,b,c,d ∈R} Definition 4.11. The trace of a matrix A∈Mn(F)is: trA=Ai i which could be also seen as a contraction of the indexes of Ai j. Example 4.12. tr(a b c d) =a+d Definition 4.13. The exponent of a matrix A∈Mn(F)is defined by the series: eA= expA=∞∑ k=0Ak k! where it can be shown that series converges for any matrix in Mn(F). The always converge property will not be proved on this report. Example 4.14. The exponent of θAwhereA=(0 1 1 0) . Notice that A2=Iwe have expθA=∞∑ k=0(θA)k k!=∞∑ k=0( Iθ2k (2k)!+Aθ2k+1 (2k+ 1)!) =Icoshθ+Asinhθ=(coshθsinhθ sinhθcoshθ) Lemma 4.15. The matrix exponent satisfies the following properties: •e0=I •eaXebX=e(a+b)X 21 •eXe−X=I • IfXY =YX(or[X,Y ] = 0 ) theneX+Y=eXeY=eYeX •( eX)t=eXt. whereX,Y∈Mn(F),Xtis the transpose of Xdefined in 2.19,Iand0are the identity matrix and the zero matrix of Mn(F)anda,b∈F. Proof. Item by item proofs: •e0=I+∞∑ k=10k k!=I •eaXebX=∞∑ k=0(aX)k k!∞∑ j=0(bX)j j!=∞∑ k=0k∑ j=0ak−jbjXk (k−j)!j!=∞∑ k=0(a+b)kXk k!=e(a+b)X • Using first two properties together we get eXe−X=e(1−1)X=e0=I. •eX+Y=∞∑ k=0(X+Y)k k!=∞∑ k=0k∑ j=0Xk−jYj (k−j)!j!=∞∑ k=0Xk k!∞∑ j=0Yj j!=∞∑ k=0Yk k!∞∑ j=0Xj j! •( eX)t=(∞∑ k=0Xk k!)t =∞∑ k=0(Xk)t k!=∞∑ k=0(Xt)k k!=eXt Definition 4.16. The adjugate matrix of any matrix A∈Mn(F)is defined by: adj(A)i j=∂detA ∂Aj i where the determinant defined in 2.33 is seen as a differentiable function of n2independent entries ofA. Example 4.17. The adjugate matrix of A=(a b c d) adj(A)i j=∂detA ∂Aj i=∂(A1 1A2 2−A1 2A2 1) ∂Aj i =∂(ad−bc) ∂Aj i=(d−b −c a) Theorem 4.18. The product of any matrix A∈Mn(F)with it’s adjugate matrix adjAisIdetA. Proof. Using 2.33 we find adj(A)i kexplicitly: ∂detA ∂Ak i=∂ ∂Ak i(1 n!εiεkAk i) =1 (n−1)!εii′εkk′Ak′ i′ where i,k,i′andk′are index arrays of sizes n,n,n −1,n−1respectively, since there are nsymbols of Aand each has to be differentiated which gives some δi ˜iδ˜k kwhere ˜iand ˜kare the indexes of the specific Asymbol and since we are moving two indexes on two different 22 Levi-Civita symbols by the same ”distance” we get no sign flip, summing up we get a factor of n. Now substituting the result into 4.18 and using 2.36,2.42 and2.32 we get: Ak jadj(A)i k=1 (n−1)!εii′εkk′Ak jAk′ i′ =1 (n−1)!εii′εji′detA=δi jdetA Lemma 4.19. Any matrix A∈Mn(F)having detA̸= 0 is invertible and given by A−1= 1 detAadjA. Proof. Since by 4.18 we getAadjA=IdetAifdetA̸= 0we divide by detAand see that A(1 detAadjA) =I as required. Lemma 4.20. For any matrix A∈Mn(F)the following holds: deteA=etrA Proof. First we will findddetA(t) dtwhereA(t)is a differentiable function from a real interval into Mn(F), by the chain rule: ddetA(t) dt=∂detA ∂Aj idAj i dt= adj(A)i jdAj i dt= tr( adjAdA dt) ifA(t)is invertible for all tin the interval then by 4.19 we have ddetA(t) dt= detAtr( A−1dA dt) now taking A(t) =etBwhich from 4.15 is invertible for any B∈Mn(F)bye−tB. NowdA dt= AB=BAand we get: ddetetB dt= detetBtr (B) which it’s solution is detetB=ettrB. Corollary 4.21. From 4.20 for anyX∈Mn(R)we get deteX>0, and for any Y∈Mn(C)we getdeteY̸= 0. Definition 4.22. The general matrix group GLn(F)is the most unrestricted subset of Mn(F) defined in ( 4.9) which could be a group where the group operation is matrix product. Lemma 4.23. The general matrix group GLn(F)defined in 4.22 is the subset of all invertible matrices inMn(F)or more concisely GLn(F) ={A∈Mn(F) : detA̸= 0}. 23 Proof. Any non-invertible (singular matrix) cannot be included in GLn(F)since any element in a group must have inverse element. Closure -GLn(F)is closed since the product of two invertible matrices in Mn(F)is an another invertible matrix in Mn(F). Associativity - the matrix product is associative. Identity element - the identity matrix Iexists inGLn(F)since it is an invertible matrix and obey∀X∈Mn(F) :IX=XI=X. Inverse element - all invertible matrices have an inverse which is invertible and hence in GLn(F). 4.1.3 Group representation Definition 4.24. A representation Πof a groupGon a vector space Vover a field F(RorC) is a group homomorphism from GtoGL(V), which is the group of linear invertible maps on V. WhenVis of finite dimension nit is possible to choose a basis for V, using this basis the elements of GL(V)can be written as elements of GLn(F)defined in 4.22. We will always work in the setting of Vbeing finite and will identify GL(V)withGLn(F), as a result we can see the representation Πas a mapping from group elements to matrices, which for any two group elements g1,g2inGhas Π(g1g2) = Π(g1)Π(g2) which means that the matrix multiplication under the representation is ”compatible” with the group operation. This type of representation is called matrix representation . We will deal exclu- sively with matrix representations. Example 4.25. The group SO(2). This is the group of rotations around the origin in the Eu- clidean plane R2with the group operation of compositing the rotations. To each rotation Rin SO(2)an angleθinRcan be assigned, which specifies the angle of rotation counterclockwise. This is a commutative group, since for any two rotations RθandRϕthe composition RθRϕ= RϕRθ=Rθ+ϕis also a rotation. The identity element is the identity transformation which can be viewed as no rotation at all, and the inverse of Rθcan be represented by R−θ. A represen- tation Πfor this group can defined by: Π(Rθ) =(cosθ−sinθ sinθ cosθ) 4.1.4GLn(F)and manifold The groupGLn(F)is not a linear space since for example A,B∈M2(R)whereA=(1 0 0 1) and B=(1 0 0−1) butA−B,A +B̸∈M2(R). This means that GLn(F)does not have a global sense of displacement, and it is very hard to work with non-linear objects. We will now develop an idea of subsets of linear space which are locally linear. This notion will be applicable to GLn(F). By locality we mean that when we ”zoom in” the space is similar to some linear space, which is related to the infinitesimal ”directions” we can ”go” from a point to it’s ”nearby” points. Definition 4.26. A function (or mapping) that has derivatives of all orders is called smooth. 24 Example 4.27. The real exponent function x∈R→ex. Any finite polynomial of maximal degreek,x∈R→Pk(x). The bump function f(x) ={ x∈[−1,1]e1 x2−1 otherwise 0 Definition 4.28. A smooth curve γis a smooth function γ:I→Sfrom an interval I⊂F (whereFisRorC) into a subset S⊂Vof a linear space VoverF. The tangent to the curve γ(t)at some point x∈Sisdγ(t) dt/vextends]ngle/vextends]ngle x=dγ(t) dt/vextends]ngle/vextends]ngle t=t0whereγ(t0) =x. Example 4.29. The pathγ:θ∈(−π,π)→S1, defined as γ(θ) = (cosθ,sinθ), whereS1is the unit circle of R2. The tangent at x= (1,0)isdγ dθ/vextends]ngle/vextends]ngle 0= (0,1). Definition 4.30. A manifold Sof dimension nin a vector space Vof finite dimension m(n≤ m), is a subset of Vsuch that for every point xinSthe set of all tangent vectors 4.28 atxis a linear subspace of Vof dimension n. The collection of all tangent vectors at xis called the tangent space at xand is noted as TxS. Note. An alternative definition for TxSis the space of all directional derivatives at xfor function defined on S. Example 4.31. TheR2unit circleS1. It is the set of all points (x,y)∈R2satisfyingx2+y2= 1. It is clearly not a linear space A= (1,0),B= (0,1)∈S1whileA+B,A−B̸∈S1. But now observe that any point of S1has a tangent, which is a line in R2and thus a 1-dimensional manifold. Theorem 4.32. GLn(F)is a manifold of Mn(F)having at each point a tangent space isomorphic toMn(F). Proof. For anyg∈GLn(F)and anyX∈Mn(F)by4.21γ(t) =getXis inGLn(F), hence the tangent at gisdγ dt/vextends]ngle/vextends]ngle 0=gX. Sincegis invertible then for any Y∈Mn(F)there exists g−1Y∈Mn(F), thus for any Y∈Mn(F)we get that Yis also inTgGLn(F). 