aa r X i v : . [ phy s i c s . p l a s m - ph ] D ec A general theory for gauge-free lifting
P. J. Morrison ∗ Department of Physics and Institute for Fusion Studies,The University of Texas at Austin, Austin, TX, 78712, USA (Dated: July 16, 2018)
Abstract
A theory for lifting equations of motion for charged particle dynamics, subject to given elec-tromagnetic like forces, up to a gauge-free system of coupled Hamiltonian Vlasov-Maxwell likeequations is given. The theory provides very general expressions for the polarization and magneti-zation vector fields in terms of the particle dynamics description of matter. Thus, as is common inplasma physics, the particle dynamics replaces conventional constitutive relations for matter. Sev-eral examples are considered including the usual Vlasov-Maxwell theory, a guiding center kinetictheory, Vlasov-Maxwell theory with the inclusion of spin, and a Vlasov-Maxwell theory with theinclusion of Dirac’s magnetic monopoles. All are shown to be Hamiltonian field theories and theJacobi identity is proven directly.Key Words: Vlasov, Maxwell, Hamiltonian, noncanonical Poisson bracket, gauge-free, constitu-tive ∗ [email protected] . INTRODUCTION In conventional treatments of electricity and magnetism phenomenological susceptibilitiesare introduced to describe material media. Concomitant with the introduction of thesesusceptibilities is the idea that charge can be separated into bound and free components,current can be similarly decomposed, and based on these separations expressions for thepolarization and magnetization of the medium are obtained. However, it is well-known toplasma physicists that such a simple characterization is not possible for plasmas, whereparticle orbits may transition from trapped to passing and, indeed, may exhibit complicatedbehavior that can only be described by the self-consistent treatment of the dynamics ofboth the particles and the fields. Because of these complications, tractable and reliableexpressions for the polarization and magnetization are not so forthcoming, particularly whenapproximations are made and/or additional physics is added.The purpose of the present paper is to construct a general theory for the coupling of chargecarrying particle dynamics, entities possibly with internal degrees of freedom described bya kinetic theory, coupled to electromagnetic-like field theories. A theory that is gauge-free and ultimately expressible without the introduction of vector and scalar potentials isconstructed. Like Maxwell’s equations, the Vlasov-Maxwell equations, and virtually everyimportant system in physics, the theory will have Hamiltonian form. This Hamiltonian formwill be noncanonical, following the program initiated in Refs. [1, 2].The construction begins in Sec. II with a set of ordinary differential equations that de-scribe the particle dynamics. This set of equations, which is the basic model of the matterunder consideration, is assumed to have a very general Hamiltonian form, possibly with anunconventional phase space and with a Hamiltonian that depends on specified electromag-netic fields including the field variables E ( x , t ) and B ( x , t ), and possibly all their derivatives.The problem then is to lift this finite-dimensional dynamical system that describes the mat-ter to a gauge-free field theory with a kinetic component that is of Vlasov type coupledto an electromagnetic component of Maxwell type. The difficulty with this lifting programlies in the coupling of the two components of the field theory. It is shown in Sec. III thatthe construction given naturally results in a field theory that is also Hamiltonian. Thisassures that there is a consistency to the coupling. Because the Hamiltonian theory requiresvariational calculus, it is most convenient to discuss constitutive relations resulting from the2atter system in this section as well. In Sec. IV several examples are presented, beginningwith the usual Vlasov-Maxwell system, followed by a general guiding center kinetic theory,a theory that includes spin and, to show the generality of our construction, a theory withmonopole charge where the Maxwell field is modified. Gaussian units are used for all exam-ples. Section V contains concluding remarks. In Appendix A of the paper there are severalsubsections with direct proofs of the Jacobi identity for Poisson brackets of the noncanonicalHamiltonian field theories. The first one describes an old calculation of the author that hasnot heretofore appeared in print, a calculation that contains several useful techniques. Theother subsections contain analyses of the other brackets of the examples of Sec. IV. II. A GENERAL ELECTROMAGNETIC KINETIC THEORY VIA LIFTING
Consider a general dynamical system with an n -dimensional phase space with coordinates z = ( z , z , . . . , z n ) and evolution determined by a Poisson bracket and Hamiltonian E asfollows: ˙ z a = [ z a , E ] = J ab ∂ E ∂z b , a, b = 1 , , . . . , n (1)where the Poisson bracket on phase space functions g and h is defined by[ g, h ] = ∂g∂z a J ab ∂h∂z b (2)and repeated indices are to be summed. The only requirements placed on the cosymplectictensor J is that it endow the Poisson bracket with the properties of antisymmetry, [ g, h ] = − [ h, g ], and the Jacobi identity, [ g, [ h, k ]] + [ h, [ k, g ]] + [ k, [ g, h ]] = 0 for all functions g, h, k (see e.g. [3, 4]). In the context of geometrical mechanics this is referred to as a flow on aPoisson manifold, but this formalism and its language will not be used here. Rather, thephysics of matter described by this finite-dimensional dynamical system, as embodied in theHamiltonian function E , is emphasized. The description of the matter in the formalism ofthis paper is contained in this function E and its associated Poisson bracket (2).Particle orbits in given E ( x , t ) and B ( x , t ) fields are usually described in terms of theelectromagnetic potentials, φ ( x , t ) and A ( x , t ), where E = −∇ φ − c ∂ A ∂t and B = ∇ × A . (3)3ollowing this usual procedure, the Hamiltonian E of the general system of (1) will berestricted for the purposes of the present lift theory to have the following form: E = ¯ K ( p − e A /c, w, E , B , ∇ E , ∇ B , . . . ) + eφ , (4)where ¯ K is an arbitrary function of its arguments. This form was proposed in the contextof the variational theory of Refs. [5–7]. Since the aim is to generalize usual charged particledynamics, the parameter e that denotes charge is included and c is the speed of light asusual. This particular form assures electromagnetic gauge invariance. Here, the phasespace has been split into two parts, z p = ( x , p , w , w , . . . , w d ), where the coordinates ( x , p )are the usual canonical six-dimensional phase space coordinates and the coordinates w =( w , w , . . . , w d ) describe additional degrees of freedom, such as might occur in the classicaldescription of a molecule with rotational or vibrational degrees of freedom. These additionalcoordinates will be referred to as internal degrees of freedom. An important, but not theonly, Poisson bracket is given by[ f, g ] = [ f, g ] p + [ f, g ] w =: ∇ f · ∂g∂ p − ∇ g · ∂f∂ p + ∂f∂w α J αβw ∂g∂w β , α, β = 1 , , . . . , d , (5)which is the sum of canonical and internal pieces, and is assumed to satisfy the Jacobiidentity. Such a bracket and Hamiltonian generate dynamics of the form of (1).Several comments on the Hamiltonian form of (4) and (5) are in order. Note that E maydepend explicitly on the fields, and all of their derivatives, and the same may be true for thetensor J of the Poisson bracket provided the Jacobi identity is satisfied for any choice of thesefields. The Poisson bracket may also have explicit w dependence, but no direct couplingof the internal degrees of freedom to the fields has been made explicit. In general, suchcoupling would need to be consistent with the symmetries of interest for these variables. Forexample, if w were a spin variable, say s , then E would depend on s · B . This case is treatedas an example in Sec. IV C. Also, observe that all of the dependence on the spatial variable x occurs through the fields. The omission of such explicit dependence in E is appropriatefor media where spatial homogeneity is broken only by the presence of the fields; however,the x dependence could be added for further generalization.Alternatively, a manifestly gauge invariant form is obtained in terms of the coordinates z v = ( x , v , w , w , . . . , w d ), where ( x , v ) denotes the usual six-dimensional velocity phase4pace coordinates, with p := m v + ec A , (6)and m denoting the mass of the charged particle. In terms of these variables (1) takes theform ˙ z v = [ z v , K ] + em I d · E , (7)where the bracket of (5) becomes the Littlejohn [8] Poisson bracket in (7),[ g, h ] L = [ g, h ] v + [ g, h ] B , (8)where [ g, h ] v = 1 m (cid:18) ∇ g · ∂h∂ v − ∇ h · ∂g∂ v (cid:19) and [ g, h ] B = em c B · (cid:18) ∂g∂ v × ∂h∂ v (cid:19) , (9) I d is a (6 + d ) × E into the force law, and K ( v , w ; E , B , ∇ E , ∇ B , . . . ) = ¯ K ( p − e A /c, w, E , B , ∇ E , ∇ B , . . . ) . (10)Thus the electric field appears as an external force in addition to any dependence on it thatmay come through the function K , and the electromagnetic potentials no longer appear inthe dynamics. Note, in general E cannot be written as a gradient in order to combine itwith the first term of (7). The dynamics of (7) with arbitrary Poisson bracket in terms of z v , possibly depending explicitly on z v , E , and B , can be taken as the starting point forlifting to a kinetic theory.Usual Lorentzian dynamics is given by K = m | v | /
