On the self-force in Bopp-Podolsky electrodynamics

In the classical vacuum Maxwell-Lorentz theory the self-force of a charged point particle is infinite. This makes classical mass renormalization necessary and, in the special relativistic domain, leads to the Abraham-Lorentz-Dirac equation of motion possessing unphysical run-away and pre-acceleration solutions. In this paper we investigate whether the higher-order modification of classical vacuum electrodynamics suggested by Bopp, Lande, Thomas and Podolsky in the 1940s, can provide a solution to this problem. Since the theory is linear, Green-function techniques enable one to write the field of a charged point particle on Minkowski spacetime as an integral over the particle's history. By introducing the notion of timelike worldlines that are"bounded away from the backward light-cone"we are able to prescribe criteria for the convergence of such integrals. We also exhibit a timelike worldline yielding singular fields on a lightlike hyperplane in spacetime. In this case the field is mildly singular at the event where the particle crosses the hyperplane. Even in the case when the Bopp-Podolsky field is bounded, it exhibits a directional discontinuity as one approaches the point particle. We describe a procedure for assigning a value to the field on the particle worldline which enables one to define a finite Lorentz self-force. This is explicitly derived leading to an integro-differential equation for the motion of the particle in an external electromagnetic field. We conclude that any worldline solutions to this equation belonging to the categories discussed in the paper have continuous 4-velocities.


Introduction
For many applications it is reasonable to model moving charges in terms of classical charged point particles. In accelerator physics., e.g., it is usually neither desirable nor feasible to model particle beams in terms of extended classical charged bodies or in terms of quantum matter. Therefore a mathematically consistent theory of classical charged point particles is of high relevance. Unfortunately, such a theory does not exist so far. Of course, there is no problem as long as we restrict to a classical charged test particle, i.e., as long as we neglect the particle's self-field. Then the equation of motion is just the ordinary Lorentz force equation in a given external field and everything is fine. If, however, the self-field is taken into account, then the theory becomes pathological. According to the standard Maxwell theory in vacuum, the electromagnetic field of a point charge becomes infinite at the position of the charge, so the particle experiences an infinite self-force. This infinity is so bad that the field energy in an arbitrarily small ball around the source is infinite which leads to an infinite term in the equation of motion of a point charge. Dirac [5] suggested to counter-balance this infinity by postulating that the point charge carries a negative infinite "bare mass" which leads to the Lorentz-Dirac equation. Even if one accepts this ad-hoc idea of an infinite bare mass, the problem has not been solved. The Lorentz-Dirac equation is known to predict absurd behaviour such as run-away solutions and pre-acceleration so that it has to be considered as unphysical. For reviews of this dilemma, including detailed accounts of the history, we refer to the books by Rohrlich [15] and Spohn [16]. Seeing that the standard vacuum Maxwell theory does not lead to a consistent equation of motion of charged point particles, one might think about modifying this theory. In the course of history at least two such modifications have been suggested, both motivated by the desire of solving the problem of an infinite selfforce, namely the Born-Infeld theory and the Bopp-Podolsky theory. The Born-Infeld theory is by far the better known of the two. This theory, which was suggested by Born and Infeld [3] in 1934, modifies the Maxwell vacuum theory by introducing non-linearities proportional to a factor b −2 where b is a new hypothetical constant of Nature with the dimension of a (magnetic) field strength. For b → ∞ the standard Maxwell vacuum theory is recovered. Therefore, the fact that the standard Maxwell theory is very well verified by many experiments is in agreement with the Born-Infeld theory as long as b is sufficiently big. By contrast, the Bopp-Podolsky theory retains linearity but it introduces higher-derivative terms proportional to a factor ℓ 2 where ℓ is a new hypothetical constant of Nature with the dimension of a length. Again, for ℓ → 0 the standard Maxwell vacuum theory is recovered. The Bopp-Podolsky theory was first suggested by Bopp [2] in 1940. It was independently rediscovered by Podolsky [10] in 1942. Both Bopp and Podolsky formulated their theory in terms of an action functional and then derived the field equation which is of fourth order in the electromagnetic potential. As noted by both Bopp and Podolsky, this fourth-order equation is equivalent to a pair of second-order equations. If rewritten in this form, the Bopp-Podolsky field equation coincides with those of a theory suggested in 1941 by Landé and Thomas [9]. Just as the Born-Infeld theory, the Bopp-Podolsky theory was first formulated as a classical field theory but with the intention of deriving a quantum version later. In particular, Podolsky further pursued both the classical and the quantum aspects of the theory in several follow-up articles with different co-authors, see Podolsky and Kikuchi [11,12] and Podolsky and Schwed [13]. In the present article we are interested only in the classical theory. In both the