4.1.5 Matrix Lie groups and Matrix Lie algebras We will define matrix Lie groups in a way which will suffice to our needs. For proper introduc- tion to Lie groups and Lie algebras see [ 5]. Definition 4.33. A matrix Lie group Gis a subgroup of GLn(F)(includingGLn(F)) which is also a manifold 4.30 ofMn(F). Definition 4.34. The matrix Lie algebra gassociated with the matrix Lie group Gis identified as the tangent space at the identity element TeG. It is closed under the Lie bracket [.,.] :g×g→ gwhich is defined for any X,Y∈gas[X,Y ] =XY−YX, where juxtaposition is matrix multiplication. The elements of a Lie algebra are also called generators. Remark. The Lie algebra captures the local structure of the Lie group. This is due to the fact that the matrix exponent is depended on the commutation relations of the matrices in it’s argument. We will deal from now on mostly with Lie algebras of Lie groups and less with the Lie group by themselves. Example 4.35. so(2)the Lie algebra of SO(2)defined in 4.25. We know that this is one parameter group given by Rθ=(cosθ−sinθ sinθcosθ) , so it has one generator Jusing 4.34,4.30 and 4.28 we find: J=dRθ dθ/vextends]ngle/vextends]ngle/vextends]ngle/vextends]ngle θ=0=(−sinθ−cosθ cosθ−sinθ) θ=0=(0−1 1 0) 25 4.2 The Lorentz group, it’s Lie algebra and their representations 4.2.1 The Orthogonal group O(p,q)and it’s Lie algebra o(p,q) The group defined in 4.25 is the rotation group of rotations in the Euclidean space R2, we called itSO(2)which is the Special Orthogonal group of the Euclidean plane, which is a length pre- serving transformation. In this section we will introduce the Orthogonal group in a generalized manner, which preserve the generalized ”length” defined through the metric. Definition 4.36. LetRp,qbeNdimensional metric space equipped with a metric gαβhaving sig- nature (p,q), meaning that in any point the metric is similar to a diagonal martix diag (+1,...,−1) withpentries of +1andqentries of −1, the signature sof the metric equals here to q. Definition 4.37. The orthogonal group O(p,q)is a matrix Lie group 4.33 associated with the space Rp,q4.36, it is the group of all linear transformations that preserves the inner-product of Rp,qinduced by the metric. It’s Lie algebra is denoted by o(p,q). Theorem 4.38. The elements of the orthogonal Lie algebra o(p,q)4.37 are anti-symmetric matrices, and it is of dimension(p+q 2) . It is closed under the matrix commutator. Proof. Let anyxa,ybinRp,qand a linear transformation La′ athen by 4.37 we must have: gabxayb=ga′b′x′a′y′b′=ga′b′La′ aLb′ bxayb sincexaandybare arbitrary we can omit them and get gab=ga′b′La′ aLb′ b usingga′b′to lowerLb′ band contracting with the inverse metric gcbwe arrive at δc a=gcbgab=gcbLa′ aLa′b=La′ aLb a′ in a matrix form this reads I=LLt whereLtis the transpose of Ldefined in 2.19. Now take a smooth curve L(ψ) =eψmwherem∈o(p,q)substituted into LLt=I, and using( eX)t=eXtfrom 4.15 we get at the identity d dψI/vextends]ngle/vextends]ngle/vextends]ngle/vextends]ngle 0= 0 =d dψLtL/vextends]ngle/vextends]ngle/vextends]ngle/vextends]ngle 0=dLt dψL/vextends]ngle/vextends]ngle/vextends]ngle/vextends]ngle 0+LtdL dψ/vextends]ngle/vextends]ngle/vextends]ngle/vextends]ngle 0=mt+m rearranging we find mt=−m as required. Since the elements of o(p,q)are anti-symmetric matrices we have p+qindexes, so the dimension is indeed(p+q 2) the number of possible choices of two different indexes. For anyA,B∈o(p,q)check that also [A,B ]∈o(p,q)we get: [A,B ]t= (AB−BA)t=BtAt−AtBt=BA−AB=−[A,B ] as required. 26 Following 4.38 An anti-symmetric tensor of rank 2 in an Ndimensional space has(N 2) inde- pendent components, this is equivalent to choosing the two directions which define the rotation plane, so we can define a generic notion to rotation generators and find their commutation re- lations. Definition 4.39. InNdimensional metric space a generator of rotation in the plane designated by the ordered indexes a,bis Rab=Rµ abν=δµλ abgνλ=δµ agνb−δµ bgνa wherea,bare labels and not indexes or can be viewed as indexes of the matrix of all rotation generators. The result of 4.35 is easily recovered from the general definition 4.39 if we identify Jas R21, and recall that in Euclidean space gab=δab, we get J=R21=δµα 21gνα=δµ 2gν1−δµ 1gν2=δµ 2δν1−δµ 1δν2=(0−1 1 0) as was expected. If we had instead identified JwithR12we would have gotten a rotation in the reversed direction. Since the commutation relations of a Lie algebra determinate it fully we now will find the commutation relations of o(p,q)from the definition 4.39: [Rab,Rcd] =RabRcd−RcdRab =δµα abgναδνβ cdgλβ−δµα cdgναδνβ abgλβ = (δµ agνb−δµ bgνa) (δν cgλd−δν dgλc)−(δµ cgνd−δµ dgνc) (δν agλb−δν bgλa) =gad(δµ bgλc−δµ cgλb) +gbd(δµ cgλa−δµ agλc) +gac(δµ dgλb−δµ bgλd) +gbc(δµ agλd−δµ dgλa) =gadδµα bcgλα+gbdδµα cagλα+gacδµα dbgλα+gbcδµα adgλα and finally using 4.39 we arrive at [Rab,Rcd] =gadRbc+gdbRca+gcaRdb+gbcRad (4.2.1.1) 4.2.2 One dimensional Lorentz transform Consider the Orthogonal group O(1,1)from the signature we get gab= diag(1,−1)using 4.39 we find that the only rotation generator has a matrix form: R10=B=(0 1 1 0) note thatB2=I, the action of Bis eψB=Icoshψ+Bsinhψ=(coshψsinhψ sinhψcoshψ) identifying γ= coshψandβ= tanhψwe get eψB=γ(1β β1) 27 putting it as a transformation from(ct x) to(ct′ x′) we get (ct′ x′) =γ(1β β1)(ct x) =γ(ct−βx x−βt) or just   t′=γ( t+vx c2) x′=γ(x+vt) which is the known formula for Lorentz transform in one dimension, where the relative velocity between frames is vin thex-direction, where the direction of the xaxis in each frame is the reverse of the direction in the other frame. A generators like Bare called boosts generators. 4.2.3 Representation π Note. We will use the same symbol for the representation of the Lorentz group and for it’s Lie algebra representation, except that the Lie algebra symbol will have additional underline. In special relativity we consider a space which has 3 directions, having a rotation around each direction and having a boost in each direction. Since we can get a boost only with a mixture of indexes with opposite signs, we must have a metric with mixed signature (p,q). We can summarize this requirement on the signature (p,q)as: (q 2) = 3 andpq= 3 the result is q= 3,p= 1, thus the Lorentz group of special relativity is O(1,3), and the space is called the Minkowski space having metric ηαβ= diag(1,−1,−1,−1), this is in agreement with what was defined in 3.1above. Note. The property of having the same number of rotations and boosts generators is unique to Minkowki space in the sense that this is the only Lorentzian signature having this property. Definition 4.40. The Lorentz group is represented by O(1,3)it is a six parameter group, it’s Lie algebra o(1,3)has generators: 3 boosts Bi=Ri0and 3 rotations Ri=1 2ϵjk 0iRjk, where Rabis the rotation generators defined in 4.39. Lemma 4.41. The rotation generator Rijdefine in 4.39 is related to the rotation generators Rk defined in 4.40 as Rij=−ϵk 0ijRk Proof. Starting with the definition of Rkin4.40 and multiplying both sides with −ϵk 0ij, and using 2.43 we get: −ϵk 0ijRk=−1 2ϵk 0ijϵab 0kRab=−1 2ϵk 0ijϵ0ab kRab =1 2δ0ab 0ijRab=1 2(Rij−Rji) =Rij as required. 