2: when K is written in terms of p ,the bracket of (5) with E yields the equations of motion for a particle of charge e and mass m subject to given electric and magnetic fields. Alternatively, the same equations are givenfrom (7) with [ , ] L .Now, the finite degree-of-freedom system of (7) will be lifted. The first step is to liftthe particle dynamics to a kinetic theory for determining a phase space density f ( z, t ) = f ( x , v , w, t ). This is easily achieved by the standard Liouville form ∂f∂t + [ f, K ] + em E · ∂f∂ v = 0 , (11)where the generalization to multiple species is straightforward. Clearly the characteristicequations of (11) correspond to the finite-dimensional matter model of (7).5he second part of lifting is to describe the coupling to Maxwell’s equations. This couplingis effected by introducing the energy functional K [ E , B , f ] := Z d x d v dw K f , (12)whence the following expressions for the charge and current densities are obtained: ρ ( x , t ) = e Z d v dw f − ∇ · δKδ E (13) J ( x , t ) = e Z d v dw ∂ K ∂ v f + ∂∂t δKδ E + c ∇ × δKδ B . (14)Inserting these expressions for the sources into the usual form of Maxwell’s equations com-pletes the lift.From (13) and (14) it is evident that the polarization, P , and magnetization, M , can beidentified as P ( x , t ) = − δKδ E and M ( x , t ) = − δKδ B , (15)which is consistent with the usual definitions of bound charge density, polarization current,and magnetization current, ρ b = −∇ · P , J p = ∂ P ∂t , and J m = c ∇ × M , (16)respectively. Although the manner of lifting embodied in (11), (13), and (14) is straight-forward, because of the functional derivatives in (15) the dependencies of P and M on thefields E and B may be very complicated and contain high order spatial derivatives.In Sec. III it is shown that this manner of lifting results in a Hamiltonian field descriptionof the coupled system. It should be emphasized that this construction does not require theexplicit introduction of the vector and scalar potentials. However, from the Hamiltonianform using E it is clear that it subsumes the description using φ ( x , t ) and A ( x , t )). To seethe explicit form, define the momentum phase space density by ¯ f ( x , p , w, t ) = f ( x , v , w, t ),which gives under the change v ↔ p the governing kinetic equation ∂ ¯ f∂t + (cid:2) ¯ f , E (cid:3) = 0 , (17)with [ g, h ] = [ f, g ] p + [ f, g ] w . The coupling to Maxwell’s equation is essentially unchanged: ρ ( x , t ) = e Z d p dw ¯ f − ∇ · δKδ E (18) J ( x , t ) = e Z d p dw ∂ K ∂ p ¯ f + ∂∂t δKδ E + c ∇ × δKδ B , (19)6s are the expressions for P and M . Using the chain rule expressions, ∂ ¯ f∂ p = 1 m ∂f∂ v , ∇ ¯ f = ∇ f − emc ∂f∂ v · ∇ A , and ∂ ¯ f∂t = ∂f∂t − emc ∂f∂ v · ∂ A ∂t , (20)it is not difficult to show that [ g, h ] p transforms to [ g, h ] L and (17) transforms into (11). III. HAMILTONIAN FORM AND CONSTITUTIVE RELATIONS
Based on past experience, viz. Vlasov-Maxwell and guiding center kinetic theories, anatural choice for the Hamiltonian functional is the following: H [ f, E , B ] = K − Z d x E · δKδ E + 18 π Z d x (cid:0) | E | + | B | (cid:1) = K + Z d x E · P + 18 π Z d x (cid:0) | E | + | B | (cid:1) (21)where P , as given by (15), is used as a shorthand in the second line, which one could rewritein terms of D := E + 4 π P . The Hamiltonian of (21) is a generalization of the energycomponent of the energy-momentum tensor first derived by variational methods in [5–7].It is straightforward to verify directly that (21) is conserved by the combined field theory,Maxwell’s equations with the sources (13) and (14) coupled to the kinetic theory of (11).One might think that the | B | term of (21) should be replaced by B · H , where H = B − π M , but this is incorrect. All polarization and magnetization effects are modeled hereby the terms involving K , i.e., they are a consequence of the particle dynamics. Rather thanrelating E and B to D and H by constitutive relations, the particle dynamics, extended andother, describes the physics that is often approximated by simplistic constitutive relations.For example, the difference between | E | and E · D arises from the K term that contains thematter dynamical information.If only ( f, E , B ) are used as dynamical variables, there is a difficulty in obtaining aPoisson bracket description for the field theory. The problem is readily encountered whenone attempts to include polarization effects, because the polarization current has a timederivative and Poisson bracket expressions such as { E , H } do not produce terms with timederivatives of the dynamical variables; i.e., in Hamiltonian theories all time derivatives areon the left hand side, so to speak. Consequently a term of the form ∂ P /∂t cannot appear.However, there is a way to circumvent this problem, a problem that does not occur in actionprinciple formulations, such as those of [5–7].7unctional differentiation of (21) gives δHδ E = − (cid:18) δ Kδ E δ E (cid:19) † · E + E π , (22) δHδ B = δKδ B − (cid:18) δ Kδ B δ E (cid:19) † · E + B π , (23)where δ K/δ E δ E and δ K/δ E δ B are second functional derivative operators that satisfy (cid:18) δ Kδ E δ E (cid:19) † = δ Kδ E δ E and (cid:18) δ Kδ B δ E (cid:19) † = δ Kδ E δ B . (24)(See e.g. [3, 4, 9] for a review of functional differentiation.) Expressions (22) and (23)reveal how polarization and magnetization effects are embodied in K . Since the functionalderivatives above have, in a sense, ‘dressed’ E and B , existing Hamiltonian structures will notbe adequate. It is clear that without some modification one cannot obtain the polarizationcurrent.If the theory were expressed in terms of D , the following bracket on functionals ¯ F [ D , B , f ]could give the correct temporal evolution of D , provided δ ¯ H/δ B = H / π : { ¯ F , ¯ G } = Z d x d v dw f (cid:0) [ ¯ F f , ¯ G f ] v + [ ¯ F f , ¯ G f ] w (cid:1) (25)+ Z d x d v dw f [ ¯ F f , ¯ G f ] B (26)+ 4 πem Z d x d v dw f (cid:0) ¯ G D · ∂ v ¯ F f − ¯ F D · ∂ v ¯ G f (cid:1) (27)+4 πc Z d x (cid:0) ¯ F D · ∇ × ¯ G B − ¯ G D · ∇ × ¯ F B (cid:1) , (28)with [ f, g ] v and [ f, g ] B given by (9), ∂ v g = ∂g/∂ v , and ¯ F f := δ ¯ F /δf , ¯ F D := δ ¯ F /δ D ,etc. The Born-Infeld term of (28) is motivated by their original theory [10] that was alsowritten in terms of D and B (see also [11]). Although this term can give something like ∂ D /∂t = ∇ × H , it remains to properly define the meaning of D and H . Thus, this bracketalone does not constitute a closed theory. Similarly the coupling term (27), a generalizationof that introduced in [2, 3] that includes the internal variable w , is written here in termsof D , but the generalization of the Marsden-Weinstein term [12] (see also [13]) of (26), andthe first term of (25), also a generalization of that given in [2, 3], are unchanged. The newinternal term here of (25) with [ ¯ F f , ¯ G f ] w does not depend on D and does not affect theJacobi identity (cf. Appendix A 2). 