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Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations in four-dimensional Minkowski… The, Dennis 2003

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N O N L O C A L SYMMETRIES A N D NONLOCAL CONSERVATION LAWS OF MAXWELL'S EQUATIONS IN FOUR-DIMENSIONAL MINKOWSKI SPACE by DENNIS THE B.Math. University of Waterloo, 2001 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 2003 © Dennis The, 2003 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for exten-sive copying of this thesis for scholarly purposes may be granted by the head of publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada my department or by his or her representatives. It is understood that copying or Abstract We derive new nonlocal symmetries and corresponding new nonlocal conservation laws for the free-space Maxwell's equations (ME) in 3+1 space-time dimensions. These results arise from a detailed point symmetry analysis of several potential systems associated with ME: (1) LPS: the standard Lagrangian potential system, (2) A-LPS: LPS augmented by the Lorentz gauge, (3) PS: a natural non-Lagrangian potential system with dual vector potentials, and (4) A-PS: PS augmented by dual Lorentz gauges. The well-known (local) space-time symmetries and (local) energy-momentum conservation laws are shown to be recovered from the point symmetries of LPS and A-LPS. The point sym-metry structure of A-PS is much richer: inversion and vector rotation / boost symmetries arise which yield nonlocal symmetries of ME. Through an embedding of the set of local PS and A-PS symmetries into the set of (local and nonlocal) ME adjoint-symmetries, an explicit formula is given for constructing ME conservation laws from PS and A-PS symmetries. The resulting ME conservation laws are completely classified with respect to their locality or nonlocality. This clas-sification relies on an important cohomology result for ME for which a (partial) proof is given using tensorial methods. The work in this thesis extends the previously known classes of symmetries and conservation laws admitted by ME, and emphasizes the utility of potential systems and gauge constraints in their construction. ii Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgements vii Chapter 1. Introduction 1 1.1 Introduction and statement of principal results 1 1.2 Symmetries of partial differential equations 2 1.3 Potential systems and potential symmetries 4 1.4 Gauge symmetries and gauge constraints 5 1.5 Conservation laws 6 Chapter 2. Symmetries of Maxwell's Equations 9 2.1 Maxwell's equations 9 2.2 Potential systems of Maxwell's equations 10 2.3 Gauge symmetries and the Lorentz gauge condition 11 2.4 Symmetries of LPS 13 2.4.1 LPS symmetry determining equations 13 2.4.2 Solution of the LPS symmetry determining equations 17 2.5 Symmetries of A-LPS 17 2.5.1 A-LPS symmetry determining equations 17 2.5.2 Solution of the A-LPS symmetry determining equations 19 2.6 Symmetries of PS 19 2.6.1 PS symmetry deterrruning equations 20 2.6.2 Solution of the PS symmetry determining equations 22 2.7 Symmetries of A-PS 23 2.7.1 A-PS symmetry determining equations 23 2.7.2 Solution of the A-PS symmetry determining equations 27 2.8 Geometrical interpretation of symmetries admitted by LPS, A-LPS, PS, and A-PS .. 29 2.8.1 Space-time symmetries 29 2.8.2 Internal symmetries 30 2.9 Induced local and nonlocal symmetries of Maxwell's equations 33 2.9.1 Local ME symmetries induced by LPS and A-LPS symmetries 33 2.9.2 Local ME symmetries induced by PS and A-PS symmetries 34 2.9.3 Nonlocal ME symmetries induced by A-PS vector rotations / boosts 35 2.9.4 Nonlocal ME symmetries induced by A-PS inversions 35 2.9.5 LPS symmetries induced by PS and A-PS symmetries 37 2.10 Chapter 2 Summary 39 iii Table of Contents iv Chapter 3. Conservation Laws of Maxwell's equations 40 3.1 Introduction 40 3.2 Local conservation laws of ME and ME potential systems 42 3.3 ME conservation laws derived from LPS and A-LPS symmetries 43 3.4 ME conservation laws derived from PS and A-PS symmetries 44 3.4.1 Locality criterion 46 3.4.2 Computation of characteristics and conservation law classification 49 3.5 LPS conservation laws derived from PS and A-PS symmetries 53 3.5.1 Locality criterion and conservation law classification 54 3.6 Chapter 3 Summary 56 Chapter 4. Cohomology 57 4.1 Introduction 57 4.2 Tensor decompositions 57 4.3 Solution of the ME cohomology equation 59 4.3.1 Case r = 0 59 4.3.2 Case r = 1 60 4.3.3 General case: r > 1 62 4.4 Chapter 4 Summary 64 Chapter 5. Future Research 65 Bibliography 66 Appendix A. Tensor Methods and Index Juggling 67 Appendix B. Conformal Killing Equation Solution 69 iv List of Tables 2.1 Well-known gauges for LPS 11 2.2 Point symmetry classification for ME potential systems 32 2.3 Classification of LPS induced symmetries of ME 33 2.4 Classification of PS induced symmetries of ME 34 2.5 Classification of A-PS induced symmetries of ME 37 2.6 Classification of PS induced symmetries of LPS 38 2.7 Classification of A-PS induced symmetries of LPS 38 3.1 Comparison of ME and ME potential systems 42 3.2 Characteristics of ME conservation laws induced by PS symmetries 52 3.3 Characteristics of ME conservation laws induced by A-PS symmetries 52 3.4 Classification of ME conservation laws induced by PS and A-PS symmetries 53 3.5 Classification of LPS conservation laws induced by PS and A-PS symmetries 56 v List of Figures 2.1 Potential systems of Maxwell's equations Acknowledgements It is with great pleasure that I express my gratitude to all who have helped me in some way with the writing of this thesis. I thank George Bluman, my thesis co-supervisor, for his introduction to the world of Lie symmetries, valuable suggestions, guidance and encouragement. I am indebted to my Stephen Anco, my thesis co-supervisor, for his guidance, patience, im-peccable attention to detail, criticisms and corrections, and his ability to keep the big picture in focus for me throughout the writing of this thesis. To friends that have been here along the way - Janice Shu, Brenda Fine, Yinan Song, and Leo Tzou. For moral support and encouragement, enlightening discussions, and of course comic relief, I thank you. To my family, especially my mother, whose strength I draw from every day. Thank you for your love and support. To my professors and mentors that have brought me here today - in particular, Roman Smirnov and Ray McLenaghan. Thank you for inspiring me and continuing to fuel my passion for mathematics. Finally, I would like to thank the University of British Columbia and NSERC for financial support during the course of my Master's studies. vii Chapter 1 Introduction 1.1 Introduction and statement of principal results Lie symmetries and their generalizations are essential tools that have arisen in group theoretic and geometric approaches to the study of differential equations since the work of Sophus Lie over 100 years ago. They are useful in reducing the order of ODE's, constructing invariant solu-tions of PDE's, finding linearizations of nonlinear PDE's, discovering conservation laws, etc. Conservation laws play an important role in physical field theories by determining con-served quantities as the fields evolve in time. The familiar laws of conservation of energy, linear momentum, and angular momentum are important physical principles which are invaluable tools for tackling many problems arising in mathematical physics. From a mathematical stand-point, conservation laws provide information on the basic properties of solutions of a system of PDE's. Over the past century, Noether's theorem [15] has been the principal symmetry-based tool for systematically finding conservation laws of PDE's involving local expressions in terms of the given field variables. Roughly speaking, it says that there is a 1-1 correspondence between (variational) symmetries and (equivalence classes of) conservation laws. However, Noether's theorem is valid only in the setting of Lagrangian systems - i.e. systems arising from a varia-tional principle. Recently, a generalization of Noether's theorem by means of adjoint-symmetries to the setting of non-Lagrangian PDE systems was provided in papers by Anco and Bluman [3,5,6]. For the free-space Maxwell's equations (abbreviated throughout by "ME") in 3+1 space-time dimensions arising in classical electromagnetic theory, all local conservation laws and local sym-metries have been recently classified by Anco and Pohjanpelto [8, 10] using a variant of the methods of Anco and Bluman coupled with spinor techniques. In this thesis we investigate non-local symmetries and generate nonlocal conservation laws of ME, purely by tensorial techniques. This work is an extension of the investigation of ME in 2+1 space-time dimensions initiated by Anco and Bluman [4], which introduced the main ideas to be used here. Local (point and generalized) symmetries admitted by a PDE system can be computed by means of an explicit algorithmic procedure, but no such procedure is known for computing all nonlocal symmetries. Bluman, Kumei and Reid [12] described a method for computing a class of nonlocal symmetries (called potential symmetries) by realizing such nonlocal symmetries as local symmetries of a potential system associated with the given PDE system. We adopt this approach for our investigations and consider several potential systems of ME: 1 Chapter 1. Introduction 2 • LPS: the standard Lagrangian potential system, • A-LPS: LPS augmented by Lorentz gauge, • PS: a natural non-Lagrangian potential system, • A-PS: PS augmented by dual Lorentz gauges. The systems LPS and A-LPS have been well-studied, while PS and A-PS have not been investi-gated as much in the literature. LPS and A-LPS introduce a single vector potential. PS and A-PS possess a natural duality between two introduced vector potentials. By comparison, for ME in 2+1 space-time dimensions, the dual potential is a scalar function [4]. A detailed point symmetry analysis of the potential systems LPS, A-LPS, PS, and A-PS is carried out in Chapter 2. Projection of these point symmetries back onto ME yields only lo-cal symmetries in the case of LPS, A-LPS, and PS. For A-PS however, projection of A-PS point symmetries generates nonlocal symmetries of ME - i.e. symmetries which involve an explicit dependence on the potentials. In Chapter 3, our approach to generating nonlocal conservation laws of ME is similar in spirit to deriving nonlocal symmetries: We realize nonlocal conservation laws of ME as local conservation laws of a potential system associated with ME. In implementing this approach we utilize a corre-spondence between symmetries of the potential systems and the adjoint-symmetries needed for producing conservation laws. For LPS, symmetries are adjoint-symmetries, and we show that the LPS conservation law formula recovers all local conservation laws of ME. For PS and A-PS, an important result is that there is an embedding of the set of local PS [A-PS] symmetries into the set of (local and nonlocal) ME adjoint-symmetries. This embedding leads to a formula which generates conservation laws of PS in terms of symmetries of PS and A-PS, and by projection this yields (local and nonlocal) conservation laws of ME. The embedding relation is one of the central new contributions of this thesis, going beyond the ideas in Anco and Bluman [4] for the 2+1 dim-ensional case. In the case of A-PS, new nonlocal ME conservation laws are generated from the additional point symmetries admitted by A-PS. A complete classification of ME conservation laws derived from local PS or A-PS conser-vation laws is established. Our main classification theorem provides necessary and sufficient conditions for the locality of ME conservation laws in terms of their characteristics with respect to PS or A-PS. To address the question of whether or not any of the new nonlocal ME conser-vation laws are genuinely nonlocal when regarded as conservation laws of LPS, we establish an analogous classification theorem for the locality of LPS conservation laws. The classification method is another new contribution of this thesis. These locality classification theorems rely on important cohomology results on the ME and LPS solution jet spaces. A partial proof of the ME cohomology result using tensorial methods is provided in Chapter 4. 1.2 Symmetries of partial differential equations The concept of symmetry is fundamental in mathematics. For PDE's, we have the following formulation: A symmetry of a PDE is a transformation of solutions of the PDE into solutions of the same PDE. Chapter 1. Introduction 3 The precise notion of a "transformation" in the context of symmetries and PDE's has signif-icantly evolved over the past century. Lie considered point symmetries, which are transforma-tions acting on the space of independent variables x = (a;1,..., xn) and dependent variables u = (u1,..., uN) of a PDE, and showed that they can be characterized by infinitesimal genera-tors X = ^ « ) ^ + ^ ( x , « ) ^ . (1.D (Here and throughout this thesis, summation convention is assumed over repeated indices.) Any infinitesimal generator can be naturally "prolonged" to a generator X^ f c' acting on higher derivatives of u with respect to x up to order k. The action of a generator (1.1) on functions is equivalent to one involving no motion on the independent variables: ±= (rja(x,u) -e(x,u)diUa)-^. (1.2) This is referred to as the evolutionary form of the point symmetry. Noether recognized the pos-sibility of considering more general transformations. She suggested a generalization of Lie's infinitesimals by allowing the coefficients £ l and r]a to depend on derivatives of the dependent variables up to any finite order. Such symmetries are commonly called generalized symmetries or local symmetries. They are local in the sense that at any point x, the infinitesimals depend only on x, u(x), and higher derivatives of u evaluated at x. Point symmetries of a PDE can be computed using the following infinitesimal criterion. Theorem 1.1 Let X = + V01-^ be an infinitesimal generator, where and rf depend locally on x,u and derivatives of u up to some finite order. Then X is a point symmetry of the system of PDE's GA(x,vS-r">) = 0 ijfX^GA(x^u(-r">) = 0 whenever u satisfies GA(x,u^) = 0. (Here represents u and its higher order derivatives up to order r.) A geometrical construction useful for the computation of local symmetries is the jet space. For an excellent introduction to these ideas, we refer the reader to Olver [15]. We summarize the main ideas behind this construction here: Given an r-th order system of PDE's, the corre-sponding jet space is the cartesian space whose coordinates represent all independent variables, dependent variables, and their derivatives up to some finite order R (where R>r typically). All coordinates are taken to be independent, and hence the PDE system and its differential conse-quences becomes a system of algebraic equations which determine some concrete, geometrical subset of the jet space, also referred to as the solution jet space. In the jet space context, a sym-metry can be visualized as a tangent vector field on the solution jet space. Consequently, any symmetry is the solution of the linearization of the given PDE system. We describe this in more detail below. Consider a general system GA(x,u,uW) = 0 (1.3) of at most first order (for simplicity, though we can easily generalize for higher order systems), where ua(x) depends on n > 2 independent variables x%. Here i = 1,..., n (coordinate index), a = 1,...,N (dependent variable index), and A = 1,..., M (equation index). We will represent the first order partial derivatives dua(x)/dx1 by the coordinates ua^ in the associated jet space. Chapter 1. Introduction 4 Definition 1.2 Local symmetries X = va^a of a first-order PDE system (1.3) are given by the solu-tions r]a(x, u, , u(2),...) of the linearized system L G [ r ? ] A = VaGA,a + ( A r ? a ) G A a = 0 (1.4) on the solution jet space. Here GAt0t = dGA/dua, GA?a = dGA/dua^ and Di is the total derivative operator with respect to x1 in the jet space. 1.3 Potential systems and potential symmetries New classes of symmetries can be investigated by considering infinitesimals that at any point x depend on the values of u = u(x) in a neighbourhood of x. These are referred to as nonlocal symmetries. The infinitesimals of nonlocal symmetries exhibit a more general dependence than do local symmetries, such as through integrals of the dependent variables or through potential variables, and hence cannot be formulated in the jet space associated with the given PDE system. Classes of nonlocal symmetries called potential symmetries were introduced in [12] by con-sidering the local symmetries of a potential system S of a given PDE system R. The ideas behind this approach are best illustrated through an example (found in [12,14]). The wave equation R{x, u} : uxx 2 7 _ T ' U « = 0 ( 1 - 5 ) C (X) may be written, by inspection, in a conserved form, Dxux - Dt ( ^ y ) = 0. (1.6) We introduce the potential v through the potential system S{x,u,v}: ( * I UZ_ (1.7) Note that by construction, the compatibility condition, vtx = vxt, is automatically fulfilled. The systems R{x, u) and S{x, u, v} have the following relationship: • If (u(x, t),v(x, t)) solves S{x, u, v}, then u(x, t) solves R{x, u}. • If u(x,t) solves R{x, u), then there exists a non-unique v(x,t) such that (u(x,t),v(x,t)) solves S{x, u, v}. We say that the R{x, u} solution space is embedded in the S{x, u, v} solution space. We emphasize that the transformation from R{x, u} to S{x, u, v} is not 1-1 since the potential v is non-unique: if ( i t , v) solves S{x, u, v}, then so does (u, v + C) for any constant C since v only appears in terms of its partial derivatives. Consider a point symmetry group Gs admitted by S{x, u, v}. The solution space of S{x, u, v} is invariant under the action of Gs, i.e. any solution of S{x, u, v} is mapped into another solution of S{x, u, v}. Since the R{x, u} solution space is embedded in the S{x, u, v} solution space, then Chapter 1. Introduction 5 any solut ion of R{x, u} is m a p p e d to another solution of R{x, u} under the action of Gs- Thus , local symmetries of S{x, u, v} induce (possibly nonlocal) symmetries of R{x, u}. Induced symmetries are described in terms of infinitesimals as follows. Suppose that X 5 is an infinitesimal generator for a local symmetry of S{x, u, v}, and hence has the fo rm d d d d X s = (s{x,t,u,v)~ + TS(x,t,u,v)— + rjS(x,t,u,v)— + Cs(x,t,u,v) — . (1.8) The corresponding induced symmetry X # of R{x, u} is consequently g iven b y d d d X « = £s(x,t,u,v)— +rs(x,t,u,v)— + ns(x, t,u,v) — . (1.9) If (£ s, TS, Vs) depends nontrivial ly o n the potential v o n the solution space of S, then we say that X.R is a potential (nonlocal) symmetry of R, or that X g projects to a potential (nonlocal) symmetry of R. M o r e generally, w h e n considering P D E systems wi th more than n = 2 independent variables, potentials can be introduced if the system can be written in a conserved (divergence) form. The most important feature in the construction of a potential system is that the solution space of the original P D E system is embedded in the solution space of the potential system. In this way, local symmetries of the potential system naturally induce (possibly nonlocal) symmetries of the original system, and we have an algorithmic w a y of searching for n e w symmetry classes admitted b y the original system. Wi th n > 3 independent variables however, the non-uniqueness of the potentials as solutions of the potential system becomes more complicated. Instead of hav ing to al low uniqueness u p to an addit ive constant, we must in general deal w i th gauge symmetries. 