28 Using 4.39 to find a component-wise formulation of the generators from 4.40BjandRj: B1= 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 B2= 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 B3= 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0  R1= 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 0 R2= 0 0 0 0 0 0 0 1 0 0 0 0 0−1 0 0 R3= 0 0 0 0 0 0 −1 0 0 1 0 0 0 0 0 0 (4.2.3.1) The Faraday tensor of mixed rank (1,1)can be written as Fµ ν=1 cEj(Bj)µ ν+Bj(Rj)µ ν(4.2.3.2) in component-wise formulation we get the same result as in ( 3.1.2.3 ) above. Theorem 4.42. The commutation relations of the generators of the rotations Rjand the boosts Bj are: [Ri,Rj] =−ϵk 0ijRk (4.2.3.3) [Bi,Rj] =−ϵk 0ijBk (4.2.3.4) [Bi,Bj] =ϵk 0ijRk (4.2.3.5) Proof. Using ( 4.2.1.1 ),4.39,4.40,2.43 and4.41 for (4.2.3.3 ) we find [Ri,Rj] =1 4ϵab 0iϵcd 0j[Rab,Rcd] =1 4ϵab 0iϵcd 0j(gadRbc+gdbRca+gcaRdb+gbcRad) =−1 4gljgeagfbδ0lcd 0ief( −gadϵk 0bc−gdbϵk 0ca−gcaϵk 0db−gbcϵk 0ad) Rk =1 4glj( −gfbδ0lc 0ifϵk 0bc+geaδ0lc 0ieϵk 0ca+gfbδ0ld 0ifϵk 0db−geaδ0ld 0ieϵk 0ad) Rk =−gljgfbδ0lc 0ifϵk 0bcRk=−( −gljglbϵk 0bi+gijgcbϵk 0bc) Rk=−ϵk 0ijRk and for ( 4.2.3.4 ) we find [Bi,Rj] =1 2ϵcd 0j[Ri0,Rcd] =1 2ϵcd 0j(gidR0c+gciRd0) =−1 2ϵc d 0j(gciBd−gidBc) =−ϵk 0ijBk and finally for ( 4.2.3.5 ) we find [Bi,Bj] = [Ri0,Rj0] =Rji=ϵk 0ijRk Remark. We can see from ( 4.2.3.3 ) that the rotations are closed under the commutator and hence generate a sub-group which is isomorphic to the rotation group SO(3). 29 Another special property of the Lorentz group O(1,3)is that since it’s total dimension is 4the Hodge dual maps the space of rank 2anti-symmetric tensors into itself. For example using 2.44, 4.39,2.45 and ( 4.2.3.6 ) we can calculate the Hodge dual of the boost and rotation generators defined in 4.40 as ⋆Bi=⋆(Ri0) =⋆(δµρ i0gνρ) =1 2ϵη ν λµδµρ i0gνρ =1 2( ϵη λi0−ϵη λ0i) =−ϵη λ0i=−ϵη 0i λ =−1 2ϵjk 0iδηρ jkgλρ=−1 2ϵjk 0iRjk=−Ri(4.2.3.6) ⋆Ri=⋆⋆Bi=Bi (4.2.3.7) summarizing ( 4.2.3.6 ) and ( 4.2.3.7 ) we get ⋆Bj=−Rj ⋆Rj=Bj(4.2.3.8) We are trying to find an invariant scalar under the Lorentz group which will be made out of the null-displacement connecting an event and the observer. We know that the only thing that the Lorentz group preserves is the inner-product, but since the displacement is null, it’s inner product yields 0 which is a useless trivial invariant. Since the Hodge dual induces an automorphism for the Lorentz generators (rank 2 anti- symmetric tensors), using the Hodge duality we might get a finer structure of the Lorentz group which will preserve non-trivial invariants. In order to do so we investigate the Hodge dual as linear operator on a vector space, where the vector space is the space of the rank 2 anti-symmetric tensors and try to find it’s eigenvalues and eigenvectors. Let Fbe a rank 2 anti-symmetric tensor which is an eigenvector of the Hodge dual operator having eigenvalue λ̸= 0thus ⋆F=λF applying the Hodge dual we find ⋆⋆F =⋆λF =λ⋆F =λ2F (4.2.3.9) on the other hand from 2.45 the square of the Hodge dual on rank 2 tensors in Minkowski space is the same tensor multiplied by −1. Substituting into ( 4.2.3.9 ) and omitting the tensor since it is arbitrary we get a condition on the eigenvalues λas λ2=−1 (4.2.3.10) clearly this equation does not have a solution under the reals so it cannot have a solution in the real Minkowski space. A remediation to this problem will be presented in the next section, which will introduce the complexified Minkowki space which will allow us to solve ( 4.2.3.10 ). 4.2.4 Complexified Minkowski space and the σ-dual subspaces o±ofo(1,3) Definition 4.43. The complexified Mankowski space Mcis a natural complex extension of the real Minkowski space 3.1, thus it is a space of 4 complex dimensions equipped with a metricηαβ= diag(1,−1,−1,−1)that acts in the same manner on the real and the imaginary parts of the vectors. The inner-product of the complex 4-vectors xandyis defined through x◦y=ηαβxαyβ=ηαβxβyα. 30 Now in the context of the complexified Minkowski space 4.43 we can find a solution to ( 4.2.3.10 ), since the complexified space Mcis a natural extension of the real Minkowski space Mall of the derivations done above for Mwill hold also in Mc. Now the eigenvalues under Mccan be found as λ=±ı (4.2.4.1) The space of rank 2 anti-symmetric tensors which coincides with o(1,3)(orofor short) theπLie algebra representation of the real Lorentz Lie algebra, has by ( 4.2.4.1 ) two different eigenvalues for the Hodge dual operator. This means that the matrix space ohas two sub-spaces each associated with one of the two eigenvalues, we define these two sub-spaces of oaso+and o−. Definition 4.44. The sub-space o±ofois the subspace consisting of eigenvectors of the Hodge dual operator having the eigenvalue ±ı. It is called a σ-dual subspace and it’s elements are calledσ-dual, where σtakes the value ±ı. Lemma 4.45. Theo±sub-spaces 4.44 are related through the complex conjugation as o∗ ±=o∓. In detail for any Fino±thenF∗is ino∓. Proof. For anyFino±by definition we have: ⋆F=±ıF taking the complex conjugation of this equation we get ⋆F∗=∓ıF∗ as required for any Fino±,F∗is ino∓. Theorem 4.46. The projection operators from ointoo±denoted asP±are given by P±=1 2(1∓ı⋆) Proof. Validating that the projection operators 4.46 indeed project into the correct sub-space ⋆P±=P±⋆=⋆1 2(1∓ı⋆) =1 2(±ı+⋆) =±ı1 2(1∓ı⋆) =±ıP± (4.2.4.2) which agrees with the definition of o±4.44. Now validate that the projection operators 4.46 are indeed projections (P±)2=1 4(1∓ı⋆)2=1 4( 1∓2ı⋆−⋆2) =1 2(1∓ı⋆) =P± which is the defining property of a projection operator. Note. We can show that the sub-spaces o±are disjoint since P±P∓=1 4(1∓ı⋆) (1±ı⋆) =1 4( 1 +⋆2) ≡0 Lemma 4.47. Any element in ocan be decomposed as a unique sum of element in o+and an element ino−. 31 Proof. AnyFinohas unique components P±Fino±, since from 4.46 we get: P++P−=1 2(1−ı⋆) +1 2(1 +ı⋆) = 1 (4.2.4.3) thus we have a unique composition for Fas: F= (P++P−)F=P+F+P−F 4.2.5 Representations ˜π+and ˜π− Using ( 4.2.3.8 ) the projections of the generators BiandRidefine in 4.40 can be calculate as: ˜B± i=P±Bi=1 2(Bi±ıRi) ˜R± i=P±Ri=1 2(Ri∓ıBi) =∓ı˜B± i(4.2.5.1) Theorem 4.48. The projection of the generators calculated in (4.2.5.1 )˜B± iand ˜R± i, each of the two sets induced by the different projection operators P±obey the same commutation relation 4.42 as the generators defined in 4.40, and the two sets completely commutes. Proof. Using 4.42 and ( 4.2.5.1 ) the first commutation relation is [˜R± i,˜R± j] =1 4[Ri∓ıBi,Rj∓ıBj] =1 4([Ri,Rj]−[Bi,Bj]∓2ı[Bi,Rj]) =−ϵk 0ij1 2(Rk∓ıBk) =−ϵk 0ij˜R± k as requried, since by ( 4.2.5.1 ) we have ˜R± i=∓ı˜B± i, the other two relations follows trivially. As for the commutation again using 4.42 and ( 4.2.5.1 ) we get [˜R± i,˜R∓ j] =1 4[Ri∓ıBi,Rj±ıBj] =1 4([Ri,Rj] + [Bi,Bj]) = 0 and again as before as consequence of ˜R± i=∓ı˜B± i, all the other combinations follows trivially. Next we introduce a very useful property of the Lorentz generators in ˜π±which ease dras- tically the calculation of the exponent map, thus finding the resulting transformation. Lemma 4.49. The projections (4.2.5.1 )of theπgenerators satisfy the relation {˜R± i,˜R± j} =1 2ηijI whereηis the Minkowski metric, {,}stands for the anti-commutator {A,B}=AB+BA, andI stands for the 2-rank delta δµ ν. 32 Proof. First we find ˜R±µ i ν˜R±ν j λusing 4.39,4.40 and2.43 ˜R±µ i ν˜R±ν j λ=1 4( ϵµ 0i ν∓ıδµα i0ηνα)( ϵν 0j λ∓ıδνβ j0ηλβ) =1 4( −ϵµ 0νiϵν 0jλ−δµα i0ηναδνβ j0ηλβ∓ı( δµα i0ηναϵν 0j λ+ϵµ 0i νδνβ j0ηλβ)) =1 4( δρπµ 0jληπiηρ0+δµ 0ηjiηλ0+δµ iηλjη00∓ı( −δµ 0ϵ0jiλ+ϵµ 0i jηλ0)) =1 4( δµ ληji−δµ 0ηλ0ηji−δµ jηλi+δµ iηλj+δµ 0ηjiηλ0∓ı( −δµ 0ϵ0jiλ+ϵµ 0i jηλ0)) =1 4( δµ ληji+δµ iηλj−δµ jηλi∓ı( −δµ 0ϵ0jiλ+ϵµ 0i jηλ0)) all terms except δµ ληjiare anti-symmetric in i,jand thus under the anti-commutator vanishes, the termδµ ληjiis symmetric in i,jand hence get multiplied by 2 under the commutator, and sinceδµ ν≡Iµ ν, we get {˜R± i,˜R± j} =1 42δµ ληji=1 2ηijI as required. Lemma 4.50. The ˜B±generators projections (4.2.5.1 )obey: ˜B±αβ i˜B± jαβ=ηij ˜B±αβ i˜B∓ jαβ= 0 Proof. Using 4.49 we get: −1 2ηijδµ λ=˜B±µ i ν˜B±ν j λ+˜B±µ j ν˜B±ν i λ contracting on µandλwe get: −2ηij=˜B±λ i ν˜B±ν j λ+˜B±λ j ν˜B±ν i λ= 2˜B±αβ i˜B± jβα=−2˜B±αβ i˜B± jαβ as required. Now using ( 4.2.5.1 ) and 2.43 we get: ˜B±αβ iηαα′ηββ′˜B∓α′β′ j =1 4( δαβ i0±ıϵαβ i0) ηαα′ηββ′( δα′β′ j0∓ıϵα′β′ j0) =1 4( 2ηijη00−2η0iηj0+ϵαβ i0ϵj0αβ+ 2ı[ϵi0j0−ϵj0i0]) =1 4(2ηijη00−2ηijη00+ 2ηi0η0j) = 0 as required. Corollary 4.51. From 4.50 we see that we can write the projection operators 4.46 in coordinates using as a basis the projected generators (4.2.5.1 )as: Pαβ ±µν=ηij˜B±αβ i˜B± jµν Definition 4.52. We define a representations of the Lorentz group ˜π±induced by the Lie alge- bra˜π±which coincides with o±the eigen-subspace of o(1,3)under the Hodge operator. These are indeed a representations since by 4.48 each subspace o±obey the same commutation rela- tions as o(1,3)and has the same dimension, more over also by 4.48,˜π±and ˜π∓commutes. 