Edmundo Capelas de Oliveira

The Many Faces of Maxwell, Dirac and Einstein Equations

Preface.- Introduction.- Multivector and Extensor Calculus.- The Hidden Geometrical Nature of Spinors.- Some Differential Geometry.- Clifford Bundle Approach to the Differential Geometry of Branes.- Some Issues in Relativistic Spacetime Theories.- Clifford and Dirac-Hestenes Spinor Fields.- A Clifford Algebra Lagrangian Formalism in Minkowski Spacetime.- Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes.- The DHE on a RCST and the Meaning of Active Local Lorentz Invariance.- On the Nature of the Gravitational Field.- On the Many Faces of Einstein Equations.- Maxwell, Dirac and Seiberg-Witten Equations.- Superparticles and Superfields.- Maxwell, Einstein, Dirac and Navier-Stokes Equations.- Magnetic Like Particles and Elko Spinor Fields.-Appendices A1-5.- Acronyms and Abbreviations.- List of Symbols.- Index.

Does Elko Spinor Field Imply the Existence of an Axis of Locality

In this paper we show that the statement in Ahluwalia, Lee, and Schritt (2011) that the existence of elko spinor fields implies in an axis of locality is equivocated. The anticommutator {{\Lambda}(x,t),{\Pi}(x,t} is strictly local.

A Clifford Algebra Lagrangian Formalism in Minkowski Spacetime

Using tools introduced in previous chapters, particularly the concept of Clifford and spin-Clifford bundles (and the representations of sections of the spin-Clifford bundles as equivalence classes of sections the Clifford bundle), extensor fields and the Dirac operator we give an unified and original approach to the Lagrangian field theory in Minkowski spacetime with special emphasis on the Maxwell and Dirac-Hestenes fields. We derive for these fields their canonical energy-momentum extensor fields and also their angular momentum and spin momentum extensor fields. In particular we show that the antisymmetric part of the canonical energy-momentum tensor is the “source” of spin of the field. Several nontrivial exercises are solved with details in order to help the reader to familiarize with the formalism and to make contact with standard formulations of field theory.

Magnetic Like Particles and Elko Spinor Fields

This chapter scrutinizes the theory of the so-called Elko spinor fields (in Minkowski spacetime) which always appears in pairs and which from the algebraic point of view are in class five in Lounesto classification of spinor fields. We show how these fields differs from Majorana fields (which also are in class five in Lounesto classification of spinor fields) and that Elko spinor fields (as it is the case for Majorana fields) do not satisfy the Dirac equation. We discuss the class of generalized Majorana spinor fields (objects that are spinor with “components” with take values in a Grassmann algebra) that satisfy Dirac equations, clarifying some obscure presentations of that theory appearing in the literature. More important, we show that the original presentation of the theory of Elko spinor fields as having mass dimension 1 leads to breakdown of Lorentz and rotational invariance by a simple choice of the spatial axes in an inertial reference frame. We then present a Lagrangian field theory for Elko spinor fields where these fields (as it is the case of Dirac spinor fields) have mass dimension 3∕2. We explicitly demonstrate that Elko spinor fields cannot couple to the electromagnetic field, that they describe pairs of “magnetic” like particles which are coupled to a short range su(2) gauge potential. Thus they eventually can serve to model dark matter. The causal propagator for the 3∕2 mass dimension Elko spinor is explicitly calculated with the Clifford bundle of (multivector) fields. Taking the opportunity given by the formalism developed in our theory we present a very nice representation of the parity operator acting on Dirac-Hestenes spinor fields.