1.4 Gauge symmetries and gauge constraints D e f i n i t i o n 1.3 A local symmetry of a system of PDE's is a gauge symmetry if it depends homoge-neously on an arbitrary function of all independent variables in the system of PDE's. In this situation, the PDE system is said to have gauge freedom. Otherwise, if the system does not admit any gauge symmetries it is said to be wel l-posed. For example, i n R3, consider the irrotational equation V x u = 0 regarded as a system of P D E ' s for the u n k n o w n functions u = ( i t 1 , u2, u 3 ) . This system admits the gauge symmetries x = ^ -where <p is an arbitrary function of x = (xl,x2,xz). The gauge freedom amounts to shifting any solution why a gradient V</>. G a u g e symmetries naturally arise if we consider systems wi th n > 3 independent variables. A n c o a n d B l u m a n [4] show that the presence of gauge f reedom rules out the existence of poten-tial symmetries. Chapter 1. Introduction 6 Theorem 1.4 Suppose that the PDE system R{x, u} is well-posed and its solution space is embedded in the solution space of a potential system S{x, u,v} that has gauge freedom. Then every local symmetry admitted by S{x, u, v} projects to a local symmetry of R{x, u}. Definition 1.5 A gauge (or gauge constraint) is any condition imposed on the dependent variables (and their derivatives) of a given system of PDE's such that the augmented system consisting of the system of PDE's and the gauge condition admits NO gauge symmetries. Arbitrary gauges might be too restrictive in the sense that the resulting augmented system either might not possess any solutions or may lose solutions of the original PDE system. A natu-ral requirement for a candidate gauge constraint is that the solution space of the original system should be embedded in the solution space of the augmented system in a suitable manner. For this purpose, solutions related by a gauge symmetry transformation are regarded as belonging to an equivalence class. Definition 1.6 A gauge is compatible with a given PDE system if for each gauge-equivalence class of solutions there exists a solution of the augmented system. (If a gauge is compatible then we also say it can be achieved.) Thus, when considering a candidate gauge constraint, we must check if (i) the gauge is compatible, and (ii) the augmented system does not admit gauge symmetries. Given a PDE system with gauge symmetries, a general systematic procedure for determining compatible gauges is still unknown. Moreover, given a compatible gauge constraint, a theoreti-cal foundation has yet to be developed for determining a priori when the resulting augmented system admits symmetries which project to potential (nonlocal) symmetries of the original sys-tem. We do not investigate these problems in this thesis, but note that they are worthy of future investigation. 1.5 Conservation laws Definition 1.7 A conservation law of a PDE system G(x,u^) = 0 is an expression $ = ($l(x, u ^ ) , $ n ( x , u^)) that is divergence-free on solutions u = f(x) of the given system, i.e. d i v $ = Da$a — 0 w h e n e v e r u — f(x) satisf ies G(x,u^) = 0. (1-10) The conservation law is local if $ is a function defined on the jet space associated with G. Otherwise, $ is a nonlocal conservation law. On the solution space of the system, conservation laws that differ only by a curl term Di,0ab, Qab _ 0{ab}^ a r e t o foe equivalent, since they yield the same conserved quantities [5, 6, 15]. Local conservation laws can also be formulated as equations holding on the entire jet space by the following standard result. Chapter 1. Introduction 7 Proposition 1.8 Under mild non-degeneracy conditions on the system, which includes ME and its po-tential systems (see Anco and Pohjanpelto [8]), every local conservation law is equivalent to one such that div ($ + R) = Q • G (1.11) holds identically. Here Ris some expression that depends linearly on the components GA of the PDE system. Q is called the characteristic of the conservation law <£. For Lagrangian systems, Noether's theorem [15] establishes a 1-1 correspondence between variational symmetries (modulo gauge symmetries, if any) and equivalence classes of conserva-tion laws. (Variational symmetries are symmetries of the corresponding Euler-Lagrange equa-tions for which the action is invariant.) Recently, a generalization of Noether's theorem to non-Lagrangian systems has been established by means of adjoint-symmetries. Specifically, all lo-cal conservation laws of a PDE system arise from multipliers (characteristics) that are adjoint-symmetries of the system subject to additional adjoint-invariance conditions [3,5,6]. Once mul-tipliers are identified, the associated conservation law can be constructed by a homotopy in-tegral expression [3, 15]. The adjoint-symmetry equations together with the adjoint-invariance conditions provide a system of determining equations for finding local conservation laws [5, 6]. Definition 1.9 For a first order PDE system (1.3), the adjoint of the linearization (1.4) is LG[u}a = uAGA,a - Di(uAGA:a) = 0 (1.12) on the solution jet space. The solutions loa(x,u,u^\u^2\ ...) of (1.12) are called the local adjoint-symmetries of (1.3). There is a systematic way of directly generating conservation laws from symmetry / adjoint-symmetry pairs, which we describe here. Lemma 1.10 For a first order PDE system (1.3), there is an important "integration by parts" formula: uALG[V}A - r)aLG[uj]a = A*Mw, m G], (1.13) where $i[u,V-G}=r1auAGAX. (1.14) Then, for any symmetry /adjoint-symmetry pair r]a, u>A, we have a conservation law $l[u, rj; G] since, on solutions of system (1.3), the left side of (1.13) vanishes and thus Di$l[u,ri; G] = 0 on solutions of (1.3). This formula holds regardless of whether the symmetries / adjoint-symmetries are local or nonlocal. The conservation law formula generated from symmetry / adjoint-symmetry pairs is due to Anco and Bluman [3, 5, 6] for general PDE systems, and has been applied to ME by Anco and Pohjanpelto [8]. It was first used for linear self-adjoint PDE systems by Anco and Bluman in [2]. Moreover, in [2] it is shown that when a scaling symmetry of the dependent variables is used, the formula recovers all local conservation laws obtainable from Noether's theorem in the case of linear PDE systems. An application to self-adjoint potential systems of Maxwell's equations in 2+1 space-time dimensions appears in [4]. Chapter 1. Introduction 8 M a n y physical ly mteresting systems are invariant under some scaling symmetry. B y fixing such a symmetry, expression (1.14) generates conservation laws in terms of adjoint-symmetries. For our purposes we wi l l only present the formula for systems admitt ing a scaling symmetry in the dependent variables. Suppose that a P D E system (1.3) admits the scaling symmetry Xua = rjsa(x, u, u^) = ua. (1.15) T h e n the expression $1[uj, rjs] generates a conservation law in terms of any adjoint-symmetry. Proposition 1.11 For a PDE system (1.3) admitting a scaling symmetry X = ua-^, every adjoint-symmetry (1.12) generates a conservation law & = uau>AG AX- (1-16) The fo l lowing key result and its proof are consequences for linear P D E ' s of the general results p roved in [1]. Theorem 1.12 IfuiA is a characteristic for a local conservation law of a scaling-invariant PDE system (1.3), then $ l = uaujAGA*a is equivalent to a non-zero scaling multiple of the conservation law deter-mined by wa-A generalization of equation (1.16) and Theorem 1.12 to higher order P D E systems, as wel l as to more general scaling symmetries invo lv ing both the independent and dependent variables, has been der ived b y A n c o in [1]. Chapter 2 Symmetries of Maxwell's Equations 2.1 M a x w e l l ' s e q u a t i o n s Maxwell's equations in a medium that is uniform in space, i.e. homogeneous, isotropic, and stationary in time, can be expressed in terms of the electric and magnetic fields E and B as: dE V - E = p, V x B - — = J, dB V - B = 0, V x E + — = 0, (2.1) (2-2) where p and J = (J 1, J 2 , J 3) are respectively scalar and vector functions of time t and position x = (a?1, a;2, a:3). Let x*= (x° = t,x\x2,x3), J = (J° = p,j\j2,J3), (2.3) be the 4-vectors for position and current density, and consider the antisymmetric 4 x 4 matrix / 0 Ei E2 E$ \ —Ei 0 S3 — B2 —E<i —Bz 0 B\ \ -E3 B2 -Bi 0 ) Then Maxwell's equations (in Lorentz-covariant form) take the form (2.5) (2.6) (jPa6) = (2.4) d{aFbc] = 0, riabdaFbc = Jc. where Jc are the components of the 4-vector J, da = g | a , r]ab = diag(-l, +1, +1, +1) is the Minkowski metric, and [ • • • ] denotes antisymmetrization over the enclosed indices. The sum-mation convention is used over repeated space-time indices 0,1,2,3 (0 denoting the time coor-dinate). The lowering or raising of indices is performed using the covariant metric 77^ or the contra variant metric rjab. The free-space Maxwell's equations (ME) in Minkowski space (M4,r]) stated in Lorentz-covariant form are d[aFbc] = 0, daFab = 0. 9 (2.7) (2.8) Chapter 2. Symmetries of Maxwell's Equations 10 2.2 Potential systems of Maxwell's equations A standard w a y of introducing potentials is through the div-curl relation f rom 3-dimensional Euc l idean vector calculus, i.e. if V • F = 0 (on a simply-connected domain) , then F = V x A , where A is some vector potential. A dua l fo rm of this fact in the context of differential r-forms is a result f rom differential geometry k n o w n as Poincare's L e m m a : Any closed r-form on a smoothly contractible manifold is exact. M i n k o w s k i space ( M 4 , rf) is a 4-dimensional manifo ld that is smoothly contractible. Thus , by Poincare's L e m m a , any r-form is exact - i.e. it can be written as the exterior derivative of some (r — l ) - form. Since Fab denotes the components of a 2-form F = Fabdxa A dxb, then (2.7) says that F is a closed 2-form, or dF = 0, where d is the exterior derivative. Hence F must be exact, and s o F = dA, where A — Abdxb is a 1-form, i.e. d[aFbc] = 0 ^ F a b = 8[aAb]. (2.9) The standard potential system associated wi th M E is LPS: Fab = d[aAb], daFab = 0. (2.10) The system (2.10) is self-adjoint, and hence has an associated Lagrangian. We call this system the Lagrangian potential system (LPS). Since this system is derivable f rom a variational pr inc i-ple, Noether 's theorem says that there is a 1-1 correspondence between variational symmetries m o d u l o gauge symmetries and equivalence classes of conservation laws. A s for (2.8), note that this equation can be rewritten in the equivalent form, dF' — 0, or d[aF'bc] = 0, (2.11) us ing F'ab := (*F)ab = l-eabcdFcd, (2.12) w h i c h is the H o d g e dua l of FC([. Here eabcd = e^abcd^ is the vo lume tensor, w i th corresponding v o l u m e f o r m eabcd associated wi th rjab. (Properties of the vo lume fo rm are g iven in A p p e n d i x A . ) Note that in M i n k o w s k i space, * satisfies * 2 = - 1 since {*2F)ab = \eabcd€cdmnFmn = —Fab. U s i n g the H o d g e dual , we see that there is a natural dual ity between F a n d F'. Equat ion (2.11) indicates that F' = F'abdxa A dxb is a closed 2-form, so appeal ing again to Poincare's L e m m a , we conclude that F' must be exact: F' = dA', where A' = A'bdxb is a 1-form, i.e. d[aF'bc] = 0 ^ F ' a b = d[aA'b]. (2.13) U s i n g the algebraic relation (2.12) combined wi th the potential equations (2.9) and (2.13), we get a potential system of LPS , and hence a further potential system of M E , PS: d[aA'b] = l-eabcddcAd. (2.14) In contrast to LPS , the potential system PS is not self-adjoint and hence non-Lagrangian. We wi l l investigate symmetries of L P S and PS and demonstrate h o w they can be used to construct s y m -metries a n d conservation laws of the original system M E . Chapter 2. Symmetries of Maxwell's Equa tions 11 2.3 Gauge symmetries and the Lorentz gauge condition Through definition (2.9) of the potential variable A, the LPS system (2.10) admits the gauge symmetries d Ab H-> Ab + dbx (2.15) where x is a n arbitrary function of all independent (space-time) variables xa. Similarly, through definitions (2.9) and (2.13) of the potential variables A and A', the system PS admits the gauge symmetries d Ab H+ Ab + dbx A'b - > A'b + dbX' X = 9bXdAb x ' = a 6 X ' 9 dA'C (2.16) (2.17) where \, x' are arbitrary functions of all independent (space-time) variables xa. Since the system ME is well-posed, then by Theorem 1.4 any local symmetry of a ME po-tential system with gauge freedom projects to a local symmetry of ME. Hence, for the purposes of our investigation, we need to impose gauge constraints on LPS and PS. In the case of LPS, some well-known gauges arising in the literature are shown in Table 2.1. In each case, the gauge restricts x m the original gauge symmetry (2.15) so that it is no longer an arbitrary function of all independent variables xa. Moreover, the gauge is compatible with the original system. Note: In the table below, repeated Greek indices indicate summation over 1,2,3 (spatial variables). Gauge Name Description Lorentz VabdaAb = 0 Coulomb S^d^Av = 0 Temporal Ao = 0 ' Axial n»A^ = 0 (n 1, n 2, n 3) =unit spatial vector Cronstrom xaAa = 0 Table 2.1: Well-known gauges for LPS Since M E is written in a manifestly covariant manner, natural gauges to investigate for sym-metry purposes should also be manifestly covariant. For LPS we will consider the standard Lorentz gauge on the potential variable A, i.e. daAa = 0. It should be noted that in 2+1 space-time dimensions, Doran-Wu [13] showed that neither the temporal nor axial gauge for LPS led to any nonlocal symmetries of Maxwell's equations. With the Lorentz gauge imposed on LPS we have the resulting augmented system A-LPS: Fab = d[aAb], daFab = 0, OaAa = 0. (2.18) Chapter 2. Symmetries of Maxwell 'sEqua Hons 12 This system, unlike LPS, is not self-adjoint, and hence is not Lagrangian. For the potential system PS, duality between A and A' motivates us to impose the Lorentz gauge on the potential variable A' as well. Hence, we have the augmented potential system A-PS consisting of PS augmented by dual Lorentz gauges A-PS: d[aA'b] = \eabcddcAd, daAa = 0, daA'a = 0. (2.19) The potential systems of ME that we will consider are displayed in Figure 2.1. ME: d[aFbc] = 0) daFab = 0 Fab=d[aAb LPS: Fab = d[aAb], O^Fab = 0 A-LPS: Fab = d[a.Ab], cPFab = 0, t^Aa = 0 Kb=d[aA'b] =d<aA', PS: d[aA'b] = \eabcdb^AA A-PS: d[aA'b] = \eabcddcAd, daAa = 0, daA'a = 0 Figure 2.1: Potential systems of Maxwell's equations We will establish compatibility of the dual Lorentz gauges with PS. The compatibility of the Lorentz gauge with LPS is established similarly. To establish compatibility of daAa = 0, daA'a = 0 with PS, it is necessary to show that for any solution Aa, A'a of PS (2.14) we can find specific functions x, x' s u c n t n a t + daX, A'a + dax' satisfy A-PS (2.19). Hence, we must have dadaX = -daAa, dadaX' = -cTA'a. (2.20) By standard existence results for the inhomogeneous linear wave equation in ( M 4 , 77 ) , a solution (x> x') to (2.20) exists. Thus, dual Lorentz gauges are compatible with PS. Observe that (2.16), (2.17) are symmetries of A-PS iff dadaX = 0, dadaX' = 0. (2.21) Thus, x a n d x' satisfy the wave equation in (M4,??). By standard existence results, a solution exists and x, X a r e n o longer arbitrary functions of the independent variables. Consequently (2.16), (2.17) are not gauge symmetries of A-PS, and A-PS does not possess any gauge freedom in the sense of Definition 1.3. The potential system A-PS will be the main focus of our investigation here. Unlike the po-tential systems LPS and A-LPS, which have been well-studied in the literature in terms of their symmetries, a detailed symmetry analysis of the potential systems PS and A-PS has not been Chapter 2. Symmetries of Maxwell 'sEqua tions 13 carried out. It will be shown that A-PS admits point symmetries that project to nonlocal (poten-tial) symmetries of ME and lead to corresponding nonlocal conservation laws of ME. This will be contrasted with the point symmetries admitted by LPS and A-LPS, and the conservation laws of ME that they generate. It has been proven that any point symmetry (in evolutionary form) of the original system ME is linear in the dependent variables [9, 10]. The same methods are expected to establish an analogous result for the potential systems (with or without gauges). Hence, this motivates us to search for linear point symmetries of the potential systems. For LPS and A-LPS, linear point symmetries take the form X = ( X A a ) ^ + ( X F o j ) ^ , where X A a and ~X.Fab are lin-ear in A, F and their (first) derivatives. For PS and A-PS, linear point symmetries take the form X = ( X A a ) + (KA'a)g^r, where X A a and XA'a are linear in A, A' and their (first) derivatives. A complete classification of the linear point symmetries of all potential systems LPS, A-LPS, PS, and A-PS will be given. These results are an extension of the results of Anco and Bluman ob-tained for ME and related potential systems in 2+1 dimensions. We turn now to the computation of the point symmetries of the potential systems. 