33 4.2.6 The projections of the Faraday tensor In this section we explore the separation of πinto ˜π±, this in order to find the necessary details of suitable definition of a Faraday tensor for each ˜π±representation. Using ( 4.2.3.2 ), (4.2.5.1 ) to calculate the projections of the Faraday tensor ( 3.1.2.2 ) we get F±=P±F=P±(1 cEjBj+BjRj) =(1 cEj∓ıBj) ˜B± j (4.2.6.1) Following ( 4.2.6.1 ) we introduce the Riemann–Silberstein vector Fj ±=1 cEj∓ıBjorF±=1 cE∓ıB (4.2.6.2) this is indeed a vector since Bjis a pseudo-vector and ıis a pseudo-scalar their product is a proper vector and can be added to Ej. Now using this we can write F±=Fj ±˜B± j (4.2.6.3) Since the projection defines a unique decomposition 4.47 we have: F=F++F−(4.2.6.4) Now after we got the separation into ˜π±we first have to prove that the separation does make sense. Meaning that each ˜π±representation is ”on it’s own” and is indeed compatible with π, this is in the sense that we can define a relation between πand ˜π±which will be useful and preserve covariance in a compatible way. 4.2.7 Lorentz covariance relation of πand ˜π± Consider the exponent map Λ =eFof some generator F∈π, using the decomposition ( 4.2.6.4 ) and the commutativity 4.48 and the commutation property of the matrix exponent map that if [A,B ] = 0 theneA+B=eAeB=eBeA4.15 to get: Λ =eF=eF++F−=eF+eF−=eF−eF+= Υ+Υ−(4.2.7.1) where F±∈˜π±are defined in ( 4.2.6.4 ) and Υ±=eF±. In order to ease calculations we use F±=Fj ±˜B± jfrom ( 4.2.6.3 ) to find the square of the Faraday tensor projections, using also the anti-commutation relations 4.49 and4.50, we get (F±)2= (Fj ±˜B± j)2=F2 ± 4I (4.2.7.2) whereIis the identity matrix and F2 ±=−ηijFi ±Fj ±is the norm of the Riemann–Silberstein vector. This result is a wonderful, because, we see that the square of the projection of the Faraday tensor is an identity matrix multiplied by scalar. This will ease the calculation of the exponent tremendously as the series of the exponent will now have just two ”matrix” values. For odd terms that will be the identity matrix. For even terms that will be the projection of the Faraday tensor. 34 To ease calculation further we can find a complex scalar η±where (η+)∗=η−such that η2 ±= (F±)2, multiplying both sides of ( 4.2.7.2 ) by( 2 η±)2 we get (2F± η±)2 =I (4.2.7.3) we just defined from F±a tensor that will square to the unit matrix, now we are ready for some calculations. Using ( 4.2.7.1 ) and ( 4.2.7.3 ) to calculate Υ±, we get Υ±=eF±=eη± 22F± η±=Icosh(η± 2) +2F± η±sinh(η± 2) (4.2.7.4) Putting the result for Υ±(4.2.7.4 ) back into Λ =eF(4.2.7.1 ) we find Λ = Υ+Υ− =( Icosh(η+ 2) +2F+ η+sinh(η+ 2))( Icosh(η− 2) +2F− η−sinh(η− 2))(4.2.7.5) which is indeed real, and we see clearly that the computation in the half-representation eased the calculation of the whole πrepresentation. Theorem 4.53. The covariance of the Faraday tensor Funderπis equivalent to the covariance of the projections of the Faraday tensor F±under ˜π±. Proof. Take any Λ∈πwhich is an exponent map of generator in πby (4.2.7.1 ) there exists Υ±∈˜π±which is an exponent map of a generator in ˜π±such that Λ = Υ+Υ−and(Υ+)∗= Υ−, now for any F∈π, the transformation Λsends it toF′asF′= ΛFΛ−1, using the decomposition (4.2.6.4 ) and the commutation 4.49 we find out: F′= ΛFΛ−1= Υ+Υ−( F++F−)( Υ+Υ−)−1 = Υ+F+( Υ+)−1+ Υ−F−( Υ−)−1=F′++F′− as required. 4.2.8 Complexification of the 4-potential The projections of the real Faraday tensor into ˜π±(byP±) as defined in ( 4.2.6.1 ) are F± αβ=P±Fαβ=P±2A[β,α] whereAβis the 4-potential of the EM field Fαβas in ( 3.1.2.5 ). Consider a complex co-vector fieldA± µsuch that F± αβ=P±2A± [β,α]. Using ( 4.2.4.2 ) we get: F± αβ=P±2A± [β,α]=P±( 2ℜA± [β,α]+ 2ıℑA± [β,α]) =P±( 2ℜA± [β,α]±2⋆ℑA± [β,α]) , (4.2.8.1) taking the real part of ( 4.2.8.1 ) we get Fαβ= 2ℜA± [β,α]±2⋆ℑA± [β,α]. (4.2.8.2) 35 This relation is the Cabibbo-Ferrari-Shanmugadhasan relation [ 2] used when considering mag- netic charges (magnetic monopoles). Question arises, when does F± αβ= 2A± [β,α]? more precisely when 2A± [β,α]∈o±? This is equivalent to stating that: P∓2A± [β,α]= 0. (4.2.8.3) In such cases we will call A±aσ-dual complex 4-potential . IfA±is aσ-dual then 2ℜA± [β,α]= ±2⋆ℑA± [β,α]and hence Fαβ=±4⋆ℑA± [β,α]. Now substituting into the Maxwell’s equation (3.1.2.4 ), using Hodge dual definition 2.44 and the anti-symmetry of the Levi-Civita tensor we get µ0√−gJα=∂βFαβ=±4∂β⋆ℑA±[β,α]=±2∂βϵαβµν∂µℑA± ν= 0. This implies that: Proposition 4.54. Outside of the sources and only there it is possible to have a σ-dual complex 4-potential. In this case 4ℜA± [β,α]=FαβandℜA± [β,α]=±⋆ℑA± [β,α]. Note. In [1] a source free potentials inducing Hodge dual fields are used. If our point of interest xis inside the sources J(x)̸= 0, substituting ( 4.2.8.2 ) the Maxwell’s equation ( 3.1.2.4 ) we get: ∂βFαβ=ℜA±β,α ,β−□ℜA±α=µ0√−gJα(4.2.8.4) ∓∂β(⋆F)αβ=ℑA±β,α ,β−□ℑA±α= 0 (4.2.8.5) wheregis the determinant of the metric and µ0is the magnetic permeability of the vacuum. This implies that: Proposition 4.55. Inside the sources the real part of any complex 4-potential coincides with the real 4-potentialAup to some source free potential AsfasℜA±=A+Asf. And the imaginary part ℑA± is a source free potential. 4.2.9 Prepotential We will define for the complex 4-potential A±a prepotential, denoted as S±. We cannot sim- ply relate A±to the prepotential as A± µ=S±,µwhich is a gauge potential, since the partial derivatives commute it would imply that F±≡0which is totally useless. To remedy this we will use a conjugation C±= 2cj˜B± j, for some given constants cjsatisfying c2=−ηijcicj= 1 implying (C±)2=I. The conjugation is an anti-symmetric tensor in o±. The relation between the complex 4-potential and the prepotential is given by: A± µ=C∓λ µS±,λ. (4.2.9.1) Lets find the condition on the prepotential inducing a σ-dual 4-potential. Using ( 4.2.8.3 ), 4.51,4.49 and the commutativity of the partial derivatives we get Pαβ ∓µν2A± [β,α]=Pαβ ∓µν2A± β,α= 4ηij˜B∓ jµν˜B∓αβ ick˜B∓λ k βS±,λα = 2ηij˜B∓ jµνck( ˜B∓αβ i˜B∓λ k β+˜B∓αβ k˜B∓λ i β) S±,λα =−ηij˜B∓ jµνckηikηαλS±,λα=−( cj□S±)˜B∓ jµν =−1 2C∓ µν□S±,(4.2.9.2) 36 which implies that Pαβ ∓µν2A± [β,α]= 0if and only if □S±= 0. Thus, we have shown Proposition 4.56. A prepotential S±has aσ-dual 4-potential if and only if it satisfies the wave equation □S±= 0. 