Born-Infeld theory and the Bopp-Podolsky theory the self-field is finite for a static point charge, i.e., for a point charge that is at rest in some inertial system on Minkowski spacetime. This was shown already in the earliest articles on these theories. Moreover, in both theories for a static charge the field energy in a ball of radius R around the charge is finite, even in the limit R → ∞. As conservation of energy holds in both theories, this implies finiteness of the field energy for all worldlines that asymptotically approach a straight timelike line on Minkowski spacetime in the past. However, very little is known for worldlines on Minkowski spacetime that may approach the light-cone. To the best of our knowledge, in the Born-Infeld theory nothing in this direction is known and in the Bopp-Podolsky theory the only result we are aware of is due to Zayats [18] who showed that the self-force is finite for a uniformly accelerated particle on Minkowski spacetime. It is the purpose of this paper to add some results on the finiteness of the self-force in the Bopp-Podolsky theory. In our view, these results give strong support to the idea that the Bopp-Podolsky theory provides a consistent theory of classical charged point particles including the self-force. In Section 2 we briefly review the basic equations of the Bopp-Podolsky theory on Minkowski spacetime, emphasising the fact that because of the linearity Green-function techniques can be used. In Section 3 we specify the equations of Section 6 to the case that the source of the electromagnetic field is a point charge with a prescribed worldline on Minkowski spacetime. We discuss various ways of writing the field strength at a point off the worldline as an integral over the particle's history. As a first example, we treat the simple case of a point charge that is at rest in an inertial system. In Section 4 we discuss the self-field on the worldline which requires applying a subtle limiting procedure to the equations of the preceding section. Knowing the field on the worldline gives us the self-force. As a second example, we treat the case of a uniformly accelerated charge. In Section 5 we present our main results on the finiteness of the field and of the self-force. We show that the self-force is finite unless the worldline approaches the light-cone in the past in a very contrived manner. As a third example, we discuss such a pathological worldline where the self-field actually diverges on a lightlike hyperplane, so the self-force diverges at one point on the worldline. However, we demonstrate that even in this case the singularity of the field is so mild that it does not cause a major problem for the equation of motion. In Section 6 we write down the equation of motion of a charged point particle in the Bopp-Podolsky theory, which is an integro-differential equation, and we discuss how the Lorentz-Dirac equation comes about as a specific limit if ℓ is sent to zero. In the body of the paper we specify to Bopp-Podolsky theory on Minkowski spacetime and we work in an orthogonal coordinate system. However, we have added an appendix where we consider the Bopp-Podolsky theory on a curved spacetime. This allows us to offer a derivation of the dynamical (Hilbert) stress-energymomentum tensor of the theory using exterior calculus. The appendix also includes a derivation of the Lorentz-force equation in the Bopp-Podolsky theory which is crucial for our reasoning in the body of the paper.

Bopp-Podolsky theory
We consider Minkowski spacetime with standard inertial coordinates x = (x 0 , x 1 , x 2 , x 3 ), where η ab = diag(−1, 1, 1, 1). Here and in the following, Einstein's summation convention is used for latin indices which take values 0, 1, 2, 3 and for greek indices which take values 1, 2, 3. Indices are lowered and raised with η ab and with its inverse η ab , respectively. We use units making c equal to 1. The higher-order electrodynamics suggested by Bopp [2] and, independently, by Podolsky [10] is based on the action functional Here A a is the electromagnetic potential, is the electromagnetic field strength, j a is the current density 4-vector field and ℓ is a hypothetical new constant of nature with the dimension of a length. Note that the action functional (2) can be equivalently replaced withS because the integrands differ only by a total divergence. The field equations of the Bopp-Podolsky theory result from varying the action functional (2) or (4) with respect to the potential. They read or, in terms of the potential if we impose the Lorenz gauge, where = ∂ b ∂ b is the wave operator. Both Bopp and Podolsky observed that the fourth-order differential equation in (6) can be reduced to a pair of second-order differential equations. More precisely, (6) is equivalent to This can be demonstrated in the following way. Assume we have a solution A a to (6). Then we definê and it is readily verified that (7) and (8) are indeed true. Conversely, assume that we have solutionsÂ a and A a to (7) and (8), respectively. Then we define and it is readily verified that (6) is true. This gives a one-to-one relation between solutions to (6) and pairs of solutions to (7) and (8) which allows to view the Bopp-Podolsky theory as equivalent to a theory based on the two equations (7) and (8). The latter was suggested, shortly after Bopp but independently of him and shortly before Podolsky, by Landé and Thomas [9]. In a quantised version of the Landé-Thomas theory, (7) describes the usual (massless) photon while (8) describes a hypothetical massive photon whose Compton wave length is equal to the new constant of nature ℓ.
