Head or tail:
the dilemma of electrodynamics
Jerzy Kijowski and Dariusz Chruściński 1
Center for Theoretical Physics
Polish Academy of Sciences
Al. Lotników 32/46
02 - 668 Warsaw, Poland
Abstract
Equations of motion for a test charged particle moving in external electromagnetic field are derived from Maxwell equations. This way a new, gauge-independent
description of motion on both Lagrangian and Hamiltonian level is obtained. The
derivation follows Einstein’s idea about the relation between equations of motion
and field equations.
1
Introduction
Classical electrodynamics splits into two parts: the field theoretical part governed by
Maxwell equations and the mechanical part governed by the Lorentz force law: maν =
euµ fµν . These two parts are – in principle – incompatible because Maxwell equations
imply at least a “1/r2 ” – singularity of the electromagnetic field in a neighbourhood of a
charged, point-like particle. Consequently, the Lorentz force acting on the particle is ill
defined. To avoid this infinity one may assume a finite size of the particle and a smooth
charge distribution inside, but then the theory of the “particle + field” system is not
closed unless we describe consistently the nature of forces (called Poincaré stresses) keeping together matter fields – constituents of the particle. Whenever such a specific model
of the particle is chosen (i. e. its microscopic structure fully specified) we deal with an
1
Institute of Physics, N. Copernicus University; ul. Grudzia̧dzka 5/7; 80-100 Toruń, Poland
infinite number of degrees of freedom of matter fields, instead of a finite number of degrees
of freedom of a point-like particle. The question arises: whether or not a mathematically
consistent description of the interacting “particle + field” system is possible, which is universal (model-independent) and approximates in a physically acceptable way this infinite
number of degrees of freedom by a finite number of collective variables.
Such a theory was proposed a few years ago (see [1] – [3]). It is based on a certain
definition of the renormalized four-momentum of the “particle + field” system and its
conservation law. As a byproduct of this theory we obtain equations of motion of charged
particles uniquely derived from field equations (modulo purely qualitative assumptions
about the stability of the particle), as proposed by Einstein in a different context (cf. [4]).
When applied to the problem of motion of test particles, the above approach provides
a completely new picture – both from mathematical and physical point of view – which
I am going to present here. Mathematically, it leads to a new, gauge-invariant, second
order Lagrangian L for the motion of charged particles in the external Maxwell field (see
[5]):
√
int
L = Lparticle + Lint = − 1 − v2 (m − aµ uν Mµν
(t, q, v)) ,
(1)
int
where aµ := uν ∇ν uµ is the particle’s acceleration. The skew-symmetric tensor Mµν
(t, q, v)
is equal to the amount of angular-momentum of the field, which is acquired by our physical
(y,u)
system, when the (boosted) Coulomb field fµν , accompanying the particle moving with
constant velocity u through the space-time point y = (t, q), is added to the background
field f . The new Lagrangian differs from the standard, first order Lagrangian
√
L = Lparticle + Lint = − 1 − v2 (m − euµ Aµ (t, q)) ,
(2)
by (gauge-dependent) boundary corrections only. Hence, both Lagrangians generate the
same equations of motion (although L is of second differential order, it depends linearly
upon the second derivatives and its Euler – Lagrange equations are of second order).
The relation between L and L is, therefore, analogous to the one well known in General
Relativity: the gauge-invariant, second order Hilbert Lagrangian for Einstein differs from
the first order, gauge-dependent Lagrangian by a boundary term.
But even more dramatic are physical consequences of the theory. Consider the tiny
region containing the total charge of the particle. We call this region the head of the
particle. It is necessarily surrounded by its own Coulomb field (which we call the tail ).