4.3 The ˜π±invariant ζ±-factor 4.3.1 The eigenvectors of ˜B± i By lemma 4.49 the matrices K± i= 2˜B± iobeys K±µ i αK±α i ν=δµ ν (4.3.1.1) by having their square equals to the identity the K± idoes not have zero eigenvalues, moreover any eigenvector vµofK± iwith eigenvalue λsatisfies K±µ i νvν=λvµ(4.3.1.2) using ( 4.3.1.2 ) and ( 4.3.1.1 ) we get: vµ=δµ νvν=K±µ i αK±α i νvν=λK±µ i αvα=λ2vµ hence any eigenvalue λofK± iwe have: λ2= 1 (4.3.1.3) which means λis+1or−1. Using ( 4.2.5.1 ) and 4.39 we rewrite ( 4.3.1.2 ) as ( δµα i0gνα±ıϵµ 0i ν) vν=λvµ using a cyclic index set i,j,k we can reformulate as system of equations λv0=vi λvi=v0 λvj=∓ıvk λvk=±ıvj(4.3.1.4) Using ( 4.3.1.4 ) we define a basis of the complexified Minkowski space Mcout of the eigenvec- tors ofK± ias ni ±=1√ 2(1 ±i) ;mi ±=1√ 2(0 j±ık) (4.3.1.5) where i,j,kis set of ordered orthonormal vectors associated with the cyclic index set i,j,k , the only non-zero inner-products are ni +·ni −=−mi +·mi −= 1 (4.3.1.6) and under complex conjugation we have ( ni ±)∗=ni ±( mi ±)∗=mi ∓(4.3.1.7) using ( 4.3.1.4 ) and ( 4.3.1.5 ) we find: K+ knk ±=±nk ± K+ kmk ±=±mk ±K− knk ±=±nk ± K− kmk ±=∓mk ±(4.3.1.8) 37 Theorem 4.57. TheK± iandK± jmatrices defined in (4.3.1.1 )act onnk ±andmk ±defined in (4.3.1.5 )as: K± ink +=mk ∓ K± ink −=mk ±K± imk +=nk ∓ K± imk −=nk ±K± jnk +=±ımk ∓ K± jnk −=∓ımk ±K± jmk += +ınk ∓ K± jmk −=−ınk ± Proof. Using ( 4.3.1.5 ) and ( 4.3.1.8 ) we prove the first relation: K± ink +=K± i1 2( ni ++ni −−ımi ++ımi −) =1 2( ni +−ni −∓ımi +∓ımi −) =mk ∓ as required. The rest of the relations in 4.57 follows in the same manner. 4.3.2 The null basis In a similar way to the treatment of spinors in 3D, we choose an arbitrary direction k, out of a positively oriented orthonormal 3D basis (i,j,k). Using this basis we can construct a null basis of the complexified Minkowski space Mc, written in 1+3D decomposition as: n=nk +=1√ 2(1 k) ; ˜n=nk −=1√ 2(1 −k) m=mk +=1√ 2(0 i+ıj) ;m∗=mk −=1√ 2(0 i−ıj), (4.3.2.1) wherenk ±,mk ±was defined in ( 4.3.1.5 ) above. This basis is also called a Bondi null-tetrad [ 3], or Newman-Penrose [ 4] basis. The only non-zero inner-product relations of the null basis ( 4.3.2.1 ) elements n,˜n,mand m∗as required by ( 4.3.1.6 ) are: n·˜n=−m·m∗= 1. (4.3.2.2) Any four-vector x=xµeµcan be decomposed in the null basis ( 4.3.2.1 ) as: x=An+D˜n+Bm +Cm∗. (4.3.2.3) Using ( 4.3.2.2 ) and ( 4.3.2.3 ) we find that the relation between the parameters A,B,C,D in (4.3.2.3 ) and the inner products of the null basis with vector xis A=x·˜n D =x·n B=−x·m∗C=−x·m. (4.3.2.4) Now substituing ( 4.3.2.4 ) into ( 4.3.2.3 ) any four-vector xis represented in the null basis ( 4.3.2.1 ) as x= (x·˜n)n+ (x·n)˜n−(x·m∗)m−(x·m)m∗. (4.3.2.5) 4.3.3 The matrix associated with 4-vector For any 4-vector xusing ( 4.3.2.3 ) and ( 4.3.2.2 ) we find the inner-product with itself as: x·x= (An+D˜n+Bm +Cm∗)·(An+D˜n+Bm +Cm∗) = 2 (AD−BC) = 2 det(A B C D) = 2 detX. (4.3.3.1) 38 Definition 4.58. We define Ψas the invertible linear mapping form any 4-vector xto a matrix Xgiven in ( 4.3.3.1 ). And also sends the inner-product as Ψ(x·y) = det (Ψ(x+y))−det Ψ(x)− det Ψ(y). We easily see that under this definition 4.58 the following identification relations hold: Ψ(n) =( 1 0 0 0) ; Ψ(m) =( 0 1 0 0) ; Ψ(m∗) =( 0 0 1 0) ; Ψ(˜n) =( 0 0 0 1) . (4.3.3.2) Lemma 4.59. The complex conjugate operation on any 4-vector xin the complexified Minkowski spaceMc, is equivalent to the conjugate transpose operation on it’s associated matrix X= Ψ(x)by 4.58 as: Ψ(x∗) =X† where the†is the conjugate transpose operation on the matrices (without notion of metric). Proof. Using ( 4.3.2.3 ) and 4.58 Ψ(x∗) = Ψ (A∗n+D∗˜n+B∗m∗+C∗m) =(A∗C∗ B∗D∗) =(A B C D)† =X† Remark. Following 4.59 any real 4-vector x=x∗the matrix X= Ψ(x)is hermitian X=X†. Now we want to describe the action of Lorentz representations ˜π±by the matrix notation as multiplication by matrices from M2(C), the space of 2×2complex matrices. We begin by introducing the Pauli matrices. Definition 4.60. The Pauli matrices are the following hermitian M2(C)matrices: σi=(0 1 1 0) ;σj=(0−ı ı0) ;σk=(1 0 0−1) (4.3.3.3) where the cyclic index set i,j,k are associated with the ordered orthonormal vectors i,j,k. Lemma 4.61. ApplyingK+ l(K− l) defined in (4.3.1.1 )on any 4-vector xis equivalent to the ap- plication of a respective Pauli matrix σl4.60 on the left (right) of X= Ψ(x)defined in 4.58, or simply: Ψ( K+x) =σlX Ψ( K−x) =Xσl Proof. The lemma 4.61 is direct outcome of 4.58, (4.3.3.2 ), (4.3.1.8 ) and 4.57. Lemma 4.62. The Pauli matrices σldefined in 4.60 satisfies the commutation and anti-commutation relations: [σa,σb] =−2ıϵc 0abσc {σa,σb}=−2ηabI Moreover they are also hermitian σ† l=σl. 39 Proof. Using 4.61,4.42,4.48 and4.49 we get [σa,σb]X= Ψ([ K+ a,K+ b] x) = Ψ( 4[˜B+ a,˜B+ b] x) = Ψ( 4ϵc 0ab˜R+ c) = Ψ( −2ıϵc 0abK+ cx) =−2ıϵc 0abσcX sinceXis arbitrary we can omit it and get the desired result. Now as for the the anti-commutation relation: {σa,σb}X= Ψ({ K+ a,K+ b} x) = Ψ( −4{˜R+ a,˜R+ b} x) = Ψ (−2ηabx) =−2ηabX sinceXis arbitrary we can omit it and get the desired result. As for the hermitian property: Ψ(( K+ lx)∗) = Ψ( K− lx∗) = (σlX)†=X†σ† l=X†σl sinceX†is arbitrary we can omit it and get the desired result. Lemma 4.63. The action of Υ+(Υ−) defined in (4.2.7.4 )on any 4-vector xis equivalent to the action ofU(U†) onXas Ψ( Υ+x) =UX Ψ( Υ−x) =XU† whereU=e1 2Fl +σland theFl +is the Riemann–Silberstein vector defined in (4.2.6.2 ). Proof. Using ( 4.2.7.4 ), (4.2.6.3 ) andK± l= 2˜B± lwe find Υ±=( Υ∓)∗=eFl ±˜B± l=Icosh(η± 2) +Fl ±K± l η±sinh(η± 2) by4.61,4.58 and4.62 we get Ψ( Υ+x) =( Icosh(η+ 2) +Fl +σl η+sinh(η+ 2)) X=e1 2Fl +σlX=UX Ψ( Υ−x) =X( Icosh(η− 2) +Fl −σl η−sinh(η− 2)) =Xe1 2Fl −σl=XU† Lemma 4.64. The matrices U,U†from 4.63 have unit determinant. detU= detU†= 1 Proof. Since Υ±are part of the orthogonal group they preserve the inner-product of any two 4-vectors, while specifically using 4.58: x′·x′=x·x↔det Ψ(x′) = det Ψ(x) and following separately for UandU†we get detX′= det (UX) = detUdetX= detX detX′= det( XU†) = detXdetU†= detX which leads directly to detU= detU†= 1as expected. 40 Theorem 4.65. The action of any Λinπdefined in (4.2.7.5 )on a four vector xis equivalent to the action ofUandU†onXby Ψ (Λx) =UXU† where Λ∈πis defined as an exponent map of a generator F∈πas in (4.2.7.1 ), andU,U†are defined as in 4.63 Proof. Since by ( 4.2.7.1 ) and 4.48 the action of Λcan be split into action of Υ±and afterwards Υ∓, using for each action 4.63 we get the desired result. Remark. From 4.65 and4.59 we can see that the full group πpreserves the real and imaginary parts of the complexified Minkowski space Mc, since X′†=( UXU†)†=UX†U† thus the (anti)-hermitian property of X′depends solely on X. 4.3.4 Decomposition of null-vectors in the null-basis Using ( 4.3.2.5 ) and ( 4.3.2.2 ) for any null-vector r= (∥r∥,r), we get r·r= 2(r·˜n)(r·n)−2(r·m∗)(r·m) = 0 (4.3.4.1) rearranging we find out 2(r·˜n)(r·n) = 2(r·m∗)(r·m) we define a scalar ´ϱas ´ϱ=√ (r·˜n)(r·n) =√ (r·m∗)(r·m) (4.3.4.2) we can now satisfy ( 4.3.4.1 ) generally using a decomposition for a realras r= ´ϱ( eθn+e−θ˜n+e−ıφm+eıφm∗) (4.3.4.3) using ( 4.3.2.2 ) we can extract from ( 4.3.4.3 )θandφ: θ=1 2lnr·˜n r·n= lnr·˜n ´ϱ=−lnr·n ´ϱ(4.3.4.4) φ=−ı 2lnr·m r·m∗=−ıln−r·m ´ϱ=ıln−r·m∗ ´ϱ(4.3.4.5) 4.3.5 Local orthonormal basis, and it’s null-vector decomposition Following ( 4.3.4.3 ) we define on the light-cone a reallocal orthonormal basis( Θ,Φ,˜Φ,˜Θ) as Θ =1√ 2( eθn+e−θ˜n) = (coshθ;ksinhθ) Φ =1√ 2(eıφm∗+e−ıφm) = (0; icosφ+jsinφ) ˜Φ =ı√ 2(eıφm∗−e−ıφm) = (0; −isinφ+jcosφ) ˜Θ =1√ 2( eθn−e−θ˜n) = (sinhθ;kcoshθ). (4.3.5.1) It is useful to note that the basis vectors ( 4.3.5.1 ) are depended on θorφand related by differ- entiation by θorφas: ˜Θ = Θ,θΘ = ˜Θ,θ ˜Φ = Φ,φΦ =−˜Φ,φ. (4.3.5.2) 41 We can view the null basis ( 4.3.2.1 ) as depended on the orthonormal basis ( 4.3.5.1 ) as: n=e−θ √ 2( Θ + ˜Θ) m=eıφ √ 2( Φ +ı˜Φ) m∗=e−ıφ √ 2( Φ−ı˜Φ) ˜n=eθ √ 2( Θ−˜Θ). (4.3.5.3) Now substituting ( 4.3.5.3 ) into ( 4.3.4.3 ) we can decompose any null vector ralso as: r=ϱ(Θ + Φ) =ϱˆr, (4.3.5.4) whereϱ=√ 2´ϱ, and ( 4.3.5.4 ) can be viewed as a ”length” and ”direction” of rin a bi-polar decomposition. 