2.4 Symmet r i e s of L P S Consider LPS: Fab = d[aAb], daFab = 0. (2.22) This potential system of ME has been well-studied in the literature. In addition to admitting the obvious scaling symmetry in the dependent variables, LPS is known to admit space-time symmetries induced by translations, dilation, rotations / boosts, and inversions of the indepen-dent variables xa, i.e. conformal isometries of Minkowski space (see [17], for example). It is also well-known that by imposing the Lorentz gauge on LPS, the resulting system A-LPS loses the in-versions as admitted point symmetries. We now carry out these calculations on the LPS solution jet space using modern tensor notation. Let X = (XAa)^^ + (XFab)-Q^ be a linear point symmetry of LPS. Then X A , = e(x)DcAa + aab(x)Ab + pabc(x)Fbc, (2.23) X F a 6 = e(x)DcFab + Kabc(x)Ac + »abcd{x)Fcd, (2.24) where w.l.o.g., we have the index symmetries (3abc = f3aybc^, Kabc = K [ a 6 ] c , p,abcd = nab[cd] = M[a6]cd/ due to antisymmetry of Fab. 2.4.1 LPS symmetry determining equations The LPS symmetry determining equations are XFab = D[aXAb], (2.25) DaXFab = 0, (2.26) on the LPS solution jet space. Substitution of (2.23), (2.24) into (2.25), (2.26) yields a system of equations linear in A, F and their derivatives. We arrange the splitting of the determining equa-tions as follows: Chapter 2. Symmetries of Maxwell's Equations 14 • The coefficients of Ac must vanish. • The (symmetrized) coefficients of the symmetrized derivatives D^Aq must vanish. • Replace antisymmetrized derivatives D[CA^ by Fcd. In the resulting equation, the (anti-symmetrized) coefficients of Fcd must vanish. • For the coefficients of DaFbc, we use Proposition 2.1. Proposition 2.1 Let Tklm = Tk^lm\ Then TklmDkFim = 0 on the LPS solution jet space if and only if rf{kl)m _ ^ ara _ ^.{l^k) (2.27) for some ak. Proof. See Section 4.2 (Proposition 4.3), where a more general result is stated and proved. • When manipulating the determining equations it will be convenient to use the following definitions: Definition 2.2 Let Oiab •= Ctab + da£b, &ab •= ®ab ~ (2.28) Decomposing DaAb = D(aAb) + D[aAb] and using (2.23), (2.24), we can write out (2.25), (2.26) more explicitly as: =0 (on LPS) 0 = c f D c j ^ ^ ^ + t / W ^ -S^a^D^Aq + {pabcd - oyfy'Vc - 6[akpb]lmDkFlm, (2.29) and =0 (on LPS) 0 = £J3T0^HdaKabc)Ac-KbWD{cAd) -k^D^Aq +(daPabcd)Fcd + (5bmdie - pbklm)DkFlm. (2.30) Using the notation defined in (2.28), we obtain the following splitting of (2.25), (2.26): • Coefficient of Ac: Kobe = d[aOJ(,]c> (2.31) t^Kabc = 0. (2.32) • Coefficient of D(cAdy. S[a{c&b]d) = 0, (2.33) H{cd) = 0. (2.34) Chapter 2. Symmetries of Maxwell's Equations 15 • Replace D^A^ by Fcd. T h e n extract the coefficient of Fcd: Vabcd-d[a(3b]cd = 6{Jcab]^, (2.35) d > 0 6 c d - ^ M = 0. (2.36) • B y Proposit ion 2.1, the coefficients of DkFim must satisfy: V F C /V ) M = rf^-V^TabV, (2.37) 6[mbdk}^ _ ^ {kl)m = v^abm _r/m(labk)j ( 2 3 g ) for some Tabm, obk w i th Tabm = T^m. Note that in the first term of (2.38), symmetr izat ion o n (Ik) is raised above antisymmetrization o n [Im]. Th is indicates that symmetr izat ion is per formed after antisymmetrization. Propos i t ion 2.3 The solution of (2.37) is j3abc = 0. Proof. We use a transvection technique. Let v G R4 be arbitrary. M u l t i p l y (2.37) b y vkvi to get v[aPb]lmvi = [(v • v)6km - vkvm]rabk, (2.39) where we use the dot to indicate the M i n k o w s k i inner product of vectors. M u l t i p l y (2.39) b y Vf a n d antisymmetrize o n [abf]. T h e n 0=[(v v)6km - vkvm]T[abkvf]. (2.40) M u l t i p l y (2.40) by v9 and antisymmetrize o n [mg] to get 0 = (v • v)v[fTab^mv9h This impl ies 0 = V[fTab] \-mv9^ for all v such that v • v ^ 0. Since v • v ^  0 defines a dense set in R4, then by continuity 0 = vlfTab][m^ (2.41) must ho ld for all v G R4. Since r is independent of v (it depends only o n x), then equation (2.41) can only h o l d for all v if rabc = 0. N o w mul t ip ly 6[Jk(3b]l^m = 0 b y 5ia. Simplif ication yields Pbkm = ±6bkpaam. (2.42) Mult ip l icat ion of (2.42) by Smb and us ing antisymmetry in the second and third indices of (3 yields f3aam = 0, and hence j3bkm = 0. • Consequently, (2.35) becomes ^bcd = S[a^ab]^, (2.43) w h i c h can be used to rewrite (2.38) as Sldbd(cka) + rjk(a6[k^ab]^ = rfcobd - Vd^coba\ (2.44) Propos i t ion 2.4 For LPS, the following hold: (i) a[o6] = 0[o&], (ii) ot(ab) = \VabUkk, (iii) 0(a&) = \Vabdk^k. Chapter 2. Symmetries of Maxwell's Equations 16 Proof. After multiplying (2.33) by 5da and simplifying, we obtain = \r}bdakk, which implies &[bd] = 0/ a n d consequently yields (i). Multiplying (2.44) by r\ac, we can solve for abd: cV = \^dbdckc + S^a^) = I (\(6\dctc - + \{2abd + 6bdacc^j . (2.45) Now multiply (2.44) by 5bd. Then + -(rjacabb+ 2a^) = rjacabb -which simplifies to 4d(a£c) _ nacdk£k = \vacakk - 2a^. (2.46) Writing aac = aac + 2da£c, and using &ac = zVac&kk in (2.46) we get (ii) and (iii). • Remark 2.5 Equation (iii) is the conformal Killing equation whose solutions are well-known: <f = aa + uxa + ^ akxk - xa<t>kxk + ^cpaxkxk, (2.47) where aa, u>, ipab = —ipba, <fia are arbitrary constants. (A self-contained proof is given in Appendix B). Note that 3acf = vab(u - 4>cxc) - *l>ab - 2^ax^. (2.48) From (2.34), we have Kakk = 0, which by (i), (ii), and (2.31) implies that 3aakk = dkaak = dk Qr/afcacc + d[a^ . (2.49) This simplifies to daakk = §<9fc<9[Q£fc] = -40 a . Hence, we have akk = Afl - A(j)kxk, where Cl is a constant of integration. For later convenience, we let Cl = Q, + u. We obtain Oiab = Vab{tt + UJ - <j)kXk) - tpab - 2(j)[aXb] = Slr}ab + da^b. (2.50) Through (2.31), (2.50), we obtain Kabc = 0. (2.51) The solution for p is obtained through (2.43): Pabcd = <J[a[c66]dl = SlSaHfl + 26{M]^ • (2-52) Chapter 2. Symmetries of Maxwell's Equations 17 2.4.2 Solution of the LPS symmetry determining equations We summarize the solution of the LPS symmetry determining equations with the following theorem. Theorem 2.6 The linear point symmetries X = (XAa)-^- + ( X F a ( , ) ^ of LPS are given by the following expressions for the coefficients in (2.23), (2.24): £a = aa + uxa + i;akxk-xa<f>kxk+ ^4>axkxk, dab = ftVab + 9a£b, Pabc = 0, Kabc = 0) where aa, u, ipab = ip^, <j>a, Q are arbitrary constants. A geometrical interpretation of these symmetries is discussed in Section 2.8. 2.5 Symmetries of A-LPS Consider A-LPS: Fab = d[aAb], daFai) = 0, daAa = 0. (2.53) As with LPS, we search for linear point symmetries X = ( X i a ) ^ - + ( X F a j ) ^ of A-LPS. We work on the A-LPS solution jet space and assume the same form for the coefficients of X as in (2.23), (2.24): XAa = e(x)DcAa + aab(x)Ab + pabc(x)Fbc, (2.54) XFab = C{x)DcFab + Kabc{x)Ac + pabcd(x)Fcd, (2.55) where w.l.o.g., we have the index symmetries f3abc - /3a [ 6 c ], Kabc = K [ a 6 ] c , pabcd = pab[cd] = P[ab}cd, due to antisymmetry of Fab. 2.5.1 A-LPS symmetry determining equations The A-LPS symmetry determining equations are X F a 6 = D[aXAb], (2.56) DaXFab = 0, (2.57) DaXAa = 0, (2.58) on the A-LPS solution jet space. The equations resulting from (2.56), (2.57) on the A-LPS solution jet space are similar to the equations resulting from (2.25), (2.26) on the LPS solution jet space. The only difference is that inclusion of the Lorentz gauge on A affects the splitting of the symmetrized derivatives D^aAby Chapter 2. Symmetries of Maxwell's Equations 18 All other equations resulting from the splitting - i.e. coefficient of Ac, coefficient of Fab (after replacing D[aA^ by Fab), etc. - remain the same. In particular, for the determining equations (2.56) and (2.57) with respect to A-LPS, the equa-tions (2.31), (2.32), (2.35)-(2.38) and Proposition 2.3 still hold. Furthermore, we will show that part (i) in Proposition 2.4 still holds in the A-LPS case, and consequently, parts (ii), (iii) in Propo-sition 2.4 hold for A-LPS as well. First we describe the changes to the splitting of D(aAy that must be implemented. We use the following result: Proposition 2.7 TabD(aAb) — 0 on the A-LPS solution jet space iff T(ab) = I^Tfc*. (2.59) Proof. For any Tab, we have Tab = + 7>6L For the symmetric part, we first obtain a decomposition into trace and trace-free parts T(ab) = t r T(ab) + t r f r T(ab) ^  £.60) where we require 77a6trfr 7>6) = 0, tr 7>6) = \rjab. (2.61) By contracting (2.60) with 7]ab, we get Tcc = 4A and hence A = \TCC. So, tr 7>6) = \r}abTcc. It is important to note here that the tr(=trace) and trfr(=trace-free) parts are linearly independent. We have TabD{aAb) = 0 on the A-LPS solution jet space iff TabD{aAb) oc DkAk iff 7>6) oc rfb. This occurs iff oc tr T^ab\ since trfr is linearly independent of tr . • Consequently, we have the following: • Coefficient of D^cAdy. Instead of (2.33), (2.34) we obtain S[a(C&b]d) = \vC%ba], (2-62) Ka(bc) = ^VbcKakk- (2-63) Proposition 2.8 Equation (2.33), (2.34), and Proposition 2.4 also hold for A-LPS. Proof. Multiplying (2.62) by 6da and lowering c yields 4<3i6c - ribc®^ = <*[««] • (2.64) Symmetrizing on (bc) yields a ^ c ) = \vbc&kk- Antisymmetrizing on [bc] yields a^] = 0- Conse-quently part (i) in Proposition 2.4 holds, and upon examining the proof, we see that parts (ii) and (iii) hold as well. Equation (2.33) holds for A-LPS, since it is satisfied by abc = \r}bcakk. Multiply (2.63) by r/ab and recall that Kabc is antisymmetric in its first and second indices. Then \{^kkc + ^ kck) = \l^ckk Kckk = 0 => K a(6c)=0. (2.65) Chapter 2. Symmetries of Maxwell 'sEqua (ions 19 Hence, (2.34) also holds for A-LPS. • We have shown that the coefficients of the symmetries of A-LPS must satisfy the same equa-tions as those for LPS. Thus, they must be of the form stated in Theorem 2.6. However for A-LPS, additional restrictions are obtained from the splitting of (2.58). Upon substitution of (2.54), the third determining equation (2.58) becomes =0 (on A—LPS) 0 = i ^ D c ^ C +(daaab)Ab + a^D{aAb) + a^D[aAb] + (da8abc)Fbc + (3abcDaFbc. (2.66) Since a^ab) = \r)abakk holds for A-LPS, then d^D^aAb) = \dkkDaAa = 0 on A-LPS. Also, since /3abc = 0, &[ab] = 0 hold for A-LPS, we obtain only one final equation from the splitting of the third determining equation (2.66): b^aab = 0. (2.67) Using aab = £lr]ab + da^b (obtained in Theorem 2.6), we get 0 = daaab = = ~<t>b ~ 2<t>[a6b]a = ~4>b ~ <t>aSba + <f>b6aa = 2</>b. (2.68) Thus, (f>a = 0, i.e. A-LPS does not admit inversion symmetries. 2.5.2 Solution of the A-LPS symmetry determining equations We have shown that all A-LPS linear point symmetries are also admitted by LPS, and have re-covered the well-known fact that the inversions are lost as admitted symmetries when imposing the Lorentz gauge constraint on A. Theorem 2.9 The linear point symmetries X = ( X A I ) M ^ + (X.Fab)-Q^ of A-LPS are given by the following expressions for the coefficients in (2.54), (2.55): e = o-a+uxa+ipakxk, dab = ttVab + da£b, Pabc = 0, Kabc = 0) Pabcd = n5jc6bd^+2S[M]^d], where o-a,u>, tpab = ip^, Q, are arbitrary constants. A geometrical interpretation of these symmetries is discussed in Section 2.8. 2.6 Symmetries of PS Consider PS: d[aA'b] = l-eabcddcAd. (2.69) In contrast to LPS, the potential system PS has not been well-studied in the literature. Using the tensor formalism and working on the PS solution jet space, we calculate the point symmetries Chapter 2. Symmetries of Maxwell 'sEqua tions 20 admitted by PS, and show that these include a scaling symmetry, the space-time symmetries, and a new scalar rotation symmetry acting "off-diagonally" on the dependent variables. Let X = (X.Aa) + (XA' a )^|r be a linear point symmetry of PS. Then X A * = Zk(x)DkAa + aak(x)Ak + pak(x)A'k, (2.70) XA'a = Zk(x)DkA'a + a>ak(x)Ak + B'ak(x)A'k, (2.71) where £ f c, aab, f3ab, o4 6 > flb are functions to be determined. 2.6.1 PS symmetry determining equations The PS symmetry determining equations are D[aXA'b] = ±eabcdDcXAd (2.72) on the PS solution jet space. Substitution of (2.70), (2.71) into (2.72) yields a system of equations linear in A, A' and their derivatives. The equations (2.69) express antisymmetrized first derivatives of A' in terms of an-tisymmetrized derivatives of A. We arrange the splitting of (2.72) as follows: • The coefficient of Ak, A'k must vanish. • The (symmetrized) coefficients of the symmetrized derivatives D^Aky D^A'^ must van-ish. • Replace antisymmetrized derivatives D[mA'n] by \emnjkD]Ak. In the resulting equation, the (antisymmetrized) coefficients of -Dy A ] must vanish. The following definitions will be useful in our calculations: Definition 2.10 Let &ab •= dab ~ da£b, Oiab •= "ab + 9a£b, (2-73) Pab == Pab - datb, Bob == Pab + da^b- (2-74) Decomposing DaAb = -D(a A ) + D[aAb\ and similarly for DaA'b, we can write out (2.72) more explicitly as: =0 (on PS) 0 = ikDk {d^^3^^^ + (d[aa'b]k - \eabcddcadk^ Ak + (d[a8'b]k - \eabcddcpdk^j A'k + (a'[ftfcV - \*^abd) D(jAk) + (a'[bk6a]j - ^adkejabd^j D{jAk] + (P\bk6a]j ~ \pdkejaba^j D(jA'k) + (p\bn6a]m - \pdnemabd^ D[mA'n]. (2.75) Using the notation defined in (2.73), (2.74), we obtain the following splitting of (2.75): Chapter 2. Symmetries of Maxwell 'sEqua tions 21 • Coefficients of Ak and A!k: d[a<*'b]k = \eabcddcad\ (2.76) d[aP'b]k = \tabcdd*Pdk. (2.77) • Coefficient of D^Ak) and D^A'ky 0 = a\bHa^ - l^^ahd, (2-78) 0 = P\b{kSa)j)-\pd{kzj)abd. (2-79) • Replace D{mA'n] by \emJkDjAk. Then extract the coefficient of D^Aky. 0 = (aW 1 - \ad[kzj]abd) + (p'[b[nSa]m] ' \pd[nem]abd^ \emJk = \ (P\bnea]njk - ad[kej]abd) + <>VV'l + ^ V^l + \PmmS^aSk]b. (2.80) Proposition 2.11 The solution of (2.78), (2.79) is (i) aab = \r)abakk, (ii) a'ab = \vaba'kk, (iii) (3ab = \vabPkk, (iv) P'ab = \vabP'kk-Proof. Let v e M4 be arbitrary. Multiply (2.78) by vkVjVa to get 0 = vaV[aa'b]kvk, which implies 0 = [(v • v)6ba — vbva]vka'ak (where the dot indicates the Minkowski inner product). Multiply by vg and antisymmetrize on [bg] to get 0 = (v • v)vkv^ga'b]k, and hence 0 = vkV[ga'b^k for all v such that v • v ^ 0. Since v • v ^ 0 defines a dense set in M4, then by continuity 0 = vkV[ga'b]k for all v € R4. Since v is arbitrary, and a' is independent of v, this is only possible if vka'bk = Cvb for some function C(x). Hence, we must have ct'bk = Cr}bk. Multiplying this last equation by rjbk yields C = \a.'kk, and (ii) follows. Substituting (ii) into (2.78) and multiplying by vkVj€mabn we obtain 0 = vkV[man]k. This is the equation 0 = vkV[ga'b]k under (a', b, g) >-> (a, n, m), which we solved previously. Hence, (i) follows. Equations (iii) and (iv) follow because (2.79) is exactly (2.78) under (a, a') i-> (/?,/r). Substitution of (i)-(iv) back into (2.78), (2.79) yields identities. • Proposition 2.12 The following hold: (v) a'ab = -0ab, (vi) aab = 0'ab, (vii) £ a = aa + uxa + ^ \xk - xa<j>kxk + \cf>axkxk, where aa,u, ipab = —4>ba, <t>a txre arbitrary constants. Note that ga^b = ^ab^ _ _ ^ab _ 2 ( ^ ] . Proof. Substitute (ii), (iii) into (2.80) to get 0 = \ (P\bnta)njk - &d[kej]abd) + \(a'mm + 8^)6^aSk\. (2.81) Chapter 2. Symmetries of Maxwell's Equations 22 For (v), mul t ip ly (2.81) b y 6a[j6bky S impl i fy ing, we obtain a'mm = —f3mm, w h i c h by (ii), (iii) in Proposit ion 2.11 yields (v). We use this to simplify (2.81). For (vi), mul t ip ly (2.81) by €jkam. S impl i fy ing, we get 0 = -2~B\h^a] - 2bimk8k\ = -p'bm ~ \Ka&mb ~ a m b + akkSmb (2.82) L o w e r m i n (2.82) and antisymmetrize o n [bm] to get = P'^y^, w h i c h implies = P\aby M u l t i p l y (2.82) b y 6mb to get akk = P'kk, wh ich implies akk = P'kk, and hence aab = P'ab b y (i), (iv). Th is yields (vi). L o w e r m i n (2.82), symmetr ize on (bm) and substitute aab = P'ab (which is a consequence of (vi)) to get P'(bm) = lvbmP'kk- Writ ing P'(bm) = p'(bm) + 2d{b£m) and us ing (iv), we get the conformal K i l l ing equation d(a£b) = \r]abdk^k. Its solutions (see A p p e n d i x B) are g iven in (vii). • Corollary 2.13 Equations (i)-(vii) are equivalent to (2.78)-(2.80). N o w we return to (2.76), (2.77). Note that multipl ication of (2.77) by eabmn a n d the replace-ment (P, p') •—• (-a', a) reduce (2.77) to (2.76). N o w write adk = adk + d4^ = \ r ] d k a m m + dd£k. Substitute this and a'bk = | r 7 h f c a ' m m into (2.76) to get: \s[bkda]a'mm = l-eabcd<? (jvdk&mm + dd^ ^ 6[bkda]a'mm = \eabckdcamm. (2.83) M u l t i p l y i n g (2.83) b y Ska and s impl i fy ing, we get 0 = dba'mm. T h e n (2.83) reduces to 0 = dcamm. Hence, amm — 4Q and a'mm = —4p where Q, p are constants of integration. N o w we can solve for a and p. In particular, dab = &ab + da6> = ^Vabdmm + da£b = £lr]ab + da£b, (2.84) Pab = -a'ab = -^Vabd'm™ = PVab- (2.85) 2.6.2 Solution of the PS symmetry determining equations F r o m (2.84), (2.85), and Proposit ion 2.12, the solution of the PS symmetry determining equations is s u m m a r i z e d b y the fo l lowing theorem. Theorem 2.14 The linear point symmetries X = (XAa)^ + {XA'a)-^r of PS are given by the following expressions for the coefficients in (2.70), (2.71): e = Oa+L0Xa + ^akXk dab = ttrjah + da£,b, Pab = PVab, a'ab = ~Pab, P'ab = Oiab, where aa, uj, ipab = i}Mb\ (j>a, p, Q are arbitrary constants. A geometrical interpretation of these symmetries is discussed in Section 2.8. Chapter 2. Symmetries of Maxwell's Equations 23 2.7 Symmetries of A-PS We augment PS w i t h Lorentz gauge constraints on A, A' to get A-PS: d[aA'b] = \eabcddcAd, daAa = Q, daA'a = 0. (2.86) Recall that w h e n the Lorentz gauge was imposed o n L P S to obtain A - L P S , the inversion s y m -metries were no longer admitted symmetries. In contrast, by impos ing dua l Lorentz gauges o n PS to obtain A-PS , all symmetries of PS are retained, inc luding the inversion symmetries (though they are slightly modif ied). In addit ion, n e w vector rotation / boost symmetries are admitted by A-PS . These symmetries have no analogues in the previously considered potential systems. We per form these calculations o n the A - P S solution jet space. Let X = (XAa)-^- + (XA'a)ojr be a linear point symmetry of A-PS. T h e n X A , = tk(x)DkAa + aak(x)Ak + 0ak(x)A'k, (2.87) XA'a = Zk(x)DkA'a + a'ak(x)Ak+p'ak(x)A'k, (2.88) where £ f c, aab, (3ab, ct'ab, 8'ab are functions to be determined. 