Clifford and Dirac-Hestenes Spinor Fields

This chapter presents an original approach to the theory of Dirac-Hestenes spinor fields. After recalling details the structure of the Clifford bundle of a Lorentzian manifold structure \((M,\boldsymbol{g})\) we introduce the concept of spin structures on \((M,\boldsymbol{g})\), define spinor bundles and spinor fields. Left and right spin-Cillford bundles are presented and the concept of Dirac-Hestenes spinor fields (as sections of a special spin-Clifford bundle denoted by \(\mathcal{C}\ell_{Spin_{1,3}^{e}}^{r}(M,\mathtt{g})\)) is investigated in details disclosing their real nature and showing how these objects can be represented as some equivalence classes of even sections of the Clifford bundle \(\mathcal{C}\ell(M,\mathtt{g})\). We introduce and obtain the connection (covariant derivative) acting on spin-Clifford bundles \(\mathcal{C}\ell_{Spin_{1,3}^{e}}^{r}(M,\mathtt{g})\) from a Levi-Civita connection acting on the tensor bundle of \((M,\boldsymbol{g})\). Associated with that connection we next introduce the standard spin-Dirac operator acting on sections of \(\mathcal{C}\ell_{Spin_{1,3}^{e}}^{r}(M,\mathtt{g})\) (not to be confused with the Dirac operator acting on sections of the Clifford bundle defined in Chap. 4). We discuss how to write Dirac equation using the spin-Clifford bundle formalism and clarifies its properties and many misunderstandings relating to such a notion that are spread in the literature. We also discuss the notion of what is known as amorphous spinor fields showing that these objects cannot represent fermion fields. Finally, the Chapter also presents a simple proof of the famous Lichnerowicz formula which relates the square of the standard spin-Dirac operator to the curvature of the (spin) connection and also we obtain a generalization of that formula for the square of a general spin-Dirac operator associated to a general (metric compatible) Riemann-Cartan connection defined in \((M,\boldsymbol{g})\).

Multivector and Extensor Calculus

This chapter is dedicated to a thoughtful exposition of the multiform and extensor calculus. Starting from the tensor algebra of a real n-dimensional vector space \(\boldsymbol{V }\) we construct the exterior algebra \(\bigwedge \boldsymbol{V }\) of \(\boldsymbol{V }\). Equipping \(\boldsymbol{V }\) with a metric tensor \(\mathring{g}\) we introduce the Grassmann algebra and next the Clifford algebra \(\mathcal{C}\ell(\boldsymbol{V },\mathit{\mathring{g}})\) associated to the pair \((\boldsymbol{V },\mathit{\mathring{g}})\). The concept of Hodge dual of elements of \(\bigwedge \boldsymbol{V }\) (called nonhomogeneous multiforms) and of \(\mathcal{C}\ell(\boldsymbol{V },\mathit{\mathring{g}})\) (also called nonhomogeneous multiforms or Clifford numbers) is introduced, and the scalar product and operations of left and right contractions in these structures are defined. Several important formulas and “tricks of the trade” are presented. Next we introduce the concept of extensors which are multilinear maps from p subspaces of \(\bigwedge \boldsymbol{V }\) to q subspaces of \(\bigwedge \boldsymbol{V }\) and study their properties. Equipped with such concept we study some properties of symmetric automorphisms and the orthogonal Clifford algebras introducing the gauge metric extensor (an essential ingredient for theories presented in other chapters). Also, we define the concepts of strain, shear and dilation associated with endomorphisms. A preliminary exposition of the Minkoswski vector space is given and the Lorentz and Poincare groups are introduced. In the remaining of the chapter we give an original presentation of the theory of multiform functions of multiform variables. For these objects we define the concepts of limit, continuity and differentiability. We study in details the concept of directional derivatives of multiform functions and solve several nontrivial exercises to clarify how to work with these notions, which in particular are crucial for the formulation of Chap. 8 which deals with a Clifford algebra Lagrangian formalism of field theory in Minkowski spacetime.

