Variational Principle for Electrodynamics of Moving Particles

Variational Principle for Electrodynamics of Moving Particles Jerzy Kijowski Centrum Fizyki Teoretycznej PAN Aleja Lotnik´ow 32/46, 02-668 Warsaw, Poland and Dariusz Chru´sci´ nski Institute of Physics, Nicholas Copernicus University ul. Grudzi¸adzka 5/7, 87-100 Toru´ n, Poland Abstract Consistent relativistic theory of the classical Maxwell field interacting with classical, charged, point–like particles, proposed in [1], is now derived from a variational principle. For this purpose a new electrodynamical Lagrangian based on fluxes is constructed. As a result, we obtain the action principle where 1) field degrees of freedom and particle degrees of freedom are kept at the same footing, 2) contrary to the standard formulation, no infinities arise, 3) energy (Hamiltonian) is obtained from the Lagrangian via the Legendre transformation, without any need of “adding a complete divergence”.

PACS: 03.50.De; 04.20.Fy; 41.10.-j; 41.70.+t

Contents 1 Introduction

2

2 Electrodynamics of moving particles: statement of results

5

3 Co-moving description of a relativistic field theory

7

4 Point particles and extended particles. Renormalization

12

5 Equations of motion from the variational principle

16

6 Variational principle based on fluxes

18

1

7 The Lagrangian in the co-moving frame

25

8 Particle in an external potential

27

Appendixes

28

A Hamiltonian structure for a 2-nd order Lagrangian theory

28

B Maxwell equations in the co-moving frame

29

C Boundary momenta

32

D Proof of the conservation laws

35

1

Introduction

Recently, one of us (J.K) proposed a consistent relativistic theory of the classical Maxwell field interacting with classical, charged, point-like particles (cf. [1]). For this purpose an “already renormalized” formula for the total four-momentum of a system composed of both the moving particles and the surrounding electromagnetic field was proposed. It was proved, that the conservation of the total four-momentum defined by this formula is equivalent to a certain boundary condition for the behaviour of the Maxwell field in the vicinity of the particle trajectories (in [1] this condition is called the fundamental equation). Field equations of such a theory are, therefore, precisely the linear, inhomogeneous Maxwell equations for the electromagnetic field surrounding the point-like sources. The new element introduced in [1], which completes this standard theory, is the above boundary condition, with the particle trajectories playing the role of the moving boundary. Together with this condition, the theory (called electrodynamics of moving particles) becomes causal and complete: initial data for both the field and the particles uniquely imply the evolution of the system. This means e. g. that the particle trajectories may also be calculated uniquely from the initial data. It was proved that the limit of this theory for e → 0 and m → 0 with their ratio being fixed, coincides with the Maxwell–Lorentz theory of test particles moving on the background described by the free field. However, for any finite value of e, the acceleration of the particle can not be equal to the Lorentz force, the latter being always ill defined, because of the field singularities implied by Maxwell equations. Physically, the “already renormalized” formula for the total four-momentum (formula (33) in the present paper) was suggested by a suitable approximation procedure applied to an extended-particle model. In such a model we suppose that the particle is a stable, soliton-like solution of a hypothetical fundamental theory of interacting electromagnetic

2

and matter fields. Assuming, that for weak electromagnetic fields and vanishing matter fields (i. e. outside of the particles) the theory coincides with linear Maxwell electrodynamics, a formula was found, which gives in a good approximation the total four-momentum of a system composed of both the moving particles and the surrounding electromagnetic field. The formula uses only the “mechanical” information about the particle (position, velocity, mass m and the electric charge e) and the free electromagnetic field outside of the particle. It turns out, that the formula is meaningful also in the case of point particles. Hence, it can be taken as a starting point for a mathematically self-consistent theory of point-like particles interacting with the linear Maxwell field. In the present paper we give the variational formulation of the above theory. The corresponding, highly nontrivial canonical (Hamiltonian) structure, will be given in the next paper. The standard variational principle used in electrodynamics cannot be extended to the theory containing also point-like particles interacting with the electromagnetic field. Such a principle is based on the following Lagrangian, written usually in textbooks (see e. g. [2], [4]): Ltotal = LM axwell + Lparticle + Lint ,

(1)

with LM axwell = −

1√ −gf µν fµν , 4

(2)

Lparticle := −mδζ ,

(3)

and the interaction term given by Lint := eAµ uµ δζ .

(4)

Here by δζ we denote the Dirac delta distribution localized on the particle trajectory ζ. The above Lagrangian may be used to derive the trajectories of the test particles, when the field is given a priori. In a different context, it may also be used to derive Maxwell equations, if the particle trajectories are given a priori. Simultaneous variation with respect to both fields and particles leads, however, to a contradiction, since the Lorentz force will be always ill defined due to Maxwell equations. But already in the context of the inhomogeneous Maxwell theory with given pointlike sources, the variational principle based on Lagrangian (1) is of very limited use, since the interaction term Lint becomes infinite. As a consequence, the hamiltonian of such a theory will always be ill defined, although the theory displays a perfectly causal behaviour. The main result of the present paper consists in removing this difficulty. Applying an appropriate Legendre transformation to the field Lagrangian LM axwell , we obtain a new, quasi-local variational principle for the Maxwell field. Already in the context of the inhomogeneous Maxwell theory with given (i. e. non-dynamical) point-like sources, our Lagrangian produces no infinities and enables us to describe the dynamics of the field influenced by moving particles as an infinite-dimensional Hamiltonian system. It turns 3

out, that adding the particle Lagrangian (3) and varying it with respect to both fields and particles is now possible and does not lead to any contradiction. As a result, we obtain precisely the “electrodynamics of moving particles” proposed in [1]. It is not unusual that the same physical theory is described by different variational principles. We derive our new variational principle, transforming the standard Maxwell Lagrangian by an appropriately chosen Legendre transformation. Each of these two variational principles is related to a specific way of controlling the boundary data of the field. Changing the variational principle means changing the physical quantities, which are kept fixed at the boundary during the variation. In the standard approach, the variation is performed with values of the electromagnetic potentials Aµ being kept on the boundary. In our approach, we keep at the boundary the value of the electric and the magnetic fluxes. Passing from the lagrangian description to the hamiltonian one, different Lagrangians lead to different field Hamiltonians, describing the field dynamics with different boundary conditions. It is worthwhile to notice that the Hamiltonian obtained from our Lagrangian is equal to the field energy: 21 (D2 + B 2 ). Other Hamiltonians, related to other boundary conditions, which may be obtained from other Lagrangians (e. g. the standard one (2)), are not positive and even not bounded from below (see [8]). The relation between different Hamiltonians, corresponding to different boundary conditions, is similar to the relation between the internal energy and the free (Helmholtz) energy in thermodynamics. The first one describes the evolution of the thermodynamic system, when insulated adiabatically from any external influence, whereas the latter describes the (completely different) evolution of the same system, when put into a thermal bath. From this point of view, controlling the electric and the magnetic fluxes on the boundary of a 3-dimensional volume V means insulating it adiabatically from any external influence, whereas the standard control of potentials still leaves the possibility of the energy exchange between the exterior and the interior of V . The paper is organized as follows. Section 2 contains the main results of the theory proposed in [1]. In Section 3 we develop a new technique, which enables us to describe at the same footing the field and the particle degrees of freedom. For this purpose we formulate any relativistic, hyperbolic field theory with respect to a non-inertial reference frame defined as a rest-frame for an arbitrarily moving observer. Such a formulation will be used as a starting point for our renormalization procedure. In Section 4 we show how to extend the above approach to the case of electrodynamical field interacting with point particles. This enables us to derive in subsequent Sections the Electrodynamics of Moving Particles from a variational principle. Finally, in Section 8 we present the lagrangian formulation for the particle interacting not only with the radiation field, but also with a fixed, external potential, produced by a heavy external device. This is a straightforward extension of our theory, where the homogeneous boundary condition for the radiation field is replaced by an inhomogeneous condition, the inhomogeneity being provided by the external field. The Appendixes contain mainly calculations. However, in Appendix A we present 4

the hamiltonian formulation of theories arising from second order Lagrangians. This formulation is rather difficult to find elsewhere, although it surely belongs to the “folklore” of classical mechanics. In the present paper we do not prove the existence and the uniqueness of the solution of equations derived here. Such a proof may be obtained using two different methods. The first one consists in analyzing directly the boundary value problem for linear Maxwell equations in a co-moving reference frame, which we describe in Section 3. The other method consists in expressing the electromagnetic field explicitly in terms of the initial data and the trajectory. This way we end up with an ordinary differential equation for the trajectory, where the field initial data play the role of parameters given a priori. Both proofs are relatively simple and will be published elsewhere.

2

Electrodynamics of moving particles: statement of results

In the present Section we briefly sketch the electrodynamics of moving particles, presented in [1]. Let y = q(t) with t = y 0 , be the coordinate description of a time-like world line ζ of a charged particle with respect to a laboratory frame, i.e. to a system (y µ ), µ = 0, 1, 2, 3; of Lorentzian space-time coordinates. The theory contains as a main part the standard Maxwell equations with point-like sources: ∂[λ fµν] = 0 , ∂µ f µν = euν δζ ,

(5)

where uν stands for the particle four-velocity and δζ denotes the δ-distribution concentrated on the smooth world line ζ: δζ (y 0 , y k ) =

q

1 − (v(y 0 ))2 δ (3) (y k − q k (y 0 )) .

(6)

Here v = (v k ) = (q˙k ) is the corresponding 3-velocity and v2 denotes the square of its 3-dimensional length (we use the Heaviside-Lorentz system of units with c = 1). In the case of many particles the total current is a sum of contributions corresponding to many disjoint world lines and the value of charge is assigned to each world line separately. For a given particle trajectory, equations (5) define a deterministic theory. This means that initial data for the electromagnetic field uniquely determine its evolution. However, if we want to treat also the particle initial data (q,v) as dynamical variables, the theory based on the Maxwell equations alone is no longer deterministic: the particle trajectory can be arbitrarily modified in the future or in the past without changing the initial data. This non-completeness of the theory is usually attributed to the fact that the particle’s equations of motion are still missing. We stress, that such an interpretation is false.