8o close the theory requires a constitutive relation, something like D = ǫ · E . Suchrelations are often appended to electromagnetic theory based on phenomenological materialproperties, but here they emerge as a consequence of the Vlasov-like dynamics and thedefinitions (15). Using (15) gives D = D [ E , B ; f ] = E + 4 π P [ E , B ; f ] (29)with both P and D linear in f , but not in E and B . In general these functionals can benonlinear and even global in nature. It is only required that there be a unique inverse E = D − [ D , B ; f ] = E [ D , B ; f ] . (30)Similarly, using (15) H = H [ B , E ; f ] = B − π M [ B , E ; f ] , (31)which is also assumed to have an inverse, i.e. B = B [ H , E ; f ] = H + 4 π M [ H , E ; f ] . (32)For given K , the expressions of (15) can be quite complicated, particularly when deriva-tives of the fields are included. However, these expressions are local in time, i.e., E , B , D ,and f are all evaluated at the same time. Because of the presence of f and the equationgoverning it, causal effects are included in a dynamical sense and do not need to be put inat the expense of breaking time-reversal symmetry. Also, there is no artificial separationof charge into bound and free components or current into magnetization or other. Rather,charges and currents are determined dynamically according to the Vlasov equation. A givencharge may behave in any manner consistent with this dynamics.When lim E → D [ E , B ; f ] = 0 the first order term in an expansion in E gives D = ǫ · E , where ǫ is the dielectric permittivity operator. Similarly, B = µ · H , where µ is thepermeability operator. If one were to replace the Vlasov dynamics by trivial dynamics oflinear response away from equilibrium, then one can recover the usual permittivity andpermeability relations, including the usual causal (see e.g. [14]) form in space and time.But, this will not be done here.To sum up, the Hamiltonian of (21) is given in terms of ( E , B , f ), the bracket of (25)-(28)in terms of ( D , B , f ), and (29) is a closure relation relating D to E . Thus, if ¯ H [ D , B , f ] = H [ E , B , f ] is defined by inserting the inverse of (29) in for E , a closed theory is obtained.9or general K this inversion cannot be done explicitly (although a series expansion may bepossible); however, the chain rule can be used to relate functional derivatives of H to thoseof ¯ H and thereby obtain equations for the time derivatives of ( D , B , f ) which can then beshown to be equivalent to those of Sec. II. Alternatively, the chain rule can be used to writethe bracket of (25)-(28) in terms of the set of fundamental variables ( E , B , f ).To understand the chain rule, suppose two functionals are related by ¯ F [ D , B , f ] = F [ E , B , f ], where the right hand side is obtained by inserting (29) in for D in the func-tional F on the left. Variation of ¯ F = F gives Z d x (cid:16) δ ¯ Fδ D · δ D + δ ¯ Fδ B · δ B (cid:17) + Z dz δ ¯ Fδf δf = Z d x (cid:16) δFδ E · δ E + δFδ B · δ B (cid:17) + Z dz δFδf δf , (33)while variation of (30) gives δ E = δ E δ D · δ D + δ E δ B · δ B + δ E δf δf , (34)where δ E /δ D etc. are the usual Fr´echet derivatives obtained by first variation. Inserting(34) into (33) and comparing the coefficients of the independent variations δ D etc., gives δ ¯ Fδ D = (cid:18) δ E δ D (cid:19) † · δFδ E , (35) δ ¯ Fδ B = δFδ B + (cid:18) δ E δ B (cid:19) † · δFδ E , (36) δ ¯ Fδf = δFδf + (cid:18) δ E δf (cid:19) † · δFδ E . (37)A more explicit expression for ( δ E /δ D ) † can be obtained by varying (29) at fixed B and f , giving δ D = (cid:18) I − π δ Kδ E δ E (cid:19) · δ E =: ε · δ E (38)or δ E = δ E δ D · δ D = ε − · δ D (39)where ε is the nonlinear permittivity operator (not to be confused with ǫ ). Evidently, E D := δ E δ D = ε − = (cid:18) I − π δ Kδ E δ E (cid:19) − , (40)and E D = ( E D ) † or ( ε − ) † = ε − . It is important to note that although δ D = ε · δ E , D = ε · E ; the correct relation between D and E is given by the nonlinear expression of (29).Similarly, E B := δ E δ B = 4 π ε − · δ Kδ B δ E and E f := δ E δf = 4 π ε − · δ Kδf δ E . (41)10nd the functional derivatives of (35)–(37) can now be calculated:¯ H D = E † D · H E = ε − · ε · E π = E π ¯ H B = H B + E † B · H E = K B − (cid:18) δ Kδ B δ E (cid:19) † · E + B π + (cid:18) ε − · δ Kδ B δ E (cid:19) † · ε · E = B π − M ¯ H f = H f + E † f · H E = K − (cid:18) δ Kδf δ E (cid:19) † · E + (cid:18) ε − · δ Kδf δ E (cid:19) † · ε · E = K . (42)Now, using the expressions of (42) in the bracket of (25)–(28) gives ∂ B ∂t = − πc ∇ × ¯ H D = − c ∇ × E (43) ∂ D ∂t = 4 πc ∇ × ¯ H B − πem Z d v dw f ∂ v ¯ H f = c ∇ × H − πem Z d v dw f ∂ v K (44) ∂f∂t = − (cid:2) f, ¯ H f (cid:3) − ∂ v · (cid:0) f ¯ H D (cid:1) = − [ f, K ] − em E · ∂f∂ v , (45)where [ , ] = [ , ] L + [ , ] w as defined by (8) and (9). Thus the bracket reproduces theVlasov-like equation of (11) and Maxwell’s equations with the polarization and magnetiza-tion currents, but remember f and [ , ] can be written in terms of p using (6) and thus theabove is also equivalent to (11).In the above, D is a convenience, a shorthand for E + 4 π P , with E being the fundamentalvariable. One could eliminate D from these equations e.g. by writing (44) as follows: ∂ D ∂t = E D · ∂ E ∂t + P B · ∂ B ∂t + P f · ∂f∂t , (46)where E D = ε , as before, and P B and P f are again operators obtained by variation of P , andthen inserting the other two equations of motion for the time derivatives. This procedurewill lead to a complicated set of equations in terms of the fundamental variables ( E , B , f ).Another way of obtaining these complicated equations is to obtain a bracket in terms of E , B , and f alone, by inserting the transformations for the functional derivatives of (35),1136), and (37) into the bracket of (25)–(28). This yields the following complicated bracket: { F, G } = Z d x d v dw f h F f + E † f · F E , G f + E † f · G E i + 4 πem Z d x d v dw f (cid:16)(cid:16) E † D · G E (cid:17) · ∂ v (cid:16) F f + E † f · F E (cid:17) − (cid:16) E † D · F E (cid:17) · ∂ v (cid:16) G f + E † f · G E (cid:17)(cid:17) +4 πc Z d x (cid:16) (cid:16) E † D · F E (cid:17) · ∇ × (cid:16) G B + E † B · G E (cid:17) − (cid:16) E † D · G E (cid:17) · ∇ × (cid:16) F B + E † B · F E (cid:17) (cid:17) , (47)where to complete the procedure the expressions of (40) and (41) are to be inserted. Thusthe bracket of (47) is quite complicated with many terms and operators. Nevertheless, byits construction it satisfies the Jacobi identity (modulo the ∇ · B = 0 obstruction discussedin Appendix A 1). With Hamiltonian H [ E , B , f ] it is easily seen from (42) that this bracketproduces the correct equations for ∂f /∂t and ∂ B /∂t , but it is less easy, yet possible, to seeit produces the complications of (46) correctly.For some theories of interest, including usual linear response theory, K has a simplifiedform, viz. K ( v , w, E , B ) = h ( v , w, B ) + P ( v , w, B ) · E + 12 E · k ( v , w, B ) · E , (48)where k † = k . For example, such a linear polarization theory is sufficient for some drift andgyrokinetic-like theories. With (48) P = − Z d v dw ∂ K ∂ E f = − Z d v dw P f − E · Z d v dw k f . (49)The first term of the second equality of (49) represents a permanent dipole moment perunit volume, which can be dropped: because (21) is a Legendre transform it will cancel outanyway. The second term of (49) defines the electric susceptibility, χ e . Thus P = χ e · E or D = ǫ · E (50)with ǫ = I + 4 πχ e . Note that unlike the ε of (38), when (48) is assumed, ǫ is independentof E but not of B . Explicitly in terms of indices ǫ ij = δ ij − π Z d v dw k ij ( v , w, B ) f . (51)Assuming ǫ − exists and has the form ǫ − = II + 4 πχ e = I − πχ e + (4 πχ e ) − . . . , (52)12here χ e stands for matrix multiplication, and so on down the line. Hence, E = ǫ − · D .Although ǫ is linear in f , this is not the case for ǫ − . Finally, for the K of (48) that isquadratic in E , the following simplified expressions are obtained: E † D = ǫ − (53) E † B = D · ∂ ǫ − ∂ B (54) E † f = D · ǫ − · k · ǫ − . (55)Note, obtaining these formulas can be facilitated by using identities obtained by varying theexpression ǫ − · ǫ = I . IV. EXAMPLES
In this section four examples are given: that of Sec. IV A is the usual Vlasov-Maxwelltheory, that of Sec. IV B is a guiding center drift kinetic theory that includes nontrivialpolarization and magnetization effects, that of Sec. IV C includes a physically perspicuousinternal variable, and that of Sec. IV D was chosen to show the generality of the lift theoryby altering Maxwell’s equations.