On the Nature of the Gravitational Field

In this chapter we investigate the nature of the gravitational field. We first give a formulation for the theory of that field as a field in Faraday’s sense (i.e., as of the same nature as the electromagnetic field) on a 4-dimensional parallelizable manifold M. The gravitational field is represented through the 1-form fields \(\{\mathfrak{g}^{\mathbf{a}}\}\) dual to the parallelizable vector fields \(\{\boldsymbol{e}_{\mathbf{a}}\}\). The \(\mathfrak{g}^{\mathbf{a}}\)’s (a = 0, 1, 2, 3) are called gravitational potentials, and it is imposed that at least for one of them, \(d\mathfrak{g}^{\mathbf{a}}\neq 0\). A metric like field \(\boldsymbol{g} =\eta _{\mathbf{ab}}\mathfrak{g}^{\mathbf{a}} \otimes \mathfrak{g}^{\mathbf{b}}\) is introduced in M with the purpose of permitting the construction of the Hodge dual operator and the Clifford bundle of differential forms \(\mathcal{C}\ell(M,\mathtt{g})\), where \(\mathtt{g} =\eta ^{\mathbf{ab}}\boldsymbol{e}_{\mathbf{a}} \otimes e_{\mathbf{b}}\). Next a Lagrangian density for the gravitational potentials is introduced with consists of a Yang-Mills term plus a gauge fixing term and an auto-interacting term. Maxwell like equations for \(F^{\mathbf{a}} = d\mathfrak{g}^{\mathbf{a}}\) are obtained from the variational principle and a legitimate energy-momentum tensor for the gravitational field is identified which is given by a formula that at first look seems very much complicated. Our theory does not uses any connection in M and we clearly demonstrate that representations of the gravitational field as Lorentzian, teleparallel and even general Riemann-Cartan-Weyl geometries depend only on the arbitrary particular connection (which may be or not to be metrical compatible) that we may define on M. When the Levi-Civita connection of \(\boldsymbol{g}\) in M is introduced we prove that the postulated Lagrangian density for the gravitational potentials differs from the Einstein-Hilbert Lagrangian density of General Relativity only by a term that is an exact differential. The theory proceeds choosing the most simple topological structure for M, namely that it is \(\mathbb{R}^{4}\), a choice that is compatible with present experimental data. With the introduction of a Levi-Civita connection for the structure \((M = \mathbb{R}^{4},\boldsymbol{g})\) as a mathematical aid we can exhibit a nice short formula for the genuine energy-momentum of the gravitational field. Next, we introduce the Hamiltonian formalism and discuss possible generalizations of the gravitational field theory (as a field in Faraday’s sense) when the graviton mass is not null. Also we show using the powerful Clifford calculus developed in previous chapters that if the structure \((M = \mathbb{R}^{4},\boldsymbol{g})\) possess at least one Killing vector field, then the gravitational field equations can be written as a single Maxwell like equation, with a well defined current like term (of course, associated to the energy-momentum tensor of matter and the gravitational field). This result is further generalized for arbitrary vector fields generating one-parameter groups of diffeomorphisms of M in Chap. 14 Chapter 11 ends with another possible interpretation of the gravitational field, namely that it is represented by a particular geometry of a brane embedded in a high dimensional pseudo-Euclidean space. Using the theory developed in Chap. 5 we are able to write Einstein equation using the Ricci operator in such a way that its second member (of “wood” nature, according to Einstein) is transformed (also according to Einstein) in the “marble”nature of its first member. Such a form of Einstein equation shows that the energy momentum quantities \(-T^{\mathbf{a}} + \frac{1} {2}T\mathfrak{g}^{\mathbf{a}}\) (where \(T^{\mathbf{a}} = T_{\mathbf{b}}^{\mathbf{a}}\mathfrak{g}^{\mathbf{b}}\) are the energy momentum 1-form fields of matter and T = T a a ) which characterize matter is represented by the negative square of the shape operator (\(\mathbf{S}^{2}(\mathfrak{g}^{\mathbf{a}})\)) of the brane. Such a formulation thus give a mathematical expression for the famous Clifford “little hills” as representing matter.