2.7.1 A - P S symmetry determin ing equations The A - P S symmetry determining equations are D[aXA'b] = \eabcdDcXAd, (2.89) DaXAa = 0, (2.90) DaXA'a = 0, (2.91) o n the A - P S solut ion jet space. The equations in (2.86) express antisymmetrized first derivatives of A' i n terms of ant isym-metr ized derivatives of A, and impose conditions o n the trace of the symmetr ized derivatives. Hence, to effect the splitting of (2.89)-(2.91), we use the fol lowing: • The coefficient of Ak, A'k must vanish. • Since the gauge constraints impose further conditions o n the trace of D^A^ and D(kA'j}, w e use the decomposit ions D^Aj) = ^VjkDaAa + trfr D(kAj)t D^A'fi = ^jkDaA,a + tvbD{kA'j). Hence, the coefficients of the trace-free parts of D^A^ and D^A'^ must vanish. We use the fo l lowing results, analogous to Proposit ion 2.7: - O n the A - P S solution jet space, TabD{aAb) = 0 iff 7 > 6 ) = \r)abTkk. - O n the A - P S solution jet space, T,abD[aA'b) = 0 iff T ' ( a 6 ) = {r)abT'kk. Chapter 2. Symmetries of Maxwell's Equations 24 • Replace antisymmetrized derivatives D[mA'n] by \emnjkD3Ak. In the resulting equation, the (antisymmetrized) coefficients of DyAk] must vanish. The following definitions will be useful in our calculations: Definition 2.15 Let &ab '•= ®-ab - da£b, Pab '•= Pab ~ 9a€b, dab •= aab + da£b, Pab '•= Pab + 9a£b-As in the PS case, we can write (2.89) explicitly as =0 (on A-PS) 0 = i k D k { ^ ^ } ^ d P ^ d + (d[ad'b]k - \eabcddcddk^j Ak + [d[aP'b]k - \eabcddcpdk^ A'k + (d\bk6a]j - ^&dkejabd^ D(jAk) + (a'[bkSa]j - ^adkejabd^ D{jAk] + (P'[bksa]j - \pdkzjabd^ D{jA!k) + [p\bn5a]m - \pdnemab^j D[mA!n]. (2.92) Explicitly, (2.90) and (2.91) become =0 (on A-PS) 0 = £Dk&>X +(8aaak)Ak + (daPak)A'k + aikD{jAk) + &kD{jAk] +PkDuA'k) + PmnD[mA'n], (2.93) and 0 = =0 (on A-PS) i i A ^ f ^ C +(dad,ak)Ak + (dap'ak)A'k + a'ikDuAk) + a'^D^ +P'jkD{jA'k) + P'mnD[mA!n]. (2.94) The splitting of (2.92) w.r.t. A-PS is similar to the PS case for those equations determined by the coefficients of Ak,A'k and the coefficients of the antisymmetrized derivatives D^A^ (after elimination of D[mA'n] terms). The difference occurs in the splitting associated with the sym-metrized derivatives D(jAk),D(jA'ty due to the presence of the gauge conditions. We obtain the following splitting of (2.92), (2.93), and (2.94). Coefficients of Ak and A'k: d[ad'b}k = ^abcddcddk, (2.95) d[aP'b)k = ^abcddcpdk, (2.96) daaab = 0, daPab = 0, (2.97) daa'ab = 0, dap'ab = 0. (2.98) Chapter 2. Symmetries of Maxwell's Equations 25 • Coefficients of D^Ak) and D^A'ky 4 / V ' - \*dikJ]aM = i 1 * cd i a[ba] + 2€abcda ) ' (2.99) P\b{k6a)j) - \pd{kej)abd = P'[ba] + ^e a 6 c d /? c d ^ , (2.100) &W = ±r?kacc, = \rfk0cc, (2.101) (2.102) • Replace D[mA'n] by \emn^kDjAk. Then extract the coefficient of D^Aky. 0 = \ (0\bnea]njk - «d[kej]abd) + a'[6[V1 + ^ V*] + ^ Anm^a<y*]6, (2-103) &W = - l e j k m n 0 m n , (2.104) <*'W = -^JkmnP'mn. (2-105) Proposition 2.16 The solution of (2.99), (2.100) is (i) a'(a6) = habd'cc, (ii) a'[a&] = -^abcda"*, (iii) «(ab) = i%t.acc, (iv) £ ' ( a 6 ) = \vabP'cC, (v) £' [ a 6 ] = -leabcd(3cd, (vi) / J ( a 6 ) = i ^ c c . Proof. Multiply (2.99) by 6jb, and lower k. Then symmetrizing on (ak), we obtain (i), while antisymmetrization on [ak] yields (ii). Substitute (i) and (ii) into (2.99) and multiply by eabjn. Simplification of the resulting expression yields (iii). Then (2.99) reduces to an identity, after ee terms are simplified. Since (2.99), (2.100) are exactly the same equation under the correspondence (a1, a) <—> 0', 0), then (iv)-(vi) follow immediately from (i)-(iii). • Corollary 2.17 The following hold: (vii) d ( a £ 6 ) = \r}abdk(k, (viii) a ( a b ) = \r)abakk, (ix) 0\ab) = \rjab0'kk. Proof. For (vii), use (iii) and the first equation in (2.101) to get d( a£ 6) = 5(6(06) - <*(ab)) = \r}ab(ot-kk - &kk) = \r)abdk£k. Substitution of (vii) into (iii) and (iv) yields (viii) and (ix). • Substitution of (i)-(ix) into (2.99)-(2.102) yields identities. Thus, we have: Corollary 2.18 Equations (i)-(ix) are equivalent to (2.99)-(2.102). Lemma 2.19 The solution of (2.104), (2.105) is (x) &[ab] = (3\aby (xi) 0[ab] = -a\ab], where a and a' satisfy (ii). Chapter 2. Symmetries of Maxwell's Equa Hons 26 Proof. Comparing (v) and (2.104) immediately gives (x). Apply \eab]k to (2.104) to get (3^ = \^abjkcVk. Compare this with (ii), to get (xi). Using (x), (xi), both (2.104), (2.105) reduce to (ii). • Proposition 2.20 Through (i)-(xi), the solution of (2.103) is (xii) pkk = -a'kk, (xiii) akk = B'kk. Proof. Note that using (i)-(xi) we can write: Pab = \vabPkk + ^ a b c d a c d , a'ab = \-qaba'kk - ^eabcdacd, &ab = \r]abCtkk + &[ab], P'ab = \rjabP'kk + d[ab]. Upon substitution of these equations into (2.103), we obtain 0 = l ( K m - d m ^ e a b 3 k + ^(a'mm+pmm)6j3S^ ( V V f c - <$d[M«6c) - \ * [ c d ] ^ J ^ ^ (2.106) The third term in (2.106) is in fact equal to \a^r]d[aebcmn]r)n3r]mk = 0. Thus, (2.103) reduces to 0 = (~P'mm - amm)tabjk + 2(a'mm + Pmm)^hk]- (2.107) Multiplication of (2.107) by Sja yields (xii). Multiplication of (2.107) by eabjk yields ~P'mm = a m m and hence (xiii). • Note that (viii), (ix), (xiii) imply that a ( a 6 ) = P\ab), while (x) implies a[o6] = p\ab]. Thus, aab = P'ab- Similarly, (i), (vi), (xi), (xii) imply that pab = —ct'ab. Then (2.95), (2.96) reduce to the single equation d[aPb]k = -\eabcddcadk. (2.108) Similarly, equations (2.97), (2.98) reduce to daaab = 0, daPab = 0. (2.109) Lemma 2.21 The following equations arise from the splitting of the A-PS symmetry determining equa-tions and are equivalent to (2.95)-(2.105). Algebraic conditions: where aab = aab - da£b. • Differential conditions: -a'ab = Pab = \vabPcC + P[ab], (2.H0) P'ab = OLab = ^VabdcC + «[a6] > (2-H1) P[ab] = Uahcd**, (2.112) 0(a&) = \vabdc£°, (2.113) 1 cd' 4 d[aPb]k = - ^ a b c a d c a d k , (2.114) daaab = 0, dapab = 0. (2.115) Chapter 2. Symmetries of Maxwell's Equa tions 27 2.7.2 Solution of the A-PS symmetry determining equations Remark 2.22 The first differential condition (2.113) is the conformal Killing equation whose solutions are well-known (see Appendix B): <f = cr a + uxa + ipakxk - xa(f>kxk + ^(j>axkxk, (2.116) where aa, uj, ipab = —i{jba, (pa are arbitrary constants. Note that ga^b = vab^ _ _ ^ ab _ ^ [a^] _ ^.117) To solve (2.110)-(2.115), we prove the following lemmas: Lemma 2.23 The coefficients a, 3 satisfy (i) 0kk = Ap, (ii) akk = -Qcjjkxk + 4fi, where p, Cl are arbitrary constants. Proof. Multiply (2.114) by rjak. Then 9[a0b]a = -^abcd^ = 0C (±ebcadaad^ byi = U) dc0[bc] + ^ > e a ^ C ^ = dc0[bc], which simplifies, using (2.115) to db0aa = 0. Thus, (i) follows. Multiplying (2.114) by eahmn, we can rewrite this equation as d[aab^k = leabcddc0dk. Multiply-ing by rjak, we get d[aab]a = \eabcddc0da = -\ebcda(?pW ^ 2 = 1 1 2 ) a cci [ 6 c ] = d ca [ 6 c ] - <?d[bt;c] by(2A17) + 2dc{(j)[bXc]) = gca[bc] + 3 ^ which simplifies, using (2.115), to dbaaa = —6(pb. Thus, (ii) follows. • Lemma 2.24 The coefficients a, 0 satisfy (iii) 0[ab] = \eabcd{lcd + <t>cxd), (iv) a[ab] = -ipab + jab - 4>[axb], where ipab = ip^, 7 c d = 7 ^ , 4>c are arbitrary constants. Proof. Note that &[ab] = -\eabcd0cd implies a[ab] = -ipab - 2(j)[axb] - ^eabcd0cd. Now write the RHS of (2.114) in terms of 0: :£abcddc (jr}dk(-WmXm + 4fi) - iJdk - 2(j)[dXk] - ^€dkmn0mn^j 2 1 1 = I {tabck4>° + 6Vk{adCPbc}) = ^ {tabck<t>C + 2<9fc/3[a6]), Chapter 2. Symmetries of Maxwell's Equations 28 since 0 = daBab = daP[ab] by Lemma 2.23. Thus, d[aPb}k = \ Kbcktf + 2dkp[ab]). (2.118) Antisymmetrize over [abk] to get d[a(3bk] = \tabck<t>C. (2.H9) Since 3(ab) = const by Lemma 2.23, we obtain dapbc = daP[bc] • Consequently, we can write 3d[a(3bk] = da3[bk] + db/3[ka] + dkp[ab] = daPbk - dbpak + dkp[ab] = 2d[aPb]k + dkp[ab], (2.120) so that (2.118) becomes 3 1 1 1 ^[aPbk] - 2^k^[ab] - ^(eabck(pC + 2dkp[ab]) =>- dk3[ab] = -eabck(j)c. (2.121) This implies 3^ — \eabck<f)cxk + rab, where rab = Tyab\ are constants of integration. W.l.o.g. let Tab = \tabc&lci, and thus (iii) follows. Finally, we use a[ab] = -tpab ~ 2<j)[axb] - \eabcd^cdmn(imn + 4>mXn), which simplifies to (iv). • For later convenience, we will write fj = fi + uj. We summarize the point symmetries found for A-PS with the following theorem. Theorem 2.25 The linear point symmetries X = ( X A a ) ^ + (XA'a)-gjr of A-PS are given by the following expressions for the coefficients as in (2.87), (2.88): £a = oa + uxa + tpakxk -xa(f)kxk + \(t>axkxk, Otab = Tjab^p + U- ^ / c ^ ~ i>ab + lab ~ <t>[aXb], Pab = PVab + ^abcd(jcd + <t>CXd), Ct'ab = ~Pab, P'ab = OLab, where oa, uj, tpab = ip^, (f>a, p, fi, 7 a b = ^ab\ are arbitrary constants. A geometrical interpretation of these symmetries is discussed in Section 2.8. Chapter 2. Symmetries of Maxwell's Equations 29 2.8 Geometrical interpretation of symmetries admitted by LPS, A-LPS, PS, and A-PS 2.8.1 Space-time symmetries For each potential system LPS, A - L P S , PS, and A-PS, the admitted point symmetries include the space-time symmetries. In the preceding calculations, these correspond to the parameters appear ing in the vector £, where in each case £ satisfied the conformal K i l l ing equation C K E : t 9 ( a £ h ) = ^vab9cC- (2-122) The solutions of C K E are called conformal Killing vectors and are of the fo rm (see A p p e n d i x B): £° = CT° + uxa + i>akxk - xa4>kxk + \4>axkxk. (2.123) We have the fo l lowing geometrical interpretation of the parameters appear ing above: aa : four translations, each in the direction xa ui : one dilation, un i fo rm in all directions ijjak : six rotations / boosts, acting in the xaxk plane <f)a : four inversions We wi l l refer to these collectively either as conformal Killing vector symmetries or space-time symmetries. The translations and rotations / boosts comprise the Poincare symmetries. Homothetic Killing vector symmetries refer to the Poincare symmetries together w i th the dilation. We can give a succinct, geometrical description of the action (on the dependent variables of each potential system) induced by the space-time symmetries us ing the L ie derivative operator. U s i n g the general definit ion in A p p e n d i x A , we get the fo l lowing formulas: £ C A = e D k A a + AkDae, (2.124) CtAa = £kDkAa - AkDk£a, (2.125) jC^Fab = tkDkFab + 2Fk[bDa]Zk, (2.126) c^Fab = ^DkFab _2Fk[bDk^a}^ (2.127) where Fab is antisymmetric. Note in particular, that multipl ication by the M i n k o w s k i metric does not in general commute wi th the Lie derivative - i.e. r]abC^Aa / C^(r)ahAa) = C^Ab. Let £ a = aa + u>xa + ipakxk — xa<f)kxk + \$axkxk, w i th associated point symmetry X . 1. LPS: The action of X o n (Aa, Fab) is g iven by XAa = C^Aa, XFab = C^Fab. (2.128) 2. A - L P S : The action is the same here as with L P S except </>a = 0, i.e. inversion symmetries are not admitted by A - L P S . 3. PS: The action of X o n (Aa, A'a) is g iven by X A * = C^Aa, XA'a = C^A'a. (2.129) Chapter 2. Symmetries of Maxwell 'sEqua tions 30 4. A-PS: In this situation, the action is more complicated. We consider two cases: (i) nomothetic Killing vector symmetries: Let £ = cra+ujxa+ipakXk, i.e. translations, di lation, rotations / boosts, w i th associated point symmetry X. We have X A = CtAa, XA'a = C^A'a. (2.130) (ii) Inversion symmetries: Let £ a = —xa(pkXk + ^(f>axkXk be an inversion, w i th associated symmetry X. Define a conformally-weighted L ie derivative via £ * : = £ € + i d i v £ and div,£ = b\£k = ~A(f>kXk, (2.131) 8 and (ab ._ _^g[ag] = (^[aa.6] j gab ._ ( < ) a 6 = l^bcd^ = ^abcd^ ^.132) T h e n X generates the action X A , = ttAa + Ca6A + C'abA'b, XA'a = C^A'a - £abAb + CabA'b. (2.133) Note these inversion symmetries involve an internal rotation (i.e. invo lv ing no mot ion o n space-time) o n the potentials through ( a b and ( ' a b . 2.8.2 Internal symmetries A l l symmetries that involve no mot ion o n space-time (i.e. £ = 0) wi l l be referred to as "internal symmetr ies". 1. Scaling: The parameter Q in all considered potential systems corresponds to a u n i f o r m scaling in the dependent variables. For L P S and A - L P S , the scaling acts as XAa = Aa, XFab = Fab, (2.134) whi le for PS and A-PS , the scaling acts as X A = A , XA'a = A'a. (2.135) 2. Scalar rotation: The parameter p i n PS and A - P S corresponds to a scalar rotation o n the dependent variables (A- A') w i th generator X A = A'a, XA'a = - A i . (2.136) We give an alternative w a y of representing the scalar rotation action as follows. Identify R 4 x K 4 ~ R 4 ® R2, and identify basis elements A - Aa ® e\ and A'a ~ A <8> e2. In this setting, w e can identify p w i th id® Re £(M 4 ® R2), where id 6 £(M 4) is the identity map , a n d R € C(R2) is a standard rotation o n R2, i.e. id(Aa) = Aa, R(ei) = e^, Rfa) = —e\. (C(V) denotes the set of linear operators o n the vector space V.) Here, the scalar rotation p acts via: p = id®R € £ ( R 4 ® R 2 ) , p(Aa®e^) = id(Aa)®R(efM) = Aa®R(ell), Chapter 2. Symmetries of Maxwell's Equations 31 and therefore acts non-trivially only o n the R2 factor. Thus , p is said to act qff-diagonally o n the space 8 4 x l 4 ~ l 4 ® l 2 , w i th matrix representation i.e. the scalar rotation holds irrespective of the tensorial nature of Aa and A'a. 3. Vector rotations / boosts: A p p e a r i n g in the A - P S case, wi th no analogues in the other consid-where jab := [*i)ab = \^abcdlcd- If w e consider as before M4 x R 4 ~ R 4 <g> R2, the vector rotations / boosts combine a nontrivial action o n the R2 factor and the R 4 factor, in contrast to the scalar rotation action. Identify Aa ~ Aa®e\ and A'a ~ Aa®e2-Define id 6 £(M4) to be the identity map , and R G £(R2) a standard rotation o n R2, i.e. R(ei) = e-i, R(e.2) = —e\. A l s o define 7 • R G £(R4) to be the standard rotation / boost operator w i th parameters lab = 7[o6]- Recall that these standard rotations / boosts act infinitesimally by ( 7 • R)(va) = labv0. Here the constant parameters 7 ^ geometrically define 2-dimensional planes in R 4 o n w h i c h the rotation / boost takes place. Wi th this notation, the vector rotations / boosts act v ia We then see that 7 is a s u m of diagonal and off-diagonal rotations/boosts o n R 4 x R 4 ~ M 4® R2: the diagonal action involves a standard rotation/boost o n R4, and the plane for the off-d iagonal rotation/boost is the (Hodge) dua l of the plane for the d iagonal rotation/boost. A s u m m a r y of the (linear) point symmetries admitted b y each of the potential systems LPS , A - L P S , PS, a n d A - P S is g iven in Table 2.2. 7 = (7 - i? )®zd+(*7 - i?)(8 ) i?G£(R 4 (8>R 2), 7 ( A a ® e M ) = (i-R)(Aa)®id{etl) + (*j-R)(Aa)®R(elx). Chapter 2. Symmetries of Maxwell's Equations 32 X = ( X A a ) , System [# Symmetries] ^ A + ( X F a 6 ) Q r , denotes a LPS [A-LPS] linear >Aa oFab Space-time symmetries point symmetry. Internal symmetries LPS [15+1] Conformal Killing vectors: £a = a a + toxa + ipakxk - xacf)kxk + \4>axkxk 1 XAa = C^Aa, \ XF„i, = C^Fab Scaling: f X A a = Aa, | XFaft = Fab A-LPS [11+1] Homothetic Killing vectors: £a = oa + uxa + tpakxk \ X A a = C(:Aa, \ XFab = ^Fab Scaling: 1 X A a = Aa, \ XFab = Fab X = (XA 0 ) -System [# Symmetries] fA +CKA'a)rfAI denotes a PS [A-PS] linear poi: Ma dA'a Space-time symmetries at symmetry. Internal symmetries PS [15+2] Conformal Killing vectors: £a = aa + uxa + ipakxk - xa$kxk + \$axkxk 1 X A a = C^Aa, \ XA'a = C^A'a Scaling: f X A 0 = Aa, 1 XA'a = A'a Scalar rotation: J X A a = A'a, 1 XA'a = ~Aa A-PS [15+8] Homothetic Killing vectors: £a = oa + uxa + i>akxk 1 XAa = C^Aa, \ XA'a = ^A'a Inversions: £a = -xa(f)kxk + \<t>axkxk [ XAa = Ct:Aa+(abAb + ('abA'b, \ XA'a = CiA'a ~ CabAb + (a*A'b (£ 5 := £ ? + ±div£, C a 6 := - ^ a c f ] = $axb\ C'ab := ^eabcdCcd) Scaling: 1 X A a = Aa, 1 X A ' a = A'a Scalar rotation: 1 X A a — A'a, 1 XA ' a = — Aa Vector rotations / boosts: f XAa = labAb + iahA'b, \ XA ' a = -iahAb + lah'A'b (lab ' = 2^abcdlC ) Table 2.2: Point symmetry classification for ME potential systems Chapter 2. Symmetries of Maxwell 'sEqua Hons 33 2.9 Induced local and nonlocal symmetries of Maxwell's equations Any symmetry X = P a ^ + Paftgfc o f L P S [A-LPS] or any symmetry X = P a ^ + P'a^r of PS [A-PS] determines a symmetry of ME through a natural projection of the potential system solution space onto the ME solution space. Strictly speaking, since ME is written completely in terms of the dependent variables Fab, any symmetry of ME is of the form X = P a 6 - ^ , (2.138) 9Fab where Pab = XFa(,. However, it will be sometimes convenient to regard F, F' as being formally independent, with ME augmented by the algebraic equation F' = *F. Then we can extend X to * = p * i k + F i t i k ' ( 2 ' 1 3 9 ) where P'ab = XF'ab = \eabcdXFcd = \eabcdPcd so that the duality between F and F' is made manifest. 2.9.1 Local ME symmetries induced by LPS and A-LPS symmetries We showed in Section 2.5 that the linear point symmetries of A-LPS are linear point symmetries of LPS. LetX = Pazfx^ + Pab ajr^ b e a symmetry of LPS, where Pa = XAa, Pab = XFab. Since the ME solution space is embedded in the LPS solution space, this symmetry projects in a natural way to a symmetry X = PabQjr; o r Mis through the mapping (F, A) i—> F, so XFab = Pab := X F a 6 . (2.140) The induced symmetry X is a local symmetry of ME iff Pab is local in F and derivatives of F. Upon examining all linear point symmetries calculated for LPS (see Table 2.2), we see that Pab is always local in F. We obtain the corresponding induced symmetries of ME shown in Table 2.3. LPS Symmetry Type Induced symmetry of ME ME classification Space-time symmetries (^Fab)dFab local Scaling TP 9 rabdFab local Table 2.3: Classification of LPS induced symmetries of ME As predicted by Theorem 1.4, only local symmetries of ME are generated through projection of point symmetries of LPS (since LPS admits gauge symmetries). Proposition 2.26 All linear point symmetries of LPS (and A-LPS) project to local symmetries of ME. Thus, for LPS and A-LPS, no nonlocal symmetries of ME are generated. Chapter 2. Symmetries of Maxwell 'sEqua Hons 34 2.9.2 Local M E symmetries induced by PS and A-PS symmetries Let X = P a ^ + P ' a b e a symmetry of PS [A-PS], where Pa = XAa, P'a = XA'a. Since the ME solution space is embedded in the PS [A-PS] solution space, X projects in a natural way to a symmetry X = Pabgjr^ o r ME through the mapping F = dA, F' = dA', so XFab = Pab := XFab = Xd[aAb] = D[aXAb] = D[aPb]. (2.141) The induced symmetry X is a local symmetry of ME iff Pab is local in F and derivatives of F. Using (2.141) and the fact that Lie derivatives commute with curls, we obtain the classification of PS induced symmetries of ME shown in Table 2.4. PS Symmetry Type Induced symmetry of ME ME classification Space-time symmetries (.