A. Vlasov-Maxwell
For Vlasov-Maxwell theory w is nonexistent and only z = ( x , p ) appears. Thus, ¯ f ( x , p , t )and, with K = | p − e A /c | / m , Eq. (11) becomes ∂ ¯ f∂t = (cid:20) | p − e A /c | m + eφ, ¯ f (cid:21) = − emc ( p − e A /c ) · ∇ A · ∂ ¯ f∂ p + e ∇ φ · ∂ ¯ f∂ p − m ( p − e A /c ) · ∇ ¯ f . (56)In terms of f ( x , v , t ), K = m | v | / ∂f∂t = − v · ∇ f − em (cid:16) E + v c × B (cid:17) · ∂f∂ v . (57)From the general Hamiltonian (21), with K = m | v | /
2, evidently the Vlasov-MaxwellHamiltonian is H = m Z d x d v | v | f + 18 π Z d x (cid:0) | E | + | B | (cid:1) , (58)13nd with [ g, h ] L the bracket of (25)–(28) becomes the Vlasov-Maxwell bracket { F, G } = Z d x d v (cid:18) f [ F f , G f ] v + f [ F f , G f ] B + 4 πem f ( G E · ∂ v F f − F E · ∂ v G f ) (cid:19) +4 πc Z d x ( F E · ∇ × G B − G E · ∇ × F B ) . (59)With this bracket and Hamiltonian, one obtains the usual Vlasov-Maxwell equation as ∂f∂t = { f, H } = − [ f, K ] L − em E · ∂f∂ v , (60)which is equivalent to (57). Similarly, since for this example D = E , the usual expres-sion for the current is obtained from { E , H } , there being no polarization or magnetizationcontributions, and Faraday’s law is given by ∂ B /∂t = { B , H } .The relativistic Vlasov-Maxwell theory similarly follows with the choice K = c p | p − e A /c | + m c and the theory can be written in manifestly covariant form [6], but this will not pursuedfurther here. B. Guiding Center Drift Kinetic Theory
A canonical Hamiltonian description for guiding center particle motion was obtained in[6, 7] by applying Dirac constraint theory to Littlejohn’s degenerate Lagrangian [15, 16](with a regularization suggested in [17]) in order to effect a Legendre transformation. Thecanonical variables of the theory are ( x , q , p , p ) and the particle Hamiltonian, Dirac’sprimary Hamiltonian for this problem (see e.g. [7, 18]), is E ( x , q , p , p , E , B , ∇ B ) = v g · ( p − e A ∗ /c ) + V p + eφ ∗ (61)= K (cid:0) p − e A /c, q , p , E , B , ∇ B (cid:1) + eφ , (62)which is of the form of (4). Here A ∗ = A + b mcv γ ( q /v ) /e , eφ ∗ = eφ + µ | B | + m (cid:0) q + | v E | (cid:1) / , (63) v E = c ( E × B ) / | B | , b = B / | B | , (64) v g = q B ∗ / ( γ ′ B ∗|| ) + c E ∗ × B /B ∗|| , V = e E ∗ · B ∗ / ( mg ′ B ∗|| ) , (65) B ∗ = ∇ × A ∗ and E ∗ = −∇ φ ∗ − c ∂ A ∗ ∂t . (66)The function γ ( z ) is an antisymmetric regularization function with z = q /v and v someconstant velocity. The Littlejohn theory is recovered if γ ( z ) = z , in which case q = v || .14n the regularized theory γ ( z ) ≈ z for small | z | , but for large | z | , γ is bounded so thatwith v >> v thermal , mv c γ ( ∞ ) << e | B | / ( b · ∇ × b ), which is accomplished, e.g., by γ =tanh( z ).This theory has an eight-dimensional phase space with the canonical bracket[ g, h ] = [ g, h ] p + ∂ q g ∂ p h − ∂ q h ∂ p g , (67)which with K of (62) completes the theory. The appropriate bracket of the form of (25)–(28) is obtained with (67), and from K the functional K can be constructed and thus theHamiltonian of (21). This bracket and Hamiltonian produces the equations of motion; thusthis system is a Hamiltonian field theory. From K the polarization and magnetization arecan be obtained straightforwardly. Since expressions are complicated, the reader is referredto [6, 7] for details. C. Spin Vlasov-Maxwell
The nonrelativistic spin Vlasov-Maxwell system is a kinetic theory generalization of theVlasov-Maxwell system that includes a semiclassical description of spin [19–21]. The Hamil-tonian description of this system [21] will be shown to fit within the gauge-free lifitingframework. The spin Vlasov-Maxwell electron distribution function, f ( x , v , s , t ), containsthe internal spin variable s = ( s , s , s ), and satisfies ∂f∂t = − v · ∇ f + (cid:20) em (cid:16) E + v c × B (cid:17) + 2 µ e m ~ c ∇ ( s · B ) (cid:21) · ∂f∂ v + 2 µ e ~ c ( s × B ) · ∂f∂ s , (68)where m and e > π ~ is Planck’s constant, µ e = gµ B / µ B , the Bohr magneton, and theelectron spin g -factor. Equation (68) is coupled to the dynamical Maxwell’s equations, ∂ B ∂t = − c ∇ × E , ∂ E ∂t = c ∇ × B − π J , (69)through the current J = J f + c ∇ × M , which has “free” and spin magnetization parts: J f := − e Z d v d s v f and M := − µ e ~ Z d v d s s f . (70)Equations (68) and (69), with (70), are to be viewed classically and consequently a full nine-dimensional phase space integration, d z = d x d v d s , is considered for f . Spin quantization15s obtained as an initial condition that constrains s to lie on a sphere (see [21]). Extensionto multiple species is straightforward, so will not be included.For this system K is chosen as follows: K ( v , s , B ) = m | v | + 2 µ e ~ c s · B , (71)however, more general forms are possible. It is not difficult to check that the characteristicequations of (68) are of the form of (11) with the Poisson bracket [ g, h ] = [ g, h ] v + [ g, h ] s ,where [ g, h ] s = s · ( ∂ s g × ∂ s h ) , (72)with ∂ s := ∂/∂ s , and K given by (71).Thus the Hamiltonian functional (21) becomes H [ E , B , f ] = Z d z (cid:18) m | v | + 2 µ e ~ c s · B (cid:19) f + 18 π Z d x (cid:0) | E | + | B | (cid:1) , (73)which can be shown to be conserved directly by using the equations of motion, and thebracket of (25)–(28) adapted to the present example is { F, G } sV M = Z d z f (cid:16) [ F f , G f ] v + [ F f , G f ] B + [ F f , G f ] s (74)+ 4 πem ( F E · ∂ v G f − G E · ∂ v F f ) (cid:17) (75)+4 πc Z d x ( F E · ∇ × G B − G E · ∇ × F B ) . (76)The last term of (74) of { , } sV M accommodates the spin, an internal variable; it is notsurprising that it has an inner bracket based on the so (3) algebra (e.g. [18]). The remainingterms of (74), (75), and (76) produce the usual terms of Vlasov-Maxwell theory: it is astraightforward exercise to show that Eqs. (68) and (69) are given as follows: ∂f∂t = { f, H } sV M , ∂ B ∂t = { B , H } sV M , ∂ E ∂t = { E , H } sV M . This is facilitated by the identity R d z f [ g, h ] = − R d z g [ f, h ], which works for each of[ g, h ] v , [ g, h ] B , and [ g, h ] s . It follows that the polarization P ≡ δK/δ B = − M asgiven by (70).In Appendix A 2 a direct proof of the Jacobi identity for { f, H } sV M is given.16 . Monopole Vlasov-Maxwell This example differs from the previous ones in that Maxwell’s equations are changed toanother field theory. In particular, Dirac’s theory of electromagnetism [22, 23] with monopolecharge will be treated in the lift framework.For Dirac’s theory, (7) is replaced by the particle orbit equations˙ x = v and ˙ v = em ( E + v c × B ) + gm ( B − v c × E ) . (77)where g and e are magnetic and electric charges, respectively. The appropriate particlePoisson bracket for insertion into (25)–(28) is[ g, h ] m = [ g, h ] v + [ g, h ] B + [ g, h ] E (78)where [ f, g ] E := − gm c E · ( ∂ v f × ∂ v g ) . (79)The particle Hamiltonian is given by K = m | v | /