5

Indeed, it was proved in [1] that the field initial data fully determine the particle’s acceleration and this is due to Maxwell equations only, without postulating any equations of motion. More precisely, there is a one-to-one correspondence between the (r−1 )-term of the field in the vicinity of the particle and the acceleration of the particle. The easiest way to describe this property of Maxwell theory is to use the particle’s rest-frame. For this purpose consider the 3-dimensional hyperplane Σt orthogonal to ζ at the point (t, q(t)) ∈ ζ. We shall call Σt the “rest frame hyperplane”. Choose on Σt any system (xi ) of cartesian coordinates centered at the particle’s position and denote by r the corresponding radial coordinate. The initial data for the field on Σt are given by the electric induction field D = (Di ) and the magnetic induction field B = (B i ) fulfilling the conditions div B = 0 (3) and div D = e δ0 . Maxwell equations can be solved for arbitrary data, fulfilling the above constraints, but the solution will be usually non-regular, even far away from the particles. To avoid singularities propagating over a light cone from (t, q(t)), the singular part of the data in the vicinity of the particle has to be equal to "

Ã

e xk 1 xi xk D = − a + ak i 3 2 4π r 2r r k

!#

+ O(1) ,

(7)

where a = (ak ) is the acceleration of the particle (in the rest frame we have a0 = 0) and O(1) denotes the nonsingular part of the field (the magnetic field B k (r) cannot contain any singular part). This gives the one-to-one correspondence between the (r−1 )-term of the field and the particle’s acceleration, which is implied by the regularity of the field outside of the trajectory ζ. ˙ B, ˙ q, ˙ v) ˙ of the Cauchy data Hence, for regular solutions, the time derivatives (D, (D, B, q, v) of the composed (fields + particles) system are uniquely determined by the ˙ and B ˙ are given by the Maxwell equations, q˙ = v and v˙ may data themselves. Indeed, D be uniquely calculated from equation (7). Nevertheless, the theory is not complete and its evolution is not determined by the initial data. This non-completeness may be interpreted as follows. Field evolution takes place not in the entire Minkowski space M , but only outside the particle, i. e. in a manifold with a non-trivial boundary Mζ := M − {ζ}. The boundary conditions for the field are still missing! To find this missing condition, the following method was used. Guided by an extended particle model, an “already renormalized” formula was proposed in [1], which assigns to each point (t, q k (t)) of the trajectory a four-vector pλ (t). It is interpreted as the total fourmomentum of the physical system composed of both the particle and the field (see Section 4 for details). For a generic trajectory ζ and a generic solution of Maxwell equations (5) this quantity is not conserved, i. e. it depends upon t. The conservation law d λ p (t) = 0 (8) dt is proposed as an additional equation, which completes the theory. It was shown that, due to Maxwell equations, only 3 among the 4 equations (8) are independent. Given a laboratory reference frame, we may take e. g. the conservation of the momentum p = (pk ): d p=0 (9) dt 6

as independent equations. They already imply the energy conservation d 0 p (t) = 0 . (10) dt In Section (5) we will show, that the above momentum p may be obtained from the variational principle as the momentum canonically conjugate to the position of the particle, whereas p0 is equal to the total Hamiltonian of the composed (particle + field) system. The four-momentum pλ (t) was defined in terms of an integral over any hypersurface Σ, which intersects the trajectory at the point (t, q k (t)). It has been proved in [1] that, due to Maxwell equations, the integral (global) condition (9) is equivalent to a (local) boundary condition for the behaviour of the Maxwell field in the vicinity of the trajectory. The condition was called the fundamental equation of the electrodynamics of moving particles. In particle’s reference frame it may be formulated as a relation between the (r−1 ) and the (r0 ) terms in the expansion of the radial component of the electric field in the vicinity of the particle: 1 D (r) = 4π r

µ

¶

e α + + β + O(r) , 2 r r

(11)

where by O(r) we denote terms vanishing for r → 0 like r or faster. For a given value of r both sides of (11) are functions of the angles (only the r−2 term is angle–independent). The relation between the acceleration and the (r−1 ) – term of the electric field given in (7) may be rewritten in terms of the component α of this expansion: xi α = −eai (12) r (it implies that the quadrupole and the higher harmonics of α must vanish for regular solutions). In this notation the fundamental equation (equivalent to the conservation law (8)) reads: DP(mα + e2 β) = 0 ,

(13)

where by DP(f ) we denote the dipole part of the function f on the sphere S 2 . Together with this condition, Maxwell theory becomes complete, causal and fully deterministic: initial data for particles and fields uniquely determine the entire history of the system.

3

Co-moving description of a relativistic field theory

To construct the variational formulation of the above theory, we will need a description of electrodynamics with respect to the particle’s rest-frame. This is not an inertial frame. In the present section we show how to extend the standard variational formulation of field theory to the case of non-inertial frames. Consider any field theory based on a first-order relativistically-invariant Lagrangian density L = L(ψ, ∂ψ) ,

(14) 7

where ψ is a (possibly multi-index) field variable, which we do not need to specify at the moment. As an example, ψ could denote a scalar, a spinor or a tensor field. We will describe the above field theory with respect to accelerated reference frames, related with observers moving along arbitrary space-time trajectories. Let ζ be such a (time-like) trajectory, describing the motion of our observer. Let y = q(t), or y k = q k (t), k = 1, 2, 3; be the description of ζ with respect to a laboratory reference frame, i. e. to a system (y λ ), λ = 0, 1, 2, 3; of Lorentzian space-time coordinates. We will construct an accelerated reference frame, co-moving with ζ. For this purpose let us consider at each point (t, q(t)) ∈ ζ the 3-dimensional hyperplane Σt orthogonal to ζ, i.e. orthogonal to the four-velocity vector U (t) = (uµ (t)): (uµ ) = (u0 , uk ) := √

1 (1, v k ) , 1 − v2

(15)

where v k := q˙k . We shall call Σt the “rest frame surface”. Choose on Σt any system (xi ) of cartesian coordinates, such that the particle is located at its origin (i. e. at the point xi = 0). Let us consider space-time as a disjoint sum of rest frame surfaces Σt , each of them corresponding to a specific value of the coordinate x0 := t and parametrized by the coordinates (xi ). This way we obtain a system (xα ) = (x0 , xk ) of “co-moving” coordinates in a neighbourhood of ζ. Unfortunately, it is not always a global system because different Σ’s may intersect. Nevertheless, we will use it globally to describe the evolution of the field ψ from one Σt to another. For a hyperbolic field theory, initial data on one Σt imply the entire field evolution. We are allowed, therefore, to describe this evolution as a 1-parameter family of field initial data over subsequent Σ’s. Formally, we will proceed as follows. We consider an abstract space-time M := T × Σ defined as the product of an abstract time axis T = R1 with an abstract, three dimensional Euclidean space Σ = R3 . Given a world-line ζ, we will need an identification of points of M with points of physical space-time M . Such an identification is not unique because on each Σt we have still the freedom of an O(3)-rotation. Suppose, therefore, that an identification F has been chosen, which is local with respect to the observer’s trajectory. By locality we mean that, given the position and the velocity of the observer at the time t, the isometry F(q(t),v(t)) : Σ 7→ Σt

(16)

is already defined, which maps 0 ∈ Σ into the particle position (t, q(t)) ∈ Σt . As an example of such an isometry which is local with respect to the trajectory we could take the one obtained as follows. Choose the unique boost transformation relating the laboratory time axis ∂/∂y 0 with the observer’s proper time axis U . Next, define the position of the ∂/∂xk - axis on Σt by transforming the corresponding ∂/∂y k – axis of the laboratory frame by the same boost. It is easy to check, that the resulting formula for F reads: 1 xl vl (t) , y 0 (t, xl ) := t + q 2 1 − v (t) 8

³

´

y k (t, xl ) := q k (t) + δlk + ϕ(v2 )v k vl xl .

(17)

Here, the following function of a real variable has been used: 1 ϕ(τ ) := τ

Ã

!

1 1 √ √ −1 = √ . 1−τ 1 − τ (1 + 1 − τ )

(18)

The function is well defined and regular (even analytic) for v2 = τ < 1. The operator (δlk + ϕ(v2 )v k vl ) acting on rest-frame variables xl comes from the boost transformation. Suppose, therefore, that for a given trajectory ζ a local isometry (16) has been chosen, which defines Fζ : M 7→ M . This mapping is usually not invertible: different points of M may correspond to the same point of space-time M because different Σt ’s may intersect. It enables us, however, to define the metric tensor on M as the pull-back Fζ∗ g of the Minkowski metric. The components gαβ of the above metric are defined by the derivatives of Fζ , i. e. they depend upon the first and the second derivatives of the position q(t) of our observer. Because (xk ) are cartesian coordinates on Σ, the space-space components of g are trivial: gij = δij . The only non-trivial components of g are, therefore, the lapse function and the (purely rotational) shift vector: √ 1 = 1 − v2 (1 + ai xi ) , −g 00 √ = g0m = 1 − v2 ²mkl ω k xl ,

N = √ Nm

(19)

where ai is the observer’s acceleration vector in the co-moving frame. The quantity ω m is a rotation, which depends upon the coordination of isometries (16) between different Σt ’s. Because ω m depends locally upon the trajectory, it may also be calculated in terms of the velocity and the acceleration of the observer, once the identification (16) has been chosen. In the case of example (17), it is easy to check that ai =

´ 1 ³ i 2 i k δ + ϕ(v )v v k v˙ , 1 − v2 k

(20)

1 ϕ(v2 )v k v˙ l ²klm , 1 − v2

(21)

ωm = √

where v˙ k is the observer’s acceleration in the laboratory frame. The metric Fζ∗ g is degenerate at singular points of the identification map (i.e. where the identification is locally non-invertible because adjacent Σ’s intersect, i. e. where N = 0), but this degeneration does not produce any difficulties in what follows. The simplest O(3)-coordination of the isometries (16) would consist in Fermi-propagating the xk – axis along ζ, i. e. in putting ω m ≡ 0. Such a coordination is, however, non-local with respect to the trajectory. Indeed, the identification Ft between Σ and Σt would be, in this case, a result of the Fermi – propagation of a given mapping Fto from the initial time t0 to the actual time t. Such a mapping cannot be described by a local 9

formula (16). We stress, however, that for our construction we do not need to specify any coordination F , provided it is local. Once we know the metric (19) on M, we may rewrite the invariant Lagrangian density L of the field theory under consideration, just as in any other curvilinear system of coordinates. The Lagrangian obtained this way depends on the field ψ, its first derivatives, but also on the observer’s position, velocity and acceleration. Variation with respect to ψ produces field equations in the co-moving system (xα ). Due to the relativistic invariance of the theory, variation of the Lagrangian with respect to the observer’s position q should not produce independent equations but only conservation laws, implied already by the field equations. For our purposes we will keep, however, at the same footing the field degrees of freedom ψ and (at the moment, physically irrelevant) observer’s degrees of freedom q k . For such a Lagrangian theory, we perform a partial Legendre transformation, and pass to the Hamiltonian description of the field degrees of freedom, keeping the Lagrangian description of the “mechanical” degrees of freedom. For this purpose we define LH := L − Πψ˙ ,

(22)

where Π is the momentum canonically conjugate to ψ: Π :=

∂L . ∂ ψ˙

(23)

The function LH plays the role of a Hamiltonian (with negative sign) for the fields and a Lagrangian for the observer’s position q. It is an analog of the Routhian function in analytical mechanics. For a mechanical system with n degrees of freedom the Routhian function obtained from the Lagrangian by a partial Legendre transformation R = R(q 1 , q 2 , ..., q n ; q˙1 , ..., q˙l , pl+1 , ..., pn ) = L −

n X

pk q˙k

(24)

k=l+1

generates the following symplectic relation dR =

l X

(p˙k dq k + pk dq˙k ) +

k=1

n X

(−p˙k dq k + q˙k dpk ) .