If we want to find the retarded solution to the fourth-order Bopp-Podolsky equations (6), for any given source j a , we can use the reduction to the second-order equations (7) and (8) and then apply standard Green-function techniques. The details were worked out already by Landé and Thomas, see Section 9 in [9]. The retarded solution to isĜ and the retarded solution to isG Here and in the following, J n is the nth order Bessel function of the first kind, y < x means that y is in the chronological past of x and D(x − y) is the Lorentzian distance between these two events, Hence, the retarded solution to (7) iŝ the retarded solution to (8) isÃ and the retarded solution to (6) is The Lorenz gauge condition is satisfied byÂ a ,Ã a and A a if the current density satisfies the continuity equation A a is the retarded potential of the standard Maxwell theory. For ℓ → 0 we haveG → 0 and henceÃ a → 0, so in this limit the standard Maxwell theory is recovered, as is obvious from (2). (18) can be viewed as a map that assigns to each current density j the corresponding retarded Bopp-Podolsky potential A. A general framework for investigating the question of whether this map is well-defined would be to assume that j is a (tempered) distribution and to ask if A is again a (tempered) distribution. As the Green function G does not satisfy a fall-off condition in all spacetime directions, this is a non-trivial question. In this paper we want to restrict to a more specific question. We will assume that j is the current density associated with a point charge, and we will investigate if the integral on the right-hand side of (18) converges for events x in the chronological future of the worldline of the charge. If this is the case, the left-hand side of (18) is, of course, well defined, not only as a distribution but even as a function.
Up to now we have discussed only how the electromagnetic field can be calculated from its source, i.e., from the current that generates the field. We will also need the equation for the Lorentz force density which, in inertial coordinates on Minkowski spacetime, reads This equation gives us the force density the field F ab exerts onto the current j b . In the case that the current is concentrated on a worldline, it gives the self-force. A derivation of (20), and of the precise form of the stress-energy-momentum tensor T ab in the Bopp-Podolsky theory, will be given in the Appendix.
3 The field of a point charge off the wordline Figure 1: The open set U is the chronological future of the worldline without the worldline itself As curves with unbounded acceleration may reach past or future infinity in a finite proper time, inextendibility does not necessarily mean that τ ranges over all of R. We denote the interval on which ξ a is defined by ] τ min , τ max [ . Consider the chronological future of this worldline, i.e., the set of all events that can be reached from the worldline along a future-oriented timelike curve. If the worldline approaches a light-cone asymptotically for τ → τ min , its chronological future is bounded by a lightlike hyperplane, see Fig. 1; otherwise it is all of R 4 . Let U be the open set of all events x that are in the chronological future of the worldline but not on the worldline. To each x ∈ U we assign the retarded time τ R (x), defined by the properties that and the retarded distance For events x off the worldline this implies that with a well-defined spatial unit vector n(x), see Fig. 2. By differentiation, we find and For a sequence x N ∈ U that approaches the worldline, For a point charge with worldline ξ(τ ), the current density reads where q is the charge. Then (16) gives the standard Liénard-Wiechert potential, and (18) reads Both (29) and (30) hold for x ∈ U . In Section 4 we will investigate what happens if x approaches the wordline.
x ξ τ R (x) Note that, in contrast to the Liénard-Wiechert potential (29), the potential (30) at an event x depends on the whole history of the charge from τ = τ min up to τ = τ R (x). (The same is true, in general, for the standard Maxwell theory on curved spacetimes.) The integrand in (30) is bounded for τ → τ R (x) because where we have used the Bernoulli-l'Hôpital rule and J ′ 1 (0) = 1/2. By contrast, for τ → τ min , the individual componentsξ a (τ ) may blow up arbitrarily. Therefore, the existence of the integral on the right-hand side of (30) is not guaranteed. We will later show that, for a fairly large class of worldlines, this integral does converge even absolutely, as a Lebesgue integral or as an improper Riemann integral, for all x in the chronological future of the wordline; however, we will also give a (contrived) example where it does not converge for some x in the chronological futrure of the worldline. On the assumption that the intergal converges, at all events x ∈ U we can differentiate (30) with respect to x b . Antisymmetrising the resulting expression gives the field strength (3), Here we have used (26) and the identities 2J 1 (z) = z J 0 (z) + J 2 (z) and 2J ′ 1 (z) = J 0 (z) − J 2 (z) of the Bessel functions. Again, we postpone the discussion of what happens if x approaches the worldline to Section 4. We observe that, keeping x fixed, we may use ζ = D x− ξ(τ ) as the parameter along the worldline. Indeed, differentiation of the equation As, by the wrong-way Schwarz inequality for timelike future-oriented vectors, ζ is monotonically decreasing along the worldline. This guarantees that the equation When proper time τ runs from −∞ to τ R (x), the new parameter ζ runs (backwards) from ∞ to 0. Hence (30) can be rewritten as Note that, if we view x as a parameter, (37) has the form of a Hankel transform which transforms a function of ζ to a function of 1/ℓ. We may use the parameter ζ in the formula for the field strength as well. If such a change of the integration variable is performed on the right-hand side of (32), the resulting equation reads We get an alternative expression for the field strength if we differentiate (37) with respect to x b and antisymmetrise, dζ .