2, just as for Vlasov-Maxwell theory.In lifting to a kinetic theory there are various kinds of multi-species dynamics, with andwithout magnetic charge, that could be considered. Here the case of a single species ofidentical particles that carry both magnetic and electric charges will be developed, alongthe lines of the quantum fluid theory considered in [24]. Inserting (78) into (25)–(28) andadding a new coupling term to account for the B that acts as an external force, gives thebracket { F, G } mV M = Z d x d v (cid:16) f [ F f , G f ] m (80)+ 4 πem f ( G E · ∂ v F f − F E · ∂ v G f ) + 4 πgm f ( G B · ∂ v F f − F B · ∂ v G f ) (cid:17) (81)+4 πc Z d x (cid:0) F E · ∇ × G B − G E · ∇ × F B (cid:1) . (82)With Hamiltonian of (58) this bracket yields ∂f∂t = − v · ∇ f − ∂f∂ v · (cid:16) em ( E + v c × B ) + gm ( B − v c × E ) (cid:17) (83) ∂ E ∂t = c ∇ × B − π J e (84) ∂ B ∂t = − c ∇ × E − π J m (85)17here J e = e Z f v d x and J m = g Z f v d x . (86)Thus monopole Vlasov-Maxwell is a Hamiltonian field theory.One reason for investigating Dirac’s model in the present context is to see if the ∇ · B = 0obstruction to Jacobi discussed in Appendix A can be removed. In Appendix A 3 the Jacobiidentity for { F, G } mV M is proved directly, and there it is discovered that the solenoidalcharacter of B is replaced by ∇ · ( e B − g E ) = 0. Thus, the space of functionals must stillbe restricted to such fields, as discussed in Appendix A. However, Dirac constraint theorycan reduce this to a boundary condition at infinity [25].Another reason for investigating monopole theories is for their utility in developing numer-ical algorithms. For example, the Gudunov numerical method for magnetohydrodynamics[26] (see also [27]) exploits a form that allows for ∇ · B = 0 that was subsequently shownto be Hamiltonian without the ∇ · B = 0 constraint in [1, 3] (see also [28–30]). However,since the monopole theory of this section requires a specific linear combination of E and B to be divergence free, adaptations of these methods in not so straightforward. In any event,the reader can rest assured if mononpoles are discovered there still will exist Hamiltonianguiding center and gyrokinetic kinetic theories, obtained with suitable choices for K withassociated generalized polarization and magnetization vectors. V. CONCLUSIONS
The main accomplishment of this work is to describe how a matter model of dynamicscan be lifted to a Hamiltonian coupled Vlasov-Maxwell system. En route to this Hamilto-nian theory, the general constitutive relations of (30) and (31) or, equivalently, the nonlinearpermittivity and permeability operators, as determined by (15), were obtained. These consti-tutive functionals are very general: as discussed in the paper, K may contain all derivativesof the fields and may even be global in nature and contain integral operators. From thegeneral constitutive functionals, it was shown how to obtain the usual linear relations. Anoncanonical Poisson bracket, another step in the program started in [1–3], was obtainedfor this general class of theories.Four examples were given, including the general class of guiding center kinetic theories ofSec. IV B. This latter example, like all systems that are in the class of variational theories of186], easily was shown to possess the Hamiltonian structure. Thus, the theory of this paperdetermines the path to follow for obtaining the Hamiltonian formulation of a consistentgyrokinetic theory by making use of the results [31, 32].Various generalizations are possible. Namely, the extension to many species of differentdynamics, relativistic theory other than the Vlasov-Maxwell example that was described,versions where particle matter models have more general finite-dimensional Poisson brackets,are all straightforward. Also, extending the matter dynamics by coupling to other generalgauge-free field theories is possible.Another application of the techniques of this paper will be used in a subsequent work [33]that will treat Hamiltonian perturbation theory in the field theory context. There it willbe shown how an exact transformation of the particle (characteristic) equations of Vlasov-Maxwell equations can be lifted to the kinetic and Maxwell equations, and how this can beused in perturbation theory for infinite-dimensional noncanonical Hamiltonian systems.In physics there are two ways of constructing new theories. The usual way is to con-struct an action principle by postulating a Lagrangian density with the desired observablesand symmetry group properties. Alternatively one can postulate an energy functional andPoisson bracket, which is essentially the approach of the present paper. With this lat-ter approach, one must prove directly the Jacobi identity {{ F, G } , H } + {{ G, H } , F } + {{ H, F } , G } ≡
0, for all functionals F , G , and H . Techniques for doing this are not gener-ally known, and this provides one reason for Appendix A. Appendix A: Direct proof of Jacobi identities
One term of the original Vlasov-Maxwell Hamiltonian formulation of [2] presented anobstruction to the Jacobi identity [34]. This term was replaced in [12] in order remove thisproblem, but it was then reported in [3] that the new term also presents an obstruction,viz. that given by (A22) below. One can rescue the Hamiltonian theory by requiring allfunctionals to depend on fields B such that ∇ · B = 0, but because the orginal programbegun in [1, 2] was to construct truly gauge-free field theories in terms of noncanonicalPoisson brackets, this taint was a disappointment. The same obstruction appeared in thecontext of magnetohydrodynamics [2], but a way to remove it was obtained in [3]. To date,the best fix for the taint of the Vlasov-Maxwell bracket is given in [25] by using Dirac19onstraint theory, which replaces ∇ · B = 0 by a boundary condition at infinity.Appendix A 1 contains the details of the onerous calculation first performed by the authorin 1981 (reported in detail here for the first time, as it was originally done), leading to theresult of (A22) that appeared in [3]. This is followed in Appendices A 2 and A 3 by a directproof of the brackets for the spin and monopole Vlasov-Maxwell theories, respectively.
