On Electromagnetic Duality Thomas B. Mieling Faculty of Physics, University of Vienna Boltzmanngasse 5, 1090 Vienna, Austria (Dated: November 14, 2018) CONTENTS III. DUAL TENSORS I. Introduction 1 This section introduces the notion of dual ten- sors, whose proper treatment requires some famil- II. Conventions 1 iarity with exterior algebra. Readers unfamiliar with this topic may skip section III A, since sec- III. Dual Tensors 1 tion III B suces for the calculations in the rest of A. The Hodge Dual 1 the paper, except parts of the appendix. B. The Complex Dual of Two-Forms 2 IV. The Free Maxwell-Field 2 A. The Hodge Dual V. General Duality Transformations 3 For an oriented vector space V of dimension VI. Coupled Maxwell-Fields 3 n with a metric tensor, the Hodge star opera- tor provides an isomorphism between the k-vectors VII. Applications 4 n − k-vectors. In this section, we discuss the gen- eral denition in Euclidean and Lorentzian vector VIII. Conclusion 5 spaces and give formulae for concrete calculations in (pseudo-)orthonormal frames. A. Notes on Exterior Algebra 6 The metric tensor g on V induces a metric tensor on ^kV (denoted by the same symbol) by bilinear B. Collection of Proofs 7 extension of the map References 8 g(v1 ^ · · · ^ vk; w1 ^ · · · ^ wk) = det(g(vi; wj)): If the metric on V is non-degenerate, so is g on I. INTRODUCTION ^kV . Let w be a k-vector. For every n − k-vector In abelian gauge theories whose action does not v the exterior product v ^ w is proportional to depend on the gauge elds themselves, but only the unit n-vector Ω, since ^nV is one-dimensional. on their eld strength tensors, duality transfor- We can thus write v ^ w = '(v; w)Ω, where ' mations are symmetry transformations mixing the is a bilinear function. As is shown in appendix eld tensors with certain dual tensors arising nat- A, there exists a unique element ∗w such that urally in symmetric formulations of the eld equa- g(v; ∗w) = s(g)'(v; w), which is called the Hodge tions. dual of w. These notes are intended to clarify what is meant Denition. For a k-vector w, the dual ∗w is the by dual tensors and to shed some light upon the unique n − k-vector such that for all n − k-vectors group of such duality transformations. v it holds that v ^ w = s(g) g(v; ∗w)Ω: (1) II. CONVENTIONS For a basis e it is convenient to introduce the This paper uses the Einstein summation conven- notation tion. The signature of the metric tensor in four eab::: = ea ^ eb ^ ::: dimensions is assumed to be (−; +; +; +). When g~ = g(ea; ea)g(eb; eb) ::: working with a general metric g, we denote by s(g) ab::: the sign of its determinant1. For the permutation Since the Hodge dual may be equally constructed symbol , we us the sign convention . 01::: = +1 on the dual space of an ordered vector space (where the volume form denes the preferred n-vector), we will make use of similar notation with indices raised. 1 Equivalently, one has s(g) = (−1)n, where n is the num- In [1] it is shown that the Hodge dual acts ber of negative signs in the signature of g. on an a right-handed orthonormal basis e in 2 the following way. For pairwise distinct indices Using the fact that ρσκλ κ λ λ κ , µνρσ = −2(δµδν − δµδν ) let be a completion such that i1; : : : ; ik j1; : : : ; jl it is readily veried that the operation F 7! F~ is (i1; : : : ; ik; j1; : : : ; jl) is a permutation of (1; : : : ; n). its own inverse: Denote by sgn(π) its sign. Then i F~ = − F~ρσ (2) µν µνρσ ∗ei :::i = sgn(π)~gi :::i ej :::j : 2 1 k 1 k 1 l 1 = − ρσκλF Example. In the three dimensional Euclidean 4 µνρσ κλ space we may choose a right-handed orthonor- E3 1 κ λ λ κ mal basis such that . We then = + (δµδν − δµδν )Fκλ = Fµν : e1; e2; e3 Ω = e123 2 have Every antisymmetric tensor F can be decom- posed into a self-dual and an self-anti-dual part ∗e12 = e3 1 1 and cyclic permutations thereof. The cross product F = F + + F − ≡ F + F~ + F − F~ : of two vectors a and b is given by 2 2 Note that self-dual two-forms exists only in a com- a × b = ∗(a ^ b): plex setting. Furthermore, F + and F − are com- plex conjugate to each other, if F is real. Let e denote a right-handed orthonormal frame We give some useful algebraic properties of this in a four-dimensional Lorentzian vector space with complex duality transformation. For two antisym- volume form 0123. It is readily seen that the Ω = e metric tensors A and B we have the identity Hodge dual acts on the basis vectors as µρ ~µρ ~ 1 µ ρσ (7) 1 A Bρν + B Aρν = − δν A Bρσ ; ∗eµ1...