4.3.6 The invariant ζ±-factor Definition 4.66. Letζ:N → C, be called the ζ-factor as a function from the null-cone to the complex plane, for any chosen null-basis ( 4.3.2.1 ), and a null-vector r∈ N, by usingθ(4.3.4.4 ) andφ(4.3.4.5 ) derived form ( 4.3.4.3 ) as ζ±=ζ±(r) =eθ∓ıφ=−˜n·r m∓·r. Theorem 4.67. Theζ±-factor 4.66 is invariant under ˜π±respectively. Proof. Sinceris null then its related matrix R= Ψ(r)as by 4.58 must have zero determinant detR≡0, since zero determinant means linearly dependent lines/rows then R can be decom- posed as an outer product of a two complex component column and row R=[w v] ⊗[u∗z∗] =(wu∗wz∗ vu∗vz∗) . By employing only one of ˜π±by4.63 we get that under ˜π+only the column is transformed, and under ˜π−only the row is transformed (more accurately they are transformed by the identity transformation) as UR=U[w v] ⊗[u∗z∗] =[w′ v′] ⊗[u∗z∗] RU†=[w v] ⊗[u∗z∗] U†=[w v] ⊗[u′∗z′∗]. Hence under ˜π+the fractionu∗ z∗is left invariant and under ˜π−the fractionw vis left invariant. Since by 4.58 and ( 4.3.2.5 ) we have R=(wu∗wz∗ vu∗vz∗) = Ψ(r) =(r·˜n−r·m∗ −r·m r ·n) . Using 4.66 we find out: u∗ z∗=wu∗ wz∗=−˜n·r m∗·r=ζ+ w v=wu∗ vu∗=−˜n·r m·r=ζ−. as required. 42 Note. Theζ±-factor is also conformally invariant since it depends on the ratio of two inner- products which are evaluated in the same point in space-time. Note. In the proof of theorem 4.67 we used complex 2-component row and column vectors, as an outer product decomposition of the matrix related to null vector, and each of them was acted upon deferent half-spin representation ˜π±while the other remained invariant. These 2- component vectors are actually objects by their own right called 2-spinors, for an introduction about 2-spinors see [ 6]. 5 The prepotential of the EM field 5.1 The prepotential of a point charge Note. We will work in this section under ˜π−, hereζis alwaysζ−andFis always F−. Definition 5.1. Letxbe a an observer location in Minkowski space M, lety(τ)be the world-line of a point charge and τit’s proper time. Define the past pointing null-sight/light-like 4-vector from the observer to the point charge as: r(x) =x−y(τ(x))wherer·r≡0 Definition 5.2. The prepotential at x∈Mof point particle with charge qis defined as the logarithm of the ζ−-factor 4.66 of the null-vector rdefine in 5.1 S(x) =q 8πϵ0clnζ−(r(x)) =q 8πϵ0c(θ(r(x)) +ıφ(r(x))) whereϵ0is the electric permittivity from the maxwell equation ( 3.0.8.3 ) , andcis the speed of light in vacuum. The units of the prepotential areJ·sec Cso when differentiating twice we get N C·m secwhich are the units of the Faraday tensor. Note. The termq 4πϵ0cin5.2willnotbe present in the derivations in the rest of the report for simplicity, and we will use instead S=1 2lnζ−(r) =1 2(θ(r) +ıφ(r)). 5.2 The gradient of the prepotential By (4.2.9.1 ) the 4-potential has to be a conjugation of the gradient of the prepotential ( 5.2.2.1 ). To calculate the gradient of the prepotential, which depends on rthe null-vector 5.1, which in turn depends on τthe parameter of the world line of the charge, each of which we have to find how to differentiate. We will start by finding the gradient of τ. Using 5.1we find the gradient of τ 0 =∂µ(r·r) = 2rνrν,µ = 2rν(xν,µ−yν(τ(x)),µ) = 2rν(gνµ−yν,ττ,µ). Definingu≡y,τ, rearranging and dividing by r·u, we finally arrive at τ,µ=rµ r·u. (5.2.0.1) Now using ( 5.2.0.1 ) we calculate the gradient of rdefined in 5.1we get: rµ ,ν=δµ ν−uµrν(r·u)−1. (5.2.0.2) 43 Since the prepotential 5.2is depended on the same θandφparameters which the local orthonormal basis ( 4.3.5.1 ) was defined with. We will develop derivatives of θandφusing this basis. Later it will be a very useful calculation tool. We use the decomposition of null vector ( 4.3.5.4 ) by the local orthonormal basis ( 4.3.5.1 ) so in this basis ( 5.2.0.2 ) the null vector ris written as rµ ,ν=ϱ,ν(Θµ+ Φµ) +ϱθ,ν˜Θµ+ϱφ,ν˜Φµ(5.2.0.3) we will use this which is equal to ( 5.2.0.2 ) to find the gradients of the null parameters θ,φand ϱ. Lemma 5.3. The gradients of θ,φandlnϱare given by: θ,ν= 2uα r·u˜Θ[αˆrν] (5.2.0.4) φ,ν= 2uα r·u˜Φ[αˆrν] (5.2.0.5) (lnϱ),ν= 2uα r·uΘ[νΦα] (5.2.0.6) Proof. Using both ( 5.2.0.2 ) and ( 5.2.0.3 ) and calculating the inner product of each equation with a different basis vector of ( 4.3.5.1 ), we can separate each parameter and find it’s gradient. Contracting with Θµwe get: ϱ,ν= Θµ( δµ ν−uµrν(r·u)−1) = Θν−rνΘ·u r·u =uα r·u(Θνrα−rνΘα) =ϱuα r·u(ΘνΦα−ΦνΘα) = 2ϱuα r·uΘ[νΦα]. Which is equivalent to the formula for ∂νlnϱwe expected. Contracting with ˜Θµwe get; −ϱθ,ν=˜Θµ( δµ ν−uµrν(r·u)−1) =˜Θν−rν˜Θ·u r·u =uα r·u(˜Θνrα−rν˜Θα) =ϱuα r·u(˜Θνˆrα−ˆrν˜Θα) = 2ϱuα r·u˜Θ[νˆrα]. After cancellation of ϱand slight rearrangement we get the desired result. Contracting with ˜Φµwe get: −ϱφ,ν=˜Φµ( δµ ν−uµrν(r·u)−1) =˜Φν−rν˜Φ·u r·u =uα r·u(˜Φνrα−rν˜Φα) =ϱuα r·u(˜Φνˆrα−ˆrν˜Φα) = 2ϱuα r·u˜Φ[νˆrα]. after cancellation of ϱand slight rearrangement we get the desired result. Now using ( 5.2.0.4 ) and ( 5.2.0.5 ) we find directly that the gradient of the prepotential Sis ∂νS=1 2(θ,ν+ıφ,ν) =uα r·u(˜Θ[αˆrν]+ı˜Φ[αˆrν]) (5.2.0.7) 44 5.2.1 Conjugation By (4.2.9.1 ) we are required to choose a conjugation so we can define the relation between the prepotential and the complex 4-potential. The chosen conjugation is related to the chosen null-basis as C=K+ 3 (5.2.1.1) the following relations arise for the local basis vectors ( 4.3.5.1 ) and the conjugation ( 5.2.1.1 ) Cν µΘν=−˜ΘµCν µΦν=ı˜Φµ Cν µ˜Φν=−ıΦµCν µ˜Θν=−Θµ(5.2.1.2) this can be viewed as operator −∂θ+ı∂ϕ. 5.2.2 Complex 4-potential of a point charge The complex 4-potential is defined as required by ( 4.2.9.1 ) as Aµ=Cν µ∂νS. (5.2.2.1) Substituting the gradient of the prepotential ( 5.2.0.7 ) into ( 5.2.2.1 ) and using the relations (5.2.1.2 ) we get Aµ=Cν µuα r·u(˜Θ[αˆrν]+ı˜Φ[αˆrν]) =uα 2r·u[ ˜Θα( −˜Θ +ı˜Φ) µ−(−Θµ) (Θ + Φ)α +ı˜Φα( −˜Θ +ı˜Φ) µ−ı(−ıΦµ) (Θ + Φ)α] =uα 2r·u( ΘµΘα−ΦµΦα−˜Φµ˜Φα−˜Θµ˜Θα+ 2Θ [µΦα]+ 2ı˜Φ[µ˜Θα]) . Since the ( 4.3.5.1 ) basis is orthonormal so ηµα= ΘµΘα−ΦµΦα−˜Φµ˜Φα−˜Θµ˜Θα, and using (5.2.0.6 ) the complex 4-potential ( 5.2.2.1 ) can be written as Aµ=1 2uµ r·u+1 2∂µlnϱ+ıuα r·u˜Φ[µ˜Θα]. (5.2.2.2) We can see that the real part is half the Liénard–Wiechert potentials with a scalar gauge (5.2.0.6 ), and we have additional complex term which cannot be described fully as a gradi- ent. 5.2.3 EM Field of a point charge Before we find the projection of the Faraday EM tensor field Fµν, we find some useful relations. Since the termuα r·uappears in ( 5.2.2.2 ) twice we need to find it’s gradient. We first will calculate the gradient of r·u, we get ∂µ(r·u) =( δα µ−uαrµ(r·u)−1) uα+rµr·u,τ r·u =uµ−rµ(r·u)−1(u·u−r·a)(5.2.3.1) 45 wherea≡u,τ. Now using ( 5.2.3.1 ) foruα r·uwe find ∂µuα r·u=aαrµ (r·u)2−uα (r·u)2( uµ−rµ(r·u)−1(u·u−r·a)) =rµuα (r·u)3−uµuα (r·u)2+rµaα(r·u)−rµuα(r·a) (r·u)3(5.2.3.2) Lemma 5.4. Outside for the sources the prepotetial defined in 5.2obeys□S= 0. Proof. Using ( 5.2.0.2 ) and ( 5.2.3.2 ) we get: □S=∂µ∂S ∂rνrν ,µ=∂2S ∂rλ∂rνrν ,µrλ,µ+∂S ∂rν□rν =∂2S ∂rλ∂rν( δν µ−uνrµ(r·u)−1)( ηλµ−uλrµ(r·u)−1) +∂S ∂rν( −3uν(r·u)−1−rµ[ aνrµ (r·u)2−uν (r·u)2( uµ−rµ(r·u)−1(u·u−r·a))]) =∂2S ∂rµ∂rµ−2uν r·u(∂2S ∂rµ∂rνrµ+∂S ∂rµ∂rµ ∂rν) =∂2S ∂rµ∂rµ−2uν r·u∂ ∂rν(∂S ∂rµrµ) . All what is left to prove is that∂2S ∂rµ∂rµ= 0and∂S ∂rµrµ= 0. Using 5.2and4.66 to differentiate Sbyrµwe get: ∂S ∂rµ=˜nµ 2˜n·r−m∗ µ 2m∗·r. This immediately gives∂S ∂rµrµ= 0as required. Now differentiating again by rµwe get ∂2S ∂rµ∂rµ=−˜nµ˜nµ 2 (˜n·r)2+m∗ µm∗µ 2 (m∗·r)2= 0. as required. Now since by 5.4the prepotential satisfies the homogeneous wave equation □S= 0, then by4.56 we know that the complex 4-potential Aµ(5.2.2.2 ) is aσ-dual potential 4.54, hence it is related to the real Faraday tensor as Fαβ= 4ℜA [β,α]. Leading to Fµν= 4ℜ∂[µAν]=δαβ µν∂αuβ (r·u)+ 2∂[µ∂ν]lnϱ (5.2.3.3) the second term of in the last hand vanishes since the partial derivatives commute, thus using (5.2.3.2 ) the real Faraday tensor is Fµν= 2r[µuν] (r·u)3+ 2r[µaν](r·u)−r[µuν](r·a) (r·u)3(5.2.3.4) and the projections of Fintoo±asF±=P±Fis given by F± µν=1 2(Fµν∓ı⋆Fαβ) (5.2.3.5) whereFµνis given by ( 5.2.3.4 ). 