1. Jacobi identity for the Vlasov-Maxwell bracket
For convenience the charge, mass, and a factor of 4 π are scaled out to obtain the Vlasov-Maxwell bracket for the fields f ( z, t ), E ( x , t ), and B ( x , t ) in the following form: { F, G } = Z d z f [ F f , G f ] c + f [ F f , G f ] B + f ( G E · ∂ v F f − F E · ∂ v G f )+ Z d x F E · ∇ × G B − G E · ∇ × F B =: { F, G } c + { F, G } B + { F, G } Ef + { F, G } EB , (A1)where F f := δF /δf , [ f, g ] c := ∇ f · ∂ v g − ∇ g · ∂ v f , ∂ v := ∂/∂v , [ f, g ] B := B · ( ∂ v f × ∂ v g ),and { F, G } c etc. are obvious from context. Also, note, boldface has been removed since theformulas are busy enough. Since charge can be scaled out in this manner, it is evident thatthe validity of the Jacobi identity is independent of the sign of the species charge.Above, the term { F, G } B , the Marsden-Weinstein term [12], has been separated outbecause it will be seen to be the source of the failure of Jacobi identity unless ∇ · B = 0.Considering the combination { F, G } c + { F, G } B together would simplify the calculationsomewhat.The Jacobi identity is {{ F, G } , H } :=: {{ F, G } c , H } + {{ F, G } B , H } + {{ F, G } Ef , H } + {{ F, G } EB , H } = {{ F, G } c , H } c | {z } + {{ F, G } c , H } B | {z } + {{ F, G } c , H } Ef | {z } + {{ F, G } c , H } EB | {z } + {{ F, G } B , H } c | {z } + {{ F, G } B , H } B | {z } + {{ F, G } B , H } Ef | {z } + {{ F, G } B , H } EB | {z } + {{ F, G } Ef , H } c | {z } + {{ F, G } Ef , H } B | {z } + {{ F, G } Ef , H } Ef | {z } + {{ F, G } Ef , H } EB | {z } + {{ F, G } EB , H } c | {z } + {{ F, G } EB , H } B | {z } + {{ F, G } EB , H } Ef | {z } + {{ F, G } EB , H } EB | {z } , { F, G } = h F ψ |J G ψ i ,with cosymplectic operator J , one must calculate the functional derivative δ { F, G } /δψ . Thisderivative has three contributions: one from F ψ , one from J , and one from G ψ . The firstand last give rise to second functional derivatives. Proofs of the Jacobi identity are greatlysimplified by the following: Bracket Theorem ([3])
To prove the Jacobi identity for generic brackets of the form { F, G } = h F ψ |J G ψ i one need only consider the explicit dependence of J on ψ when takingthe functional derivative δ { F, G } /δψ . Proof.
The formal proof uses the anti-self-adjointness of J and the self-adjointness ofthe second functional derivative. With these symmetries it can be shown that all secondfunctional derivative terms cancel. (cid:3) In what follows δ { F, G } /δψ ˙= . . . denotes the functional derivative modulo the secondderivative terms. Equation (A1) gives δ { F, G } c δf ˙= [ F f , G f ] c , δ { F, G } c δB ˙= 0 , δ { F, G } c δE ˙= 0 , (A2) δ { F, G } B δf ˙= [ F f , G f ] B , δ { F, G } B δB ˙= Z dv f ( ∂ v F f × ∂ v G f ) , δ { F, G } B δE ˙= 0 , (A3) δ { F, G } Ef δf ˙= ( G E · ∂ v F f − F E · ∂ v G f ) , δ { F, G } Ef δB ˙= 0 , δ { F, G } Ef δE ˙= 0 , (A4) δ { F, G } EB δf ˙= 0 , δ { F, G } EB δB ˙= 0 , δ { F, G } EB δE ˙= 0 . (A5)The following are immediate: • Term 1 vanishes because { F, G } c is Lie-Poisson; i.e., using the first of (A2), δ { F, G } c /δf ˙= [ F f , G f ],which, when inserted into {{ F, G } c , H } c and cyclicly permuting, vanishes by virtue ofthe Jacobi identity of [ , ] c . • Term 4 vanishes by the Bracket Theorem because of the second two equations of (A2). • Term 12 vanishes by the Bracket Theorem because of the second two equations of (A4) • Terms 13-16 vanish by the Bracket Theorem because { F, G } EB has no explicit depen-dence on f , E , or B , i.e. because of (A5).21 emark. One can organize Jacobi identity calculations at the outset by grouping togetherall like terms that can possibly cancel. For example terms with the same functional deriva-tives of F , G , and H must be considered together. Sometimes other considerations can aidin the grouping of terms. When terms are grouped appropriately, failure of a class of termsto cancel is a proof of the failure of Jacobi. In the heading below f f f means that onlyfunction derivatives with respect to f occur, etc. Term 6 ( f f f ) Using the first equation of (A3) gives6 :=: Z f [ B · ( ∂ v F f × ∂ v G f ) , H f ] B :=: Z f B · (cid:16) ∂ v (cid:0) B · ( ∂ v F f × ∂ v G f ) (cid:1) × ∂ v H f (cid:17) :=: Z f B i B r ǫ ijk ǫ rst ∂∂v j (cid:18) ∂F f ∂v s ∂G f ∂v t (cid:19) ∂H f ∂v k :=: Z f B i B r ǫ ijk ǫ rst (cid:18) ∂ F f ∂v j ∂v s ∂G f ∂v t ∂H f ∂v k + ∂F f ∂v s ∂ G f ∂v j ∂v t ∂H f ∂v k (cid:19) (A6):=: Z f B i B r ǫ ijk ǫ rst (cid:18) ∂ F f ∂v j ∂v s ∂G f ∂v t ∂H f ∂v k − ∂H f ∂v k ∂ F f ∂v s ∂v j ∂G f ∂v t , (cid:19) :=: 0 (A7)where (A7) follows from (A6) by permuting the second term of (A6), shifting the indicesaccording to s → k , k → t , t → j , j → s , and i ↔ r , and using the antisymmetry of theLevi-Civita symbol.Note, the above procedure is common in this game and of general utility, so it is recordedin the following: Lemma 1
If two terms can be made to cancel by permuting one of them, then all termscancel.
Proof.
By writing out all six terms by permuting
F GH → GHF → HF G , one observesthey cancel in pairs. (cid:3)
Remark.
Term 6 vanishes without any assumptions on B , i.e. ∇ · B = 0 is not required. Term 11 (
EEf ) Z f H E · ∂∂v δ { F, G } Ef δf :=: Z f H E · ∂ v ( G E · ∂ v F f − F E · ∂ v G f ):=: Z f (cid:16) H E · ∂ v ( G E · ∂ v F f ) − G E · ∂ v ( H E · ∂ v F f ) (cid:17) :=: 0 , (A8)where the last equality follows because com ( G E · ∂ v , H E · ∂ v ) = 0, where com means com-mutator. Terms 2 and 5 ( f f f )Remark.