(25)

k=l+1

This means that R is a Lagrangian in variables (q 1 , ..., q l ; q˙1 , ..., q˙l ) and a Hamiltonian (with negative sign) in variables (q l+1 , ..., q n ; pl+1 , ..., pn ). We can choose the lagrangian or the hamiltonian mode for any degree of freedom in a completely independent way. For l = 0, R becomes the complete Hamiltonian and for l = n it becomes the complete Lagrangian. In case of formula (22), the Lagrango-Hamiltonian LH generates the hamiltonian field evolution with respect to the accelerated frame, when the “mechanical degrees of freedom” q k are fixed. Due to (19), this evolution is a superposition of the following three transformations: 10

• time-translation in the direction of the local time-axis of the observer, • boost in the direction of the acceleration ak of the observer, • purely spatial O(3)-rotation ω m . It is, therefore, obvious that the numerical value of the generator LH of such an evolution is equal to ³ ´ √ LH = − 1 − v2 H + ak Rk − ω m Sm , (26) where H is the rest-frame field energy, Rk is the rest-frame static moment √ and Sm is the rest-frame angular momentum, all of them calculated on Σ. The factor 1 − v2 in front of the generator is necessary, because the time t = x0 , which we used to parameterize the observer’s trajectory, is not the proper time along ζ but the laboratory time. Now, it is easy to prove that the Euler-Lagrange equations obtained when varying LH with respect to the observer’s position q(t) are satisfied identically if the field equations are satisfied. The proof follows directly from the conservation laws of the total four-momentum P α and the total angular momentum Mαβ of the field, implied by Noether’s theorem. Indeed, the four-momentum conservation reads: ∇0 P α = P˙ α + Γα0β P β = 0 ,

(27)

where Γαβγ are the Christoffel symbols of the metric gαβ , calculated on the trajectory, i.e. at xk = 0. Putting P 0 = H and calculating Γ from (19), one immediately obtains the following “accelerated-frame version” of Noether conservation laws: √ (28) H˙ = − 1 − v2 ak Pk , ³ ´ √ ml 2 P˙ k = 1 − v −ak H − ² ωm Pl . (29) k

Putting Mk0 = Rk and Mij = ²ijk Sk , the angular momentum conservation ∇0 Mαβ = 0 may be rewritten in a similar same way: ³ ´ √ R˙ k = 1 − v2 Pk − ²kim ai S m − ²kil ω i Rl , (30) ³ ´ √ S˙ m = 1 − v2 ²mil ai Rl − ²mij ωi Sj . (31) The above conservation laws are implied by field equations only. It is a matter of simple calculations (see Section 5), that the Euler-Lagrange equations obtained by varying (26) with respect to the observer’s position q(t) are satisfied identically if equations (28) – (31) are satisfied. To perform such a calculation, an explicit formula for the mapping Fζ is ˙ In Section 5 necessary, which implies an explicit formula for ak and ω k in terms of (v, v). we use for this purpose formulae (20) and (21), corresponding to the embedding (17). The fact, that the Euler-Lagrange equations of the theory are not independent, is typical for a gauge theory. This property may be nicely described in the hamiltonian picture. Considering LH as the generator of the evolution of both the field degrees of freedom and the observer’s degrees of freedom, we may perform the Legendre transformation 11

also with respect to the latter, and find this way the complete Hamiltonian of the entire (observer + field) system (see again Section 5 for details). It may be proved, that q plays the role of a gauge parameter: momenta canonically conjugate to observer’s position are not independent but subject to constraints. Reducing the theory with respect to these constraints we end up with the “true” degrees of freedom, namely those describing the field. Fixing the trajectory plays the role of “gauge fixing” and the “evolution equations” of the observer are automatically satisfied if the field equations are satisfied. The formalism introduced in this Section cannot be a priori used for the description of the electromagnetic field, since the naive Legendre transformation (22) from the Lagrangian to the Hamiltonian picture does not lead to the correct local expression for the field energy (we obtain the “canonical Hamiltonian” which differs from the field energy by a complete divergence). However, we may take the electrodynamical Routhian (Lagrango - Hamiltonian) (26), where H, Rk and Sm are the conventional energy, static moment and the angular momentum of the electromagnetic field. They are defined as appropriate integrals of the components of the Maxwell energy-momentum tensor 1 T µν = f µλ fνλ − δνµ f κλ fκλ . 4

(32)

Of course, Maxwell equations derived from such an LH imply conservation laws (28) (31). Hence, variation of LH with respect to the observers position q(t) produces EulerLagrange equations which are automatically satisfied by virtue of the field equations. We will show in the sequel, that the renormalized version of the above Lagrangian (26) really implies the Electrodynamics of Moving Bodies. For this purpose no further justification of LH is necessary. However, in Section 6, we will construct a new Lagrangian for electrodynamics, which is directly related with LH via the simple Legendre transformation (22). This way, finally, the entire content of the present Section may be applied to electrodynamics.

4

Point particles and extended particles. Renormalization

In this Section we briefly sketch the renormalization procedure proposed in [1]. It enables us to extend the definition of “Lagrango-Hamiltonian” (26) to the case of electromagnetic field interacting with point-like particles. Given a solution of inhomogeneous Maxwell equations (5), we define the total energymomentum of the composed (particle + field) system by the following “already renormalized” formula: pλ (t) := muλ (t) + P

Z ³ Σ

´

T µλ − T(t) µλ nµ dΣ ,

(33)

where Σ is any hypersurface which intersects the trajectory at the point (t, q(t)), and T(t) µν is the energy-momentum tensor of the “uniformly moving particle”, i. e. of the Coulomb field, boosted in such a way that the position and the velocity of its singularity 12

match the velocity of our particle at (t, q(t)). By “P” we denote the principal value of the singular integral, defined by removing from Σ a sphere K(0, r) around the particle and then passing to the limit r → 0. We assume that Σ fulfills standard asymptotic conditions at infinity (this means that all the surface integrals at infinity, appearing in the energy-momentum conservation laws, vanish – see [7]). The parameter m denotes the dressed (i. e. already renormalized) mass of the particle. It was proved in [1] that the above integral is well defined. Moreover, it is invariant with respect to changes of Σ, provided the intersection point with the trajectory does not change. Hence, the total four-momentum of the composed (particle + field) system may be defined this way, at each point (t, q(t)) of the trajectory separately. Similarly, we define the total angular-momentum Mµν of the composed (field + particle) system as the sum composed of 1) an integral containing appropriate components of T and 2) an integral containing the difference T − T. The renormalization consists in replacing the first integral by the particle’s angular-momentum. In the present paper we assume the particle to be of scalar type, i.e. that its angular momentum vanishes (a generalization to the case of particles with non-vanishing internal angular-momentum is relatively straightforward and will be given in the next paper). The above definition is a result of the following physical idea, concerning the particle’s structure. Consider a general field theory describing the electromagnetic field interacting with a hypothetical multi-component matter field φ = (φK ). We assume that the dynamical equations of this “super theory” may be derived from a gauge-invariant variational principle δL =0 δA

(34)

δL =0, δφ

(35)

where A = (Aµ ) is the electromagnetic potential, i.e. fµν := ∂µ Aν − ∂ν Aµ . As an example one may consider the complex (charged) scalar field or the classical spinorial Dirac field, interacting with electromagnetism. For our purposes, however, no further assumptions about the geometric character of the field φ are necessary. Moreover, we assume Maxwell equations as a limiting case of the above field equations, corresponding to sufficiently weak electromagnetic fields and vanishing matter fields. We will suppose that the particles, whose interaction with the electromagnetic field we are going to analyze, are simply global solutions of the above field theory. Each solution of this type is characterized by a tiny “strong field region”, concentrated in the vicinity of a time-like trajectory ζ, which we may call approximate trajectory of an extended particle. Outside of the strong field region the matter fields vanish (or almost vanish in the sense, that the following approximation remains valid) and the electromagnetic field is sufficiently weak to be described by Maxwell equations. To be more precise, we imagine the “particle at rest” as a stable, static, solitonlike solution of our hypothetical “super theory”. The solution is characterized by two 13

parameters: its total charge e and its total energy m. We stress that m is an already renormalized mass, (or dressed mass), including the energy of the field surrounding the particle. Within this framework questions like “how big the bare mass of the particle is and which part of the mass is provided by the purely electromagnetic energy?” are meaningless. In the strong field region (i. e. inside the particle) the energy density may be highly non-linear and there is probably no way to divide it consistently into such two components. Due to relativistic invariance of the theory, there is a 6 parameter family of the “uniformly moving particle” solutions obtained from our soliton via Poincar´e transformations. An arbitrarily moving particle is understood as a “perturbed soliton”. This means that it is again an exact solution of the same “super theory”, with its strong-field-region concentrated in the vicinity of a time-like world line ζ, which is no longer a straight line, as it was for “uniformly moving particles”. Let us calculate the total four-momentum pλ of such a solution. For this purpose we choose any Σ and integrate the total energy-momentum tensor T µλ of our “super theory” over Σ. It is usefull to decompose T µλ : T µλ = Tµλ + (T µλ − Tµλ ) .

(36)

Here, by Tµλ we denote the total energy-momentum of the “super theory”, corresponding to the “uniformly moving particle” solution, which matches on Σ the position and the velocity of our particle. Integrating the first term we obtain obviously muµ , where uµ is the four-velocity of our particle on Σ. Consequently, pλ may be decomposed as a sum of two terms: 1) the total four-momentum of the uniformly moving particle (and its surrounding field) and 2) the difference between the two, given by the integral of the last term in (36) over Σ. The exact value of this integral cannot be found without knowing exactly the internal structure of both solutions. But it is easy to find a good approximation, which is based on the following observation: the contribution of the interior of the particles is small because of the stability of the soliton. Indeed, stability means that the soliton is a local minimum of energy in the space of the field initial data. Hence, the variation of T µλ inside the particle (corresponding to the perturbation of the soliton) is small. This means that the purely electromagnetic contribution, corresponding to the surrounding Maxwell fields, approximates with a good accuracy the above quantity. We take the above observation, which is true for extended particles, as a starting point for our definition of the total four-momentum in the case of point particles. We decompose energy-momentum tensor of the field surrounding a point particle as a sum of two terms: 1) the energy-momentum tensor corresponding to the uniformly moving particle and 2) the difference between the two. Our renormalization procedure consists, therefore, in replacing the integral of the first (non-integrable) term by the dressed quantity muµ whereas, miraculously, the second term is already integrable. This way we obtain the formula (33). Let us calculate the renormalized total four-momentum and the total angular-momentum components in the particle’s rest frame. Decomposing the electric induction field on 14

the rest-frame surface Σt into the sum D = D0 + D

(37)

of the Coulomb field er D0 = . 4πr3

(38)

and the remaining part D, we obtain the following formulae for the renormalized restframe quantities: 1Z 1 Z 2 H = m + P (D2 + B2 − D20 ) d3 x = m + (D + B2 ) d3 x , Σ Z 2 Z Z2 Σ Pl = P Rk = Sm =

Σ

1 P Z2 Σ

(D × B)l d3 x =

Z Σ

Σ

(D × B)l d3 x + 1 Z2

xk (D2 + B2 ) d3 x =

²mkl xk (D × B)l d3 x =

Σ

Z

Σ

(D0 × B)l d3 x ,

2

Σ

xk (D + B2 ) d3 x +

²mkl xk (D × B)l d3 x .