In the standard description, based on (2), the free (kinetic) four-momentum of the particle: pkin
= muµ must be supplemented by the (gauge-dependent!) interaction term
µ
“−eAµ (t, q)”. Both quantities are entirely localized within the particle’s head. We conclude that in this picture only the head feels the external field and is entirely responsible
for the interaction, whereas the tail has been amputated a priori. This is a flagrant violation of the Gauss Law. (As we know, attempts to stitch the tail back to the particle lead
to a plethora of infrared problems in both classical and quantum electrodynamics.) In our
description, which we are going to present here, no amputation is performed. Moreover,
the interaction is entirely due to the tail (the sensitivity of the head to the external field
has been neglected!). It is also entirely gauge-independent and uniquely implied by the
Maxwell field dynamics. This agrees with Einstein arguments about the relation between
field equations and equations of motion. Hence, the Lorentz force is not an arbitrary supplement to Maxwell field theory, but is its consequence. Because of the above, extremely
simple conceptual structure, the new description should, in my opinion, be seriously taken
into account when constructing the theory of electrodynamical interactions on microscopic
level.
2
Renormalization and Interaction Lagrangian
Our description of test particles is based on the following ideas: to each trajectory γ we will
assign uniquely a field f(γ) , which approximates the particle’s own field. Given an external
field f , we calculate the Maxwell lagrangian density Λ for the total electromagnetic field
ftotal := f + fγ . As a bi-linear form, Λ(ftotal , ftotal ) splits into three parts: the purely
background part Λ(f, f ), the “particle part” Λ(f(γ) , f(γ) ) and the cross term 2Λ(f(γ) , f ).
A physically admissible trajectory should correspond to a stationary point of the action.
But the background part is constant when varying the particle’s trajectory. Hence, it can
be omitted in the test particle theory. The particle part is divergent. It represents the
action density carried by the tail of the point particle, far away from the particle’s head.
To obtain a good approximation of the total value of the action, we must replace, near to
the head and inside it, this singular distribution by the actual distribution, corresponding
to the continuous particle. Even if we do not know it explicitly (it is model-dependent!),
its integral over a 3-dimensional particle’s rest-frame hyperplane must give us “minus the
total energy” (mass) carried by both the head and the tail of the free particle (we take
into account the fact that for static situations all the time derivatives vanish, i. e. the
lagrangian density equals minus the energy density). As a result of this renormalization
procedure, the integral of the particle term Λ(f(γ) , f(γ) ) must be replaced by the free
particle Lagrangian Lparticle in (2), with m representing the total (already renormalized)
mass of the free particle. This is, in fact, the idea of our renormalization procedure,
defined mathematically in [1].
We stress, that the mass m is not localized within the particle’s head. Because there
is no sharp boundary between the head and the tail, we cannot even decide meaningfully
in which proportion does it split into the contributions of each of these two regions. (The
head’s contribution would correspond to what is usually called “bare mass”, which –
according to standard recipes – gets “dressed” by the contribution from tail). In our
approach such a splitting is not only meaningless but also useless since only the total,
renormalized mass m enters into the game.
The interaction comes exclusively from the cross term 2Λ(f(γ) , f ). Actually, its integral
may be interpreted as a value of the singular tensor density f(γ) := ∗fγ (“distribution” in
the sense of L. Schwartz) on the smooth “test function” f . Here, ∗ denotes the Hodge
operator. The distribution f(γ) is assigned to the particle’s trajectory γ (for a mathematical
definition see next Section) in such a way, that its divergence equals to the Dirac “delta-
function” localized on γ:
µν
∂ν f(γ)
= euµ δ(γ) .
(3)
Hence, substitution fµν = ∂µ Aν −∂ν Aµ and integration by parts gives exactly the term Lint
from (2) modulo boundary terms. This proves that our new Lagrangian gives precisely
the Lorentz equations of motion for test particles.