£tFab)dFab local Scaling T? d rabdFab local Scalar Rotation FabdFab local Table 2.4: Classification of PS induced symmetries of ME As predicted by Theorem 1.4, only local symmetries of ME are generated through projection of point symmetries of PS (since PS admits gauge symmetries). Proposition 2.27 All linear point symmetries of PS project to local symmetries of ME. Remark 2.28 The induced symmetry of ME in the scalar rotation case, X = F'abQp-, is referred to as a (local) duality rotation symmetry since by appending the algebraic relation F' = *F to ME, we can formally write X = F'abQ§ Fab^-, which describes a rotation between F and F'. The same calculations reveal that for A-PS, the scaling, scalar rotation, and translation, dila-tion, rotation / boost space-time symmetries project to local symmetries of ME. The remaining symmetries that must be investigated are the • A-PS vector rotation / boost symmetries, • A-PS inversion symmetries. It will be convenient to use the following notation: Definition 2.29 Let Aak := d{aAk) - \vakdbAb, ^ := d{aA'k) - \vakdbA'b, (2.142) i.e. the trace-free parts of the symmetrized derivatives ofAa, A'a. Chapter 2. Symmetries of Maxwell's Equations 35 2.9.3 N o n l o c a l M E symmetr ies induced b y A - P S vector rotations / boosts Propos i t ion 2.30 The internal vector rotation /boost symmetries X = Pa + P'a^r of A-PS, given by Pa = lakAk + iakA'k, P'a = ~iakAk + 7a*' A'k, (2.143) project to the nonlocal symmetries X = PabQ§^ + P'abQpr^ of ME, with Pab = l[bkAa]k + 7\bkA'a]k, P'ab = -i[bkAa]k + j[bkA'a]k, (2.144) where F'ab = ^eabcdFcd, j'ab = \tabcdlcd, and ^ ab = 7 [ a 6 ] are arbitrary constants. Proof. U s i n g (2.143), we have Pab = libkda}Ak + ^cde[bkcdda]A'k = $[am7b]kd{mAk) + l[bkFa]k + ls[ameb]kcdlcdd(mA'k) + \lcde[bkcdF'a]k = c 5 [ a - 7 6 ] ^ ( m A ) + ^ [ a -e b ] f c c d 7 c cic9( m ^), (2.145) since \lcdZ[bkcdF'a]k = \lcde[bkcdea]kmnFmn = - f ^ V ^ T c d = -Vl^a^cd = -V^a]*-Similarly, P'ab = -^ [ a m e 6 ] f c cS C^(mA)+^[am76] f e5 ( m^)- ( 2 - 1 4 6 ) N o w use the fo l lowing decomposit ions, taking into account the gauge constraints: d(mAk) = ^VmkdcAc + trfr d{mAk) = trfr d{mAk) = Amk, d(mA'k) = ^Vmkd°A'c + trfr d{mA'k) = trfr d[mA'k) = A'mk. Thus , (2.145), (2.146) become (2.144). • 2.9.4 N o n l o c a l M E symmetr ies i n d u c e d b y A - P S invers ions L e m m a 2.31 Let £ a = —xa(j)kxk + \<$>axkxk (i.e. an inversion). For any covector function fb, £$[afb} = d[a£t;fb] + \(t>[afb] (2.147) where — + |div £. Proof. N o t e that d i v £ = da£a = -4<j>kxk. T h e n ^9[afb] = (cs + ^div d[afb] = d[a£(:fb] + ^d[a (/ 6 ] div £) - ^f[bda] (div £) d[a£zfb] + \f[bda)((l>kxk) = d[a£tfb] + ^4>[afb]-• Chapter 2. Symmetries of Maxwell's Equations 36 Propos i t ion 2.32 Let £ a = —xa(f)kXk + \<j>axkxk. The inversion symmetries, X = Pa-^ + P'a-gjr of A-PS, given by Pa = C^Aa + CakAk + CakA'k, Pa = CiA'a-C'akAk + CakA'k, (2.148) project to nonlocal symmetries X = Pab-gjrj + P'abQpr^ of ME, with Pab = C^Fab-<P[aAb] + ^eabcdcl>cA'd + CibkAa]k + C\bkA'a]k, (2-149) P'ab = tiF'ab-\eabcAcAd-cj)[aA!b]-C'[bkAa]k + C[^ (2.150) where F'ab = \eabcdFcd, (ab = -±dHb] = <t>[axb\ Cab = \eabcdQcd. Proof. We have d[a(b]k = d[a{nb]i4>[lxk]) = m[b<P[l6k]a] = -\<t>iaSkb], d[aC'b]k = ld[a(eb]kcd<t>cxd) = ^<t>ce[bkcdVa]d = \tabk<j>c-O n the A - P S solut ion space, Fab = d[aAb], F'ab = d[aA'b]. So o n A-PS, we have Pab = d[aPb] = 0 ^ 6 ] + d[a(b]kAk + C[bk9a]Ak + d[a('b]kA'k + {'[bkda]A'k = ^d[aAb] - ^[aAb^j - l<f>[aSkb]Ak + C[bkda]Ak + ^eabck^cA'k + t'[bkda]A'k = ^ F a f c - ^ a ^ + ^ ^ ^ + C^^Afc + C'^ a^A'fc, (2.151) P'ab = d[aP'b] = d[a^A'b] - d[a('b]kAk - (\bkda]Ak + d[a(b]kA'k + C[bkda]A'k = ^d[aA'b] - ^4>[aA'b^ - ^eabck(t>cAk - (\bkda]Ak - ^[aSkb]A'k + ([bkda]A'k = hF'ab - \eabcd^cAd - <j>[aA'b] - C'[bkda]Ak + C.[bkda]A'k. (2.152) Notice that under the correspondence £hfc <-> %k, C,'bk <-> \ebkcdricd, the last two terms in both (2.151) a n d (2.152) can be evaluated similarly to the vector rotation / boost symmetries in the proof of Proposit ion 2.30. Hence, (2.151), (2.152) reduce to (2.149), (2.150). • B y subtracting the local M E inversion symmetries f rom (2.149), (2.150), we obtain a purely nonlocal M E symmetry. C o r o l l a r y 2.33 ME admits the nonlocal symmetries X = Pabd§^ + P'abw^' where Pab = -l^Fab-^aA^ + ^eabcd^A^ + C^Aa^ + C'^A^, (2.153) P'ab = -\4>kXkF'ab-\eabcAcAd-^[aA"b]-C\bkAa]k+ (2.154) Proof. The local invers ion symmetries of M E are Pab = C^Fab, P'ab = C^F'ab. (2.155) Recall that C^Fab = C^Fab + ±(div £)Fab = C^Fab - \4>kxkFab, and similarly for C^F'ab. Hence , by subtraction of (2.155) f rom (2.149), (2.150), we obtain (2.153), (2.154). • We summar ize the M E symmetries induced by the A - P S point symmetries in Table 2.5. Chapter 2. Symmetries of Maxwell 'sEqua tions 37 A-PS Symmetry Type Induced symmetry of ME ME classification Homothetic Killing vector symmetries local Inversions (-\(f)kxkFab - (j)[aAb] + \eabcd(t)cAld + C[bkAa]k + C'[bkA'a]k) ^ + (-±(j)kxkFab - \eabcd(f>cAd - <f>[aA'b] C[bkAa]k + C[bkAa]k) nonlocal Scaling p d 1 pi d r^dFab 1 ra*>dFab local Scalar Rotation pi d p d rab9Fab r ^ d F ' a b local Vector Rotations / Boosts (7[6fe^a]fc+7[6fc^a]fc)5fc + ( i{bkK\k + i[bkA'a]k)d^,ab nonlocal (Here Fab = d[aAb], Aab = d{aAb) - \r)abdkAk, jab = l[ab], Cab = <P[aXb], F'ab = d[aA'b], A'ab = d(aA'b)-\r)abdkA'k, j'ab = \eabcdlcd, Cab = \eabcd(j)cxd.) Table 2.5: Classification of A-PS induced symmetries of ME 2.9.5 LPS symmetries induced by PS and A-PS symmetries Since PS and A-PS can be considered as potential systems of LPS, we can also determine the symmetries of LPS induced by projection of PS and A-PS symmetries. Let X = Pa-^ + Pag^r be a symmetry of PS [A-PS], where Pa = XAa, P'a = XA'a. By the embedding of the LPS solution space in the PS [A-PS] solution space, this symmetry projects in a natural way to a symmetry X = P a ^ + pabSrb of LPS through the mapping F = dA, F' = dA', so XAa = Pa := XAa = Pa, (2-156) X F a b = P a 6 := XFab = X8[aAb] = D[aXAb] = D[aPb] = Pab- (2.157) Note that Pa and Pab are the same as those listed in Tables 2.4 and 2.5. The induced symmetry X is a local symmetry of LPS if and only if Pa, Pab are local in A, F and derivatives of A, F. In the case of LPS symmetries induced by PS or A-PS, the scalar rotation projects to a non-local symmetry. Though PS admits gauge symmetries, the nonlocality of the induced LPS scalar rotation symmetry does not contradict Theorem 1.4 since LPS is not well-posed (it admits gauge symmetries). The other PS symmetries project to local LPS symmetries (see Table 2.6). In the A-PS inversion case, we can again subtract the local inversion symmetry to generate a purely nonlocal symmetry (see Corollary 2.34). A summary of A-PS induced symmetries of LPS is shown in Table 2.7. Chapter 2. Symmetries of Maxwell's Equations 38 PS Symmetry Type Induced symmetry of LPS LPS classification Space-time symmetries local Scaling A d | p d A a d A a -tfab9Fab local Scalar Rotation Ai d \ T?I . d nonlocal Table 2.6: Classification of PS induced symmetries of LPS Corollary 2.34 LPS admits the nonlocal symmetries X = (C^Aa)-^ + (C^Fab)-^-b, where Pa = -l^XkAa + CakAk + C'akA'k, (2.158) Pab = -l<f>kxkFab-4)[aAb] + \eabcd4>cA'd + <:[bkAa]k+ (2.159) Proof. The local inversion symmetries of LPS are X — (C^Aa)^- + (C.^Fab)^p-^. We obtain (2.159) as in the proof of Corollary 2.33. Recall that t^Aa = C^Aa + ^(div £)Aa = C^Aa-\(f)kxkAa. Hence, by subtraction of C^Aa from (2.148), we obtain (2.158). • A-PS Symmetry Type Induced symmetries of LPS LPS classification Homothetic Killing vector symmetries (cZAa)dAa +(£(:Fab)dFab local Inversions (-\<j)kXkAa + (akAk + C'akA'k) ^ + (-\<t>kXkFab ~ <t>[aAb] + \eabc&<$>CXd + C[bkAa]k + C\bkA'a]k) ^ nonlocal Scaling A d i T? . d AadAa ^r<>*>dFab local Scalar Rotation AI d i T?I d A*dAa -*-rabdFab nonlocal Vector Rotations / Boosts {lakAk+iakA'k)^-a + (l[bkAa]k+i[bkA'a}k)dFab nonlocal (Here Fab = d[aAb], Aab = d{aAb) - \rjabdkAk, 7a6 = 7[ab]> Cab = <P[a^b), F'ab = d[aA'b], A'ab = d(aA'b) - \r)abdkA'k, j'ab = \eabcdlcd, Cab = \tabcd<l>cxd.) Table 2.7: Classification of A-PS induced symmetries of LPS Chapter 2. Symmetries of Maxwell 'sEqua tions 39 2.10 Chapter 2 Summary In this chapter we have investigated the linear point symmetries of four potential systems asso-ciated with the free-space Maxwell's equations. A complete symmetry classification is provided in Table 2.2. In particular, we note that: • We have recovered the classical result that the four inversion point symmetries admitted by LPS are lost when imposing the Lorentz gauge to obtain A-LPS. • PS admits a scalar rotation symmetry which acts internally on the potentials. The symmetry structure of A-PS is richer than the other potential systems. The main features that should be highlighted are: • A-PS admits a scalar rotation symmetry which acts internally on the potentials. • Applying dual Lorentz gauges to PS to obtain A-PS does not lose the inversions as ad-mitted point symmetries. Their form however does change - their induced action now involves an internal rotation on the potentials and a conformally weighted Lie derivative. In contrast, with PS the inversions induce a natural Lie derivative action on the potentials. • Additionally, a 6-parameter subgroup of vector rotations / boosts acting internally on the potentials is admitted by A-PS. No analogue of these symmetries appears in the other considered potential systems. Since the ME solution space is embedded in the solution spaces of its potential systems, all symmetries of the potential systems induce symmetries of ME. The symmetries of LPS, A-LPS, and PS all induce local symmetries of ME. (See Tables 2.3 and 2.4.) In the case of A-PS, the inversions and vector rotations / boosts induce nonlocal symmetries of ME. (See Table 2.5.) We can also consider PS and A-PS as potential systems of LPS. Since the LPS solution space is embedded in the solution spaces of PS and A-PS, then PS and A-PS symmetries induce LPS symmetries. In this case, the scalar rotation symmetry for PS or A-PS induces a nonlocal sym-metry of LPS. The other PS and A-PS induced symmetries are summarized in Tables 2.6 and 2.7. Chapter 3 Conservation Laws of Maxwell's equations 3.1 Introduction The approach that we will adopt for generating nonlocal conservation laws of ME is similar in spirit to that used for deriving nonlocal symmetries: We realize nonlocal conservations laws of ME as local conservations laws of a potential system associated with ME. The conservation laws of a ME potential system project to conservation laws of ME since the solution space of ME is embedded in the solution space of a ME potential system. There are essentially two issues to tackle here: 1. Generate local conservation laws of a ME potential system. 2. Project these to conservation laws of ME and classify them with respect to their locality or nonlocality. For almost a century, Noether's theorem has been the principal symmetry-based tool for gen-erating local conservation laws. However, it can only be used when the system under considera-tion is Lagrangian - i.e. it admits a variational principle, and consequently the linearized system is self-adjoint. More generally, for any given PDE system, it is well-known that all local conserva-tion laws arise through multipliers [15], and that these multipliers are the adjoint-symmetries of the system subject to additional adjoint-invariance conditions [3]. (In the Lagrangian situation, an adjoint-symmetry is a symmetry, the adjoint-invariance conditions reduce to the conditions for which the action is invariant under the symmetry, and the conservation law arising from Noether's theorem is obtained.) Once multipliers are identified, the associated conservation law can be generated by means of a homotopy integral expression involving just the system equa-tions and the multipliers [3,5,6,15]. In the case of Maxwell's equations and its potential systems, we are dealing with linear PDE's, and consequently these systems admit a scaling symmetry in the dependent variables. For such systems, the additional adjoint-invariance conditions and homotopy integral expression can be by-passed by using the simple algebraic formula presented in Section 1.5 which generates local conservation laws in terms of adjoint-symmetries of the given system. We apply this formula to ME and each of its potential systems in Section 3.2. In principle, if the adjoint-symmetry equations for each of the ME potential systems were solved, we could use these simple algebraic formulas to construct local conservation laws for these potential systems and consequently conservation laws for ME. However, we will focus on generating local and nonlocal conservation laws of ME using symmetries of the ME potential 40 Chapter 3. Conservation Laws of Maxwell's equations 41 systems, thereby making use of our work in Chapter 2. The main thrust behind this approach is to utilize a correspondence between symmetries of the potential systems and the adjoint-symmetries needed for producing conservation laws. From LPS, we recover the standard local conservation laws of ME obtained through the LPS conservation law formula listed in Table 3.1 (which is equivalent to the Noether conservation law when a symmetry is variational). For PS and A-PS, we emphasize that these systems are non-Lagrangian. (Hence the classical Noether's theorem does not apply.) We will show that there is an embedding of the set of local PS [A-PS] symmetries into the set of (local and nonlocal) ME adjoint-symmetries. Consequently, we show that the ME conservation law formula listed in Table 3.1 is able to recover all local ME conservation laws from PS symmetries. We will use this formula to generate nonlocal ME conservation laws from PS point symmetries that do not project (via the embedding) to local ME adjoint-symmetries. In the case of A-PS, the formula generates addi-tional nonlocal ME conservation laws from the additional point symmetries admitted by A-PS. We also address the question of whether or not any of these newly generated nonlocal ME con-servation laws are local LPS conservation laws. This leads to a classification of LPS conservation laws derived from PS and A-PS symmetries. Since we are realizing local and nonlocal conservation laws of ME [LPS] as local conservation laws of a ME [LPS] potential system, we can calculate the characteristics of these conservation laws with respect to the potential system. We prove two main classification theorems which establish necessary and sufficient conditions for locality of ME [LPS] conservation laws in terms of their characteristics with respect to PS or A-PS. These classification theorems rely on two important cohomology results for ME and LPS which we now state. Theorem 3.1 (ME 2-form cohomology) Let Qab be a closed, local 2-form (in F and derivatives of F) on the ME solution jet space (of any finite order). Then Qa0, to within a linear combination of Fab and F'ab = \tabcdFcd, is an exact 2-form, i.e. Qab = D[aQb] + CiFab + c2F'ab, (3.1) where c\,c2 are constants and Qb is a local 1-form on the ME solution jet space. Theorem 3.2 (LPS 2-form cohomology) Let Qab be a closed, local 2-form (in F, A and derivatives of F, A) on the LPS solution jet space (of any finite order). Then Qab, to within a constant multiple of F'ab = \zabcdFcd, is an exact 2-form, i.e. Qab = D[aQb] + cF'ab, (3.2) where c is a constant and Qb is a local 1-form on the LPS solution jet space. These results have a direct proof using spinor techniques, as shown by Anco and Pohjan-pelto [7]. In Chapter 4, we will discuss these cohomology aspects in further detail and provide a (partial) proof of Theorem 3.1, relevant for the classification of ME conservation laws, using tensorial methods. Chapter 3. Conservation Laws of Maxwell's equations 42 3.2 Local conservation laws of ME and ME potential systems Definition 3.3 A ME symmetry is a 2-form P = Pabdxa A dxb depending on x, F(x) such that dP = 0, d(*P) = 0 on all ME solutions, i.e. whenever dF = 0, d(*F) = 0 and their differential consequences hold. IfPab are functions defined in the jet space (x, F, F^,...), then the ME symmetry is local. Otherwise it is nonlocal if it has any other dependence such as through a potential. Definition 3.4 A ME adjoint-symmetry is a pair of 1-forms Q = Qbdxb, Q' = Q'bdxb depending on x,F(x) such that dQ' = *dQ on all ME solutions. IfQb,Q'b are functions defined in the jet space (x,F,F^\ ...), then the ME adjoint-symmetry is local. Otherwise it is nonlocal if it has any other dependence, such as through a potential. The symmetries and adjoint-symmetries of the systems LPS, A-LPS, PS, and A-PS are de-fined in an analogous manner using the appropriate linearized equations and the corresponding adjoint-equations respectively. Locality is defined with respect to variables in the corresponding jet space, while nonlocality refers to a more general dependence, such as through potential vari-ables. We summarize the symmetry determining equations, adjoint-symmetry equations, and the corresponding conservation law formulas (given in terms of adjoint-symmetries) for ME and its potential systems in a jet space setting in Table 3.1. System Symmetry Equations Adjoint-Symmetry Equations Conservation Law ME DaPab = 0, DaP'ab = 0, (P'ab = habcdPCd) D[aQ'b\ = habcdDcQd $a = QbFab + Q'bF'ab LPS DaPab = 0, Pab = D[aPb] DaQab = 0, Qab — D[aQb] = QbFab - AbQab A-LPS DaPab = 0, Pab = F)[aPb], DaPa = 0 DaQab - DhQ = 0, Qab = D[aQb] $a = QbFab - AbQab + AaQ PS D[aP'b] = \eabcdDcPd DaQab = 0, DaQ'ab = 0, (Q'ab = {"eabcdQCd) $a = A'bQab - AbQ'ab A-PS D[aP'b] = \eahcdDcPd, DaPa = 0, DaP'a = 0 DaQab + DbQ' = 0, DaQ'ab - DbQ = 0, (Qab — 2^abcdQ° ) $a = A'bQab - AbQ'ab +AaQ + A'aQ' Table 3.1: Comparison of ME and ME potential systems The conservation law formulas hold for any adjoint-symmetries of the corresponding sys-tem. For those adjoint-symmetries that are characteristics (multipliers) for a conservation law, Chapter 3. Conservation Laws of Maxwell's equations 43 these formulas recover a multiple of the local conservation law determined by the characteristic. We turn now to the problem of generating nonlocal conservation laws of M E using symmetries of the potential systems we have found in Chapter 2. 