From (37) we can derive another form of the potential by performing an integration by parts and using the identity −J 1 = J ′ 0 of Bessel functions. The resulting equation gives the deviation of the potential from the Liénard-Wiechert potential. With the help of standard asymptotic formulas for the Bessel function J 0 the right-hand side can be written as a power series in ℓ. Such asymptotic (i.e., in general non-convergent) expansions have been used, e.g., by Frenkel [7] (also see Frenkel and Santos [8]) and Zayats [18].

Example 1: Charge at rest
The simplest case one can consider is a charge that is at rest in an appropriately chosen inertial system; this is equivalent to saying that the worldline of the charge is a straight timelike line, with a constant four-vector V satisfying V a V a = −1. In this case (33) is a quadratic equation, which can be easily solved for τ , Then (37) simplifies to which is finite for all x, By differentiation of (44), or equivalently by evaluation of (38), we get the field strength This expression is not defined on the worldline; if a point on the wordline is approached, the limit depends on the direction. We may say that the field displays a directional singularity. By contrast, (39) gives F ab (x) = 0 for events x on the worldline. This is the value that results from averaging the limits of (46) over all directions. For events off the worldline, (39) is in agreement with (46). In the rest system of the charge we have V µ = 0 for µ = 1, 2, 3 and r R (x) = (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 =: r is just the ordinary radius coordinate. In this coordinate system (46) gives a radial electrostatic field with modulus In contrast to the Coulomb field of the standard Maxwell theory, the Bopp-Podolsky E(r) stays finite if the worldline is approached. This result played a crucial role already in the original work of Bopp [2] and Podolsky [10]. It has the consequence that, at least for a charge at rest, the field energy in a ball surrounding the charge is finite. Note, however, that the electric vector field cannot be continuously extended into the origin, because of the above-mentioned directional singularity, see Fig. 3. The two expressions (30) and (37) for the potential are equivalent on U , i.e., off the worldline, and so are the two expressions (38) and (39) for the field strength. We will now discuss what happens with these expressions on the wordline. This is crucial because the value of the field strength on the wordline determines the selfforce. As to the potential, it is easy to see that both (30) and (37) can be directly evaluated at points x on the worldline and that they coincide at such points as well. Moreover, the resulting potential is continuous.
If it were differentiable, its derivative would give us the field strength on the wordline without any ambiguity. However, the potential is not differentiable on the worldline. This is the reason why the expression (38), which results from differentiating (30), and the expression (39), which results from differentiating (37), behave differently on the worldline. We discuss now these two different behaviours and then decide which value should be assigned to the field strength on the worldline. We first observe that (38) is not defined at events on the wordline because it involves the derivative ∂ b τ R (x) which is not defined at such events. By contrast, (39) does not involve this derivative and will actually give a unique value for the field strength on the worldline, provided that the integrals converge. However, this value does not result from differentiating the potential on the wordline because in order to differentiate (37) we require passing the derivative through the integral; this is only valid if the integrand has a continuous derivative, which does not occur if x is on the worldline. Correspondingly, the field given by (39) is not continuous at events on the worldline; if a point x on the worldline is approached, the limit depends on the direction. We can calculate this limit either with (39) or with (38). If we work with (38), the direction dependence comes from the first term. If we average over all directions, this term is killed and we are left with the same value for the field that we get from evaluating (39) at the event x on the worldline. This procedure gives us a unique value for the field strength on the worldline. The averaging can be done most easily if we introduce the retarded spheres To average the first term on the right-hand side of (38) over S(τ 0 , r 0 ) we choose at the event ξ(τ 0 ) an orthonormal tetrad e 0 , e 1 , e 2 , e 3 with e 0 =ξ(τ 0 ). Then n a (x) = cos ϕ sin ϑ e a 1 + sin ϕ sin ϑ e a 2 + cos ϑ e a 3 and averaging of n a (x) over S(τ 0 , r 0 ) is tantamount to averaging over the angles, n a = 1 4π 2π 0 π 0 cos ϕ sin ϑ e a 1 + sin ϕ sin ϑ e a 2 + cos ϑ e a 3 sin ϑ dϑ dϕ = 0 .