Another nice feature of this description is that also the interaction four-momentum
pint acquires a nice structure: instead of the gauge-dependent quantity “−eAµ (t, q)”
we obtain here the (gauge-independent) amount of four-momentum which is acquired
by our physical system, when the (boosted) Coulomb field accompanying the particle
moving with constant velocity u through the space-time point y = (t, q), is added to
the background field. It is obtained by integrating the cross term in the total energymomentum tensor of the field (cf. [5]).
3
A model for the particle’s own field
The only missing element is now the definition of the field fγ (or f(γ) ). The construction
which follows is entirely new. It simplifies considerably the results of [5].
We want f(γ) to approximate as much as possible the electrostatic Coulomb field. For
this purpose take a point y(t) of the trajectory γ (t is any parameter along γ) and denote
by Σt the “rest-frame hypersurface”, i. e. the 3-dimensional flat hyperplane in Minkowski
space-time, orthogonal to γ at this point. Consider the electric Coulomb field Dk on Σt
corresponding to the point charge e at y(t) and the vanishing magnetic field B l ≡ 0 (k and
l are p
three-dimensional indices on Σt ). Together with the 3-dimensional volume density
λ = det(gkl ) on Σt , these two fields define uniquely a four-dimensional tensor density
ϕµν along Σt (µ and ν are four-dimensional indices). The local coordinate expression
for this quantity is: ϕ0k = −ϕk0 = λDk and the remaining components of ϕ vanish in a
coordinate system (x0 , xk ), such that x0 is constant on Σt and (dx0 |dxk ) = 0.
We define the field f(γ) as the collection of all these fields ϕ over all hyperplanes
Σt corresponding to the trajectory. Locally, in a neighbourhood of γ, where different
hyperplains Σt do not intersect, this is a meaningful definition. To have a global definition,
valid also at points of intersection, where f(γ) should feel contributions from different
(intersecting) hyperplains Σt , we must be more careful and use the “push-forward” (or
“direct image”) of a Schwartz distribution (see e. g. [6]). More precisely: we observe that
the collection of all Σ’s cannot be identified with the physical space-time M , but can be
f := Σ × R1 , where Σ is a 3-dimensional
organized into an abstract “model space-time” M
f.
Euclidean vector space and R1 is a time axis. In fact, we have defined our field ϕ on M
f → M having the
But a parameterized trajectory γ enables us to define a mapping Fγ : M
following property: each Σ × {t} is mapped isometrically onto the corresponding Σt ⊂ M
in such a way that (~0, t) goes to the trajectory point y(t). We define f(γ) as the direct
image of ϕ with respect to Fγ :
f(γ) := (Fγ )∗ ϕ .
(4)
It is easy to check that the above quantity does not depend upon all the arbitrary choices,
which we have left in the above definition, i. e. on the parameterization of γ and on an
arbitrary modification of Fγ consisting in independent rotations of each Σt separately.
f to M, implies equation (3).
The obvious identity ∂ν ϕµν = eδ0µ δ (3) , pushed-forward from M
This means that the field f(γ) fulfills the second pair of Maxwell equations.
We stress, that it usually does not fulfill the first pair: df(γ) = 0, where f(γ) := − ∗ f(γ) .
The error (i. e. the non-vanishing quantity df(γ) ) is of the order of a(t)/r2 , where a(t) is
the acceleration of γ at y(t) and r is the distance from y(t) on Σt . Hence, for trajectories
which are not too wild, and outside of the nearest neighbourhood of γ, the field f(γ)
almost fulfills Maxwell equations. We must remember, however, that the test particle
theory is only an approximation of the complete theory, because it does not describe the
radiation field. The approximative character of the test particle theory is precisely due to
the approximative character of the field f(γ) used in its derivation. To remove errors due
to this approximation one has to use the complete theory defined in [1].
Acknowledgments
This work was supported in part by the Alexander-von-Humboldt Stiftung (Germany)
and by the Polish KBN Grant Nr. 2 P03A 047 15. The authors are very much indebted
to I. BiaÃlynicki-Birula and G. Rudolph for helpful discussions.
References
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