µk = µ1...µk eν1...νl : (3) 2 ν1...νl (n − k)! which shows that the contraction of two dual ten- Example. A two-form F in such a space has the sors can be expressed as contractions of the origi- decomposition nal tensors. A proof of this identity can be found in appendix B . Contracting the remaining pair of 1 indices, one nds F = F eµ ⊗ eν = F eµν : µν 2 µν ~µν ~ µν (8) A Bµν = A Bµν Using the fact that the Hodge star operator is lin- ear, we can compute the dual of F as From the previous equations it also follows that if both A and B are self-dual or anti-self dual 1 ρσ ρσ µν 1 2(∗F ) = Fρσ (∗e ) = Fρσ µν e ; A (µBν)ρ = − ηµν A Bρσ: 2 ρ 4 ρσ We thus have Furthermore, if one of them is self-dual and the other one anti-self-dual it holds that µν 1 µνρσ (4) ∗F = Fρσ : µν 2 A Bµν = 0; [µ ν]ρ Aρ B = 0: B. The Complex Dual of Two-Forms In general, the Hodge star operator is its own IV. THE FREE MAXWELL-FIELD inverse up to a sign, which depends (among other things) on the rank of the form acted upon. For Recall that the electromagnetic tensor F is de- two-forms, such as the electromagnetic eld tensor, rived from a covector potential A by means of it holds that F = dA, where d denotes the exterior derivative (in components ). Due to this Fµν = @µAν − @ν Aµ ∗ ∗ F = −F: fact, F satises a Bianchi identity In order to have a self-inverse operation, the au- @ν Fρσ + @ρ Fσν + @σ Fνρ = 0: thors of [2] introduce a complex operator Contracting with µνρσ, one obtains the equivalent equation F~ := −i(∗F ); (5) µν @ν F~ = 0: in components (with respect to a right-handed or- thonormal frame) In the absence of sources, the eld equations take the form i F~µν = − µνρσF : (6) µν 2 ρσ @ν F = 0; 3 for which we will write from now on div F = 0. Indeed, the variation of the action with respect to The pair of equations the potentials reads div F = 0 Z (9) δS = d4x δL div F~ = 0 i Z = d4x G~ µν δF a are clearly invariant under the substitution 2 a µν Z ~ 4 ~ µν a F 7! cos(α)F + i sin(α)F; = i d x (@ν Ga )δA µ; which transforms the electric and magnetic eld as so the eld equations for the potentials are equiv- E cos α − sin α E alent to 7! : B sin α cos α B ~ (13) div Ga = 0: Recall that the eld energy density W and the Poynting vector S are given by We dene duality transformations as symmetry transformations of the eld equations, which are 1 W = E2 + B2 ; linear transformations among the eld strengths 2 F and their associated tensors G (see [3]). S = E × B: Theorem 1. In four spacetime dimensions, a lin- Since both expressions are invariant, so is the ear transformation of the tensors F and G leaves energy-momentum tensor. the eld equations (including those of χ) invariant Remark. One might consider adding terms of the if, and only if, the transformation is symplectic. In form ~ ~µν or ~µν to the action of electrody- other words, the group of duality transformations Fµν F Fµν F namics. But due to equation (8), the rst option is Sp(2n; R). does not add anything new, and due to the fact A proof of this theorem can be found in [3]. Fol- that div F~ = 0 we have lowing [2], we do not give the general proof, but ~µν ~µν restrict our attention to a certain form of the La- Fµν F = (@µAν − @ν Aµ)F grangian, which is common in supersymmetry and ~µν = 2(@µAν )F supergravity. µν = 2@µ(Aν F~ ); so second option simply adds a divergence, which VI. COUPLED MAXWELL-FIELDS does not contribute to the eld equations. a. The Lagrangian of Coupled Maxwell Fields V. GENERAL DUALITY We consider a collection of n covector potentials TRANSFORMATIONS A1; :::; An and a complex, symmetric n×n matrix- valued function with components (which may f fab The symmetry between the eld equations and depend on other elds), providing a coupling be- the Bianchi identities is a general feature of the- tween the various covector potentials by virtue of ories whose action depends on covector potentials the Lagrangian only via their eld strength tensors.