Asξ a τ R (x) =ξ a (τ 0 is constant on S(τ 0 , r 0 ), this shows that the first term on the right-hand side of (38) averages to zero. Hence its limit for r 0 → 0 is zero as well, so this term gives no contrbution to the field at ξ(τ 0 ). This fact is illustrated by the example of a static charge which was treated at the end of the preceding section.
We mention that the averaging can be done, as an alternative, over spheres in the orthocomplement ofξ(τ 0 ), rather than over spheres in the retarded light-cone. In this caseξ a τ (x) is not constant on a sphere of finite radius r 0 , so the calculation is less convenient; however, in the limit of r 0 tending to zero one finds, again, that the first term on the right-hand side of (38) averages to zero. The two averaging methods are related to surrounding the worldline either by a Bhabha tube or by a Dirac tube, see Ferris and Gratus [6] for a discussion in the context of the standard Maxwell theory. We summarise our observations in the following way. Provided that the integrals exist, we can calculate the field on the worldline in two equivalent ways: We may either evaluate (39) directly at the point on the worldline we are interested in, or we may consider the limit of (38) if the worldline is approached and retain all terms that survive angle-averaging. The idea that angle-averaging gives the physically correct value for the field on the worldline, and thereupon for the self-force, is intuitively convincing if one thinks of the point charge as the limiting case of a charged sphere whose radius tends towards zero. Angle-averaging is formulated as an axiom in the living review on the self-force by Poisson, Pound and Vega, see Section 24.1 in [14]. If we know the field F ab x at an event x = ξ(τ 0 ) on the worldline, we know the self-force f s a (τ 0 ). The latter is given just as the Lorentz force exerted by the field F ab ξ(τ 0 ) onto the point charge that produces the field, i.e.
In the standard Maxwell theory, the self-force is infinite; therefore, it is necessary to perform a mass renormalisation, introducing a "bare mass" of the particle that is negative infinite. We will demonstrate in the following that in the Bopp-Podolsky theory there is no need, and actually no justification, for introducing an infinite bare mass. If we use (38), the self-force reads . (53) The first term on the right-hand side of (38) has dropped out because of the averaging. The self-force may also be calculated from (39) which does not require any averaging. Then we get (52) can be reproduced from (54) by integrating by parts.
Example 2: Uniformly accelerated motion For a particle with constant acceleration a, the worldline is given by In this case, for a point x = ξ(τ 0 ) on the worldline (33) reads The self-force (52) reduces to which can be expressed in terms of the Bessel functions I 1 and K 1 , This result was recently found by Zayats [18]. So the self-force is finite.

Finiteness of the field of a point charge and of the self-force
In the standard Maxwell theory, the (Liénard-Wiechert) potential of a point charge and the corresponding field strength are singular on the worldline of the source. We will now show that, by contrast, in the Bopp-Podolsky theory there is a large class of worldlines for which the self-force is given by an absolutely converging integral. Asξ(τ ) has unit length, and x − ξ(τ ) has Lorentz length ζ, we may writė and where ν(ζ, x) and µ(ζ, x) are spatial unit vectors, We introduce the following terminology. Geometrically, ξ is not bounded away from the past light-cone of x if and only if there is a sequence τ k such that x − ξ(τ k ) / x 0 − ξ 0 (τ k ) approaches a lightlike vector for τ k → τ min , see Fig. 4. The notion of being bounded away from the light-cone implicitly refers to a particular inertial coordinate system chosen, and it refers to a particular event x. However, the notion is actually independent of these choices, as the following proposition shows. Proposition 1. The property of the worldline being bounded away from the past light-cone of x is preserved if we change the inertial coordinate system by an orthochronous Lorentz transformation. If this property is true for one event x in the chronological future of the worldline, then the chronological future of the worldline is all of R 4 and the property is true for all other events y ∈ R 4 as well.