Terms 2 and 5 have been grouped together because both give rise to terms in theJacobi identity involving F f , G f , and H f .Using (A2) in 2 + 5 :=: {{ F, G } c , H } B + {{ F, G } B , H } c gives2 + 5 :=: Z f B · ( ∂ v [ F f , G f ] c × ∂ v H f ) + f [ B · ( ∂ v F f × ∂ v G f ) , G f ] c :=: Z f B i ǫ ijk ∂H f ∂v k (cid:18)(cid:20) ∂F f ∂v j , G f (cid:21) c + (cid:20) F f , ∂G f ∂v j (cid:21) c (cid:19) + Z f ǫ ijk (cid:18) ∂∂x ℓ (cid:18) B i ∂F f ∂v j ∂G f ∂v k (cid:19) ∂H f ∂v ℓ − B i ∂∂v ℓ (cid:18) ∂F f ∂v j ∂G f ∂v k (cid:19) ∂H f ∂x ℓ (cid:19) :=: Z f B i ǫ ijk (cid:18) ∂H f ∂v k (cid:20) ∂F f ∂v j , G f (cid:21) c + ∂H f ∂v k (cid:20) F f , ∂G f ∂v j (cid:21) c + (cid:20) ∂F f ∂v j ∂G f ∂v k , H f (cid:21) c (cid:19) (A9)+ Z f ǫ ijk ∂B i ∂x ℓ (cid:18) ∂F f ∂v j ∂G f ∂v k ∂H f ∂v ℓ (cid:19) . (A10)Upon defining F = ∂ v F f , G = ∂ v G f , and H = ∂ v H f , and using the Leibniz rule for [ , ] c ,Line (A9) can be rewritten as Z f B i ǫ ijk (cid:0) H k [ F j , G f ] c + H k [ F f , G j ] c + G k [ F j , H f ] c + F j [ G k , H f ] c (cid:1) :=: 0 , (A11)where upon permutation the first and fourth terms cancel, as do the second and third. Thus,2 + 5 :=: Z f ǫ ijk ∂B i ∂x ℓ (cid:18) ∂F f ∂v j ∂G f ∂v k ∂H f ∂v ℓ (cid:19) . (A12)From (A12), the only remaining term is from Line (A10). This term can be rearranged toyield 2 + 5 + cyc = Z f ∇ · B [( ∂ v F f × ∂ v G f ) · ∂ v H f ] , (A13)which is a consequence of the following: 23 emma 2 For any three vectors F , G , and H , and vector field B in R , H · ∇ B · ( F × G ) +
F · ∇ B · ( G × H ) +
G · ∇ B · ( H × F ) = ( ∇ · B ) (cid:0) H · ( F × G ) (cid:1) (A14) where H · ∇ B · C = H i C j ∂ i B j . Remark.
Terms 2 and 5 could have been combined with Term 6. They were consideredseparately to pinpoint, as will be seen, that they are the sole terms that violate the Jacobiidentity without ∇ · B = 0. Terms 7 and 10 (
Ef f ) Inserting the first and last equations of (A3) and the first equation of (A4) into 7 + 10 :=: {{ F, G } B , H } Ef + {{ F, G } Ef , H } B gives7 + 10 :=: Z f H E · ∂ v (cid:0) B · ( ∂ v F f × ∂ v G f ) (cid:1) + f B · (cid:0) ∂ v ( − G E · ∂ v F f + F E · ∂ v G f ) × ∂ v H f ) (cid:1) :=: Z f H Eℓ ∂ vℓ (cid:0) B i ǫ ijk ∂ vj F f ∂ vk G f (cid:1) − f B i ǫ ijk (cid:0) − ∂ vj ( G Eℓ ∂ vℓ F f ) ∂ vk H f + ∂ vj ( F Eℓ ∂ vℓ G f ) ∂ vk H f (cid:1) :=: Z f B i ǫ ijk H Eℓ (cid:0) ∂ vk G f ∂ vℓ ∂ vj F f + ∂ vj F f ∂ vℓ ∂ vk G f (cid:1) (A15) − f B i ǫ ijk (cid:0) − G Eℓ ∂ vk H f ∂ vj ∂ vℓ F f + F Eℓ ∂ vk H f ∂ vj ∂ vℓ G f (cid:1) . (A16)Now upon permutation, the first term of (A15) is seen to cancel the second of (A16) andthe second term of (A15) cancels the first term of (A16). Thus 7 + 10 :=: 0. Terms 3, 8, and 9 (
Ef f ) Using the first and last equations of (A2) and (A4) and the first and second equations of(A3) in 3 + 9 gives3 + 9 :=: − Z f (cid:0) H E · ∂ v [ F f , G f ] c + [ H E · ∂ v G f , F f ] c − [ H E · ∂ v F f , G f ] c (cid:1) (A17)Using Lemma 3 below in (A17) with C equal to H E gives3 + 9 :=: Z f ( ∇ × H E ) · ( ∂ v F f × ∂ v G f ) . (A18) Lemma 3
For any vector field C ( x ) and phase space functions f and g , C · ∂ v [ f, g ] c = [ C · ∂ v f, g ] c + [ f, C · ∂ v g ] c + ( ∇ × C ) · ( ∂ v g × ∂ v f ) . (A19)24 roof. With ∂ vi := ∂/∂v i and ∂ xi := ∂/∂x i , C · ∂ v [ f, g ] c = C i (cid:0) [ ∂ vi f, g ] c + [ f, ∂ vi g ] c (cid:1) = [ C · ∂ v f, g ] c − [ C i , g ] c ∂ vi f + [ f, C · ∂ v g ] c − [ f, C i ] c ∂ vi g = [ C · ∂ v f, g ] c − ( ∂ vi f )( ∂ vj g ) ∂ xj C i + [ f, C · ∂ v g ] c + ( ∂ vi g )( ∂ vj f ) ∂ xj C i = [ C · ∂ v f, g ] c + [ f, C · ∂ v g ] c + ( ∂ xj C i ) (cid:0) ( ∂ vi f )( ∂ vj g ) − ( ∂ vi g )( ∂ vj f ) (cid:1) = [ C · ∂ v f, g ] c + [ f, C · ∂ v g ] c + ( ∂ vi g )( ∂ vj f ) (cid:0) ∂ xj C i − ∂ xi C j (cid:1) = [ C · ∂ v f, g ] c + [ f, C · ∂ v g ] c + ( ∇ × C ) · ( ∂ v g × ∂ v f ) . (cid:3) (A20)Now consider Term 88 :=: − Z d x H E · ∇ × (cid:18)Z d v f ∂ v F f × ∂ v G f (cid:19) :=: − Z d z f ( ∇ × H E ) · ( ∂ v F f × ∂ v G f ) . (A21)Equations (A18) and (A21) imply 3 + 8 + 9 + cyc = 0. Remark.
Observe the terms here, like Terms 7 and 10, are
Ef f terms. However, they havebeen grouped separately because there are ‘other considerations’ as mentioned above. TheTerms 7 and 10 vanish with B , but the Terms 3, 8, and 9, do not. Thus terms of one kindcannot cancel terms of the other.Finally, from all of the above, the following is concluded: Main Theorem ([3])
For the Vlasov-Maxwell bracket of (A1) {{ F, G } , H } + cyc = Z d z f ∇ · B ( ∂ v F f × ∂ v G f ) · ∂ v H f . (A22) Remark.
It is interesting to note that the other constraint, ∇ · E = 4 πρ , need not besatisfied for the Jacobi identity to hold. It turns out to be a Casimir invariant.
2. Jacobi identity for the spin Vlasov-Maxwell bracket
Writing { F, G } sV M = { F, G } V M + { F, G } s and using :=: as defined in Appendix A 1 {{ F, G } sV M , H } sV M :=: {{ F, G } V M , H } V M + {{ F, G } s , H } V M ++ {{ F, G } V M , H } s + {{ F, G } s , H } s :=: {{ F, G } s , H } V M + {{ F, G } V M , H } s , (A23)25here the second equality follows because of the Jacobi identity for Vlasov-Maxwell (assum-ing solenoidal B ) and the fact that { F, G } s is a Lie-Poisson bracket (see e.g. [4, 35]). Thusit only remains to show that the cross terms cancel, which is facilitated by a the brackettheorem [3] stated in Appendix A; viz., when functionally differentiating { F, G } V M and { F, G } s , which are needed when constructing the cross terms, one can ignore the secondfunctional derivative terms. Using the symbol ˙= again to denote equivalence modulo thesecond variation terms, δ { F, G } V M δf ˙= [ F f , G f ] c + [ F f , G f ] B + F E · ∂ v G f − G E · ∂ v F f (A24) δ { F, G } s δf ˙= [ F f , G f ] s (A25)while all other needed functional derivatives vanish. Thus {{ F, G } V M , H } s :=: Z d z (cid:16) f (cid:2) [ F f , G f ] c + [ F f , G f ] B , H f (cid:3) s + f (cid:2) F E · ∂ v G f − G E · ∂ v F f , H f (cid:3) s (cid:17) (A26) {{ F, G } s , H } V M :=: Z d z (cid:16) f (cid:2) [ F f , G f ] s , H f (cid:3) c + f (cid:2) [ F f , G f ] s , H f (cid:3) B (cid:17) + f H E · ∂ v [ F f , G f ] s (cid:17) (A27)The first lines of (A26) and (A27) cancel by virtue of the Jacobi identities for the brackets[ , ] c,B,s on functions, while the second line of (A26) cancels upon permutation of the secondterm. Similarly, the second term of (A27) vanishes.