Maxwell, Einstein, Dirac and Navier-Stokes Equations

In the previous chapters we exhibit several different faces of Maxwell, Einstein and Dirac equations. In this chapter we show that given certain conditions we can encode the contents of Einstein equation in Maxwell like equations for a field \(F = dA \in \sec \bigwedge \nolimits ^{2}T^{{\ast}}M\) (see below), whose contents can be also encoded in a Navier-Stokes equation. For the particular cases when it happens that F2 ≠ 0 we can also using the Maxwell-Dirac equivalence of the first kind discussed in Chap. 13 to encode the contents of the previous quoted equations in a Dirac-Hestenes equation for \(\psi \in \sec (\bigwedge \nolimits ^{0}T^{{\ast}}M + \bigwedge \nolimits ^{2}T^{{\ast}}M + \bigwedge \nolimits ^{4}T^{{\ast}}M)\) such that \(F =\psi \gamma ^{21}\tilde{\psi }.\) Specifically, we first show in Sect. 15.1 how each LSTS \((M,\boldsymbol{g},D,\tau _{\boldsymbol{g}},\uparrow )\) which, as we already know, is a model of a gravitational field generated by \(\mathbf{T} \in \sec T_{2}^{0}M\) (the matter plus non gravitational fields energy-momentum tensor) in Einstein GRT is such that for any \(\mathbf{K} \in \sec TM\) which is a vector field generating a one parameter group of diffeomorphisms of M we can encode Einstein equation in Maxwell like equations satisfied by F = dK where \(K =\boldsymbol{ g}(\boldsymbol{K},\) ) with a well determined current term named the Komar current\(J_{\boldsymbol{K}}\), whose explicit form is given. Next we show in Sect. 15.2 that when K=A is a Killing vector field, due to some noticeable results [Eqs. (15.28) and (15.29)] the Komar current acquires a very simple form and is then denoted \(J_{\boldsymbol{A}}\). Next, interpreting, as in Chap. 11 the Lorentzian spacetime structure \((M,\boldsymbol{g},D,\tau _{\boldsymbol{g}},\uparrow )\) as no more than an useful representation for the gravitational field represented by the gravitational potentials \(\{\mathfrak{g}^{a}\}\) which live in Minkowski spacetime (here denoted by \((M = \mathbb{R}^{4},\boldsymbol{ \mathring{g} }, \mathring{D} ,\tau _{\boldsymbol{ \mathring{g} }},\uparrow )\)) we show in Sect. 15.3 that we can find a Navier-Stokes equation which encodes the contents of the Maxwell like equations (already encoding Einstein equations) once a proper identification is made between the variables entering the Navier-Stokes equations and the ones defining \( \mathring{A} =\boldsymbol{ \mathring{g} }(\boldsymbol{A},)\) and \( \mathring{F} = d \mathring{A} \), objects clearly related [see Eq. (15.49)] to A and F = dA. We also explicitly determine also the constraints imposed by the nonhomogeneous Maxwell like equation \(\mathop{\delta }\limits_{\boldsymbol{g}}F = -J_{\boldsymbol{A}}\) on the variables entering the Navier-Stokes equations and the ones defining A (or \( \mathring{A} \)).

Some Issues in Relativistic Spacetime Theories

The chapter has as main objective to clarify some important concepts appearing in relativistic spacetime theories and which are necessary of a clear understanding of our view concerning the formulation and understanding of Maxwell, Dirac and Einstein theories. Using the definition of a Lorentzian spacetime structure \((M,\boldsymbol{g},D,\tau _{\boldsymbol{g}},\uparrow )\) presented in Chap. 4 we introduce the concept of a reference frame in that structure which is an object represented by a given unit timelike vector field \(\boldsymbol{Z} \in \sec TU\) (\(U \subseteq M\)). We give two classification schemes for these objects, one according to the decomposition of \(D\boldsymbol{Z}\) and other according to the concept of synchronizability of ideal clocks (at rest in \(\boldsymbol{Z}\)). The concept of a coordinate chart covering U and naturally adapted to the reference frame \(\boldsymbol{Z}\) is also introduced. We emphasize that the concept of a reference frame is different (but related) from the concept of a frame which is a section of the frame bundle. The concept of Fermi derivative is introduced and the physical meaning of Fermi transport is elucidated, in particular we show the relation between the Darboux biform \(\Omega \) of the theory of Frenet frames and its decomposition as an invariant sum of a Frenet biform \(\Omega _{F}\) (describing Fermi transport) and a rotation biform \(\Omega _{\mathbf{S}}\) such that the contraction of \(\star \Omega _{\mathbf{S}}\) with the velocity field v of the spinning particle is directly associated with the so-called Pauli-Lubanski spin 1-form. We scrutinize the concept of diffeomorphism invariance of general spacetime theories and of General Relativity in particular, discuss what meaning can be given to the concept of physically equivalent reference frames and what one can understand by a principle of relativity. Examples are given and in particular, it is proved that in a general Lorentzian spacetime (modelling a gravitational field according to General Relativity) there is in general no reference frame with the properties (according to the scheme classifications) of the inertial referenced frames of special relativity theories. However there are in such a case reference frames called pseudo inertial reference frames (PIRFs) that have most of the properties of the inertial references frames of special relativity theories. We also discuss a formulation (that one can find in the literature) of a so-called principle of local Lorentz invariance and show that if it is interpreted as physical equivalence of PIRFs then it is not valid. The Chapter ends with a brief discussion of diffeormorphism invariance applied to Schwarzschild original solution and the Droste-Hilbert solution of Einstein equation which are shown to be not equivalent (the underlying manifolds have different topologies) and what these solutions have to do with the existence of blackholes in the “orthodox”interpretation of General Relativity.