(39) (40)

Z Σ

xk DD0 d3 x ,

(41) (42)

Other contributions of the Coulomb field were killed by the principal value operator P. Let us come back to our “super theory”. For a “moving particle solution” choose the observer, which moves along the approximate particle trajectory and take the corresponding co-moving Lagrango-Hamiltonian (26). Its value may be well approximated if we replace the exact value of the quantities H, Rk and Sm by the above “renormalized” quantities (39), (41) and (42), containing only the external Maxwell field. We know that the variation with respect to the observer’s trajectory vanishes automatically when the complete “super theory” is taken into account. But this is no longer true if we approximate the extended particles by point-like particles, surrounded by the Maxwell field. We conclude that only those solutions of (5) may approximate the dynamics of the true extended particles, governed by the field equations of the “super theory”, for which the variation of the above renormalized Lagrangian LH with respect to the particle’s trajectory does vanish. The reason why the variational principle obtained this way gives now a non-trivial equation is that the conservation laws (28) - (31) are not necessarily satisfied for the solutions of the inhomogeneous Maxwell equations. Indeed, they were implied by the Noether invariance of an autonomous, Lagrangian field theory. Noether theorem does not apply to the inhomogeneous Maxwell theory with given sources. But, after all, the situation is not so bad. We prove in Appendix D that, for any regular solution of (5), the renormalized quantities (39) - (42) satisfy necessarily three among the conservation laws, namely (28), (30) and (31). This implies (see Section 5) that Euler-Lagrange equations obtained from Lagrangian LH are equivalent to the remaining momentum conservation law (29), which in turn is equivalent to the fundamental equation (13) of the electrodynamics of moving particles. This proves that the renormalized LH is really a correct Lagrangian for Electrodynamics of Moving Particles. 15

5

Equations of motion from the variational principle

In this Section we will prove explicitly the equivalence between the Euler-Lagrange equations derived from LH and the momentum conservation laws. Let us take LH given by (26), with H, Rk and Sm given by formulae (39), (41) and (42), as the Lagrango-Hamiltonian of our theory of point-like particles interacting with the Maxwell field. Being the Hamiltonian (with opposite sign) for the fields, it obviously generates the inhomogeneous Maxwell equations (5) (written in the particle’s co-moving frame). The remaining equations of the theory are obtained from the variation with respect to the particle trajectory. Observe that LH is a 2-nd order Lagrangian in the particle variables: ˙ fields) . LH = LH (q, v, v; Varying LH with respect to q we obtain the following Euler-Lagrange equations (see Appendix A): ∂LH p˙k = , (43) ∂q k where the momentum pk canonically conjugate to the particle’s position q k is defined as: ∂LH d pk := − k ∂v dt

Ã

∂LH ∂ v˙ k

!

.

(44)

Using formula (26) we obtain: √ √ ∂LH vk ∂( 1 − v2 al ) ∂( 1 − v2 ω m ) =√ H− Rl + Sm . ∂v k ∂v k ∂v k 1 − v2

(45)

The momentum canonically conjugate to the velocity v k equals (see Appendix A): à ! l m √ ∂a ∂LH ∂ω πk := = − 1 − v2 Rl − Sm . (46) ∂ v˙ k ∂ v˙ k ∂ v˙ k To calculate the time derivative of πk we use conservation laws (30) and (31) which are proved in Appendix D. This way we obtain the following formula for pk : ³ ´ vk l 2 l l m H + + δ ϕ(v )v v (47) pk = √ k Pl + A k Rl + B k Sm , k 2 1−v where Al k and B mk are given by the following expressions: √ à ! l √ d ∂a ∂( 1 − v2 al ) 1 − v2 k − − Al k = dt ∂ v˙ ∂v k ! à m ∂ω m 2 il ∂a ωi + ai , − (1 − v )²m ∂ v˙ k ∂ v˙ k √ à ! d √ ∂( 1 − v2 ω m ) ∂ω m m 2 Bk = − + + 1−v dt ∂ v˙ k ∂v k à ! ∂al ∂ω l 2 im + (1 − v )²l (48) ai + k ωi . ∂ v˙ k ∂ v˙ 16

The quantities ai and ω l have to be expressed in terms of v and v˙ via formulae (20) and (21). Using the following properties of the function ϕ(τ ) 2ϕ(τ ) − (1 − τ )−1 + τ ϕ2 (τ ) = 0 , 2ϕ0 (τ ) − (1 − τ )−1 ϕ(τ ) − ϕ2 (τ ) = 0 ,

(49)

and the identity v i (²ikl vm + ²ilm vk + ²imk vl ) = v2 ²klm ,

(50)

one easily shows that Al k ≡ B l k ≡ 0 . Thus, we finally obtain the following formula for the momentum pk canonically conjugate to the particle’s position: pk = √

³ ´ vk l 2 l H + δ + ϕ(v )v v k Pl , k 1 − v2

(51)

which we immediately recognize as the space-like component of the total four-momentum (H, Pk ), Lorentz-transformed from the particle’s co-moving frame to the laboratory frame. Since the Lagrangian LH does not depend explicitly on the particle position q we conclude that the Euler-Lagrange equation (43) is equivalent to the conservation law of total momentum in the laboratory frame: d pk = 0 . dt

(52)

To calculate the time derivative of pk Ã

p˙k

!

´ v d ³ d 2 l √ k + P ϕ(v )v v = H l k + dt dt 1 − v2 ³ ´ vk ˙ + δ lk + ϕ(v2 )v l vk P˙ l , + √ H 1 − v2

(53)

we need to know time derivatives of H and Pl . In Appendix D we show that the rest-frame energy H fulfills the conservation law (28). However, as we mentioned in the previous Section, the conservation law (29) for the rest-frame momentum Pl is not necessary satisfied for the solutions of inhomogeneous Maxwell equations. Let us denote by Xl the deviation from this law. This means that we define Xl by the formula: ³ ´ √ P˙ l = 1 − v2 −al H − ²l mk ωm Pk + Xl . (54) The following identities are easy to prove: d dt

Ã

v √ k 1 − v2

!

=

√

´

³

1 − v2 δ lk + ϕ(v2 )v l vk al ,

(55)

and ´ ´ ³ √ d ³ lj 2 m + ϕ(v )v v ϕ(v2 )v l vk = al vk − 1 − v2 δ m k ² k dt

17

m ωj

.

(56)

Inserting them, together with (28) and (54), into (53) we finally obtain: ³ ´ √ p˙k = 1 − v2 δ lk + ϕ(v2 )v l vk Xl .

(57)

It is easy√to see that the matrix (δ mk + ϕ(v2 )v m vk ) is non-singular, its inverse being equal to (δ kl − 1 − v2 ϕ(v2 )v k vl ). Hence, Euler-Lagrange equations p˙ k = 0, derived from the Lagrangian LH , are equivalent to Xl = 0, i. e. to the conservation law (29). This proves that the electrodynamics of moving particles may indeed be derived from the Lagrangian LH . It is worthwhile to notice that (29) implies also the conservation of the remaining component p0 of the total four-momentum of the system. Observe, that p0 is numerically equal to the complete Hamiltonian of the composed system, obtained from the complete Legendre transformation (formula (46) is used to calculate πk ): √ H := pk q˙k + πk v˙ k − LH = pk v k + 1 − v2 H ³ ´ 1 l H + v P . (58) = √ l 1 − v2 Indeed, the last expression is equal to the p0 – component of the total four-momentum (H, Pk ), Lorentz-transformed to the laboratory frame. We have, therefore, H = p0 . Using the same methods as before we easily obtain p˙0 = v l Xl

(59)

which ends the proof. We conclude, that in the case, when conservation laws (28), (30) and (31) are satisfied, the Euler-Lagrange equation derived from the Lagrangian (26) is equivalent to the remaining conservation law (29). In Appendix D we show that for any regular solution of (5) the non-conservation vector Xk equals Xk = mak − eβk ,

(60)

where βk is the dipole part of the function β defined by expansion (11), i. e. xk DP(β) = βk , r

(61)

Due to relation (12), vanishing of Xk is therefore equivalent to the fundamental equation (13) of the electrodynamics of moving particles imposed on the solutions of (5). It implies the total four-momentum conservation (8). In particular, it guarantees that the total Hamiltonian H = p0 remains constant during the evolution.

6

Variational principle based on fluxes

In this Section we finally derive a quasi-local Lagrangian for electrodynamics, related to field energy via the Legendre transformation (22). 18

The discrepancy between the “canonical” and the “symmetric” energy-momentum tensors was often interpreted as an argument against defining the field Hamiltonian via the Legendre transformation. Such a conclusion is false. It was shown (see [8] and [9]) that this problem (also the problems of defining gravitational energy in General Relativity), is related to the fact, that there is no unique way to represent the field evolution as an (infinite dimensional) Hamiltonian system. Each such representation is based on a specific choice of controlling the boundary value of the field, and corresponds to a specific choice of the Hamiltonian. This non-uniqueness is implied by the non-uniqueness of the evolution of the portion of the field, contained in a finite laboratory V . Indeed, the evolution is not unique because external devices may influence the field through the open windows of our laboratory. To choose the Hamiltonian uniquely, we have to insulate the laboratory or, at least, to specify the influence of the external world on it. One may easily imagine an unsuccessful insulation, which does not prevent the external field from penetrating the laboratory. From our point of view, an insulation is sufficient if it keeps under control a complete set of field data on the boundary ∂V in such a way, that the field evolution becomes mathematically unique. For relatively simple theories (e.g. scalar field theory) the Dirichlet problem may be treated as a privileged one among all possible mixed (initial value + boundary value) problems which are well posed. This means that there is a natural way to insulate the laboratory V adiabatically from the external world. But already in electrodynamics (and even more in General Relativity) any attempt to define the field Hamiltonian leads immediately to the question: how do we really define our Hamiltonian system? It was shown in [9] that the “canonical energy” obtained from the Maxwell Lagrangian via the ordinary Legendre transformation is a legitimate field Hamiltonian, describing the evolution of the field closed in a metal shell in such a way, that the potential A0 on the shell is controlled (e. g. the shell is grounded). On the contrary, the “symmetric” energy 1 (D2 + B 2 ) is related to the control of the electric and the magnetic flux on the boundary. 2 Below, we construct a Lagrangian which is directly related to the latter energy. This way, starting from the variational picture, we obtain the Hamiltonian picture by an ordinary Legendre transformation. Field equations of any (linear or nonlinear) electrodynamics may be written as follows: δL(Aν , Aνµ ) = ∂µ (F µν δAν ) = (∂µ F µν )δAν + F µν δAνµ ,