3. Jacobi identity for the monopole Vlasov-Maxwell bracket
For this case the Gaussian units of the text are used, i.e. the factors of 4 π are reinsertedand both the usual and monople charges are manifest. Let { F, G } m = { F, G } V M + { F, G } M ,where { F, G } M = Z d z f [ F f , G f ] E + 4 πgm f ( G B · ∂ v F f − F B · ∂ v G f ) , (A28)and [ , ] E is defined by (79). Thus, the Jacobi identity has four terms to consider {{ F, G } m , H } m :=: {{ F, G } V M , H } m + {{ F, G } M , H } m = {{ F, G } V M , H } V M | {z } + {{ F, G } M , H } V M | {z } + {{ F, G } V M , H } M | {z } + {{ F, G } M , H } M | {z } , (A29)26here the symbol :=: is defined in Appendix A 1. Term 1
From the [3] (cf. Appendix A 1) {{ F, G } V M , H } V M + cyc = em Z d z f ∇ · B [( ∂ v F f × ∂ v G f ) · ∂ v H f ] . (A30)As in Appendix A 1, δ { F, G } /δψ ˙= . . . denotes the functional derivative modulo thesecond derivative terms. The following will be needed: δ { F, G } V M δf ˙= [ F f , G f ] c + [ F f , G f ] B + 4 πem ( G E · ∂ v F f − F E · ∂ v G f ) (A31) δ { F, G } M δf ˙= [ F f , G f ] E + 4 πgm ( G B · ∂ v F f − F B · ∂ v G f ) (A32) δ { F, G } V M δB ˙= em Z d v f ∂ v F f × ∂ v G f , δ { F, G } V M δE ˙= 0 (A33) δ { F, G } M δE ˙= − gm Z d v f ∂ v F f × ∂ v G f , δ { F, G } M δB ˙= 0 . (A34)The Poisson bracket that generates the ‘ v × ’ part of the generalized Lorentz force is[ f, g ] = [ f, g ] c + [ f, g ] B + [ f, g ] E . Because of[[ f, g ] E , h ] E + cyc = 0 and [[ f, g ] E , h ] B + [[ f, g ] B , h ] E + cyc = 0 . (A35)the following holds:[[ f, g ] , h ] + cyc = [[ f, g ] c , h ] B + [[ f, g ] B , h ] c + [[ f, g ] c , h ] E + [[ f, g ] E , h ] c + cyc= ( e ∇ · B − g ∇ · E ) [( ∂ v f × ∂ V g ) · ∂ V h ] /m (A36)The first term of the above is the source of the RHS of (A30).Now consider the remaining terms of (A29). Term 4
Equations (A32) and (A28) give {{ F, G } M , H } M :=: Z d z f (cid:20) [ F f , G f ] E + 4 πgm ( G B · ∂ v F f − F B · ∂ v G f ) , H f (cid:21) E + 4 πgm f H B · ∂ v (cid:18) [ F f , G f ] E + 4 πgm ( G B · ∂ v F f − F B · ∂ v G f ) (cid:19) . (A37)Equation (A37) has three kinds of terms. Consider first the f f f -terms: Z d z f [[ F f , G f ] E , H f ] E = g m Z d z f E · (cid:16) ∂ v (cid:0) E · ( ∂ v F f × ∂ v G f ) (cid:1) × ∂ v H f (cid:17) = g m Z d z f ǫ ijk ǫ rst E i E r ∂ vk H f (cid:0) ∂ vj ∂ vs F f ∂ vt G f + ∂ vj ∂ vt G f ∂ vs F f (cid:1) (A38)27pon permutation and reindexing, the two terms of (A38) cancel. This, of course, followsimmediately from (A35) – the above serves as a proof that [[ f, g ] E , h ] E + cyc = 0. Nextconsider the BBf -term:16 π g m Z d z f H B · ∂ v ( G B · ∂ v F f − F B · ∂ v G f ) (A39)This vanishes upon permutation because H B · ∂ v and F B · ∂ v commute. Now all that remainsof Term 4 is the Bf f -term:4 πgm Z d z f [( G B · ∂ v F f − F B · ∂ v G f , H f ] E + f H B · ∂ v [ F f , G f ] E . (A40)This term is of the same form as the EF F term of the MV-bracket (terms 7 and 10), andvanishes for the same reason. Therefore, Term 4 vanishes.Now consider the two cross terms.
Terms 2 and 3
Term 2 is {{ F, G } M , H } V M :=: Z d z f (cid:20) [ F f , G f ] E + 4 πgm ( G B · ∂ v F f − F B · ∂ v G f ) , H f (cid:21) c (A41)+ f (cid:20) [ F f , G f ] E + 4 πgm ( G B · ∂ v F f − F B · ∂ v G f ) , H f (cid:21) B (A42)+ 4 πem f H E · ∂ v (cid:18) [ F f , G f ] E + 4 πgm ( G B · ∂ v F f − F B · ∂ v G f ) (cid:19) (A43)+ em f ∂ v H f · gm Z d v f ∂ v F f × ∂ v G f (A44) − π Z d x ∇ × H B · gm Z d v f ∂ v F f × ∂ v G f , (A45)while Term 3 is {{ F, G } V M , H } M :=: Z d z f h [ F f , G f ] c + [ F f , G f ] B (A46)+ 4 πem ( G E · ∂ v F f − F E · ∂ v G f ) , H f i E (A47)+ 4 πgm f H B · ∂ v (cid:16) [ F f , G f ] c + [ F f , G f ] B (A48)+ 4 πem ( G E · ∂ v F f − F E · ∂ v G f ) (cid:17) (A49) − gm f ∂ v H f · em Z d v f ∂ v F f × ∂ v G f . (A50)Upon comparing Terms 2 and 3 some cancellations are immediate.28 Using (A36), 1st term of (A41) + 1st term of (A46) gives − gm ( ∇ · E ) ( ∂ v F f × ∂ v G f ) · ∂ v H f • Using (A35), 1st term of (A42) + 2nd term of (A46) = 0 • Lines (A44) + (A50) =0 • The terms of (A49) vanish because H B · ∂ v and G E · ∂ v commute. Likewise the lasttwo terms of (A43)Applying the following: Lemma
For any vector field C ( x ) and phase space functions f and g , C · ∂ v [ f, g ] c = [ C · ∂ v f, g ] c + [ f, C · ∂ v g ] c + m − ( ∇ × C ) · ( ∂ v g × ∂ v f ) , (A51)which is not difficult to prove, to the last terms of (A41) + Line (A45) + the last first termof (A48) =0.There are six remaining terms. The last two terms of (A42) cancel the last term of (A48),and the first term of (A43) cancels the two terms of (A47). Thus in this case, the obstructionbecomes {{ F, G } mV M , H } mV M + cyc = (A52)1 m Z d z f ( e ∇ · B − g ∇ · E ) ( ∂ v F f × ∂ v G f ) · ∂ v H f , and it is concluded that the Jacobi identity still requires a solenoid constraint on eB − gE .Upon transforming to new variables ˜ e ˜ E = eE + gB and ˜ eB = eB − gE , where ˜ e = e + g ,reproduces the Poisson bracket for Vlasov-Maxwell theory, which is possible for this singlespecies case of Dirac’s theory. In retrospect, the existence of this transformation precludesthe necessity for the proof of the Jacobi identity; however, consistent with the goal of thisentire appendix, viz. to demonstrate techniques of general utility rather than to present themost efficient proofs, we retain it here. ACKNOWLEDGMENT
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