Green functions and propagation in the Bopp-Podolsky electrodynamics
aa r X i v : . [ phy s i c s . c l a ss - ph ] M a y Green functions and propagation in theBopp-Podolsky electrodynamics
Markus Lazar ∗ Department of Physics,Darmstadt University of Technology,Hochschulstr. 6,D-64289 Darmstadt, Germany
Wave Motion (2019), 102388; https://doi.org/10.1016/j.wavemoti.2019.102388 Abstract
In this paper, we investigate the so-called Bopp-Podolsky electrodynamics. TheBopp-Podolsky electrodynamics is a prototypical gradient field theory with weaknonlocality in space and time. The Bopp-Podolsky electrodynamics is a Lorentzand gauge invariant generalization of the Maxwell electrodynamics. We derive theretarded Green functions, first derivatives of the retarded Green functions, retardedpotentials, retarded electromagnetic field strengths, generalized Li´enard-Wiechertpotentials and the corresponding electromagnetic field strengths in the frameworkof the Bopp-Podolsky electrodynamics for three, two and one spatial dimensions.We investigate the behaviour of these electromagnetic fields in the neighbourhoodof the light cone. In the Bopp-Podolsky electrodynamics, the retarded Green func-tions and their first derivatives show fast decreasing oscillations inside the forwardlight cone.
Keywords:
Bopp-Podolsky electrodynamics; Green function; propagation; retar-dation; retarded potentials; Li´enard-Wiechert potentials
Generalized continuum theories such as gradient theories and nonlocal theories are ex-citing and challenging research fields in physics, applied mathematics, material scienceand engineering science (see, e.g., [1, 2, 3, 4, 6, 5, 7, 8, 9, 10]). Gradient theories and ∗ E-mail address: [email protected] (M. Lazar). ℓ , the so-called Bopp-Podolsky parameter. Bopp [1] and Podolsky[3] have proposed such a gradient theory representing a classical generalization of theMaxwell electrodynamics towards a generalized electrodynamics with linear field equa-tions of fourth order in order to avoid singularities in the electromagnetic fields and tohave a finite and positive self-energy of point charges (see also [4, 11, 12]). Due to itssimplicity, the Bopp-Podolsky theory can be considered as the prototype of a gradienttheory. Therefore, the Bopp-Podolsky electrodynamics represents the simplest, physicalgradient field theory with weak nonlocality in space and time.Nowadays there is a renewed interest in the Bopp-Podolsky electrodynamics (e.g., [13,14]), in particular to solve the long-outstanding problem of the electromagnetic self-forceof a charged particle present in the classical Maxwell electrodynamics which goes back toLorentz, Abraham and Dirac trying to formulate a classical theory of the electron. Theequation of motion in the classical theory of the electron, often called the Lorentz-Diracequation, is of third order in the time-derivative of the particle position, and as result itshows unphysical behaviour such as run-away solutions and pre-acceleration (see, e.g., thebooks by Rohrlich [15] and Spohn [16]). Therefore, the classical Maxwell electrodynamicsin vacuum does not lead to a consistent equation of motion of charged point particles anda generalized electrodynamics could solve this problem.In the static case, gradient electrostatics with generalized Coulomb law was givenby Bopp [1], Podolsky [3], Land´e and Thomas [12] and gradient magnetostatics includingthe generalized Biot-Savart law was given by Lazar [17]. Such generalized electrostaticsand generalized magnetostatics have a physical meaning if the classical electric and mag-netic fields are recovered in the limit ℓ →