Proof. From (60) we read that The worldline is bounded away from the light-cone of x if and only if the right-hand side is bounded for ζ → ∞, i.e., if and only if there exists δ > 0 such that for τ min < τ < τ 0 with some τ 0 . To prove the first part of the proposition, we assume that this condition holds in the chosen inertial system. Under a Lorentz transformation,x a = Λ a b x b , the denominator on the left-hand side of (63) is unchanged, while the numerator changes according tõ As x − ξ(τ ) is timelike and future-pointing, for µ = 1, 2, 3. As a consequence, (65) implies that with some positive constant K. Here we have assumed that the Lorentz transformation is orthochronous, for τ min < τ < τ 0 which proves that the condition of the wordline being bounded away from the light-cone of the chosen event holds in the twiddled coordinate system as well. To prove the second part of the proposition, we observe that (60) implies Here and in the following, we write a = δ µν a µ a ν for any a = (a 1 , a 2 , a 3 ). The worldline is bounded away from the light-cone of x if and only if the right-hand side of (69) is bounded away from 1 for ζ → ∞, i.e., if and only if there is a λ with 0 < λ < 1 such that for τ min < τ < τ 0 with some τ 0 . Let us assume that this condition holds for some particular event x. Let y be any other event and choose a constant µ such that λ < µ < 1. Define Then we have, for all τ such that ξ 0 (τ ) < t, for τ min < τ <τ 0 with someτ 0 . This inequality demonstrates that y is in the chronological future of the worldline and that the wordline is bounded away from the light-cone of y as well.
Because of this result, we may simply say that a worldline is bounded away from the past light-cone, without any reference to a specific event x.
We will now show that the field of a point charge is finite if its worldline is bounded away from the past light-cone. Using the notation of (59) and (60), the potential (37) reads For expressing the field strength tensor at a chosen event x using the notation of (59) and (60), we may choose the inertial coordinate system such thatξ a τ R (x) = δ a 0 . Then the electric and magnetic components of the field strength tensor (38) read, respectively, In a coordinate system withξ a (τ 0 ) = δ a 0 the self-force (52) is given by We can now prove the following result.

Proposition 2.
If the worldline ξ is bounded away from the past light-cone, the integrals on the right-hand sides of (74), (75) and (76) are absolutely convergent for all x ∈ R 4 ; the integral on the right-hand side of (77) is absolutely convergent for all τ min < τ 0 < τ max .
Proof. As the integrands in (74), (75), (76) and (77) stay finite for ζ → 0, we only have to verify that they fall off sufficiently quickly for ζ → ∞. We first observe that tanh α increases from −1 to 1 if α varies from −∞ to ∞, and that |µ ρ ν ρ | ≤ 1. Therefore, the condition of ψ being bounded implies that tanh ψ tanh χ µ ρ ν ρ is bounded away from 1. Thus, in each of the four equations (74), (75), (76) and(77) the modulus of the integrand is bounded by a term of the form K J k ζ/ℓ /ζ where K is independent of ζ and k is either 1 or 2. As J k ζ/ℓ falls off like ζ −1/2 for ζ → ∞, this guarantees absolute convergence of the integral.
This proposition demonstrates that, in particular, the self-force is finite for a large class of wordlines. Actually, the requirement of the worldline being bounded away from the past light-cone is sufficient but not necessary for finiteness of the self-force. The following proposition shows that there is another class of worldlines, including ones which are not bounded away from the past light-cone, for which the self-force is finite.
Proposition 3. Assume that the worldline ξ is confined to a two-dimensional timelike plane P in Minkowski spacetime. Then the integral on the right-hand side of (77) is absolutely convergent for all τ min < τ 0 < τ max .
Example 3: A worldline with diverging self-force integral From Propositions 2 and 3 we know that the self-force is finite if the particle's worldline is bounded away from the light-cone or if it is contained in a timelike plane. Actually, the proof of Proposition 3 can be generalised to the case that the worldline, rather than being confined to P , approaches P sufficiently quickly for τ → τ min . This leaves only a class of rather contrived motions for which the self-force integral could The worldline must approach the light-cone for τ → τ min with a sufficiently large tangential velocity component. In this section we present such an example for which the self-force integral, indeed, diverges at one point. We find it convenient to give the worldline in terms of a past-oriented curve parameter γ which is not proper time, where and Here C F and S F are the Fresnel-C and Fresnel-S functions. Note that ξ is an analytic timelike curve that approaches the past light-cone with an oscillatory tangential velocity component, see Fig. 5. The curve is, indeed, everywhere timelike as can be seen from It can be shown that the 4-acceleration of ξ is bounded and that its chronological future is all of Minkowski spacetime.
We will now show that for this worldline ξ the electromagnetic field (38) diverges on a hyperplane. This will imply that the self-force becomes infinite at one instant. To that end we have to rewrite the integral in (38) as an integral over γ and to investigate the behaviour of the integrand for γ → ∞. We first observe that, if γ tends to ∞, for all n ∈ N. Moreover, from the standard asymptotic formulas for the Fresnel functions we find With the help of these formulas we find that the parameter ζ which is used as the integration variable in (38) is related to our curve parameter γ by hence Inserting this expression into the well-known asymptotic formula for the Bessel function J 2 yields After these preparations, we are ready to evaluate the integral in (38). If we use γ as the integration variable, writing γ R (x) for the parameter value that corresponds to τ R (x) and thus to ζ = 0, this integral reads We evaluate this equation for a = 2 and b = 0. As we find with our asymptotic formula for J 2 (ζ/ℓ) from above Here and in the following, the ellipses indicate a term that is finite for all x. The integral over the second term is finite for all x. If we decompose the remaining integral into an integration from γ R (x) to ℓ and an integration from ℓ to infinity we find Here Ci and Si denote the cosine integral and the sine integral, respectively. As the cosine integral diverges logarithmically if its argument approaches zero, we have found that the (20)-component of the electromagnetic field according to (38) is given by i.e., that this field component diverges on the lightlike hyperplane x 0 + x 1 = 0. This may be interpreted as a shock front propagating at the speed of light. The same divergence is found, by a completely analogous calculation, for the component F 21 = −F 12 , while all other components are finite everywhere. As a consequence, the self-force becomes infinite at the instant when the charged particle crosses the hypersurface x 0 + x 1 = 0 which happens at the origin of the coordinate system. This means that at this instant an infinite external force is necessary to keep the particle on its prescribed worldline. Note that the divergence is logarithmic and thus rather mild. In particular, the electromagnetic field is a locally integrable function, i.e., a regular distribution. It is true that a charged test particle would experience an infinite Lorentz force at the instant when it crosses the hypersurface x 0 + x 1 = 0. However, as f (x) = log|x| dx is finite-valued and continuous everywhere, the particle's velocity would still be finite-valued and continuous, i.e., the particle's worldline would still be a C 1 curve.

The Lorentz-Dirac limit
The equation of motion of the charged particle is where m 0 is the (bare) mass of the particle, f s a (τ ) is the self-force (52) and f e a (τ ) is an external force. If f e a (τ ) is given, (95) is an integro-differential equation for the worldline ξ(τ ). In the standard Maxwell theory, i.e., for ℓ → 0, the self-force becomes infinite. Dirac's solution to give a meaning to the equation of motion (95) in this case was to assume that the bare mass is negative infinite such that it cancels the infinite contribution from the self-force. After this cancellation, one ends up with the Lorentz-Dirac equation which involves a renormalised (or dressed) mass which is positive and finite. It is interesting to see how the Lorentz-Dirac equation is reproduced from the Bopp-Podolsky theory in the limit ℓ → 0. To that end we substitute in (54) the integration variable ζ = ℓσ, where With two times integrating by parts yiels The first term diverges for ℓ → 0. If we assume, following Dirac's idea of mass renormalisation, that the bare mass m 0 becomes negative infinite in this limit such that the "dressed mass" remains finite, the ℓ → 0 limit of the equation of motion reads If the integral is bounded, the last term vanishes and we get the Lorentz-Dirac equation. From (97) we find, with the help of the well-known asymptotic formula for the Bessel function J 1 , that χ(σ) = O(σ −3/2 ) for σ → ∞. So the integral in (101) is certainly bounded if ∂ 3 W ab /∂ζ 3 is bounded for ζ → ∞. A sufficient (but not necessary) condition is that all componentsξ a ,ξ a , ... ξ a , and .... ξ a are bounded.

Conclusions
In this paper we have demonstrated that in the Bopp-Podolsky theory the self-force of a charged point particle is given by an integral that absolutely converges for a large class of worldlines on Minkowski spacetime. We have also provided a (contrived) example where the electromagnetic field diverges on a lightlike plane, so the self-force diverges at one point of the worldline. However, even in this case the electromagnetic field is a locally integrable function (i.e., a regular distribution), so there is no major problem for the equation of motion. This is to be contrasted with the standard Maxwell vacuum theory where the self-field of a charged point particle is infinite at every point of the particle's worldline, and the singularity is so bad that the field energy in an arbitrarily small ball around the charge is infinite which makes mass renormalisation necessary.