(62)

where Aνµ := ∂µ Aν and L is the Lagrangian density of the theory. The above formula (see [5]) is a convenient way to write down the Euler-Lagrange equations ∂µ F νµ =

∂L , ∂Aν

(63)

together with the relation between the electromagnetic field fµν = Aνµ − Aµν and the electromagnetic induction density F νµ describing the momenta canonically conjugate to the potential: F νµ =

∂L ∂L = −2 . ∂Aνµ ∂fνµ

(64) 19

For the linear Maxwell theory, the Lagrangian density is given by the standard formula √ (2) and relation (64) reduces to F µν := −gg µα g νβ fαβ . We will integrate both sides of (62) over a 3–dimensional volume V belonging to the hyperplane Σt and consisting of the exterior of the sphere S(r0 ): Z

δ

V

Z

L =

Z

V

∂0 (F k0 δAk ) +

∂V

F ν⊥ δAν

(65)

(by ⊥ we denote the component orthogonal to the boundary). To describe the boundary term it is convenient to use spherical coordinates (ξ a ), a = 1, 2, 3; adapted to ∂V . We choose ξ 3 = r as the radial coordinate and (ξ A ), A = 1, 2; as angular coordinates: ξ 1 = Θ, ξ 2 = ϕ. The Euclidean metric gab is diagonal: g11 = r2 ,

g33 = 1 ,

g22 = r2 sin2 Θ ,

(66)

and the volume element λ = (det gab )1/2 is equal to r2 sin Θ. With this notation we have: δ

V

Z

Z

Z

L =

V

∂0 (F

B0

30

δAB + F δA3 ) +

∂V

(F B3 δAB + F 03 δA0 ) .

(67)

On each sphere S(r) = {r = const} the 2-dimensional covector field AB splits into a sum of the “longitudinal” and the “transversal” part: AB = u,B +²B C v,C ,

(68)

where the coma denotes partial differentiation and ²AB is a sqew-symmetric tensor, such that λ²AB is equal to the Levi-Civita tensor-density (i. e. λ²12 = −λ²21 = 1). The functions u and v are uniquely given by the field AB up to additive constants on each sphere separately. Inserting this decomposition into (67) and integrating by parts we obtain: Z

δ

V

Z

L = +

ZV

∂0 (−F B0 ,B δu + F 30 δA3 − F B0 ,C ²BC δv) +

∂V

(−F B3 ,B δu + F 03 δA0 − F B3 ,C ²BC δv) .

(69)

Using identities ∂B F B0 + ∂3 F 30 = 0 and ∂B F B3 + ∂0 F 03 = 0, implied by equations (63) in the case of a gauge-invariant theory, and integrating again by parts we finally obtain: Z

δ

V

Z

L = +

h

i

∂0 F 30 δ(A3 − u,3 ) − (F B0||C ²BC )δv +

ZV h ∂V

i

F 03 δ(A0 − u,0 ) − (F B3||C ²BC )δv .

(70)

Here, by “||”we denote the 2-dimensional covariant derivative on each sphere S(r). The quantities (A0 − u,0 ) and (A3 − u,3 ) are “almost” gauge invariant: only their monopole part (mean-value) on each sphere may be affected if we change the additive constant in the definition of u (the choice of an additive constant in the definition of v is irrelevant, because it is always multiplied by quantities which vanish when integrated over a sphere). 20

The sum of the volume and surface integrals in (70) is however gauge invariant. It may be easily checked that 4(A3 − u,3 ) = r2 B C||D ²CD ,

(71)

where 4 denotes the 2-dimensional Laplace-Beltrami operator on S(r) multiplied by r2 (the operator 4 does not depend on r and is equal to the Laplace-Beltrami operator on the unit sphere S(1)). The operator 4 is invertible on the space of monopole–free functions (functions with vanishing mean value on each S(r)). This functional space will play an important role in further considerations and all the dynamical quantities of the theory will belong to this space. To fix both terms in (70) uniquely we choose u in such a way that the mean value of (A3 − u,3 ) vanishes on each sphere. Hence, with the above choice of the additive constants the quantity A3 − u,3 becomes gauge invariant: A3 − u,3 = r2 4−1 (B C||D ²CD ) .

(72)

Let us observe that the function v is also gauge invariant (up to an additive constant, which does not play any role and may also be chosen in such a way that its mean value vanishes on each sphere). Indeed, we have: B 3 = (curl A)3 = AC||D ²DC = −r−2 4v .

(73)

Due to the Maxwell equation div B = 0, the function B 3 is monopole–free and the Laplasian 4 may again be inverted: v = −r2 4−1 B 3 .

(74)

The formula (70) could be also obtained directly from (67) by imposing the following gauge conditions: Z

AB||B = 0 ,

(75)

λA3 = 0 .

(76)

S(r)

The above condition does not fix the gauge uniquely: we still may add to Aµ the gradient of a function of time f = f (t). This residual gauge changes only the monopole part of A0 , but both the volume and the surface integrals in (70) remain invariant with respect to such a transformation. Assuming the above gauge, we have u ≡ 0 and AB = ²B C v,C . To simplify the notation we will, therefore, replace our invariants (A0 − u,0 ) and (A3 − u,3 ) by the values of A0 and A3 , calculated in this particular gauge. Formula (70) is analogous to the lagrangian formula in classical mechanics dL(q, q) ˙ =

´ d ³ pk dq k = p˙k dq k + pk dq˙k . dt

(77)

In the case of electrodynamics, we have a system with infinitely many degrees of freedom described by four functions: F 30 , A3 , F B0||C ²BC and v. Two of them will play the role of 21

field configurations and the remaining two will be the conjugate momenta. But there is also a boundary term in (70), typical for field theory. Killing this term by an appropriate choice of boundary conditions is necessary for transforming the field theory into an (infinite dimensional) dynamical system (see [5], [8]). From this point of view, the quantity v (or, equivalently B 3 ) is a good candidate for the field configuration, since controlling it at the boundary will kill the term δv in the boundary integral. On the contrary, δA0 can not be killed by any simple boundary condition imposed on A3 . We conclude, that it is rather F 03 = λD3 than A3 , which has to be chosen as another field configuration. Hence, we perform the Legendre transformation in formula (70), both in the volume F 30 δA3 = −F 03 δA3 = −δ(F 03 A3 ) + A3 δF 03 ,

(78)

and on the boundary ∂V : F 03 δA0 = δ(F 03 A0 ) − A0 δF 03 .

(79)

This way, using (72) and (74), we obtain from (70) the following result: δ

Z h V

= +

i

L + ∂0 (λD3 A3 ) − ∂3 (λD3 A0 ) =

Z

h

λr2 ∂0 4−1 (B A||B ²AB )δD3 − 4−1 (DA||B ²AB )δB 3

ZV h ∂V

i

−λA0 δD3 − r2 4−1 (F 3A||B ²AB )δB 3 .

i

(80)

In the above formula, the function v has been replaced by the right-hand side of (74). Moreover, the operator 4−1 , which is self-adjoint on the functional space of monopolefree functions on a sphere, was moved from δB 3 to functions which multiply it under the integral sign. We see that (D3 , B 3 ) play the role of field configurations, whereas the remaining functions (B A||B ²AB , DA||B ²AB ) describe the conjugate momenta. Controlling the configurations at the boundary we kill the surface integral over ∂V and obtain this way an infinite dimensional dynamical system describing the field evolution. There is, however, a problem with such a control, because the electric induction D3 cannot be controlled freely on the boundary. The reason is that the total electric flux through both components of ∂V (i. e. through S(r0 ) and through the sphere at infinity) must be the same: Z

Z S(r0 )

F

03

=

S(r∞ )

F 03 = e ,

(81)

where e is the electric charge contained in S(r0 ). Hence, we have to separate the monopolefree (“radiative”) part of D3 (which can be independently controlled on both ends of V ) from the information about the electric charge. For this purpose we split the electric induction D3 into e 3 +D , (82) D3 = 2 4πr 22

3

where D is again a monopole-free function. It follows from (81) that the monopole part of D3 (equal to e/4πr2 ) is nondynamical and drops out from the volume integral of (80) 3 because it is multiplied by a monopole-free function B A||B ²AB . The remaining part D (which does not carry any information about the charge e), together with B 3 , can be taken as the true, unconstrained degrees of freedom of the electromagnetic field. In the same way we split the scalar potential A0 : A0 = φ + A0 ,

(83)

where φ(r) is the mean value of A0 on the sphere S(r) (monopole part) and A0 is a monopole–free function (“radiative” part of A0 ). Now, the boundary term A0 δD3 in (80) reads Z ∂V

λA0 δD3 =

Z 1 Z 3 λA0 δD . λr−2 φδe + 4π ∂V ∂V

(84)

Let us consider the first term in the r.h.s. of (84). In Appendix C we show that the monopole function φ(r) satisfies the following equation (cf. (C.19)) ∂3 φ(r) =

√

µ

1 − v2

¶

e ˜ + ∂3 φ(r) 4πr2

,

(85)

˜ where φ(r) is regular in r = 0. The first term in the above √ formula corresponds to the Coulomb potential of the particle in its rest frame (factor 1 − v2 corresponds to the fact that we use laboratory time instead of proper time). Therefore, we have à ! √ 1 Z e2 −2 2 ˜ ˜ 0 )δe . λr φδe = δ 1−v + φ(∞)δe − φ(r 4π V 32π 2 r0

(86)

Finally, we perform the Legendre transformation between φ˜ and the monopole part of D3 at infinity. Hence, we control the total charge contained in S(r0 ) and the monopole function φ˜ at infinity. Since the latter does not contain any physical information and is ˜ used only to fix the residual gauge, we may use the simplest possible choice: φ(∞) ≡ 0. We denote Ψ1 = rB 3 ,

(87)

3

Ψ2 = rD , χ1 = −r4−1 (DA||B ²AB ) , χ2 = r4−1 (B A||B ²AB ) .