0. In gradient electrostatics, for a point chargethe electric potential is finite and non-singular, but the electric field strength is finite anddiscontinuous at the position of the point charge.The Bopp-Podolsky theory has many interesting features. It solves the problem ofinfinite self-energy in the electrostatic case, and it gives the correct expression for theself-force of charged particles at short distances eliminating the singularity when r → / ℓ is in the order of ∼ − m, that means femtometre(fm), or even smaller. From the mathematical point of view, the length parameter ℓ playsthe role of the regularization parameter in the Bopp-Podolsky electrodynamics. This2ength scale is associated to the massive mode, m BP , of the Bopp-Podolsky electrodynam-ics through m BP = ~ / ( cℓ ). Moreover, it is interesting to note that the Bopp-Podolskyelectrodynamics is the only linear generalization of the Maxwell electrodynamics whoseLagrangian, containing second order derivatives of the electromagnetic gauge potentials,is both Lorentz and U (1)-gauge invariant [22].The Bopp-Podolsky electrodynamics is akin to the Pauli-Villars regularization proce-dure used in quantum electrodynamics (see, e.g., [21, 23, 13, 24]). Therefore, the Bopp-Podolsky electrodynamics provides a regularization of the Maxwell electrodynamics basedon higher order partial differential equations. On the other hand, Santos [25] analyzed thewave propagation in the vacuum of the Bopp-Podolsky electrodynamics and two kinds ofwaves were found: the classical non-dispersive wave of the Maxwell electrodynamics, anda dispersive wave reminiscent of wave propagation in a collisionless plasma with plasma(angular) frequency ω p = c/ℓ , described by a Klein-Gordon equation.In the Maxwell electrodynamics, quantities like the retarded potentials, the retardedelectromagnetic field strengths, the Li´enard-Wiechert potentials and the electromagneticfield strengths in the Li´enard-Wiechert form are the basic fields and quantities for theclassical electromagnetic radiation (see, e.g., [26, 27, 28]). In particular, the Li´enard-Wiechert form of the electromagnetic field strengths is important for the calculation ofthe self-force of a charged point particle. In the Bopp-Podolsky electrodynamics, only alittle is known for such fields necessary for the electromagnetic radiation and radiationreaction in the generalized electrodynamics of Bopp and Podolsky and their behaviour onthe light cone (see, e.g., [29, 13, 14]). Only, the three-dimensional generalized Li´enard-Wiechert potentials were given by Land´e and Thomas [29] and the corresponding three-dimensional electromagnetic fields of a point charge have been recently given by Gratuset al. [14], for the first time. The aim of the present work is to close this gap and togive a systematic derivation and presentation of all important quantities in three, twoand one spatial dimensions (3D, 2D, 1D). In particular, this work gives, for the first time,the analytical expressions for the retarded potentials and retarded electromagnetic fieldsin 2D and 1D, and