46 5.3 Prepotential of charge distribution Definition 5.5. The pre-potential of a charge distribution is defined through the super-position principle based on the result for charged point particle. We define the scalar distribution ρ(x) =√ jµ(x)jµ(x)and use it to define S(x) =∫ R3[lnζ(r)]ρ(x−r)d3r (5.3.0.6) or =∫ K−(x)[lnζ(x−y(x))]ρ(y(x))d3y (5.3.0.7) wherer(r) = (∥r∥,r)while r∈R3,K−(x)is the hyper-surface representing the past light-cone in the position x= (x0,x)andy(x) = (x0− ∥x−y∥,y)while y∈R3. 5.4 Riemann–Silberstein vector F−by the prepotential In this subsection we will find the direct relation of the Fields from some prepotential S, which by superposition principle is well defined. By (4.2.6.3 ) the Faraday projection F=F−has a decomposition by the generators ˜B− jas F−=Fj −˜B− j. We will extract a direct formula for Fj, using 4.50 we can multiply ( 4.2.6.3 ) by ˜B−αβ iwhich gives: ˜B−αβ iF− αβ=˜B−αβ iFj −˜B− jαβ=ηijFj − lowering the jindex we finally get: F−i=˜B−αβ iF− αβ(5.4.0.8) Since anywhere we can consider a complex potential as in ( 4.2.8.1 ), using ( 5.4.0.8 ) we get: F−i=˜B−αβ iF− αβ=˜B−αβ i1 2( A[β,α]+ı1 2ϵµν αβA[ν,µ]) =1 2( ˜B−αβ iA[β,α]+ı1 2ϵµν αβ˜B−αβ iA[ν,µ]) =1 2( ˜B−αβ iA[β,α]+ı(−ı)˜B−µν iA[ν,µ]) =˜B−αβ iA[β,α]=˜B−αβ iAβ,α summarizing F−i=˜B−αβ iAβ,α (5.4.0.9) substituting the complex 4-potential induced by the prepotential ( 5.2.2.1 ) into the relation (5.4.0.9 ) we get F−i=˜B−αβ iCλ βS,αλ=−Cλ β˜B−β i µηµαS,αλ=1 2αµν iS,µν (5.4.0.10) where theαimatrices coincides in the null-basis with the Dirac alpha matrices in the Weyl or chiral basis [ 13]. 47 Using ( 5.4.0.10 ), (4.2.5.1 ) and ( 4.2.3.1 ) we get a system of equations relating the Riemann– Silberstein vector ( 4.2.6.2 ) to the prepotential 5.2as: F1=−F−1=Cα ν˜B−ν 1µηµβS,αβ =1 2  0 0 0 1 0 0 −ı0 0ı0 0 1 0 0 0  0 1 0 0 1 0 0 0 0 0 0 ı 0 0 −ı0  1 0 0 0 0−1 0 0 0 0 −1 0 0 0 0 −1  αβ S,αβ =1 2 0 0ı0 0 0 0 −1 ı0 0 0 0−1 0 0 αβ S,αβ=−S,13+ıS,02 F2=−F−2=Cα ν˜B−ν 2µηµβS,αβ =1 2  0 0 0 1 0 0 −ı0 0ı0 0 1 0 0 0  0 0 1 0 0 0 0 −ı 1 0 0 0 0ı0 0  1 0 0 0 0−1 0 0 0 0 −1 0 0 0 0 −1  αβ S,αβ =1 2 0−ı0 0 −ı0 0 0 0 0 0 −1 0 0 −1 0 αβ S,αβ=−S,23−ıS,01 F3=−F−3=Cα ν˜B−ν 3µηµβS,αβ =1 2  0 0 0 1 0 0 −ı0 0ı0 0 1 0 0 0  0 0 0 1 0 0ı0 0−ı0 0 1 0 0 0  1 0 0 0 0−1 0 0 0 0 −1 0 0 0 0 −1  αβ S,αβ =1 2 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 αβ S,αβ=1 2(S,00+S,11+S,22−S,33) or summarizing: F1=−S,13+ıS,02 F2=−S,23−ıS,01 F3=1 2(S,00+S,11+S,22−S,33). (5.4.0.11) 48 5.5 Maxwell’s equation, 4-current condition and gauge conditions 5.5.1 Maxwell’s Equation Now we are about to find Maxwell’s equations in the prepotential formalism. Using ( 4.2.6.1 ), 4.46 and ( 3.1.2.4 ) we get: ∂βF±αβ=∂βP±Fαβ=∂β1 2( Fαβ∓ı(⋆F)αβ) =1 2µ0jα, now substituting ( 4.2.9.2 ), (4.2.9.1 ), and using ( 4.2.4.3 ) and the anti-symmetry of the conju- gation we get: 1 2µ0jα=∂βF±αβ= 2∂βP±A±[β,α]= 2∂βA±[β,α]−2∂βP∓A±[β,α] = 2∂βC±λ[βS,α] ±,λ−2∂βP∓C±λ[βS,α] ±,λ =−C±λα∂λ□S±+1 2∂βC±αβ□S± =−1 2C±λα∂λ□S±. This implies that: Proposition 5.6. Maxwell’s equation for the prepotential is Cλα∂λ□S=−µ0jα. 5.5.2 Current condition We can rewrite the prepotential Maxwell equation 5.6as pair of equations □S=ϑ Cλ µϑ,λ=−µ0jµ(5.5.2.1) whereϑis a complex scalar field. Taking the second equation and multiplying both sides by Cµ α we get −µ0Cµ αjµ=Cµ αCλ µϑ,λ=δλ αϑ,λ=ϑ,α, differentiating by xβwe get −µ0Cµ αjµ,β=ϑ,αβ, multiplying both sides by δαβ πρwe get −µ0δαβ πρCµ αjµ,β=δαβ πρΦ,αβ= 0, relabeling π,ρ→α,βwe getCµ αjµ,β=Cµ βjµ,α, multiplying again by Cα νand relabeling α,ν→ν,αwe finally get jα,β=Cν αCµ βjµ,ν. (5.5.2.2) This is a condition on the 4-current in order to have a prepotential, which is equivalent to the non-trivial conditions j0,0−j3,3= 0j1,1+j2,2= 0 j0,1−ıj2,3= 0j1,0−ıj3,2= 0 j0,2+ıj1,3= 0j2,0+ıj3,1= 0. This raise a question whatever any real 4-current can have complex extension that satisfies (5.5.2.2 )? If the answer is no then not every real 4-current has a prepotential. 5.5.3 Prepotential Gauge The gauge equations for the prepotential by ( 5.4.0.11 ) is: 0 =−S,13+ıS,02 0 =−S,23−ıS,01 0 =1 2(S,00+S,11+S,22−S,33). 49 SinceSis a function of the position xµasS=f(x0,x1,x2,x3), we want to find the conditions on the functions that could satisfy the equations. Factoring the first equation into linear operators we get: 0 =−S,13+ıS,02 =−1 2([∂1+∂0] [∂3−ı∂2] + [∂1−∂0] [∂3+ı∂2])S =−1 2([∂1+∂2] [∂3−ı∂0] + [∂1−∂2] [∂3+ı∂0])S =−1 2([∂3+∂0] [∂1−ı∂2] + [∂3−∂0] [∂1+ı∂2])S =−1 2([∂3+∂2] [∂1−ı∂0] + [∂3−∂2] [∂1+ı∂0])S. This are four separate options to factor the first equation. For each option respectively we have a linear solution as: S=g1(x1−x0,x1+x0) +g2(x1−x0,x3+ıx2) +g3(x1+x0,x3−ıx2) +g4(x3+ıx2,x3−ıx2) S=g5(x1−x2,x1+x2) +g6(x1−x2,x3+ıx0) +g7(x1+x2,x3−ıx0) +g8(x3+ıx0,x3−ıx0) S=g9(x3−x0,x3+x0) +g10(x3−x0,x1+ıx2) +g11(x3+x0,x1−ıx2) +g12(x1+ıx2,x1−ıx2) S=g13(x3−x2,x3+x2) +g14(x3−x2,x1+ıx0) +g15(x3+x2,x1−ıx0) +g16(x1+ıx0,x1−ıx0), where any g#is an arbitrary function in two variables. Under similar argument the linear factorization of the second equation is: 0 =−S,23−ıS,01 =−1 2([∂2+∂0] [∂3+ı∂1] + [∂2−∂0] [∂3−ı∂1])S =−1 2([∂2+∂1] [∂3+ı∂0] + [∂2−∂1] [∂3−ı∂0])S =−1 2([∂3+∂0] [∂2+ı∂1] + [∂3−∂0] [∂2−ı∂1])S =−1 2([∂3+∂1] [∂2+ı∂0] + [∂3−∂1] [∂2−ı∂0])S. And the solutions to the four options are respectively: S=h1(x2−x0,x2+x0) +h2(x2−x0,x3+ıx1) +h3(x2+x0,x3−ıx1) +h4(x3+ıx1,x3−ıx1) S=h5(x1−x2,x1+x2) +h6(x1+x2,x3+ıx0) +h7(x1−x2,x3−ıx0) +h8(x3+ıx0,x3−ıx0) S=h9(x3−x0,x3+x0) +h10(x3+x0,x2+ıx1) +h11(x3−x0,x2−ıx1) +h12(x2+ıx1,x2−ıx1) S=h13(x3−x1,x3+x1) +h14(x3+x1,x2+ıx0) +h15(x3−x1,x2−ıx0) +h16(x2+ıx0,x2−ıx0), where any h#is an arbitrary function in two variables. The only solution that satisfies the first equation and the second equation simultaneously is of the form S=f1(x0+x3,x0−x3) +f2(x0+x3,x1−ıx2) +f3(x0−x3,x1+ıx2) +f4(x1+ıx2,x1−ıx2), which coincides with the third solution for both the first and second equations given above. Moreover this solution also satisfy the third equation, hence, this implies that: Proposition 5.7. The prepotential gauge is of the form S=f1(x0+x3,x0−x3) +f2(x0+x3,x1−ıx2) +f3(x0−x3,x1+ıx2) +f4(x1+ıx2,x1−ıx2). 