(88) (89) (90)

Together with the value e of the electric charge contained in S(r0 ), they contain the entire (gauge invariant) information about the electromagnetic field. The first two quantities play the role of field configurations. The corresponding Lagrangian has to be considered as a function of these and their derivatives. The fields χ will appear as corresponding momenta. They are equal to derivatives of the Lagrangian with respect to the time derivatives of the fields. 23

The response of the system to the control of the boundary values of the configurations is given by the boundary momenta: 1 χr1 = −r4−1 ( F 3A||B ²AB ) , λ χr2 = −r−1 A0 ,

(91) (92)

equal to the derivatives of the Lagrangian with respect to the radial derivatives of the fields. Finally, formula (80) reads: Z

δ

Z V

L=

Z V

∂0 (λχA δΨA ) +

∂V

˜ ˜ 0 )δe λχrA δΨA + eδ φ(∞) + φ(r

(93)

where the new Lagrangian L equals Z µ

Z V

¶ 3

L=

V

3

L + ∂0 (λD A3 ) − ∂3 (λD A0 ) +

√

e2 . 32π 2 r0

1 − v2

(94)

We stress that that L remains finite, even if we put r0 = 0. Indeed, Maxwell Lagrangian L may be decomposed into the regular part ³ 2 ´ 1 Lreg = λN D − B2 + λN DD0 2

(95)

and the remaining part 21 λN (D0 )2 . This way we have Z V

Z

L=

V

Lreg +

√

1 − v2

e2 . 32π 2 r0

(96)

Finally, L is given by µ reg

L=L

A

+ ∂0 (λχA Ψ ) +

∂3 (λχrA ΨA )

− ∂3

¶

e ˜ φ λ 4πr2

.

(97)

Formula (93) defines a generalized lagrangian system describing the evolution of the electromagnetic field, when the dynamical variables ΨA and the electric charge e are controlled at the boundary ∂V . ¿From the lagrangian relation (93) we immediately obtain the hamiltonian one performing the Legendre transformation: Z

− δH = where H :=

Z V

˙ A δχA ) + λ(χ˙ A δΨ − Ψ

Z

A

V

∂V

˙ A − L) . (λχA Ψ

˜ ˜ 0 )δe , + φ(r λχ3A δΨA + eδ φ(∞)

(98)

(99)

It is easy to check (see [8]) that H is equal to the amount of electromagnetic energy contained in V . 24

7

The Lagrangian in the co-moving frame

We apply the above construction to the case of renormalized electrodynamics formulated in the particle’s co-moving frame. Hence, the left-hand side of (99) has to be replaced by LH , expressed in terms of the renormalized generators H, Rk and S m . The Lagrangian is given by the inverse Legendre transformation Z

Z

L=

Σ

Σ

˙ A + LH . λχA Ψ

(100)

To simplify the notation we define a complex-valued configuration quantity Ψ := Ψ1 + iΨ2

(101)

and a complex-valued momentum χ := χ2 − iχ1 = −i(χ1 + iχ2 ) .

(102)

Observe, that Z Σ

˙ A = Im λχA Ψ

Z Σ

˙ , λχ∗ Ψ

(103)

where “Im” denotes the imaginary part and “*” denotes complex conjugation. To express L given by (100) in terms of the field configurations Ψ and its derivatives we rewrite Maxwell equations in the co-moving frame (see the Appendix B for the proof): ½ µ ¶ ¾ √ k −1 m 2 ˙ 1 − v 4χ + a rKk χ + i4 Vk (rΨ),3 + ω Vm Ψ , iΨ = (104) √

µ

½

h 1 1 −1 k 1 −1 −1 Ψ + 4 (rΨ) + a 4 ∂ iχ˙ = − 1 − ,33 3 rKk 4 (rΨ),3 − r2 r r ¶ ¾ i 1 e m − ir2 Vk χ + 2 Ψxk + i x − ω V χ k m r 4πr3

v2

(105)

(operators Kk and Vk are defined by (B.22) and (B.23); together with 4, they are selfadjoint operators on the space C of monopole-free functions on the unit sphere S 2 ). Now, we rewrite the quantities H, Rk and S m in terms of the field configuration Ψ and the corresponding momentum χ. Using the same techniques as in Appendix B we obtain: ½

¾

1Z 1 1 λ 2 Ψ∗ Ψ − 2 (rΨ∗ ),3 4−1 (rΨ),3 − χ∗ 4χ , 2 Σ r r ½ Z 1 xk ∗ 1 = Re λ 2 Ψ Ψ − (rΨ∗ ),3 4−1 Kk 4−1 (rΨ),3 − rχ∗ Kk χ− 2 r r Σ ¾ e ∗ −1 − 2χ i4 Vk (rΨ),3 − i xk Ψ , 2πr3 Z

H = m+ Rk

Sm =

Σ

Re (λΨ∗ Vm χ) + ∂3 (bm ) ,

(106)

(107) (108)

where the boundary term in the last equation reads µ

bm = λΨ2 Km 4−1 −

¶

xm e Ψ1 − λ xm Ψ1 . r 4πr2 25

(109)

Inserting (104) into (100) and using (106)–(108) we derive the following formula for the Lagrangian of the electrodynamics of moving particles: e := L

Z Σ

¶ Z µ √ √ L = −m 1 − v2 + 1 − v2 L0 + ak Lk + eak lk + ∂3 (ω m bm ) , Σ

(110)

where ½

L0 =

¾

λ 1 1 − 2 Ψ∗ Ψ + 2 (rΨ∗ ),3 4−1 (rΨ),3 − χ∗ 4χ 2 r r

λ Lk = 2 lk = −

½

,

(111) ¾

xk 1 − 2 Ψ∗ Ψ + (rΨ∗ ),3 4−1 Kk 4−1 (rΨ),3 − rχ∗ Kk χ r r

,

λ xk Ψ2 . 3 4πr

(112)

(113)

Let us notice that due to the asymptotic behaviour of the field Ψ, the boundary term in (110) vanishes. Nevertheless, we keep this term to secure the consistency of the theory. This is typical situation in the field theory. Functionals with and without this term are numerically the same but functionally they are different. To complete the inverse Legendre transformation (100) one has to express the mo˙ From (104) we obtain immediately mentum χ in terms of the “velocity” Ψ. " k

−1

χ = (4 + ra Kk )

#

i ˙ − ak iVk 4−1 (rΨ),3 − ω m Vm Ψ . √ Ψ 2 1−v

(114)

e may be written in a similar form as L Observe that L total defined in (1): e =L e e L particle + LM axwell + Lint + boundary term ,

(115)

where e L

M axwell

¶ Z µ √ k 2 := 1 − v L0 + a Lk , Σ

and the interaction term Z Z √ k e 2 Lint := 1 − v ea lk = − N DD0 d3 x , Σ

Σ

(116)

(117)

e but contrary to (1) the interaction term L int is finite and, as we shall show in the next paper, the Hamiltonian for the electrodynamics of moving particles is always well defined. This way we have constructed a consistent Lagrangian structure for our theory. Now, e with respect to both fields and particles is well defined and does not lead to variation of L e with respect to field configuration Ψ we reproduce Maxwell any contradiction. Varying L equations (105) and (114). Variation with respect to the particle trajectory gives the fundamental equation (13).

26

8

Particle in an external potential

˙ t). Suppose now that the particle moves in an external (generalized) potential U = U (q, q, Then the Lagrange function is given by: e = −m L

√

1−

v2

+

√

Z µ

1−

v2

¶ k

k

m

L0 + a Lk + ea lk + ∂3 (ω bm ) − U .

Σ

(118)

e with respect to the particle trajectory ζ we obtain the following “equations of Varying L motion”: 2

DP(mα + e β) = − √

³ ´ √ e xi k 2 k 2 δ i − 1 − v ϕ(v )v vi Qk , r 1 − v2

(119)

where Qi = −

d ∂U ∂U + i ∂q dt ∂ q˙i

(120)

is a vector of the generalized force in the laboratory frame. Formula (119) is obviously equivalent to the laboratory-frame equation d pi (t) = Qi . dt

(121)

The influence of the external potential is manifested in the non-homogeneous boundary condition (119). As an example consider the particle interacting with an external electromagnetic field ext fµν . The generalized potential is given by: ˙ t) = eAext ˙ ext (q, t) , U (q, q, 0 (q, t) − eqA

(122)

where Aext and Aext stand for the four-potential of the external field in the laboratory 0 frame. The generalized force (120) in terms of the laboratory-frame components Ei and Bi of the external field now reads: ³

´

Qi = e Ei (q, t) + ²ijk v j B k (q, t) .

(123)

In the next paper we show that also in this case Maxwell equations for the radiation field, together with the non-homogeneous boundary condition (119), define an infinite-dimensional hamiltonian system. This means that initial data (Ψ, χ; q, v) for the radiation field and for the particle uniquely determine the entire history of the system if the external potential is given.

27

Appendixes A

Hamiltonian structure for a 2-nd order Lagrangian theory

Consider a theory described by the 2-nd order lagrangian L = L(q, q, ˙ q¨). Introducing an auxiliary variable v = q˙ we can treat our theory as a 1-st order one with a lagrangian constraint φ := q˙ − v = 0 on the space of lagrangian variables (q, q, ˙ v, v). ˙ Dynamics is generated by the following relation: d (p dq + π dv) , (A.1) dt where µ is a Lagrange multiplier corresponding to the constraint φ = 0 and p, π are momenta canonically conjugated to q and v respectively. From (A.1) we immediately obtain: ∂L p = µ, π= , ∂ v˙ ∂L ∂L p˙ = , π˙ = − p. (A.2) ∂q ∂v d (L(q, v, v) ˙ + µ(q˙ − v)) =

From the last equation we get the formula for p ∂L ∂L d p= − π˙ = − ∂v ∂v dt

Ã

∂L ∂ v˙

!

,

(A.3)

and, consequently, d p˙ = dt

Ã

!

∂L d2 − 2 ∂v dt

Ã

∂L ∂ v˙

!

.

(A.4)

It is equivalent to the Euler-Lagrange equation: δL d2 := 2 δq dt

Ã

!

∂L d − ∂ v˙ dt

Ã

!

∂L ∂L + = 0. ∂v ∂q

(A.5)

The hamiltonian description is obtained from the Legendre transformation applied to (A.1): − dH = p˙ dq − q˙ dp + π˙ dv − v˙ dπ ,

(A.6)

where H(q, p, v, π) = p v + π v˙ − L(q, v, v). ˙ In this formula we have to insert v˙ = v(q, ˙ v, π), ∂L calculated from equation π = ∂ v˙ . Let us observe that H is linear with respect to the momentum p. This is a characteristic feature of the 2-nd order theory. Euler-Lagrange equation (A.5) is of 4-th order. The corresponding 4 hamiltonian equations have, therefore, to describe the evolution of q and its derivatives up to third order. Due to Hamiltonian equations implied by relation (A.6), the information about succesive derivatives of q is carried by (v, π, p): 28

• v describes q˙ q˙ =

∂H ≡v ∂p

(A.7)

hence, the constraint φ = 0 is reproduced due to linearity of H with respect to p, • π contains information about q¨: v˙ =

∂H , ∂π

(A.8) ...