In the Bopp-Podolsky theory there is no need for mass renormalisation; the equation of motion (95) is an integro-differential equation which makes sense with a finite bare mass m 0 . It should also be emphasised that in the Born-Infeld theory, which is to be viewed as a natural rival to the Bopp-Podolsky theory, virtually nothing is known about finiteness of the self-force of an accelerated particle whose worldline may approach the light-cone. So it seems fair to say that, at least in view of the motion of charged point particles, the Bopp-Podolsky theory is in a more promising state. Some questions are still open. It would be desirable to have a proof that for any worldline the Bopp-Podolsky self-force is a locally integrable function of the curve parameter. Moreover, it would be crucial to demonstrate that the equation of motion (95) with vanishing external force is free of run-away solutions. Partial results in this direction have been found by Frenkel and Santos [8], but a general proof is still missing. If such a proof can be given, it will be justified to say that the Bopp-Podolsky theory successfully regularises the dynamics of classical charged point particles. This observation is independent of whether a quantised version of the Bopp-Podolsky theory is viable at a fundamental level.
The formulation of a stress-energy-momentum tensor for the theory discussed in this paper appears to have had a chequered history. Bopp [2] writes down a stress-energy-momentum tensor that is obviously based on the decomposition (10) of the potential, but no derivation is given. Podolsky in [10] also writes down a stress-energy-momentum tensor with a promise to derive it in [11] from arguments based on a canonical approach. In our view this did not succeed and the further derivation in [13] lacked transparency. To our knowledge there has been no subsequent attempt to derive any stress-energy-momentum tensor appropriate to the theory under discussion. In view of these comments it may be of value to put the matter into a modern perspective by offering a derivation based on metric variations of the Bopp-Podolsky action. This requires formulating the theory on a curved spacetime manifold. The natural tools for this purpose exploit the exterior calculus of differential forms using properties of the Hodge map ⋆ associated with the spacetime metric g and the nilpotency of the exterior derivative d. For background material on exterior calculus we refer to Straumann [17] whose sign and factor conventions we adopt. The Bopp-Podolsky action reads where F = dA and G = d⋆ F . Here we assume that J is a given current 3-form that satisfies the conservation law dJ = 0.
In the body of the paper we have restricted ourselves to the case of a flat metric and we have used orthogonal coordinates. Then the action (102) reduces to (2) where the current 4-vector j = j a ∂ a is related to the current 3-form by g(j, · ) = ⋆J.
A direct route to the symmetric dynamical (Hilbert) stress-energy-momentum tensor is obtained by making compact variations of the metric tensor in (102). Such variations can be induced by making independent variationsė a in a local g−orthonormal coframe {e a }, a = 0, 1, 2, 3 since in such a basis g = η ab e a ⊗ e b . Such variations give rise to a set of 3-forms τ a defined by and a stress-energy-momentum tensor T = T ab e a ⊗ e b where the components T ab = η bc ⋆ (τ a ∧ e c ) .
The covariant divergence of T then follows as where D denotes the covariant exterior derivative, see e.g. Benn and Tucker [1]. If we make compact variations of the potential, rather than of the metric, we get the field equation of the Bopp-Podolsky theory. We can derive the τ a and the field equation in one go if we allow for variations of the potential and of the metric simultaneously. (Note that the current J is assumed to be given and will be kept fixed during the variation.) Then the derivative of the Lagrangian 4-form with respect to the variational parameter is given byΛ For calculating the first two terms on the right-hand side we use the formula [4] (⋆Ψ)˙=ė c ∧ i c (⋆Ψ) − ⋆(ė c ∧ i c Ψ) + ⋆Ψ (108) that holds for any p−form Ψ, where i c denotes the contraction operator (or interior derivative) with the vector field X c which is defined by e a (X c ) = δ a b . Moreover, we use standard rules of exterior calculus, such as α ∧ ⋆β = β ∧ ⋆α for any p−forms α, β, the anti-derivative property and the nilpotency of d, and the commutativity of d with the variational derivative. Then we find that and Inserting (109) and (110) into (107) yieldṡ and The field equations are determined by requiring that the action is stationary for variations of the potential only (i.e.,ė c = 0) that are compactly supported i.e M dΦ = 0 . This yields The τ c from (114) give us, via (105), the dynamical stress-energy-momentum tensor of the Bopp-Podolsky theory. A routine calculation of Dτ a then shows, with the aid of the field equations (115), that the divergence (106) satisfies the Lorentz force law, ∇ · T = F ( · , j) .
If the background metric is flat and orthogonal coordinates are used, (115) reduces to (5); in this case the components of the stress-energy-momentum tensor can be calculated from (114) to be which is the stress-energy-momentum-tensor of Podolsky [10], cf. Zayats [18]. As a cross-check, we can derive with the help of the field equations (5) from (117) the equation ∂ c T cd = F db j b which is the flat-space coordinate version of (116).