3.3 M E conservation laws derived from LPS and A-LPS symmetries We showed in Chapter 2 that all A-LPS linear point symmetries are admitted by LPS. For LPS variational symmetries, Noether's theorem could be used to generate corresponding conserva-tion laws. Instead, since symmetries are the same as adjoint-symmetries for LPS, the formula in Table 3.1 yields conservation laws for any LPS symmetries (local or nonlocal). Proposition 3.5 Let ~X.be a symmetry of LPS. Write Pa = XAa, Pab = XFab, and define $Q = PbFab - AbPab. (3.3) Then Da$a = 0 on all solutions of LPS. Proof. Note that DaFab = 0 and D[aAb] = Fab on LPS solutions. Thus, Da$a = (DaPb)Fab + PbDaFab - (DaAb)Pab - AbDaPab = 0 on LPS solutions, since D[aPb] = Pab and DaPab = 0 are satisfied by symmetries of LPS. • Remark 3.6 If a LPS local symmetry is a conservation law characteristic (i.e. variational sym-metry), then (3.3) is equivalent to the resulting Noether conservation law. We apply (3.3) to the admitted point symmetries of LPS to obtain the following conservation laws. • Scaling: Here Pa = Aa, Pab = Fab. So, § a s c = AbFab - AbFab = 0. (3.4) Thus, we obtain a trivial conservation law in this case. (Consequently, we note that the scaling is not a variational symmetry of LPS.) • Conformal Killing vector symmetries: Here Pa = C^Aa, Pab — C^Fab. We have pab = jf^r^c^Fcd = CiFab + FabDk£k, and hence, HPS-CKV = {^Ab)Fab - Ab(C^Fab + FabDKCK) (3.5) are local conservation laws of LPS. Since the M E solution space is embedded in the LPS solution space, we can regard (3.5) as a M E conservation law. Proposition 3.7 Up to a constant multiple, the conservation laws ^LPS-CKV a r e equivalent to the well-known energy-momentum (local) conservation laws of ME, $EM = i\FkbFab + F'kbF'ab). (3.6) Chapter 3. Conservation Laws of Maxwell's equations 44 Proof. Recall that conservation laws are equivalent if they differ by curl terms Db8ab, 8ab = 9^ on the solution space of the system. We have nps-CKV = 2(C^Ab)Fab-^(FabAb)-FabAbDk(k = 2(2^kFkb + Db(ehAk))Fab - {eDk{FabAb) - FkbAbDke) - FabAbDk£k curl =0 (on LPS) = AikFkbFab+ 2Db{ikAkFab)-2ikAk{ DbFab )-eDk{FkbAb) -2^kDk(F^bAb) + FkbAbDkia - FabAbDk£k =0 (on LPS) curl = 4tkFkbF ab-e( DkFkb ) A - C F f e ^ F f e 6 - 2 £ f c ( £ [ f e ^ ^ +2iJ\^F^% = AChFkbFab -iaFkbFkb. Note that 2tk(FkbFab + F'kbF'ab) = 2£kFkbFab + \ekbcdeabmnF"1 Fmn = 2^kFkbFab - 3^aFm^Fmn = 2£kFkbFab - (£aFmn + 2£>Fn^a)Fmn = 4£kFkbFab - eFmnFmn. Hence, $ a L P S - c K V = 2$EM- D Thus, we have derived the well-known fact that all (linear) point symmetries of LPS generate local conservation laws of ME. (See [16] for a survey of these results.) 3.4 M E conservation laws derived from PS and A-PS symmetries Definition 3.8 Let 3>a be a conservation law of PS [A-PS]. Let Fab — D[aAb] and F'ab = \eabcdFcd = D[aA'by Then $ a is called a local conservation law of M E if on the PS [A-PS] solution space, $° is local in F and derivatives of F (i.e. it has no essential dependence on A, A') to within curl terms Db9ab, 9ab = 9^ab\ Otherwise $° is called a nonlocal conservation law of ME. We have the following conservation law formula in terms of symmetries of PS [A-PS]. Proposition 3.9 Let Xbea symmetry of PS [A-PS]. Write Pa = XAa, P'a = XA'a, Fab = D[aAb], F'ab = D[aA'b] and define $a = PbFab + P'bF'ab. (3.7) Then Da$a = 0 on all solutions of PS [A-PS]. Proof. Note that DaF'ab = \eabcdDaDcAd = 0 and DaFab = -\eabcdDaDcA'd = 0 on PS [A-PS] solutions. Thus, Da$a = (DaPb)Fab + (DaP'b)F'ab = (-leabcdDcPd + DaP'b) F'ab = 0 on PS [A-PS] solutions, since DyaP'^ = \eabcdDcPd is satisfied for symmetries of PS [A-PS]. • The motivation for considering the formula (3.7) comes from an important correspondence between PS [A-PS] symmetries and ME adjoint-symmetries, together with the M E conservation law formula (in terms of M E adjoint-symmetries) given in Table 3.1. This correspondence is described in Theorems 3.10 and 3.12. Chapter 3. Conservation Laws of Maxwell's equations 45 Theorem 3.10 The mapping F = dA, F' = dA' induces a natural embedding of: (i) the set of local ME adjoint-symmetries into the set of local PS symmetries, and (ii) the set of local PS [A-PS] symmetries into the set of (local and nonlocal) ME adjoint-symmetries. Proof. Given a ME local adjoint-symmetry (Q, Q'), we use the embedding equations F = dA, F' = *dA = dA' (which were set up in Chapter 2) to project (Q, Q') to (Q, Q') on the jet space (x, A, A', A^\ Al(A),...) (projection increases the differential order by one) via DkF = DkdA, DkF' — DkdA' for all orders k > 0. This projection is well-defined on the PS solution jet space since we can write F' = dA' = *dA = *F. Since dQ' = *dQ holds on the ME solution jet space, it follows from the embedding between the solution jet spaces of ME and PS (explained in Sections 1.3 and 2.2) that dQ' = *dQ holds on the PS solution jet space. Hence (Q, Q') is a local PS symmetry, and (i) is proved. For (ii), let (P, P') be a local PS [A-PS] symmetry. We must show that for any solution of ME, the adjoint-symmetry equation dP' = *dP holds. Let Fab(x) be a solution of ME. There exist A0(x) and A'0(x) such that F = dA and F' = dA'. (In the case of A-PS, we can also assume A(x) and A'(x) satisfy the Lorentz gauge because this gauge is compatible.) By the construction of potential system PS [A-PS] (described in Sec.2.2), A(x) and A'(x) satisfy PS [A-PS], and hence the PS [A-PS] syrnmetry determining equation dP' = *dP holds. • Remark 3.11 The set of local PS symmetries is NOT embedded in the set of local ME adjoint-symmetries. In particular, the local PS scaling symmetry X = Aa-^ + A'a-^r obviously cannot be mapped to any local ME adjoint-symmetry. (If it could, then under the projection F — dA, F' = dA', we would get back a local PS symmetry of at least first order in the deriva-tives of A, A'.) Theorem 3.12 If a local PS [A-PS] point symmetry X = PAS^ + yields a characteristic (Pa, P'a) of a local ME conservation law then (3.7) recovers a multiple of the ME conservation law deter-mined by this characteristic. Proof. This follows from the general results stated in Section 1.5 and proved in [1]. • An important consequence of Theorems 3.10 and 3.12 is that all local ME conservation laws can be recovered through (3.7) from local PS symmetries. Moreover, any local PS symmetry that does not project to a local ME adjoint-symmetry may yield a nonlocal ME conservation law due to its essential dependence on A, A'. In the case of A-PS, there is no embedding of local ME adjoint-symmetries into local A-PS symmetries. Nevertheless, nonlocal ME adjoint-symmetries and nonlocal ME conservation laws can also arise through (3.7) applied to A-PS symmetries. We will formulate simple criteria to determine the locality or nonlocality of ME conservation laws derived from PS [A-PS] symmetries using (3.7) in terms of their characteristics w.r.t. PS [A-PS]. A brief overview of characteristics of conservation laws was given in Section 1.5, where the general characteristic equation (1.11) was also stated. For clarity, we state this result more explicitly for PS and A-PS below. Chapter 3. Conservation Laws of Maxwell's equations 46 Proposition 3.13 (PS characteristics of conservation laws) Let 3>a be a conservation law for PS. In the associated jet space, let Gab '•= D[aA'b] - leabcdD cAd. Then Da($a + Ra) = QabGab, (3.8) for some expressions Qab = Q}ab\ and moreover Ra depends linearly on Gab and its total derivatives. Here Qab is the characteristic o / $ a w.r.t. PS. Proposition 3.14 (A-PS characteristics of conservation laws) Let <J>° be a conservation law for A-PS. In the associated jet space, let Gab := D[aA'b] - \eabcdD cAd, G := DkAk, G' := DkA'k. Then D a ( $ a + Ra) = QabGab + QG + Q'G', (3.9) for some expressions Qab = Q^ab\Q, Q', and moreover Ra depends linearly on Gab, G, G' and their total derivatives. Here (Qab, Q, Q') is the characteristic of®a w.r.t. A-PS. The characteristic equations (3.8), (3.9) determine the characteristics uniquely on the solution space of the respective systems. 3.4.1 Locality criterion For each of the nontrivial conservation laws of ME generated by (3.7) using symmetries of PS JA-PS], we will determine if the conservation law is local or nonlocal. To perform this classification, we will derive a criterion for locality of ME conservation laws generated from PS [A-PS] local conservation laws in terms of their characteristics w.r.t. PS [A-PS]. L e m m a 3.15 If Pa = ~XAa, P'a = XA'a are local in F and derivatives of F to within a gradient, then $a = PbFab + P'bF'ah is a local conservation law of ME. Proof. For any scalar expressions x, the gradients Pa = Dax, P'a = Dax' generate from (3.7) the trivial conservation law $ a = {DbX)Fab + {DbX')F'ab = Db6ab, where 9ab = XFab + x'F'ab. Thus, locality of 3>a is immediate when Pa, P'a have no essential dependence on A, A' to within gradients. • The space-time symmetries Pa = C^Aa, P'a = C^A'a, for any conformal Killing vector £ a , yield the PS conservation laws ^PS-CKV = (£sAb)Fab + (£^A'b)F'ab. (3.10) We can apply Lemma 3.15 to the space-time symmetries and obtain the following result. Proposition 3.16 Up to a constant multiple, the conservation laws §PS-CKV a r e equivalent to the well-known energy-momentum (local) conservation laws of ME, *EM = Zk(FkbFab + F'kbF' ab), (3.11) where F' ab = \ e a b c d F c d . Chapter 3. Conservation Laws of Maxwell's equations 47 Proof. Note that Pb = £^Ab = £kDkAb + AkDbf* = 2£kD[kAb] + Db(£kAk) = 2£kFkb + DbX, where x = £kAk. Similarly, P'b = 2£kF'kb + Dbx', where x' = £kA'k. Hence, curl &PS-CKV = ^EM + DbOab = 2$%M, where 9ab = xFab + x'F'ab. • The conservation law generated by the scaling symmetry Pa = Aa, P'a = A'a is also easy to classify. In this case, we get a trivial conservation law: $asc = AbFab + A'bF'ab = Db6ab = 0, (3.12) where 9ab = \eabcdAcA'd. To establish the locality or nonlocality of the remaining conservation laws generated by the scalar rotation [PS / A-PS], vector rotations / boosts [A-PS], and inversions [A-PS], we need a stronger result than Lemma 3.15. We will develop the theory for A-PS conservation laws first, and then the results for PS conservation laws will immediately follow. Proposition 3.17 Let $a be a local conservation law of A-PS. Then $a is a local conservation law of ME iff the characteristic (Qab, Q, Q') of $a w.r.t A-PS satisfies Q = Q' = 0 and Qab = D^aQbl Qfah ._ ieahcdQcd _ r)[aQ'b]jor S Q m e Qb^ Q'b ^ a r e iocai j n F ma> derivatives ofF. Proof. If 3>a is a local conservation law of ME, then it is local in F and derivatives of F, up to a total curl on the ME solution space. Since $ a has no essential dependence on A, A', we can calculate its characteristic (Qa, Q'a) w.r.t. ME using (1.11), which reduces to Da($a + Ra) = Qb(F)DaFab + Q'b(F)DaF'ab, (3.13) where D^aQ'b^ = \eabcdDcQd on the ME solution jet space (and hence, on the A-PS solution jet space), and Ra depends linearly on DaFab, DaF'ab and their derivatives. We will use (3.13) to calculate the characteristic of $ a w.r.t. A-PS. Let Gab := D[aA'b] - l-eabcdDcAd, G := DaAa Hab := Fab - D[aAb], (3.14) G'ab := \eabcdGcd = \eabcdDcA,d + D[aAb], G' := DaA'a H'ab := F'ab - D[aA'b]. (3.15) The auxiliary equations Hab = 0, H'ab = 0 specify the relationship between the solution jet spaces of ME and A-PS. Note that DaGab = Da(F'ab - H'ab), DaG'ab = Da(Fab - Hab). (3.16) Substitution into (3.13) yields Da{$a + Ra) = Qb(F)Da(G'ab + Hab) + Q,b(F)Da(Gab + H'ab) = Da(- • •) - (G,ab + Hab)DaQb - {Gab + H'ab)DaQ'b = Da(- ••) + Gab (-DaQ'b - l-eabcdDcQd\ - HabDaQb - H'abDaQ'b, Chapter 3. Conservation Laws of Maxwell's equations 48 and thus we obtain the A-PS characteristic (Qab, Q = 0, Q' = 0) with Qab{F) = -7>Q / 61 - \eabcdDcQd °n = P S -2D^aQ'b^. (3.17) Zi In addition, we obtain Q'ab(F) = ^eabcdQcd = 2 ^-^eabcdDcQ'd^j 011 A=PS 2/>Q 6l . (3.18) Setting Qa := -2Q'a, Q'a := 2Qa yields the result. Conversely, suppose that $ a has characteristic (Qab, Q = 0, Q' = 0) w.r.t. A-PS with Qab = D^aQb\ Q'ab = yabcdQcd = D^aQ'b\ with Qb, Q'b local in F. Then $ a is equivalent to a multiple of the conservation law $'a = A'bQab - AbQ'ab = ~eabcd(A'bDcQ'd + AbDcQd). (3.19) Integration by parts on both terms yields curl #/a = cXP+^eabcd(Q'dDcA'b + QdDcAb) = Q'dFad - QdF,ad, (3.20) where 9ac = ^eacbd(A'bQ'd + AbQd). Thus, $ a is equivalent to a multiple of the ME local conser-vation *"» = Q'bFab - QbF'ab. • When does the characteristic (Qab, Q, Q') have the form given in Proposition 3.17? Since characteristics are adjoint-symmetries, they must satisfy the A-PS adjoint-symmetry equations, DaQab + DbQ' = 0, DaQ'ab - DbQ = 0. Since Q = Q' = 0, and Qab is local in F, then DaQab = 0, DaQ'ab = 0 must hold on the solution jet space of ME. Thus, Qab satisfies the ME symmetry deter-mining equations, or equivalently via Q'ab = \eabcdQcd, D[aQbc] = 0, D[aQ'bc] = 0, (3.21) on the ME solution jet space. Geometrically, these conditions assert that Qbc, Q'bc represent closed, local 2-forms on the ME solution jet space. From the ME 2-form cohomology theorem (Theorem 3.1), and taking into account the relation Q'ab = ^abcdQ^t we have Qab = D[aQb] + ciFab + c2F'ab, Q'ab = D[aQ'b] - c2Fab + ciF'ab, (3.22) for some constants ci, c2 and some expressions Qb, Q'b that are local in F and derivatives of F which satisfy D[aQ'b] = ^€abcdDcQd. Putting together these results, we obtain the following main theorem. Theorem 3.18 (ME Locality Criterion for A-PS Conservation Laws) A nontrivial conservation law of A-PS is a local conservation law of ME iff its characteristic w.r.t. A-PS is of the form (Qab, Q — 0, Q' = 0) where Qab is local in F, and derivatives ofF, and contains no term proportional to Fab, F'ab. A similar result follows for PS conservation laws. Chapter 3. Conservation Laws of Maxwell's equations 49 Corollary 3.19 (ME Locality Criterion for PS Conservation Laws) A nontrivial conservation law of PS is a local conservation law of ME iff its characteristic Qab w.r.t. PS is local in F, and derivatives of F, and contains no term •proportional to Fab, F'ab. These results give simple criteria to check and classify ME conservation laws derived from PS [A-PS] symmetries by examining their characteristics w.r.t. PS [A-PS]. 3.4.2 Computation of characteristics and conservation law classification Throughout this section Fab denotes D[aAb], and similarly F'ab := D[aA'by Recall that to calculate the characteristics of PS [A-PS] conservation laws we use Propositions 3.13 and 3.14. Thus, we need to calculate Da$a for $ a = PbFab + P'bF'ab off the PS [A-PS] solution jet space. Let Gab '•= D[aA'b] - -eabcdDcAd, G'ab = -eabcdGcd = DyaAb^ + -eabcdDcA'd. (3.23) Note first that DaGab = DaF'ab, DaG'ab = DaFab. (3.24) Thus, we have Da$a = Da(PbFab + P'bF,ab) = FabDaPb + PbDaFab + F'abDaP'b + P'bDaF'ab = (G'ab - \eabcdF'cd^j DaPb + PbDaG'ab + (cab + ^eabcdFcd^ DaP'b + P'bDaGab = Da(PbG'ab + P'bGab) + A, where A := ±eabcd(FcdDaP'b - F'cdDaPb). (3.25) Thus, it suffices to examine A to determine the characteristic of $ a . We now compute A for each PS and A-PS symmetry. 1. Homothetic Killing vector symmetries [PS / A-PS]: Here Pa = C^Aa, P'a = £^A'a, where £ is a homothetic Killing vector. We evaluate (3.25) for *%KV = (£^Ab)Fab + (£^A'b)F'ab (3.26) by direct calculation: AHKV = \eabcd{FcdDAC^A!b - F'cdDa£tAb) M = 1 3 ) l-eabcd{F^F'a, - F'cd£^Fab) = {F'ab _ Gab)Cip>ab _ F>d ^ (±eabcdFab^ ~ ^ 6 % ° ^ ) by (A.15) ( i ? / o 6 _ G a b ) c ^ a b _ _ Q*) + ^F'cdFab(-CabcdD^) = -2Gab£^F'ab + £s(GabF'ab) + F ^ r F ^ - F a b { r £ ^ ^ d - F'abDk(k) -\F'cdFabeabcdDke -2Gab£^F'ab + ikDk(GabF'ab) + F'abGabDk^k -2Gab£^F'ab + Dk{CkGabFab). Chapter 3. Conservation Laws of Maxwell's equations 50 With respect to PS, we have the characteristic Qab = -2C^F'ab. With respect to A-PS, we obtain the characteristic (Qab, Q, Q') = {-^C^F'ab, 0,0). One can show by direct calculation that the locality criteria Qab = D[a(4Fi]ke), Q'ab = D[a(-4Fb]k£k) (3.27) hold on the PS [A-PS] solution jet space. 2. Scaling [PS / A-PS]: As proved in the previous section, we obtain a trivial conservation law in this case. (Hence, it has trivial characteristic w.r.t. PS or A-PS.) 3. Scalar rotation [PS / A-PS]: Here Pa = A'ar P'a = -Aa. The equation (3.25) for ®ROT = A'bFab - AbF'ab (3.28) reduces to 1 nhrH / T-i T-I 7-7 n/ \ / s-inh mfnh\ TH / s~ifah Tnn,b\ 7Y . ab A R O r = \eabcd(-FcdFab-FcdFab) = (G^-F'ab)Fab-(G'ab-F = Gab [ F, 1 ^  J?'cd\ on A-PS o r t a b j-, ab — ^abcd-f I — Z(_r fab. With respect to PS, we get Qab = 2Fab. With respect to A-PS, we get (Qab, Q, Q') = (2Fab,0,0). Since Qab is proportional to Fab, then by Theorem 3.18, <&ROT i s a nonlocal conservation law of ME. 4. Vector rotation / boost symmetries [A-PS]: Here Pb = ^bkAk + ibkA'k, P'b = -j'bkAk + jbkA'k. Note that ±eabcd-y'bk = \ e a b c d e b k m n ^ m n = \6ak^cd + 5^ck^a. The equation (3.25) for * V A B = (lbkAk + "f'bkA'k)Fab + (-ibkAk + %kA'k)F'ab (3.29) reduces to &VRB 2€ abed (Fcd{-ibkDaAk + "ibkDaA'k) - F'cd(lbkDaAk + ibhDaA'k)) DaAk Fcd ( ^ T i * ) - ^ahcdF'cd^ lbk {^eabcdFcd^j l b k - F'cd Q e a 6 c d 7 ' 6 f e ) DaA'k = --FcdlcdDkAk - Fcpf DaAc - (G'ab - Fab)lbkDaAk + ( F/afc _ Gab)lhkDaA>k _ lFcdlCdDkA'k - F'^DaA'0 = -\FcdlcdDkAk-~F'cd^DkA'k + 2 F S ^ ^ +Gab ^ a D b A ' k - \eabcd^cDdAk^ , since Fah^bkFak = 0 = F'abjbkFak due to antisymmetry of 7 a 6 , Fab, F'ab. With respect to A-PS, we get the characteristic Q = -lfcdi ,cd Qf = - ^ c d 7 c d , Qab = -1k[aDb]A'k - \eabcdlkcF)dAk. Since Q, Q' ^ 0, then by Theorem 3.18, § v R B are nonlocal conservation laws of ME. Chapter 3. Conservation Laws of Maxwell's equations 51 5. Inversion symmetries [PS / A-PS]: Here £ a = —xa4>kxk + \(j)axkxk. In the case of PS, we get a conservation law ^>jNV = (C(.Ab)Fab + (C^A'b)Flab of the same form as (3.26), with similar characteristic. By Proposition 3.16, this yields an energy-momentum conservation law which is a local conservation law of ME. In the case of A-PS, the situation is very different. The symmetries of A-PS induced by the inversions are given by Pb = C^Ab + CbkAk + <Z'bkA'k, P'b = C^A'b - C'bkAk + (bkA'k, where C,bk = 4>[bxk], Cbk = ^ebkmn<f>mxn, and = + gdiv£. We obtain the conservation law $aINV = {tiAb + tbkAk + £bkAlk)Fab + {CiAb-c:'bkAk + tf = *INV + I ( d i v 0 (AbFab + A'bF'ab) + (CbkAk + CbkA'k) Fab + (-CbkAk + CbkA'k)F'ab. (3.30) Since ^jNV is a local ME conservation law (generated by the inversion symmetries of PS), we have by subtraction the conservation law *INV = -\<pkxk(AbFab + A'bF,ab) + (CbkAk + ('bkA'k)Fab + (-C,'kAk + (bkA'k)F'ab. (3.31) The equation (3.25) reduces to 1 „, A INV 2 £ 16 FcdDa (-^</>kXkA'b - ^ ebkmn4>mxnAk + (f>[bxk]A'k^ -F'cdDa (^-^<t>kxkAb + (j)[bxk]Ak + ^ebkmn(t)mxnA 1 Dke^L&*r^^ \eabcd Fcd (-\<t>aA'b - l-ebkmact>mAk -^ebkmn4>mxnDaAk + <t>[bT)k]aA'k + (j)[bxk]DaA,k^ - F'cd (-^<M* +(t>[brik]aAk + <f>[bxk]DaAk + ^ebkma(j)mA'k + ^ebkmn(TxnDaA'k^ = \eabcd Fcd (-<f>aA'b - \ebkma4>mAk^ - F'cd (-<j>aAb + \ebkma<TA'^j +\eabcd [Fcd (-C'bkDaAk + (bkDaA'k) - F'cd (c,bkDaAk + (bkDaA'k)' = A i + A 2 . To evaluate A i , note that A i = 1 eabcd (F>cd(j)aAb _ l-FcdehkmAmAk^ - \eabcd (Fcd<j>aA'b + \F'cdebkma<pmA'k = ( G ' « 6 _ Fab)(t>aAb + Fab<f>aAb + (Gab - Flab)<t>aAb + F'abcl)aA'b = G'abcj)aAb + Gab(t>aA'b = Gab <j>aA!b+l-eabcdct>cAd Chapter 3. Conservation Laws of Maxwell's