• p contains information about q π˙ = −

∂H ∂L = − p, ∂v ∂v

(A.9)

• the true dynamical equation equals p˙ = −

B

∂H ∂L = . ∂q ∂q

(A.10)

Maxwell equations in the co-moving frame

Formally, the Maxwell equations in the co-moving frame look identically as in Lorentzian coordinates: ∂µ F νµ = 0 ,

(B.1)

but the relation between the tensor f and the tensor density F: F νµ =

q

− det g g να g µβ fαβ ,

(B.2)

is given by the non-trivial metric tensor gµν on M. The components of the tensor gµν are given by equations (19). Using them and formula (B.2), we obtain the following expressions for the components of F µν : F 0j = Dj , F mn =

√

(B.3) i

h

1 − v2 (Dm ²nk l − Dn ²mkl )ωk xl + ²mnk (1 + ai xi )B k .

(B.4)

Therefore, the Maxwell equation ∂0 F 0n + ∂m F mn = 0 implies: D˙ n =

√

1 − v2

i ∂ h mk n i k m l mn nk . (1 + a x )B D )ω x − ² D − ² (² i k k l l ∂xm

29

(B.5)

Changing D to B and B to −D we obtain the remaining equation B˙ n =

√

1 − v2

i ∂ h mk n nk m l mn i k (² B − ² B )ω x + ² (1 + a x )D . k i l l k ∂xm

(B.6)

√ The factor 1 − v2 corresponds to the fact, that the dot means the derivative with respect to the laboratory time t (which we have used to parameterize the trajectory) and not with respect to the proper time τ on ζ. Defining the complex vector field F := B + iD ,

(B.7)

we may rewrite equations (B.5) and (B.6) in a more compact way F˙ n =

√

1 − v2

i ∂ h mk n i k nk m l mn F )ω x − i² (1 + a x )F . F − ² (² i k k l l ∂xm

(B.8)

Now, we rewrite Maxwell equations (B.8) in terms of unconstrained degrees of freedom (by F we denote, as usual, the monopole-free part of F): 3

Ψ = rF , ³ ´ χ = r4−1 F A||B ²AB .

(B.9)

The time derivative ψ˙ of any scalar quantity ψ may be decomposed as the sum ψ˙ =

√

⊥

1 − v 2 ψ + N k ∂k ψ ,

(B.10)

⊥

where by ψ we denote the time derivative of ψ with respect to the proper time τ along ζ in the Fermi-propagated frame. This frame is characterized by vanishing of the shift vector ˜ k = 0 (actual shift N k is given by (19)). The lapse function in the Fermi-propagated N ˜ = 1 + ak xk . Hence, we will first rewrite the Maxwell equations in the frame equals N ˙ and χ. Fermi-propagated frame and then, using (B.10), we will calculate Ψ ˙ Equation (B.8) applied to the Fermi-propagated frame gives us ³

⊥

˜ ² lm Fm F k = i∂l N k

´

.

(B.11)

We will use the spherical coordinates (ξ a ) = (ξ A , ξ 3 = r) introduced in the Section 6. On each sphere S(r) the 2-dimensional complex covector field FA may be decomposed into its “longitudinal” and “transversal” part: FA = Z,A + ²AB W,B .

(B.12)

The functions Z and W are defined up to additive constants and fulfil the following identities: 2

r F

r2 F A||A = 4Z ,

(B.13)

A||B

(B.14)

²AB = 4W 30

(we remind that 4 denotes the 2-dimensional Laplacian normalized to the unit sphere it contains only the derivatives over angles and, therefore, commutes with ∂3 ). Due to the Gauss law µ

¶

3

0 = div F = r−2 (r2 F ),3 + 4Z

(B.15) 3

the longitudinal part of FA is fully determined by the radial part F . Therefore, using (B.9) and (B.15) we obtain FA = −[4−1 (rΨ),3 ],A + r²AC χ,C .

(B.16)

Moreover, according to this “2 + 1” decomposition we have (curl F)3 = ²BA FA||B , (curl F)A = ²BA (FB ,3 −F 3 ,B ) .

(B.17) (B.18)

Now, from (B.11) we have ⊥

⊥

⊥ 3

3

µ k

−i Ψ= −ir F = −irF = x ∂l

¶

˜ ² lm Fm = xk al ² lm Fm + rN ˜ (curl F)3 . N k k

(B.19)

The last term may be calculated from (B.17). In the first term we may replace Fm by the following covector Gm := −[4−1 (rΨ),3 ],m + xn ²nms χ,s .

(B.20)

Indeed, due to (B.16), both covectors F and G differ only by the radial component, which is anihilated by the term xk ²klm . Finally, we obtain µ

⊥

k

Ψ = −i4χ − a

¶ −1

irKk χ − 4 Vk (rΨ),3

,

(B.21)

where we have defined following r-independent operators µ

¶

µ

xm xk xk AB xk r2 4 + δkm − 2 λg ∂B r ∂ m = ∂A r r λ r := i²klm xl ∂ m .

Kk := Vk

¶

,

(B.22) (B.23)

Let C denote the space of complex functions on the unit sphere S 2 . Observe that for any ψ ∈ C both Kk ψ and Vk ψ belong to C (the space of monopole-free complex functions on S 2 ). Therefore, the dynamics given by (B.21) lives on C. We will see in the sequel, that the same is true for the remaining Maxwell equations. Moreover, one can easily prove that 4, Kk and Vk are self-adjoint operators on C with respect to the following scalar product: Z

< ψ1 |ψ2 >:=

S(1)

λ ψ1∗ ψ2 ,

(B.24)

31

for any ψ1 , ψ2 ∈ C. It is obvious that the generator of rotations Vk commutes with the Laplace-Beltrami operator 4. All the three operators commute with ∂3 , because they contain only differentiation over angles. √ Finally, using (B.21), (B.10) and observing that N k ∂k = −i 1 − v2 ω m Vm we obtain ˙ = Ψ

√

½

1−

µ k

v2

¶

¾

−1

−i4χ − a

m

irKk χ − 4 Vk (rΨ),3 − iω Vm Ψ

.

(B.25)

Using again (B.11) we have ⊥

−i χ = −ir4

−1

Ã⊥

F

! A||B

h

= r4−1

²AB

½h

iA||B

˜ F) curl(N

³

˜ FC ),3 − (N ˜ F 3 ),C = r4−1 ²AB ²CA (N h

˜ FB ),3 = −r4−1 (N µ

−1

−1

= −r 4 ∂3

||B

´i||B

i

¾

²AB =

=

˜ F 3) = − r−2 4(N ¶

˜ F A )||A + r−1 (N ˜ F 3) . r (N 2

(B.26)

The factor r2 appears when we change the order of lowering and rising of the indices under the differentiation ∂3 . This is due to the fact that gAB is proportional to r2 and g AB is proportional to r−2 . The last term in the above equation denotes the monopole-free part ˜ F 3 , which we obtain as the result of the operator 4−1 4 acting on it. of the function N We have µ

˜ F 3 = 1 Ψ + ak Ψxk + i e ak xk N r 4πr

¶

.

(B.27)

Moreover, ˜ F A )||A = ak (∂A xk )F A + N ˜FA (N ||A

(B.28)

The first term may be calculated as follows µ k

A

(∂A x )F = g

AB

k

(∂A x )FB = g

Ã

¶

mn

!

xk x n xm xn (∂m xk )Fn = g kn − 2 Fn . (B.29) − 2 r r

In the last expression, we may again replace Fn by Gn . This way we finally obtain √

½

µ

h 1 1 −1 k 1 −1 rKk 4−1 (rΨ),3 − iχ˙ = − 1 − Ψ + 4 (rΨ) + a 4 ∂ ,33 3 2 r r r ¶ ¾ i 1 e m 2 − ir Vk χ + 2 Ψxk + i xk − ω V m χ . r 4πr3

C

v2

(B.30)

Boundary momenta

In this Appendix we compute boundary momenta χrA , which describe the response of the system to the control of the boundary values of the configurations ΨA . ¿From the 32

definition (91) we have µ

χr1

−1

= −r4

¶

·

¸

1 3A||B F ²AB = −r4−1 (D3 N A − DA N 3 + N ²AC BC )||B ²AB = λ · ¸

= −r4−1 (D3 N A )||B ²AB + (N B A )||A ,

(C.1)

since the radial component of the shift vector vanishes. The second term in the above formula we have already computed (cf. (B.28)). To compute the first term let us observe that √ √ (C.2) NA = N k (∂A xk ) = 1 − v2 ²klm ωl xm (∂A xk ) = 1 − v2 r²AB ∂B (xm ωm ) and N A||B ²AB =

√

1 − v2

1 4(xm ωm ) . r

(C.3)

Hence, we obtain (D3 N A )||B ²AB = (D3||B N A + D3 N A||B )²AB = ¾ ½ √ 1 3 m 3||A m 2 = 1 − v rD (x ωm )||A + D 4(x ωm ) = r ½ ¾ √ x 1 m = 1 − v 2 ω m Km D 3 − 4D3 + D3 4xm = r r ½ ¾ √ 1 x 1 e m 4Ψ2 − 2 Ψ2 xm − x , = 1 − v2 ω m Km Ψ2 − m r r r 2πr2

(C.4)

due to 4xm = −2xm . It is easy to prove using the definition (B.22) that the operator Kk satisfies the following identities: 1 4Kk + Kk 4 − 2 4xk 4 = 0 , r xk 4 + 4xk − 2rKk + 2xk = 0 .

(C.5) (C.6)

¿From (C.6) we obtain immediately (D3 N A )||B ²AB =

√

½

1 − v2

µ

¶

1 m xm e ω 4 Ψ2 − Km Ψ2 − xm r r 2πr2

¾

.

(C.7)

Thus, using (B.28) and (C.7) we get finally χr1

√

½

µ

¶

1 4−1 (rΨ1 ),3 + ak 4−1 Kk 4−1 (rΨ1 ),3 − r2 iVk χ2 − 1− = r µ ¶¾ e m −1 2 2 − ω Ψ xm − 4 rKm Ψ + . xm 4πr2 v2

(C.8)

To compute χr2 = −r−1 A0 observe that ∂3 A0 = A˙ 3 − f03 ,

(C.9) 33

and f0k = −N Dk + N m fmk .

(C.10)

Since in our gauge A3 = A3 = rχ2 , thus ∂3 A0 = rχ˙ 2 + N D3 − N A (A3,A − AA,3 ) = = rχ˙ 2 + N D3 − rN A ∂A χ2 + ∂3 (N A AA )

(C.11)

due to the fact that N A does not depend on r. To calculate the last term in the above formula let us remind that in our gauge AA = −r2 ²AB ∂B (4−1 B 3 ) .

(C.12)

Therefore, using (C.2) we have √ N A AA = − 1 − v2 r2 ²AB (rxm ωm )||B ²AC (4−1 B 3 )||C = √ = − 1 − v2 ω m [rKm 4−1 Ψ1 − Ψ1 xm ] .