for the generalized Li´enard-Wiechert potentials and correspondingelectromagnetic fields of a non-uniformly moving charge in 2D and 1D in the frameworkof the Bopp-Podolsky electrodynamics. This completes the library of all important fieldsolutions needed in the Bopp-Podolsky electrodynamics in 3D, 2D, and 1D, which is anecessary step towards completing the study of the Bopp-Podolsky electrodynamics. Inparticular, we investigate the behaviour of these fields near and on the light cone.The purpose of this paper is to add relevant results of the Green functions, retardationand wave propagation in the Bopp-Podolsky electrodynamics. In Section 2, we review thebasic equations of the Bopp-Podolsky electrodynamics. In Section 3, we give a systematicderivation and collection of the (dynamical) Bopp-Podolsky Green function and its firstderivatives in 3D, 2D and 1D in the framework of generalized functions. The retardedpotentials and retarded electromagnetic field strengths are given in Section 4 for 3D,2D and 1D. In Section 5, we present the generalized Li´enard-Wiechert potentials andelectromagnetic field strengths in generalized Li´enard-Wiechert form. The paper closeswith the conclusion in Section 6. 3 Basic framework of the Bopp-Podolsky electrody-namics
In the Bopp-Podolsky electrodynamics [1, 3], the electromagnetic fields are described bythe Lagrangian density L BP = ε (cid:16) E · E + ℓ ∇ E : ∇ E − ℓ c ∂ t E · ∂ t E (cid:17) − µ (cid:16) B · B + ℓ ∇ B : ∇ B − ℓ c ∂ t B · ∂ t B (cid:17) − ρφ + J · A , (1)with the notation ∇ E : ∇ E = ∂ j E i ∂ j E i and E · E = E i E i . Eq. (1) corresponds to Bopp’sform of the Lagrangian [1]. Here φ and A are the electromagnetic gauge potentials, E isthe electric field strength vector, B is the magnetic field strength vector, ρ is the electriccharge density, and J is the electric current density vector. ε is the electric constantand µ is the magnetic constant (also called permittivity of vacuum and permeability ofvacuum, respectively). The speed of light in vacuum is given by c = 1 √ ε µ . (2)Moreover, ℓ is the characteristic length scale parameter in the Bopp-Podolsky electro-dynamics, ∂ t denotes the differentiation with respect to the time t and ∇ is the Nablaoperator. From the mathematical point of view, the characteristic length parameter ℓ plays the role of a regularization parameter in the Bopp-Podolsky theory. In addition tothe classical terms, first spatial- and time-derivatives of the electromagnetic field strengths( E , B ) multiplied by the characteristic length ℓ and a characteristic time T = ℓ/c , re-spectively, appear in Eq. (1) which describe a weak nonlocality in space and time. Thelimit ℓ → E , B ) can be expressed in terms of the electro-magnetic gauge potentials (scalar potential φ , vector potential A ) E = −∇ φ − ∂ t A , (3) B = ∇ × A . (4)Due to their mathematical structure, the electromagnetic field strengths (3) and (4) satisfythe two electromagnetic Bianchi identities (or electromagnetic compatibility conditions) ∇ × E + ∂ t B = 0 , (5) ∇ · B = 0 , (6)which are known as homogeneous Maxwell equations.The Euler-Lagrange equations of the Lagrangian (1) with respect to the scalar poten-tial φ and the vector potential A give the electromagnetic field equations (cid:2) ℓ (cid:3) (cid:3) ∇ · E = 1 ε ρ , (7) (cid:2) ℓ (cid:3) (cid:3)(cid:16) ∇ × B − c ∂ t E (cid:17) = µ J , (8)4espectively. The d’Alembert operator is defined as (cid:3) := 1 c ∂ tt − ∆ , (9)where ∆ is the Laplace operator. Eqs. (7) and (8) represent the generalized inhomoge-neous Maxwell equations in the Bopp-Podolsky electrodynamics. In addition, the electriccurrent density vector and the electric charge density fulfill the continuity equation ∇ · J + ∂ t ρ = 0 . (10)If we use the variational derivative with respect to the electromagnetic fields ( E , B ), we obtain the constitutive relations in the Bopp-Podolsky electrodynamics for theresponse quantities ( D , H ) in vacuum D := δ L BP δ E = ε (cid:2) ℓ (cid:3) (cid:3) E , (11) H := − δ L BP δ B = 1 µ (cid:2) ℓ (cid:3) (cid:3) B , (12)where D is the electric displacement vector (electric excitation), H is the magnetic ex-citation vector. The second terms in Eqs. (11) and (12) describe the polarization ofthe vacuum present in the Bopp-Podolsky electrodynamics. The vacuum in the Bopp-Podolsky electrodynamics is a classical vacuum plus vacuum polarization that behaveslike a plasma-like vacuum [25].Using the constitutive relations (11) and (12), the Euler-Lagrange equations (7) and(8) can be rewritten in the form of inhomogeneous Maxwell equations ∇ · D = ρ , (13) ∇ × H − ∂ t D = J . (14)From Eqs. (7) and (8), inhomogeneous Bopp-Podolsky equations, being partial differ-ential equations of fourth order, follow for the electromagnetic field strengths (cid:2) ℓ (cid:3) (cid:3) (cid:3) E = − ε (cid:16) ∇ ρ + 1 c ∂ t J (cid:17) , (15) (cid:2) ℓ (cid:3) (cid:3) (cid:3) B = µ ∇ × J . (16)Using the generalized Lorentz gauge condition [30] (cid:2) ℓ (cid:3) (cid:3) (cid:18) c ∂ t φ + ∇ · A (cid:19) = 0 , (17)the electromagnetic gauge potentials fulfill the following inhomogeneous Bopp-Podolskyequations (cid:2) ℓ (cid:3) (cid:3) (cid:3) φ = 1 ε ρ , (18) (cid:2) ℓ (cid:3) (cid:3) (cid:3) A = µ J . (19)5ote that the generalized Lorentz gauge condition (17) is as natural in the Bopp-Podolskyelectrodynamics as the Lorentz gauge condition is in the Maxwell electrodynamics [30]. Asshown by Galv˜ao and Pimentel [30], the usual Lorentz gauge condition, c ∂ t φ + ∇ · A = 0,does not satisfy the necessary requirements for a consistent gauge in the Bopp-Podolskyelectrodynamics: it does not fix the gauge, it is not preserved by the equations of motion,and it is not attainable. The generalized Lorentz gauge condition is also necessary inthe quantization of the Bopp-Podolsky electrodynamics leading to a generalized quantumelectrodynamics [31, 32]. Bufalo et al. [31] found that in such a generalized quantumelectrodynamics, using the one-loop approximation, the electron self-energy and the vertexfunction are both ultraviolet finite. The Bopp-Podolsky electrodynamics is a linear theory with partial differential equations offourth order. Therefore, the powerful method of Green functions (fundamental solutions)can be used to construct exact analytical solutions.The Green function G BP of the Bopp-Podolsky equation, which is a partial differentialequation of fourth order, is defined by (cid:2) ℓ (cid:3) (cid:3) (cid:3) G BP ( R , τ ) = δ ( τ ) δ ( R ) , (20)where τ = t − t ′ , R = r − r ′ and δ is the Dirac δ -function. Therefore, the Green function, G BP , is the fundamental solution of the linear hyperbolic differential operator of fourthorder, [1 + ℓ (cid:3) (cid:3) (cid:3) , in the sense of Schwartz’ distributions (or generalized functions) [33].Because we are only interested in the retarded Green function, the causality constraintmust be fulfilled G BP ( R , τ ) = 0 for τ < . (21)As always for hyperbolic operators, the Green function G BP ( R , τ ) is the only fundamentalsolution of the (hyperbolic) Bopp-Podolsky operator with support in the half-space τ ≥ (cid:2) ℓ (cid:3) (cid:3) G BP ( R , τ ) = G (cid:3) ( R , τ ) , (22) (cid:3) G BP ( R , τ ) = G KG ( R , τ ) , (23) (cid:3) G (cid:3) ( R , τ ) = δ ( τ ) δ ( R ) , (24) (cid:2) ℓ (cid:3) (cid:3) G KG ( R , τ ) = δ ( τ ) δ ( R ) , (25)where G (cid:3) is the Green function of the d’Alembert equation (24) and G KG is the Greenfunction of the Klein-Gordon equation (25). It can be seen that the Bopp-Podolskyequation (20) is a Klein-Gordon-d’Alembert equation. Finally, the Green function G BP of the Bopp-Podolsky equation can be written in terms of the Green function G (cid:3) of the6’Alembert equation and the Green function G KG of the Klein-Gordon equation (see also[35]) G BP = G (cid:3) − ℓ G KG , (26)or in the (formal) operator notation using the partial fraction decomposition (cid:2)(cid:0) ℓ (cid:3) (cid:1) (cid:3) (cid:3) − = (cid:3) − − ℓ (cid:2) ℓ (cid:3) (cid:3) − . (27)Using Eq. (26), the Green function of the Bopp-Podolsky equation can be derived bymeans of the expressions of the Green function of the d’Alembert equation (see, e.g., [36,37, 38, 39]) and the Green function of the Klein-Gordon equation (see, e.g., [11, 39, 40]).Therefore, the Bopp-Podolsky field is a superposition of the Maxwell field and the Klein-Gordon field. On the other hand, the Green function of the Bopp-Podolsky equation canbe written as convolution of the Green function of the d’Alembert operator and the Greenfunction of the Klein-Gordon operator G BP = G (cid:3) ∗ G KG , (28)satisfying Eqs. (20), (22) and (23). The symbol ∗ denotes the convolution in space andtime. It can be seen in Eq. (28) that the Green function G KG of the Klein-Gordon oper-ator plays the role of the regularization function in the Bopp-Podolsky electrodynamics,regularizing the Green function G (cid:3) of the d’Alembert operator towards the Green func-tion G BP of the Bopp-Podolsky operator. On the other hand, the limit of G BP as ℓ tendsto zero reads (see Eq. (26)) lim ℓ → G BP = G (cid:3) . (29)In this work, we only consider the retarded Green functions which are zero for τ < The three-dimensional Green functions (fundamental solutions) of the wave (d’Alembert)operator (24), the Klein-Gordon operator (25) and the Bopp-Podolsky (Klein-Gordon-d’Alembert) operator are the (generalized) functions ( τ > G (cid:3) (3) ( R , τ ) = 14 πR δ (cid:0) τ − R/c (cid:1) , (30) G KG(3) ( R , τ ) = 14 πℓ (cid:20) R δ (cid:0) τ − R/c (cid:1) − cℓ H (cid:0) cτ − R (cid:1) √ c τ − R J (cid:18) √ c τ − R ℓ (cid:19)(cid:21) , (31) G BP(3) ( R , τ ) = c πℓ H (cid:0) cτ − R (cid:1) √ c τ − R J (cid:18) √ c τ − R ℓ (cid:19) , (32)where R = p ( x − x ′ ) + ( y − y ′ ) + ( z − z ′ ) , H is the Heaviside step function and J is the Bessel function of the first kind of order one. Eq. (32) is obtained from Eq. (26)using the Green functions (30) and (31). The Green function (32) is in agreement withthe expression given earlier in [41, 18, 19]. 