equations 52 Notice that under the correspondence (bk +-> 76fc, we can evaluate A 2 in a similar manner as in the vector rotation / boost case. Hence, we have A 2 = -\FcdCcdDkAk - ±F'cd(cdDkA'k + Gab [~CkaDbA'k - \eabcd{kcDdAkSj . Thus, the characteristic corresponding to ^fNV is Q = -\FcdC,cd, Q' = -\F'cdCd, Qab = </>[aA'b] + \eabcd<fAd - Ck[aDb]A'k - \eabcd(kcDdAk. Since Q, Q' ^ 0, then by Theorem 3.18, ^J^v a r e nonlocal conservation laws of ME. We give a concise summary of these calculations in Tables 3.2 and 3.3. We omit $ g C = AbFab + A'bF'ab (generated by the scaling symmetry) from the classification since this is equiva-lent to a trivial conservation law. The remaining conservation laws are all nontrivial since they have nontrivial characteristics. Theorem 3.18 was used to classify their locality or nonlocality, and we summarize the results of this classification in Table 3.4. ME Conservation Law <3>a PS Characteristic Qab ®CKV = i^b)Fab + (C^A!b)F'ab (£ f c a conformal Killing vector) -2C^Fab ®ROT = A'bFab - AbF'ab 2Fab Table 3.2: Characteristics of ME conservation laws induced by PS symmetries ME Conservation Law A-PS Characteristic Q Q' Qab * W = (^b)Fab + (CsA'b)F'ab (£ f c a homothetic Killing vector) 0 0 -2C^F'ab * B O T = A'bFab ~ AbF'ab 0 0 2Fab nRB = hbkAk+ibkA'k)Fab +(-ibkAk+~fbkA'k)F'ab - ^ a 6 7 a 6 -lk[aDb]A'k - \eabcdikcDdAk $INV = -\4>kxk{AbFab + A'bF'ab) +(CbkAk + C'bkA'k)Fab +(-(bkAk + (bkA'k)F'ab -\FabC,ab -\F'abCab <t>[aA'b] + \eabcd<t>cAd -Ck[aDb]A'k - \eabcdCkcDdAk (Here (j)k,jbk = 7[6fc],7b/e = \tbkmnlmn are constants, and Cbfc = 4>[bXk],Cbk = ^ebkmn<f>mxn.) Table 3.3: Characteristics of ME conservation laws induced by A-PS symmetries Chapter 3. Conservation Laws of Maxwell's equations 53 ME Conservation Law $ a Classification $ aEM = Zk(FkbFab + F'kbF'ab) (£ f c a conformal Killing vector) local *AOT = A'bFab - AbF'ah nonlocal nRB = hbkAk + j'bkA'k)Fab + (-j'bkAk + ibkA'k)F'ab nonlocal $ aiNV = ~\4>kXk{AbFab + A'bF'ab) +((bkAk + CbkA'k)Fab + (-C'bkAk + CbkA'k)F'ab nonlocal (Here <f>k,ybk = l[bk\,l'bk = \c-bkmnlmn are constants, and Cbfc = 4>[bXk},Cbk = ^bkmn4>mxn.) Table 3.4: Classification of ME conservation laws induced by PS and A-PS symmetries 3.5 LPS conservation laws derived from PS and A-PS symmetries The system LPS has historically been of importance in physical applications, and in this section we consider LPS conservation laws that can be generated from PS and A-PS symmetries. A central question that we will answer is whether or not any of the nonlocal conservation laws of ME obtained in the previous section are local conservation laws of LPS. Definition 3.20 Let $a be a conservation law of PS [A-PS]. Let F'ab = ±eabcdFcd = D[aA'b]. Then $a is called a local conservation law of LPS if, on the PS [A-PS] solution space, $a is local in A,F and derivatives of A,F (i.e. it has no essential dependence on A') to within curl terms Db6ab, 9ab = 0^ab\ Otherwise $ a is called a nonlocal conservation law of LPS. Given any local symmetry X = Pa-^ + Fag^r of PS [A-PS], we can obtain its projection onto LPS through Pa = X A a , (3.32) Pab = XFab = XD[aAb]=D[aXAb]=D[aPb]. (3.33) Since LPS is self-adjoint, any symmetry (possibly nonlocal) is also an adjomt-symmetry, and so we expect to obtain a LPS conservation law (possibly nonlocal) through (3.3). Proposition 3.21 Let Xbea symmetry of PS [A-PS]. Write Pa = XAa, Pab = D{aPb)> Fab = D[aAb], and define $a = PbFab - AbPab. (3.34) Then Da$a = 0 on all solutions of PS [A-PS]. Proof. Note that on the PS [A-PS] solution space, Pa = XAa and P'a = XA'a satisfy the sym-metry determining equation D[aPb] = -^eabcdDcP'd, and hence DaPab = 0. Thus, on PS [A-PS] solutions, we have Da$a = DaPbFab + PbDaFab - DaAbPab - AbDaPab = 0. • Chapter 3. Conservation Laws of Maxwell's equations 54 Propos i t ion 3.22 IfX.— - P a ^ + ^ g ^ r projects to a characteristic (i.e. variational symmetry) of LPS, then (3.34) recovers a multiple of the conservation law determined by the characteristic (i.e. Noether conservation law). Since the L P S solution space is embedded in the PS [A-PS] solution space, w e can generate L P S conservation laws f rom PS [A-PS] symmetries through (3.34). 3.5.1 Loca l i ty cr iterion a n d conservation law classification In order to classify the L P S conservation laws arising through (3.34), w e establish a criterion for locality in terms of characteristics w.r.t. PS [A-PS], in analogy wi th T h e o r e m 3.18. Propos i t ion 3.23 Let 3>a be a local conservation law of A-PS. Then 3>a is a local conservation law of LPS iff the characteristic [Qah,Q,Q') o / $ a w.r.t A-PS satisfies Q = Q' = 0, Q'ab := \ekcdQcA = D^Q'^ on the A-PS solution space and Q'a are local in derivatives ofF, A. Proof. If <3>a is a local conservation law of LPS, then it is local i n F, A and derivatives of F, A, u p to a total curl . Since $ a has no essential dependence o n A', we can calculate its characteristic (Qa, Qab) w.r.t. L P S us ing (1.11), w h i c h reduces to Da($a + Ra) = Qb(F, A)DaFab + Qab(F, A)(Fab - D[aAb]), (3.35) where Qab = D[aQq o n the L P S solution jet space (and hence, o n the A - P S solut ion jet space), and Ra depends l inearly o n DaFab, Fab — D[aAb] and their derivatives. We wi l l use (3.35) to calculate the characteristic of $ a w.r.t. A-PS . Let Gab := D[aA'b] - ±eabcdDcAd, G'ab := ^eabcdGcd = \eabcdDcA'd + D[aAb], and Hab := Fab - D[aAb]. Note that DaG'ab = DaD[aAb], and so DaFab = DaHab + DaG'ab. A l s o , Hab = 0 specifies the relation between the solution jet spaces of L P S and A-PS. Thus , w e obtain Da($a + Ra) = QbDa(Hab + G'ab)+HabQab = Da(Qb(Hab + G'ab)) - (G,ab + Hab)DaQb + HabQab = Da(- ••) + Gab ( - i e a 6 c ^ c Q d ) + Hab(Qab - DaQb). (3.36) Hence , the characteristic of $ a w.r.t. A - P S is (Qab,Q,Q') = (-^abcdDcQd, 0,0), and Q'ab •= \tabcdQcd = D[aQb], where Qb is local in F,A. Setting Q'b := Qb y ields the re-sult. Conversely, suppose that $ a has characteristic (Qab, Q = 0, Q' = 0) w.r.t. A - P S wi th Q'ab = \eabcdQcd = D[aQ'b], and Q'b local i n F, A. T h e n is equivalent to a mult iple of the conservation law $/a = A'bQab - AbQ'ab = --eabcdA'bDcQ'd - AbQ'ab. Integration b y parts o n the first term yields curl $'a = Dr (- l-eabcdA'bQ+ Q'd (^eabcdDcA'b^j - AbQ'ab - 1\P +Q'bFab - AbQ'ab, (3.37) Chapter 3. Conservation Laws of Maxwell's equations 55 where 9ac = — \eabcdA'bQ'd. Thus, 3>a is equivalent to a multiple of the LPS local conservation law $'a = Q'bFab - AbQ,ab. • When does the characteristic (Qab, Q, Q') have the form given in Proposition 3.23? Since characteristics are adjoint-symmetries, they must satisfy the A-PS adjoint-symmetry equations, DaQab + DbQ' = 0, DaQ'ab-DbQ = 0. Since Q = Q' = 0, and Qab is local in F, A, then DaQab = 0, DaQ'ab = 0 must hold on the solution jet space of LPS. Using Qab = — \eabcdQlcd, we can state this equivalently as D[aQbc] = 0, D[aQ'bc] = 0, (3.38) on the LPS solution jet space. Geometrically, these conditions assert that Qbc, Q'bc represent closed, local 2-forms on the LPS solution jet space. By the LPS 2-form cohomology theorem (Theorem 3.2), we have Qab = D[aQb]+ClFab, Q'ab = D[aQ'b]+c2Fab, (3.39) for some constants c i , c<i and some expressions Qb, Q'b that are local in F, A and derivatives of F, A. By Proposition 3.23, a nontrivial conservation law of A-PS is a local LPS conservation law if and only if its characteristic (Qab, Q = 0, Q' = 0) w.r.t. A-PS has Q'ab exact in F, A. This occurs if and only if Q'ab contains no term proportional to F'ab, which happens if and only if Qab = - l € a b c d Q , c d contains no term proportional to Fab = -\tabcdF'cd- Putting together these results, we obtain the following main result. Theorem 3.24 (LPS Locality Criterion for A-PS Conservation Laws) A nontrivial conservation law of A-PS is a local conservation law of LPS iff its characteristic w.r.t. A-PS is of the form (Qab, Q = 0, Q' = o) where Qab is local in F, A, and derivatives of F, A, and Qab contains no term proportional to pab Corollary 3.25 (LPS Locality Criterion for PS Conservation Laws) A nontrivial conservation law of PS is a local conservation law of LPS iff its characteristic Qab w.r.t. PS is local in F, A, and derivatives ofF, A, and Qab contains no term proportional to Fab. These results give simple criteria to check and classify LPS conservation laws derived from PS [A-PS] symmetries by examining their characteristics w.r.t. PS [A-PS]. Before setting out to compute the characteristics of the PS [A-PS] conservation laws gener-ated through (3.34), recall the conservation law formula that was used to generate conservation laws of ME directly from PS [A-PS] symmetries, i.e. $a = PbFab + P'bFlab. (3.40) Both (3.34) and (3.40) are local PS [A-PS] conservation laws. We note that the difference between (3.34) and (3.40) on the PS [A-PS] solution space is AbPab + P'bF'ab = AbPab + \eabcdP'bFcd = AbPab + ^eabcd(Dc(P'bAd) - AdDcP'b) = AbPab + Dc (^tabcdP'bAd\ - AiD^pQ = Dc6ac, Chapter 3. Conserva tion La ws of Maxwell 'sequa tions 56 where 9ac = 9^ = -±eacbdP'bAd. Thus, (3.34) and (3.40) differ b y a total curl o n the PS [A-PS] solut ion space, a n d hence are equivalent. Consequently, the PS [A-PS] characteristics of the con-servation laws generated through (3.34) are the same as the characteristics of the conservation laws generated through (3.40) - i.e. we can recycle the computations per formed in Section 3.4.2. We app ly T h e o r e m 3.24 and Corol lary 3.25 to the conservation laws whose characteristics are listed in Tables 3.2 and 3.3. Since Q, Q' ^ 0 for § y R B and ^fNV, these are nonlocal conservation laws of LPS . The conservation law § R O T has characteristic (Qab, Q, Q') = (2Fab, 0,0). Since Qab is proport ional to FAB, then QRQT * s A^SO a nonlocal L P S conservation law. We summar ize this classification in Table 3.5. We remark that all nonlocal M E conservation laws generated in the previous sections are genuinely nonlocal w h e n considered as L P S conservation laws. L P S Conservat ion L a w $ a Classification $|M = ik(FkbFab + F'kbF'ab) (£ f c a conformal K i l l ing vector) local ®ROT = A'bFab - AbF'ab nonlocal n R B = (lbkAk + ibkA'k)Fab + (-ibkAk + lbkA'k)F'ab nonlocal *INV = -&kxk(AbFab + A'bF'ab) +(CbkAk + CbkA'k)Fab + (-CbkAk + (bkA'k)F'ab nonlocal (Here cf>k,jbk = j[bk],j'bk = \ebkmnlmn are constants, and (bk = <j>[bxk],Cbk = %tbkmn<Pmxn.) Table 3.5: Classification of L P S conservation laws induced b y PS and A - P S symmetries 3.6 C h a p t e r 3 S u m m a r y In this chapter w e generated both local and new nonlocal conservation laws of M E us ing the symmetries of the M E potential systems calculated in the previous chapter. This was done b y real izing conservation laws of M E as local conservation laws of a M E potential system. The point symmetries of L P S were s h o w n to generate the classical local M E energy-momen-t u m conservation laws. For PS, the scalar rotation symmetry generated a n e w nonlocal M E con-servation law. In the case of A-PS , n e w nonlocal M E conservation laws arose f rom the scalar rotation, vector rotations / boosts, and the inversions. These nonlocal M E conservation laws were also s h o w n to be nonlocal w h e n regarded as L P S conservation laws. S imple criteria were developed to verify the locality or nonlocality of M E [LPS] conservation laws in terms of their characteristics w i th respect to PS or A-PS . These results relied o n two important cohomology results for M E and L P S stated in Theorems 3.1 and 3.2 w h i c h wi l l be further investigated in the next chapter. Chapter 4 Cohomology 4.1 Introduction Any exact r-form is always closed, but the converse is not always true. Cohomology refers to any obstructions for having closed forms being exact. When the free-space Maxwell's equations are written as d[aFbc] = 0, d[aF'bc] = 0, (4.1) where F'ab = ^eabcdFcd, we can interpret this as saying that Fab(x) and F'ab(x) are closed 2-forms in Minkowski space. By Poincare's Lemma, they are exact, and so there exist 1-forms Ab(x) and A'b(x) such that F — dA and F' = dA'. In general, these 1-forms A,A' have nonlocal dependence on F,F'. In the jet space (x, F, F^,...), the condition (4.1) becomes D[aFbc] = 0, D[aF'bc] = 0, (4.2) which again implies that Fab and F'ab are closed. However, we cannot immediately conclude they are exact since this would require Fab = D^A^, F'ab = D^A'^ to hold for some Ab, A'b defined on the ME solution jet space, i.e. such that Ab, A'b have local dependence on F,F' and their differential consequences. Our main classification theorems (Theorem 3.18 and Corollary 3.19) for ME conservation laws derived from PS or A-PS symmetries relied on the statement (see Theorem 3.1) that Fbc, F'bc generate all the nontrivial cohomology of local 2-forms on the ME solution jet space. Our goal in this section is to provide a partial proof of this fact. We will investigate the ME cohomology equation XFab + \F'ab = D[aQb} on the ME solution jet space, (4.3) and show that no solutions Qb exist that are linear in F and its derivatives up to order r when (A, A) / (0,0). When A = A = 0, solutions of (4.3) are called trivial solutions. 4.2 Tensor decompositions Before proceeding along these lines, we establish some general results involving tensors. Proposition 4.1 Let T f c l " W m = r ( f c i - f c r ) [ M . 77^ p c a n decomposed as rpki---krlm _ ^rp(k\---[kTHm\ _|_ ^rp{k-i---krh)rin} (4 4) 57 Chapter 4. Cohomology 58 where a = ^ and A* = 2^+2^• ^n inefirsi i e r m above, symmetrization on (ki---kr) is performed after antisymmetrization, while in the second term, antisymmetrization on [Im] is performed after sym-metrization.) Proof. E x p a n d i n g (4.4), we obtain rpk\--krlm ~~ Tkl kr^m + 2T^kl '"^r-i\[lm]\kr)^j _|_ P ^ rpki--krlm _ rrp(ki---kr-i\[lm]\kr)^j . f-t _ °i _ _JL— \ rpki-krlm _ (^°~_ _ r M \ rp{k1---kr-l\{lm}\kr) V 3 r + l) \3 r + l) Since we want this decomposit ion to be an identity, the above coefficients must vanish. Conse-quently, cr = p = ^ q ^ , and (4.4) holds for arbitrary T. • In the sections to fol low we wi l l need to investigate equations of the f o r m 0 = Tkl'"krlmDk1 • • • DkrFim o n the M E solution jet space, and determine what condit ions these equa-tions impose o n T. L e m m a 4.2 The following equations hold on the ME solution jet space: (i)DaFab = 0, (ii) D[aFbc] = 0, (in) DaDaFbc = 0. Proof. Equat ions (i) and (ii) are the equations def ining the M E solution jet space. To obtain (iii), expand (ii) to get DaFbc = 2D^cFb^a. A p p l y i n g Da, we f ind that the R H S vanishes due to (i). • Propos i t ion 4.3 Let Tkl"'krlm = r ( f c l ' " M M . Then Tkv~krlmDkl • • • DkrFlm = 0 on the ME solution jet space if and only if rp(ki---krl)m _ ^(Ikipk2---kr)m _ ^m(lpki---kr) (4 ^ holds for some (5 with pk*-kr™ = Jg(fa-*r)mi Proof. If Tkv"krlmDkl • • • DkrFim = 0 o n the M E solution jet space, then f rom (4.4), w e have 0 = o T t k i - M " ^ . . . Dkr_,D[kFlm] + pT^kv"krV>mDkl • • • DkrFlm. (4.6) O n the M E solution jet space, the first term vanishes. So we obtain 0 = Tk^lmD(kl---DkrFl)m. (4.7) Because of its index symmetrization, expression (4.7) vanishes o n the M E solution jet space on ly if Dfa • • • F)krF^m oc tr D(kl • • • DkrF^m and hence the coefficient must be trace-free. Thus , trfr T^kl"'krV>m — 0, w h i c h immediately yields jr(fci-M)m _ ^m^Qjfci-fcr) _|_ ^(Ikipk2-kr)m^ (4.8) for some a, (3, where w.l.o.g. akv"kr = a^kl' "kr"> and /3 f e2-fcr"i _ ^ (fe2-fcr)m_ Symmetr izat ion of (4.8) over ( fc i • • • krlm) y ields 0 = rflmakl"'kr"> + 7^(im/gfci-fcr)_ (49) Chapter 4. Cohomology 59 Let v E R 4 be arbitrary. T h e n mult ip ly ing the above equation by vkl - • • vkrvivm, we obtain 0 = (v • v)vkl • . . V k r { a k ^ + Bk^), (4.10) where the dot in the first pair of parentheses indicates the M i n k o w s k i inner product . Thus , we have 0 = vkl • • • vkr(akl"'kr + Qki~kr) for all v such that v • v / 0. Since v • v ^ 0 is a dense set in R 4 , then b y continuity we get 0 = vkl • • • vkr(akl-kr + /?*!-*»•) for a l l v € R 4 . Thus , c ^ 1 " ^ = _p(ki-kT) a n c j t n e r e s u i t (4.5) follows. For the converse, suppose that (4.5) holds. B y (4.4), to show that T f c l " W m . D f c l • • • DkrFlm = 0 o n the M E solution jet space, it suffices to show that T^-k^m]Dkl • • • DkrFim = 0 o n the M E solution jet space. Contract (4.5) w i th Dkl • • • DkrFim. Note that the resulting expression vanishes, since the 77-factors y ie ld terms of the form DaFao, DaDaFbc, or rjlmFim, all of w h i c h vanish (by L e m m a 4.2) on the M E solution jet space. • 4.3 S o l u t i o n o f t h e M E c o h o m o l o g y e q u a t i o n We start w i th the cases where Qb depends linearly on F (r = 0 case), l inearly o n F and F^ (r = 1 case) and then proceed to tackle the general case. 4.3.1 C a s e r = 0 Suppose that Qb depends linearly on F, i.e. Qb = qblmFlm, (4.11) where qblm = qb[lm](x)-T h e n (4.3) is equivalent to 0 = 6[akqb]lmDkFlm + d[aqb]lmFlm - XFab - \F'ab. (4.12) For the first term in (4.12) to vanish o n the M E solution jet space, we obtain the condit ion trfr T a 6 ( f e ' ) m = 0, (4.13) where "tr fr" refers to upper indices, and Tabklm = 8\akqb]lm. We use the decomposit ion (4.5) to get S[a{kqb]l)m = VklPabm - r]^l3abk\ (4.14) for some 8abk. W.l.o.g. we have 8abk = 8[ab]k- We n o w go through the details for so lv ing the algebraic equation (4.14) b y tensorial methods. (Hereafter we regard x as fixed.) We use a transvection technique. Let v e R 4 be an arbitrary vector. Contract (4.14) w i th vkvi to get viv[aqb]lm = [(v • v)8pm - vpvm](3abp. (4.15) Next take the product of (4.15) w i th Vf and antisymmetrize [abf]. This yields 0 = [(v v)6pm - vpvm]v[f3abf. (4.16) Chapter 4. Cohomology 60 M u l t i p l y (4.16) b y v9 and antisymmetrize o n [mg] to get 0 = (v • v)vy0ab^mv9y This impl ies 0 = V{ff3ab^mv9} for all v such that v • v ^ 0. Since v • v / 0 defines a dense set in R 4 , then by continuity, O = v[f0ab]^vti (4.17) must h o l d for all v G M4. Since 0 is independent of v (it depends only o n x), this can only h o l d for all v if Pab = 0. (4.18) T h e n (4.15) yields 0 = viV[aqbfm. Since q is independent of v, this equation impl ies qb lmvi = Tvb (4.19) for some 7 independent of v. Contract ing (4.19) wi th vm y ields 0 = ^ mvmvb/ since qb(lm) = 0. T h e n 7 m u m = 0 a n d hence 7™ = 0 since v m is arbitrary. So (4.19) reduces to qb lmvi = 0, and w e conclude that qblm = 0. (4.20) F r o m (4.12), we obtain 0 = XFab + \F'ab. Since Fab and F a b are linearly independent, then A = A = 0. T h u s we have s h o w n Qb = 0 and hence A = A = 0, so (4.3) has only trivial solutions for the order r = 0 case. 4.3.2 Case r = 1 Suppose that Qb depends linearly o n F a n d F^, i.e. Qb = 9 6 W m D f c F , m + { l l m F l m ^ (421) where ? 