(C.13)

Now, since N k ∂k = N A ∂A and using (B.30) we obtain ½ µ ¶ √ ∂3 A0 = − 1 − v2 ∂3 4−1 (rΨ2 ),3 + ak 4−1 rKk 4−1 (rΨ2 ),3 + ir2 Vk χ1 + µ

+ ω

m

¶¾ −1

1

1

+ N D3 − N D3 .

rKm 4 Ψ − Ψ xm

(C.14)

Taking the monopole–free part of ∂3 A0 we obtain finally ½ µ ¶ √ 1 χr2 = 1 − v2 4−1 (rΨ2 ),3 + ak 4−1 Kk 4−1 (rΨ2 ),3 + r2 iVk χ1 + r ¾ ³

+ ω m Km 4−1 Ψ1 − Ψ1 xm

´

.

(C.15)

If we define the complex boundary momentum χr := χr1 + iχr2 ,

(C.16)

then one can easily prove that χr =

δL . δΨ∗,3

(C.17)

We stress that the above formula is not true without keeping the boundary term in (110). This term is responsible for terms linear in ω m in (C.8) and (C.15). Using complex variables Ψ, χ and χr and the identity (C.5) we may rewrite the formulae (C.8) and (C.15) in a compact form χr =

√

1 − v2 µ

1 r

µ

½

¶

4−1 (rΨ),3 + ak 4−1 Kk 4−1 (rΨ),3 − r2 iVk χ +

+ iω m rKm 4−1 Ψ − Ψxm + i

e xm 4πr 34

¶¾

.

(C.18)

Observe that from (C.14) we may compute the monopole part of the scalar potential φ = mon(A0 ). Namely ∂3 φ = mon(N D3 ) +

√

·

¸

1 − v2 ∂3 ω m mon(Ψ1 xm ) =

·

√

¸

e 1 = 1− + ak mon(Ψ2 xk ) + ω m ∂3 mon(Ψ1 xm ) ≡ 2 4πr r ¸ · √ e ≡ 1 − v2 + ∂3 φ˜ . 4πr2

D

v2

(C.19)

Proof of the conservation laws

To calculate the time derivative of H, Pk , Rk and S m given by (39)–(42) we first do it for the integrals extended over the region {r > r0 } and then finally go to the limit r0 → 0. 1) Conservation of the energy H: ˙ 0) = H(r

Z

n

{r>r0 }

˙ D + B˙ n B ) d3 x = (D n n

Z {r>r0 }

(D˙ n Dn + B˙ n Bn ) d3 x .

(D.1)

Here, we used the fact that the time derivative of D0 vanishes and that the scalar product ˙ vanishes when integrated over any sphere S(r) (the field D is angleof D0 with D 0 ˙ independent, whereas D contains the dipole and higher harmonics only). Using equations (B.5) and (B.6) we obtain ˙ 0) = H(r −

√

Z

1−

v2

²mnk (1

{r>r0 }

½

·

∂m i

1 n (D Dn + B n Bn )²mkl ωk xl − 2 o

+ ai xi )Dn B k − ²nkm Dn B k am d3 x .

(D.2)

Using the Gauss theorem and calculating the limit r0 → 0 we obtain H˙ =

√

Z (

1−

v2

Z m

Σ

a Pm + lim

r0 →0 S(r0 )

i xm h ²mnk (1 + ai xi )Dn B k dσ r

)

,

(D.3)

where dσ denotes the surface measure on the sphere S(r0 ). Observe that the contribution from D0 vanishes since it is parallel to xn . The remaining field D behaves like r−1 and the surface element dσ like r2 . Therefore, the surface integral vanishes in the limit and we have: √ H˙ = 1 − v2 am Pm , (D.4) which proves (28). 2) Conservation of the momentum Pj : √

Z

½

1 (Di Bj − Bi Dj )ω i − aj (Di Di + Bi B i )− 2 {r>r0 } ·µ ¶ ¸¾ 1 − ∂m Dm Dj + B m Bj − δ mj (Di Di + Bi B i ) (1 + ak xk ) d3 x 2

P˙ j (r0 ) =

1−

v2

35

(D.5)

and P˙ j =

√

³

µ

Z

+

´

1 − v2 ²mkj Pm ωk − aj (H − m) −

S(r0 )

√

½

1 − v2 lim

r0 →0

¶

1Z {r > r0 }|D0 |2 d3 x 2 )

xm 1 Dm Dj + B m Bj − δ mj (Di Di + Bi B i ) (1 + ak xk ) dσ r 2

.

(D.6)

The contribution of the non-singular part of the fields to the surface integral in (D.6) vanishes in the limit r0 → 0. Hence ( ´ √ ³ √ e2 mk aj + P˙ j = 1 − v2 ² j Pm ωk − aj (H − m) − 1 − v2 lim r0 →0 8πr0 ) µ ¶ Z 1 m 1 + x Dm Dj + Dm Dm xj (1 + ak xk ) dσ . (D.7) 2 S(r0 ) r Inserting (7) into (D.7) one can easily calculate the limit on the right hand side: Ã ! Z √ e 1 ˜ j dσ , P˙ j = 1 − v2 ²mkj Pm ωk − aj (H − m) − lim D 4π r0 →0 S(r0 ) r2

(D.8)

˜ stands for the regular part of D. To calculate the surface integral let us decomwhere D ˜ into the radial component D ˜ 3 and the 2-dimensional field D ˜ A tangent to pose the field D the sphere S(r): ˜ 3 + ∂ A xj D ˜A . ˜ j = xj D (D.9) D r The contribution from the radial part equals Z Z xj ˜ 3 xj 4π lim D dσ = lim (β + O(r))dσ = βj . (D.10) r0 →0 S(r0 ) r 3 r0 →0 S(r0 ) r 3 3 The tangent components give: Z Z Z xj ˜ A 1 1 A ˜ ∂ x D dσ = − D dσ = xj (r2 D˜3 ),3 dσ = j A ||A 2 2 4 r r r S(r0 ) S(r0 ) S(r0 ) Z xj 8π = (2β + O(r)) dσ = βk + O(r0 ) . (D.11) 3 3 S(r0 ) r Finally in the limit r0 → 0 we obtain ³ ´ √ P˙ j = 1 − v2 ²mkj Pm ωk − aj H + (maj − eβj ) . (D.12) This way we proved that the conservation of momentum (29) is equivalent to the fundamental equation maj − eβj = 0. 3) Conservation of the static moment Rk : ½ Z √ 2 ˙ ²kij xi ω j (Dn Dn + B n Bn ) − (1 + ai xi )²mnk Dn Bm − 1−v Rk (r0 ) = {r>r0 }

− al xk ²mnl Dm Bn + ∂l +

·

1 lij xk ² xi ωj (Dn Dn + B n Bn )+ 2 ¸¾

xk (1 + ai xi )²nlm Dn Bm

d3 x 36

(D.13)

and R˙ k =

√ √

µ

¶

1 − v2 Pk − ²kim ai S m − ²kil ω i Rl + ·

Z

¸

xl (1 + ai xi )xk ²nlm Dn Bm dσ . (D.14) r0 →0 S(r0 ) r The contribution from D0 to the surface integral vanishes since it is parallel to xn . Due to the asymptotic behaviour of the fields this integral vanishes in the limit r0 → 0 which proves (30). 4) Conservation of the moment of momentum S m : ½ Z i h √ m m 2 ˙ ² lk ²lrs xi Bs Dr ²kit + xk (Bs Dj ²rjt + Dr Bj ²sjt ) ωt + S (r0 ) = − 1 − v +

1 − v2 lim

{r>r0 }

·

¸

1 1 + ²mkl al xk (Dn Dn + B n Bn ) + ∂ l xk (1 + ai xi )(Dn Dn + B n Bn ) − 2 · 2 ¸¾ − ∂j xk (1 + ai xi )(B j B l + Dj Dl ) + xk ²lrs ²jit Bs Dr xi ωt and S˙ m =

√

µ

1−

√

v2

d3 x

(D.15)

¶ mkl

²

a k Rl − ²

Z

mkl

ωk Sl +

·

¸

xj 1− lim (1 + ai xi )xk ²mkl (B j B l + Dj Dl ) dσ . (D.16) + r0 →0 S(r0 ) r To calculate the surface integral let us observe that the contribution from the non-singular part of the fields vanishes in the limit r0 → 0. The only possibility to obtain nonzero value of this integral is to integrate the Coulomb component of Dj and r−1 component of Dl . Then, due to (7) we have v2

Z k xk e e m lZ i x m l − lim (1 + ai x ) 3 ² kl a dσ = − ² kl a dσ ≡ 0 , 8π r0 →0 S(r0 ) r 8π S(1) r 3

(D.17)

which ends the proof of (31).

Acknowledgments The authors are very much indebted for the financial support, which they got from the European Community (HCM Contract No. CIPA–3510–CT92–3006) and from the Polish National Committee for Scientific Research (Grant No. 2 P302 189 07).

References [1] J. Kijowski, Electrodynamics of Moving Particles, Gen. Rel. and Grav. 26, p. 154 (1994) J. Kijowski, On Electrodynamical Self-interaction, Acta Phys. Polon. A85, p. 771-787 (1994) 37

[2] C. Misner, K.S. Thorne, J. A. Wheeler Gravitation, W.H. Freeman and Co., San Francisco (1973) [3] J. D. Jackson, Classical electrodynamics, 2-nd Ed., Wiley, New York 1975 [4] F. Rohrlich, Classical Charged Particles, Addison–Wesley, Reading 1965 [5] J. Kijowski and W.M. Tulczyjew, A Symplectic Framework for Field Theories, Lecture Notes in Physics vol. 107, Springer-Verlag, Berlin, 1979 [6] L. Landau and E. Lifshitz, The Classical Theory of Fields, Pergamon, New York, 1962 [7] I. BiaÃlynicki-Birula and Z. BiaÃlynicka-Birula, Quantum Electrodynamics, Pergamon, Oxford, 1975 [8] J. Jezierski and J. Kijowski, Gen. Rel. and Grav. 22, pp. 1283-1307 (1990) [9] J. Kijowski: Asymptotic Degrees of Freedom and Gravitational Energy, in: Proceedings of Journees Relativistes 1983, Torino, ed. S. Benenti et al., pp.205-211, Pitagora Editrice, Bologna, 1985 J. Kijowski, Unconstrained degrees of freedom of gravitational field and the positivity of gravitational energy, in Gravitation, Geometry and Relativistic Physics, Lecture Notes in Physics vol. 212, Springer-Verlag, Berlin, 1984 J. Jezierski, J. Kijowski, Quasi-local hamiltonian of the gravitational field, in Proceedings of the Sixth Marcel Grossmann Meeting on General Relativity, Kyoto 1991, Part A, editors H. Sato and T. Nakamura, World Scientific (1992), p. 123-125

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