7 - X (a) (b) τ - - - - - - X (c) τ (d) - - - Figure 1: Plots of the three-dimensional Bopp-Podolsky Green function for c = 1, ℓ = 0 . G BP(3) ( R , τ = 1) for Y = Z = 0, (b) G BP(3) ( R = 0 , τ ), (c) regular part of ∂ τ G BP(3) ( R , τ = 1)for Y = Z = 0, (d) regular part of ∂ τ G BP(3) ( R = 0 , τ ).Using lim z → z J ( z ) = 12 , (33)on the light cone, cτ = R , the Green function (32) is discontinuous (see Fig. 1a) and readsas G BP(3) ( R , τ ) ≃ c πℓ H (cid:0) cτ − R ) . (34)Furthermore, the Green function (32) shows a decreasing oscillation (see Fig. 1b) and doesnot have a δ -singularity unlike the Green function (30). One can say, Eq. (32) describesa wake in a plasma-like vacuum. The two-dimensional Green functions (fundamental solutions) of the wave (d’Alembert)operator (24), the Klein-Gordon operator (25) and the Bopp-Podolsky (Klein-Gordon-8 - X (a) (b) τ - - - - X (c) τ (d) - Figure 2: Plots of the two-dimensional Bopp-Podolsky Green function for c = 1, ℓ = 0 . G BP(2) ( R , τ = 1) for Y = 0, (b) G BP(2) ( R = 0 , τ ), (c) ∂ τ G BP(2) ( R , τ = 1) for Y = 0, (d) ∂ τ G BP(2) ( R = 0 , τ ) (red dashed curves are the classical Green function G (cid:3) (2) and ∂ τ G (cid:3) (2) ).d’Alembert) operator are the (generalized) functions ( τ > G (cid:3) (2) ( R , τ ) = c π H (cid:0) cτ − R (cid:1) √ c τ − R , (35) G KG(2) ( R , τ ) = c πℓ H (cid:0) cτ − R (cid:1) √ c τ − R cos (cid:18) √ c τ − R ℓ (cid:19) , (36) G BP(2) ( R , τ ) = c π H (cid:0) cτ − R (cid:1) √ c τ − R (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19)(cid:21) , (37)where R = p ( x − x ′ ) + ( y − y ′ ) . Eq. (37) is obtained from Eq. (26) using the Greenfunctions (35) and (36). The Green function (37) has been derived by Lazar [35] in theframework of dislocation gauge theory. The Green function (37) of the Bopp-Podolskyoperator is zero on the light cone (see Fig. 2a), sincelim z → z cos( z ) = 1 z . (38)Furthermore, the Green function (37) shows a decreasing oscillation around the classicalGreen function (35) (see Fig. 2b). 9 - X (a) (b) τ - - - X (c) τ (d) - - Figure 3: Plots of the one-dimensional Bopp-Podolsky Green function for c = 1, ℓ = 0 . G BP(1) ( X, τ = 1), (b) G BP(1) ( X = 0 , τ ), (c) ∂ τ G BP(1) ( X, τ = 1), (d) ∂ τ G BP(1) ( X = 0 , τ ) (reddashed curves are the classical Green function G (cid:3) (1) ). The one-dimensional Green functions (fundamental solutions) of the wave (d’Alembert)operator (24), the Klein-Gordon operator (25) and the Bopp-Podolsky (Klein-Gordon-d’Alembert) operator are the (generalized) functions ( τ > G (cid:3) (1) ( X, τ ) = c H (cid:0) cτ − | X | ) , (39) G KG(1) ( X, τ ) = c ℓ H (cid:0) cτ − | X | (cid:1) J (cid:18) √ c τ − X ℓ (cid:19) , (40) G BP(1) ( X, τ ) = c H (cid:0) cτ − | X | ) (cid:20) − J (cid:18) √ c τ − X ℓ (cid:19)(cid:21) , (41)where X = x − x ′ and J is the Bessel function of the first kind of order zero. Eq. (41) isobtained from Eq. (26) using the Green functions (39) and (40). The Green function (41)of the Bopp-Podolsky operator approaches zero on the light cone (see Fig. 3a), sincelim z → J ( z ) = 1 . (42)Due to the Bessel function term J , the Green function (41) shows a decreasing oscillationaround the classical Green function (39) (see Fig. 3b).10 .4 Derivatives of the Bopp-Podolsky Green function In this subsection, we derive the first time-derivative and first gradient of the Bopp-Podolsky Green function.