6 ' m = 9 6 P m l ( s ) and ? 6 W m = qbkW(x). U s i n g (4.5), the first order terms of the cohomology equation (4.3) lead to 6[a(p <»b]kl)m = vdPpabk)m _ rjmdpjp), ( 4 . 2 2 ) for some Pabhm = P[ab]km- N o w we parallel the analysis of (4.14). Contract ion of (4.22) w i th vkvrvp yields v[a %}klmvkvi = [(v • v)5pm - vpvm]vkpabkv. (4.23) N e x t take the product of (4.23) w i th Vf and antisymmetrize o n [abf] to get 0=[(v v)6pm - vpvm]vkv[fPab]kP. (4.24) Take the product of (4.24) wi th v9 and antisymmetrize o n [mg]. This yields 0 = (vv)vkv[fpab]k^mv9\ (4.25) w h i c h impl ies 0 = vkvifPab^mv9^ for all v such that v • v ^ 0. Since v • v ^ 0 defines a dense set in R 4 , then by continuity w e must have 0 = vkv[fPab]k^ (4.26) Chapter 4. Cohomology 61 for arbitrary v. The only way (4.26) can hold is if vkBob km = PabV m + v{Jb]m (4.27) for some J3ab, (3bm independent of va with f3ab = (3\aoy Now substitution of (4.27) back into (4.23) cancels the ft term, and yields v[a %]klmvkvi = [(v • v)6™ - vpvm]v{Jbf. (4.28) This equation has the structure of an outer product involving va. Consequently, we must have q b k l m v k V l = [(v • v)6pm - vpvmW + i k m v k v b (4.29) for some 7 f c m independent of v. Contract (4.29) with vm to get 0 = 7kmVkVmVb ^ 7(*"») = 0 . (4.30) Since v is arbitrary, then (4.29) implies By the tensor decomposition (4.4), we have =0 (onME) (i) %klmDkFlm = ( + f ) DkFlm = l k l m D [ k F l m ] + | %W™DkFlm / =0 (onME) \ PbmDkFkm - $b{kDkFP +5b^k)mDkFlm \ J \ l ^ D k F l m = \ = yBH)mDkFlm. Using the antisymmetry of 7, we find that on the ME solution jet space the 1-form Qb reduces to Qb = 7film{DbFim + DmFlb) + lower order terms = Db% + lower order terms, x = llmFim, since DymF^b = \DbFim from expanding D[bFlm] = 0. Hence, after an integration by parts, equation (4.3) now reduces to the order r — 0 case and we obtain our main result. T h e o r e m 4.4 IfQb is linear in F and F^ then the solution of the cohomology equation (4.3) is Qb = DbX, X = YdFcd, A = A = 0. Thus, we conclude that there are only trivial solutions for the order r = 1 case. Chapter 4. Cohomology 62 4.3.3 G e n e r a l case: r > 1 Suppose that Qb depends l inearly o n F, F^l\ . . . , F^T\ i.e. Qb =qbk'-k~-lmDkl • • • DkrFlm + ...+ qbk'lmDklFlm+ qblmFlm, (4.32) where g6 f ci-**'"» =g 6(*i-<y)[M( x) for j = 1 , . . . , r . U s i n g (4.5), the r-th order terms of the cohomology equation (4.3) lead to $[a{p qb]k^k^m = r](lp6abkl-kr)m - r]m^lpabkv-krp). (4.33) N o w w e parallel the analysis as in the previous cases wi th the transvection technique. Let v G K 4 be arbitrary. T h e n contraction of (4.33) w i th vkl • • • vkrvivp y ields v[a %]kl-krlmvkl • • • vkrvt = [(v • v)6pm - vpvm]vkl • • • v k r 8 a b k ^ p . (4.34) Next take the product of (4.34) w i th Vf and antisymmetrize o n [abf] to get 0=[(v v)6pm - vpvm]vkl • • • v k r v [ f 8 a b ] k ^ p . (4.35) Take the product of (4.35) w i th v9 and antisymmetrize on [mg]. This yields 0 = (v • v)vkl • • • v k r v [ f p a b ] k ^ m v 9 l (4.36) T h i s i m p l i e s O = vkl • • • vkrv [f/3ab}kl '"kr^-mv9^ for a l l u € M 4 such that v • v ^ O.Sinceu-w ^ Odefines a dense set in R 4 , then b y continuity we have for arbitrary v, 0 = vkl---vkrv[f8ab]k--k^mv9y (4.37) The on ly w a y (4.37) can h o l d is if Vkl--- V k J a b k V - k r m = Vkl--- (pabk^~Hm + V V 1 ' ' ^ " 1 " 1 ) ( 4 3 8) for some ft, $ independent of va. W.l.o.g. we have J3abkv"kr~1 = J3[ab}('kl'"kr~^ a n d /3bkl"-kr-im = ^ ( f e i fcr_i)m N o w substitution of (4.38) back into (4.34) cancels the /3 term, and yields v[a qbkv~krlmvk, • • • vkrvt = [(v • v)5pm - vpvm]vkl • • • v k r _ l V [ J b ] k ^ k ^ p . (4.39) This equation has the structure of an outer product invo lv ing va. Consequently, we must have q b k l - k r l m v k l • • • vkTvt = [(v • v)6pm - vpvm]vkl • • • vkr_Jbkl-kr~lP + j k ^ k r m v k l • • • vkrvb (4.40) for some 7 independent of v w i th 7 f c l " k r T n — 7 ( f c i - f c r ) " \ Contract (4.40) w i t h vm to get 0 = 7 f c i - f c r m U f c i . . . V k r y m V b =^ 7 ( f e i - f c r m ) = 0 ^ 7 f c i - f c r m = _ r 7 m ( f c i - f c r ) _ ( 4 4 1 ) L e t 7 f c l ' " f c r m := - 2 ^7 f c l ' ' f c r - 1 f f c ' " T n ] . Note that ^(fci---fc r)m _ r ^ki—krm• _ ^m(ki—kr)^ by (AAI) ^k\--kTra (4 42) Chapter 4. Cohomology 63 Thus, ^ki-krm _ (^fci-fcr)m.^  where 7 satisfies 7 f ei- f c»-m _ (^fci-fcr_i)[ferm]_ Since i> is arbitrary and after taking into account the symmetries over the indices hi,..., kr, I, equation (4.40) yields qb(ki—krl)m _ (ikrpbki—kr-i)m _ ^(fei-fer^O"1 _|_ 8b^l^kl-kO™1. (4-43) Further relations between ft and 7 may arise, but their determination is unnecessary for the main result to be proved here. Contract (4.43) with Dkl • • • DkrFim (upon which the $ terms will cancel) and simplify the resulting expression using Lemma 4.2. By the tensor decomposition (4.4), we have qbkl"-k*lmDkl---DkrFlm =0 (cmME) = a % k - ^ l m D k l • • • Dkr_xD[kFlM +p q ^ - ^ D k , • • • DkrFlm = ^ q \ ^ v - k r l ) r n D k i . . . D k r F l m =0 (onME) = P (ri(lkrPbkv"kr-l)m ~ /Vfcl"N°m) Dkl--- DkrFlm + p 5 6 ( ' 7 f c l " ' f c r ) m ^ i '"' = ^t±2lsb^-k^Dkl---DkrFlm r + 2 r + 2 _J_^ki-krm fDki... D^Fbm + (r- l)Dkl • • • Dkr^2DbDkrFkr_im r + 2 +Dkl---Dkr_1DbFkrm) 2 „ f c l . . . f c r m ,_D^ _ _ _ D [ k r F m ] b _ ( r _ i ) D f e l • • • D f c r _ a D b D [ f e r i i , m ] f c r _ 1 r + + D 6 7 J f c l - - - D f c r _ 1 F f c r m ) kv-krm (1 + ( L _ l ) + 1) . . . Dkr^FkrT r + 2 [using D[ f c r F m ] 6 = -1/2 DbFkrTn, obtained from -D[6F f c r.m ] = 0] = lkl-krmDbDkl---Dkr_1Fkrm = Db (jkl"krmDkl • • • D f c r - i - F f c r m ) + lower order terms. [integration by parts] We have proven the following result. T h e o r e m 4.5 IfQb is linear in F, F^\ F^ then the solution of the cohomology equation (4.3) is Qb = FJbX + lower order terms , x = l k l " ' k r m D k l • • • D ^ F ^ , and hence the cohomology equation (4.3) reduces to the order r — 1 case. (Here ^k\-krm j s ar})[iraryi and depends only on x.) Chapter 4. Cohomology 64 Using a descent argument, we get Qb = Dbx + 0th order terms, where x is a linear combination of F, F^\ ..., F ( r _ 1 ) . Since there are only trivial solutions for the order r = 0 case, we conclude that there are only trivial solutions for any order r. T h e o r e m 4.6 On the ME solution jet space, the cohomology equation (4.3) has only trivial solutions A = A = 0, Qb = DbX when Qb is linear in F and its derivatives up to some finite order. Thus, modulo the linearity assumption, Fab and F'ab are cohomology elements, i.e. they are obstructions to having exact solutions of the equation D^aQbc) = 0 on the ME solution jet space. 4.4 Chapter 4 Summary In this chapter, using tensorial methods we have shown that the ME cohomology equation XFab + \F'ab = D[aQb] (4.44) has only trivial solutions A = A = 0, Qb = F>bX when Qb is assumed to be linear in F and its derivatives up to some finite order. This approach falls short of proving the ME 2-form coho-mology theorem on two accounts: (i) The linearity assumption should be removed. (ii) It must be proved that no other nontrivial cohomology elements other than Fab and F'ab exist. We note that Proposition 3.17 is sufficient for classifying all ME conservation laws derived from PS or A-PS symmetries with the exception of the scalar rotation conservation law $ROT-Since § R O T has characteristic (Qab, Q, Q') = (2Fab, 0,0) with respect to A-PS, then proving the nonlocality of this ME conservation law amounts to demonstrating that Fab is n ° t exact on the ME solution jet space. This fact has been established in this chapter modulo the linearity as-sumption. We have not investigated the LPS cohomology result in this thesis, but a similar analysis could be carried out. The complete proof of the ME and LPS 2-form cohomology results using spinorial techniques has been established by Anco and Pohjanpelto in [7]. Chapter 5 Future Research Here we pose some open problems that require further investigation: 1. Prove that the addit ional nonlocal conservation laws of M E der ived in this thesis are func-tionally independent of the wel l -known (local) energy-momentum conservation laws. 2. D o alternative choices for gauge constraints o n A (and A') i n the M E potential systems, e.g. C ronst rom gauge, lead to addit ional nonlocal symmetries of M E ? 3. Solve the adjoint-symmetry equations, construct all local conservation laws for each of the M E potential systems, and project them back to M E . In this way, obtain a more complete list of nonlocal conservation laws arising f rom the M E potential systems. 4. A n c o and Pohjanpelto [10] have s h o w n that M E admits a r ich structure of generalized (lo-cal) symmetries. Th is suggests the possibil ity that M E potential systems m a y also admit general ized symmetries. W h i c h nonlocal symmetries and nonlocal conservation laws of M E arise f rom generalized symmetries admitted b y an M E potential system? 5. Construct potential systems for other P D E systems and investigate the n e w symmetry classes and conservation laws arising f rom symmetries admitted b y the potential systems. 6. Suppose that a P D E system R is given, and its solution space is e m b e d d e d in the solut ion space of a potential system S. G i v e n a compatible gauge constraint C, a theoretical foun-dat ion has yet to be developed for determining a pr ior i w h e n the resulting augmented system (S, C) admits symmetries w h i c h project to potential (nonlocal) symmetries of the original system R. 7. O u r construction of M E conservation laws f rom PS (or A-PS) symmetries relied o n an e m -b e d d i n g of local PS (or A-PS) symmetries into (local and nonlocal) M E adjoint-symmetries. For w h i c h P D E systems and w h i c h potential systems does such an e m b e d d i n g exist? 8. The cohomology of local 2-forms o n the M E and L P S solution jet spaces p layed an i m -portant role in der iv ing efficient criteria for the classification of conservation laws. Does cohomology o n the solution jet space p lay a larger role in the study of properties of solu-tions of P D E systems? A r e there more efficient ways of comput ing the cohomology o n a solut ion jet space associated wi th a g iven P D E system? 65 Bibliography [1] Stephen C . A n c o , "Conservat ion laws of scaling-invariant field equations", To appear in J. Phys. A : M a t h . Gen. , 2003. [2] Stephen C . A n c o and George B luman, "Der ivat ion of conservation laws f rom nonlocal s y m -metries of differential equations", J. M a t h . Phys. 37 (1996), 2361-2375. [3] , "Direct construction of conservation laws f rom field equations", Phys. Rev. Lett. 78 (1997), 2869-2873. [4] , "Non loca l symmetries and nonlocal conservation laws of Maxwel l ' s equations", J. M a t h . Phys. 38 (1997), 3508-3532. [5] , "Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation laws classifications", Eur. J. A p p l . M a t h . 13 (2002), 545-566. [6] , "Direct construction method for conservation laws of partial differential equations. Part II: General treatment", Eur. J. A p p l . Math . 13 (2002), 567-585. [7] Stephen C . A n c o and Juha Pohjanpelto, Unpub l i shed . [8] , "Classif ication of local conservation laws of Maxwel l ' s equations", A c t a A p p l i c . M a t h . 69 (2001), 285-327. [9] , "General ized symmetries of massless free fields o n M i n k o w s k i space", Preprint, 2003. [10] , "Symmetries and currents of massless neutrino fields, electromagnetic a n d gravi-ton fields", To appear in C R M Proceedings and Lecture Notes (P. Winternitz, J. H a r n a d , C.S. L a m , and J. Patera eds.), vo l . 34,2003. [11] George W. B l u m a n and Stephen C . A n c o , Symmetry and Integration Methods for Differential Equations, A p p l i e d Mathematical Sciences, vo l . 154, Spr ing Verlag, 2002. [12] George W. B l u m a n , Sukeyuk i K u m e i , and Gregory J. Re id , " N e w classes of symmetries for partial differential equations", J. M a t h . Phys. 29 (1988), 806-811. [13] Patrick D o r a n - W u , Unpub l i shed , private communications. [14] A l e x Y. M a , Extended group analysis of the wave equation, Master 's thesis, U B C , 1988. [15] Peter J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. , Graduate Texts in Mathematics, vo l . 107, Spr ing Verlag, 2000. [16] Juha Pohjanpelto, "Symmetries, conservation laws, and Maxwel l ' s equations", A d v a n c e d Electromagnetism: Foundations, Theory and Appl icat ions (T.W. Barrett and D . M . Gr imes, eds.), W o r l d Scientific, Singapore, 1995, pp . 560-589. [17] Robert M . Wald , General Relativity, Univers i ty of Ch icago Press, 1984. 66 Appendix A Tensor Methods and Index Juggling The tensor notion used here is standardized notation to which we refer the reader to [17] for a thorough introduction. We mention here the notations and operations commonly used through-out this thesis. • Symmetrizations / antisymmetrizations: Square brackets [ • • • ] denotes antisymmetrization over the enclosed indices, while round brackets (• • •) denotes symmetrization over the enclosed indices. Occasionally, vertical bars | • • • | within a symmetrization (or antisym-metrization) are used - these indicate that the operation only applies to those indices out-side the bars. Some examples: T{ab) = \(Tab + Tba), (A.l) S[abc] = gC'S'abc + Sbca ~r" Scab — Sacb — Sbac — Scba) = ^(Sa[bc] + Sb[ca] + Sc[ab}), (A.2) g[a\b\rjpc] _ -(sabTc - ScbTa). (A.3) • Kronecker delta (identity map): 6ab denotes the Kronecker delta symbol. It is symmetric in its two indices, so that 6ab = Sba. Multiplication by the covariant metric r)ac lowers the first index and yields SabVac = $cb = Vcb- (A.4) We have a similar result with multiplication by the contravariant metric. Contraction on a, b yields 5%6ba = S\ = 6aa = A (A.5) in Minkowski space. • Index "juggling": Repeated indices - one upper index and one lower index - can be inter-changed using the covariant and contravariant metrics. For example, Taa = VabVacTbc = SbcTbc = Tbb = T\. (A.6) • Volume tensor / form: In Minkowski space (M 4,T]), eabcd = e^abcd^ denotes the completely antisymmetric volume tensor with corresponding volume form eabcd, which satisfies the 67 Appendix A. Tensor Methods and Index Juggling 68. property: abed AI r[a rt rc rd] / A J\ In particular, contractions of (A.7) result in the following important relations: eabcd€anPq = -SSKt^q, (A.8) eabcdeabpq = - 4 ^ V \ (A.9) eabcdeabca = -6<5dg, (A.10) e a 6 c de a 6 c d = -24. (A.ll) Definition A . l The Lie derivative £x (with respect to a vector Xk) of an (r, s) tensor Tn "'lrjv--js is given by CxTu-irjv..j« := XkDkTi'"''r- Tki2"'irj1...jsDkXil - ... - T^-^h...J3DkXir +Ti^kJ2...jsDjlXk + ...+ T^jv..js_lkDJ3Xk. (A.12) The Lie derivative satisfies the following well-known properties: Proposition A.2 Let Xa be a vector. Then 1. The Lie derivative commutes with curls: For any 1-form ub, £xD[au)b] = D [ a £ x u i b ] . (A.13) 2. The Lie derivative is a derivation on tensor fields. In particular, it is a linear operator and satisfies a Leibniz rule - e.g. Sab£xTab + Tab£xSab = £x(TabSab) = £kDk(TabSab). Proposition A.3 Let nab be the Minkowski metric, and let £fc satisfy the conformal Killing equation p)(a£b) _ ljjabjj^k in Minkowski space (M4,r?). Then the following equations hold: CtVab = -\vabDke, (A.14) c^eabcd = _€abcdDk£k^ ( A 1 5 ) £t_Tab = rjacribd£^Tcd-TabDk^k, (A.16) ^Tab = VacVbdCtT^ + TabDkZ". (A.17) Proof. (A.14), (A.16), and (A.17) follow from the definition of the Lie derivative, its Leibniz property, and the conformal Killing equation. We now prove (A. 15): £^eabcd = i i E ^ g ^ - ekbcdDk£a - eakcdDk£,b - eabkdDk£c - eabckDk£d = D^bj^F^- eabcd£k) = -eabcdDk£k. • Appendix B Conformal Killing Equation Solution The conformal K i l l ing equation is g iven by C K E : d^a^ = ^ vabdctc- (B.1) L e m m a B . l We have (i) dbdc? = -2dcd%\ (ii) dadadb^b = o. Proof. A p p l y da to (B.l) and s impl i fy us ing symmetry of mixed partials to get (i). A p p l y db to (i) a n d s impl i fy to get (ii). • T h e o r e m B.2 The solution of the conformal Killing equation is £a = aa+ uixa + ipakxk - xa4>kxk + \(paxkxk, (B.2) where aa, u>, ipab = , 4>a are arbitrary constants. Proof. A p p l y dmdn to (B.l), symmetrize over m, n to get: . d(mdn)d(ag) CKE l^ab^mga)Q^C (J _ \ ^ B Q C Q C Q { m g i ) CKE A ^ m n ^ Q C g ^ d («) Q ( B 3 ) 4 2 8 Hence, 0 = dmdnda^b — dndadm£b ^B=^ dmdnd^a^ - dndad^m^ = I (dmdn(da^b - dbC) - dnda{dmib - db£m)) = ^ dndb{daC - dmC) = dndbd^at,m}. Therefore, Q(ag) = Qab _ ^ab^c^ d[ag] = _^ab + ^ab^ ( B 4 ) for some constants uiab = u^ab\ (j>abc = ^ a b \ , ipab = ifiab\ u,abc = p\ab]c- B y (B.l) w e must have uab - <j)abcXc = \rfb(Qdd - ¥dcxc), and hence uab = ?f 6 w, 4>abc = r]ab4>c, where w := \uidd, 4>c := \4>ddc. Thus , gag = v a b ^ _ _ ^ab + ^ab^c^ ( B g) 69 Appendix B. Conformal Killing Equation Solution 70 We impose integrability conditions - i.e. we check equality of mixed partials dcda£b = dadc^0: -Vba<Pc + Pabc = -Vbc<Pa + Pcba- (B.6) U s i n g ant isymmetry in the first two indices of p,, the above equation becomes Pbca - Pbac = Vbafic ~ Vbc4>a- (B.7) Symmetr iz ing o n b, c we get -^(Pbac + Pcab) = 7}{riba<t>c + Vca4>b) ~ Vbc&a- (B.8) S w a p p i n g a, b in (B.6) leads to pcab = r\ac4>b — rjab^c + Pbac- Substitution into (B.8) yields Pbac = Vbc4>a ~ Vac^b = 2r?c[6^)a] [iabc = -2(f>[arib]c- (B-9) Thus , we have da£b = nab(uj - <j)cxc) - ^ab - 2^axy (B.10) ^da£b = Sab(uj-(l>cxc)-ijab-4>axb + ^bxa = gab + f{ac)bxc, (B . l l ) where gab := 6abu — ipab, / ( a c ) 6 :— Vac4>b ~ ^{a4>c) - Since we have already imposed the integra-bil ity condit ions for £, the R H S of (B. l l ) is indeed a gradient, and line integration of (B . l l ) is independent of path-curve. R e m a r k B.3 Let fac be constant. T h e n facxc is a gradient, i.e. facxc — daf for some scalar function /, if and only if /[a c] = 0. (Check integrability conditions.) R e m a r k B.4 Let ga, fac be constants w i th /[a cj = 0. T h e n J 9adxa = gaxa, J facxcdxa = \facX°X a, (B.12) to wi th in a constant of integration. These are gradient line integrals, w h i c h are thus independent of path-curve. Hence , us ing the above formulas to integrate (B. l l ) , we obtain (B.2). • 

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