50 5.6 Examples of calculations involving the prepotential 5.6.1 Infinitely long charged rod In [12, p. 11] Friedman found that the pre-potential of an infinitely long charged rod placed on thex3-axis, as S(x) =−λx3lnϱ (5.6.1.1) whereλis the charge density, and ϱ=√ (x1)2+ (x2)2is the distance form the x3-axis in the x1-x2plane. Using ( 5.4.0.11 ) we find the Riemann–Silberstein vector that corresponds to this prepotential: F1=−S,13+ıS,02=λx1 ϱ2 F2=−S,23−ıS,01=λx2 ϱ2 F3=1 2(S,00+S,11+S,22−S,33) =−2πλx3δ(x1)δ(x2)(5.6.1.2) the 4-current is jµ=−Cα µ∂α□S=Cα µ∂α2πλx3δ(x1)δ(x2) = 2πλ( δ(x1)δ(x2);ıx3δ(x1)δ′(x2);−ıx3δ′(x1)δ(x2); 0) (5.6.1.3) whereδ′is the distributional derivative of the dirac delta δ. Notice that imaginary part of the current is inside the sources and is a complex extension of the real current so it satisfies ( 5.5.2.2 ). A factor of 2πwas introduced which is because lnϱis a green function of ∆2D=∂1∂1+∂2∂2, this shows that there are still fine subjects to differentiation and integration in relation to the prepotential which has to be understood. 5.6.2 Infinite charged plate The current of an infinite charged plate positioned in the x1-x2plane, thus the current density is jµ=( σδ(x3); 0; 0; 0) , which satisfies the current condition ( 5.5.2.2 ). We will use ( 5.5.2.1 ) in or- der to find the prepotential. The conjugation of the current density is Cλ µjλ=( 0; 0; 0; −σδ(x3)) , thus it is a gradient of ϑ=σ( h(x3)−1 2) wherehis the Heaviside function and the prepotential is S=1 4σx3/vextends]ngle/vextends]nglex3/vextends]ngle/vextends]ngle. (5.6.2.1) the force is F1=F2= 0 F3=1 2σSign(x3) =σ(h(x3)−1 2) which is the known result for an infinite charged plate. 5.6.3 Infinitely long solenoid In [7, p. 14-5] the current density of a long solenoid is considered as circumferential surface current, were the pitch is neglected, so the current of solenoid of radius Ris jµ=J0δ(R−ϱ) ˆφµ (5.6.3.1) 51 whereϱ=√ (x1)2+ (x2)2and ˆφµ= (0;−x2 ϱ;x1 ϱ; 0) =∂µφ=ıCν µ∂ν(lnϱ). This distribution satisfies ( 5.5.2.2 ) so there is no need to add imaginary currents to satisfy this condition. The 4-potential is Aµ=µ0J0 4ˆφµ( h(ϱ−R)R2 ϱ+h(R−ϱ)ϱ) . Since∂νS=Cµ νAµwe get ∂νS=Cµ νAµ=ıµ0J0 4( h(ϱ−R)R2 ϱ+h(R−ϱ)ϱ) ∂µlnϱ =ıµ0J0 4( h(ϱ−R)R2 ϱ2+h(R−ϱ)) ∂µϱ. Which leads to S=ıµ0J0 4( h(ϱ−R)[ 2R−R2 ϱ] +h(R−ϱ)ϱ) . (5.6.3.2) This prepotential does not indicate that a closed loop will change the phase by 2π. If instead we try to find a ”direct” prepotential ϕas∂µϕ=Aµ, we see this cannot be done or else it would imply that there is no force or charge in the configuration, on the other hand we can see that Aµdxµ=µ0J0 4( h(ϱ−R)R2 ϱ+h(R−ϱ)ϱ) ˆφµdxµ=f(ϱ)∂φ ∂xµdxµ=f(ϱ)dφ (5.6.3.3) which shows why we can’t find a ”direct” prepotential, and also why this 4-potential gets a winding-number when line-integrated around the z-axis. 6 Discussion We have proposed a prepotential model for the classical EM theory which provides less degrees of freedom and still have conformal invariance. Some progress and advances has been made over previous works, but there are still open problems especially in the understanding of the physical meaning of our results. 6.1 Open problems 1. Finding the physical meaning of the complexified Minkowski space Mc. 2. Finding the physical meaning of ˜π±representations. 3. Finding the physical meaning of the current density condition ( 5.5.2.2 ). 4. Does any real 4-current density can be complexly extended to density that satisfies ( 5.5.2.2 )? 6.2 Advances with respect to previous works on the subject In [9] and [ 8] the prepotential and the complexified Minkoski space was firstly introduced, and the field of non-accelerating point charged particle was calculated. In [ 10] it was argued that the prepotential might account for the Aharonov-Bohm effect. In [11] a gauge of the prepotential was found and it was proved that the prepotential of stationary point particle satisfies the wave equation, also the prepotenial was calculated for an infinite charged rod and for current of an infinite rod. The advances in this report are: 52 • We re-defined the Lie algebra representations ˜π±4.52 as theσ-dual subspaces of Lie algebra representation π, which corresponds to the two eigenvalues ( 4.2.4.1 ) of the Hodge dual operator. We also the introduce the projection operators P±4.46. • We have shown in 4.54 that outside of the sources a σ-dual complex 4-potential can be defined. As for the prepotential we showed in 4.56 that□S±= 0if and only if it has a σ-dual complex 4-potential. • We generalized the definition of the null-basis ( 4.3.2.1 ) and the scalar invariant 4.66, and shown how they are related to the 2-spinor formalism. This simplified greatly the proof that theζ±-factor is invariant under ˜π±. • We introduced the θ,φandϱparameters of the decomposition of a null vector ( 4.3.4.3 ), using them we introduced the local orthonormal basis ( 4.3.5.1 ), (4.3.5.4 ). • We used the local orthonormal basis to simplify greatly the calculation of the gradient of the prepotential ( 5.2.0.7 ) and the complex 4-potential ( 5.2.2.2 ). • We showed that 5.4outside of the sources the prepotential always satisfies the wave equa- tion. This allowed using 4.56 which made the calculation of the field of an accelerating point charged particle ( 5.2.3.4 ) very simple. • We calculated Maxwell’s equations in the prepotential formalism 5.6. • The condition on the 4-current density ( 5.5.2.2 ) in order to have a prepotential was found. • We found the gauge freedom of the prepotential 5.7. • We calculated the prepotential of infinite charged plate ( 5.6.2.1 ) and of an infinitely long solenoid ( 5.6.3.2 ). 53 References [1] Konstantin Y. Bliokh, Aleksandr Y. Bekshaev, and Franco Nori. Dual electromagnetism: He- licity, spin, momentum, and angular momentum . arXiv:1208.4523; [2] Fryberger, D. ”On Generalized Electromagnetism and Dirac Algebra” , Foundations of Physics Vol. 19, p. 125 (1989). [3] David McMahon. Relativity Demystified - A Self-Teaching Guide . Chapter 9: Null Tetrads and the Petrov Classification . New York: McGraw-Hill, 2006. [4] Ezra T. Newman and Roger Penrose (1962). ”An Approach to Gravitational Radiation by a Method of Spin Coefficients”. Journal of Mathematical Physics 3(3): 566–768. [5] R. D. Richtmeyer ”Principles of Advanced Mathematical Physics” V2. Ch 25, Springer- Verlag (19810) [6] O’Donnell, Peter J. Introduction to 2-spinors In General Relativity . Singapore: World Scien- tific, 2003. [7] Feynman, Richard P, Robert B Leighton, and Matthew L Sands. The Feynman Lectures on Physics, Desktop Edition Volume II: The New Millennium Edition . Basic Books, 2013. [8] Friedman Y and Gwertzman S 2009 The scalar complex potential of the electromagnetic field arXiv:0906.0930 [9] S. Gwertzman 2009, JCT project report, Jerusalem College of Technology, Israel [10] Y Friedman and V Ostapenko 2010 The complex pre-potential and the Aharonov–Bohm effect doi:10.1088/1751-8113/43/40/405305 [11] M. Eliyahu 2012, JCT project report, Jerusalem College of Technology, Israel [12] Yaakov Friedman 2013 The wave-function description of the electromagnetic field doi:10.1088/1742-6596/437/1/012018 [13] Michio Kaku, Quantum Field Theory, ISBN 0-19-509158-2, appendix A 54 תקציר דו”חזההואהמשךפיתוחנושאהקדם-הפוטנציאלשלשדהאלמ”ג. שלחבורתלורנץעלמרחב Lie עלהצגההסטנדרטיתשלאלגברת Hodge-הגדרנופעולתדואליותה o± לתתי-מרחביםעצמיים π .הראנוכיפעולהזויוצרתפירוקשל π-מינקובסקי,המסומנתבדו”חזהכ .הגדרנוהצגות ±ıמקבלערך σ--דואלייםכךש σo± .קראנולאיבריםשל ±ıהמתאימיםלערכיםהעצמיים .F±-דואליσ טנזור F .התאמנולטנזורשדההאלמ”ג o± עלכלאחדמתתי-המרחבים ˜π± שלחבורתלורנץ -דואלי.σ הראנושמחוץלמקורותניתןלהגדירארבע-פוטנציאל ,התלוירקבמיקוםהצופהוהמקורשדההאלמ”ג,אינווריאנטיתחתההצגה ζ± עבורכלצופהמצאנוסקלר שלשדההאלמ”ג.הראנושקדם-פוטנציאלשל S± הגדרנואתהקדם-פוטנציאל ζ± .בעזרתהאינווריאנט ˜π± .-דואליσ מגדירשדההאלמ”גהידוע.הראנושקדם-פוטנציאלמחוץלמקורתמיד כלשהימטעןנקודתיבתנועה פיתחנואתמשוואותמקסוולעבורהקדם-פוטנציאל.מצאנותנאיםעלהתפלגותהזרםלקיוםקדם-פוטנציאל. חישבנואתחופשהכיולעבורהקדם-פוטנציאל.הצגנודוגמאותלקשרביןהקדם-פוטנציאל,התפלגותהזרם ושדהאלמ”געבור:תילטעוןאינסופי,משטחטעוןאינסופיוזרםבסלילאינסופי. 55