The first time-derivative and first gradient of the three-dimensional Bopp-Podolsky Greenfunction (32) read for τ > ∂ τ G BP(3) ( R , τ ) = c τ πℓ (cid:20) cτ δ (cid:0) cτ − R (cid:1) − H (cid:0) cτ − R (cid:1) ( c τ − R ) J (cid:18) √ c τ − R ℓ (cid:19)(cid:21) , (43) ∇ G BP(3) ( R , τ ) = − c R πℓ (cid:20) R δ (cid:0) cτ − R (cid:1) − H (cid:0) cτ − R (cid:1) ( c τ − R ) J (cid:18) √ c τ − R ℓ (cid:19)(cid:21) , (44)using H ′ ( z ) = δ ( z ), δ ( z ) f ( z ) = δ ( z ) f (0), and ( J ( z ) /z ) ′ = − J ( z ) /z . J is the Besselfunction of the first kind of order two. Thus, Eqs. (43) and (44) consist of two terms,namely a Dirac δ -term on the light cone plus a Bessel function term inside the light cone.The second parts (regular parts) of Eqs. (43) and (44) are discontinuous and show adecreasing oscillation.On the light cone, the derivatives of the Green function G BP(3) possess a singularity ofDirac δ -type. This is exhibited by the first term in Eqs. (43) and (44). The second termin Eqs. (43) and (44) is discontinuous on the light cone (see Fig. 1c), sincelim z → z J ( z ) = 18 . (45)In the neighbourhood of the light cone, Eqs. (43) and (44) have the form ∂ τ G BP(3) ( R , τ ) ≃ c τ πℓ (cid:20) cτ δ (cid:0) cτ − R (cid:1) − ℓ H (cid:0) cτ − R (cid:1)(cid:21) , (46) ∇ G BP(3) ( R , τ ) ≃ − c R πℓ (cid:20) R δ (cid:0) cτ − R (cid:1) − ℓ H (cid:0) cτ − R (cid:1)(cid:21) . (47)It can be seen in Fig. 1d that the second parts (regular parts) of Eqs. (43) and (44) showa decreasing oscillation. The first time-derivative and first gradient of the two-dimensional Bopp-Podolsky Greenfunction (37) read for τ > ∂ τ G BP(2) ( R , τ ) = − c τ π H (cid:0) cτ − R (cid:1) ( c τ − R ) (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19) − √ c τ − R ℓ sin (cid:18) √ c τ − R ℓ (cid:19)(cid:21) , (48) ∇ G BP(2) ( R , τ ) = c R π H (cid:0) cτ − R (cid:1) ( c τ − R ) (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19) − √ c τ − R ℓ sin (cid:18) √ c τ − R ℓ (cid:19)(cid:21) . (49)11n the light cone, the derivatives of the Green function G BP(2) possess a 1 /z -singularity(see Fig. 2c), since lim z → z cos( z ) = 1 z − z (50)and lim z → z sin( z ) = 1 z , (51)they are discontinuous. Of course, the 1 /z -singularity is weaker than the non-integrable1 /z -singularity. In the neighbourhood of the light cone, Eqs. (48) and (49) have the form ∂ τ G BP(2) ( R , τ ) ≃ c τ πℓ H (cid:0) cτ − R (cid:1) √ c τ − R , (52) ∇ G BP(2) ( R , τ ) ≃ − c R πℓ H (cid:0) cτ − R (cid:1) √ c τ − R . (53)Furthermore, Eqs. (48) and (49) show a decreasing oscillation around the classical singu-larity (see Fig. 2d). The first time-derivative and space-derivative of the one-dimensional Bopp-Podolsky Greenfunction (41) read for τ > ∂ τ G BP(1) ( X, τ ) = c τ ℓ H (cid:0) cτ − | X | ) √ c τ − X J (cid:18) √ c τ − X ℓ (cid:19) , (54) ∂ X G BP(1) ( X, τ ) = − cX ℓ H (cid:0) cτ − | X | ) √ c τ − X J (cid:18) √ c τ − X ℓ (cid:19) , (55)using J ′ = − J .On the light cone, the derivatives of the Green function G BP(1) , Eqs. (54) and (55), havea jump discontinuity, due to Eq. (33) (see Fig. 3c), namely ∂ τ G BP(1) ( X, τ ) ≃ c τ ℓ H (cid:0) cτ − | X | ) , (56) ∂ X G BP(1) ( X, τ ) ≃ − cX ℓ H (cid:0) cτ − | X | ) . (57)Furthermore, Eqs. (54) and (55) show a decreasing oscillation (see Fig. 3d) unlike thederivative of the Green function of the d’Alembert equation given in terms of δ (cid:0) cτ − | X | ). Solutions based on retarded Green functions lead to retarded fields (like retarded po-tentials and retarded electromagnetic field strengths) in the form of retarded integrals.Retarded integrals are mathematical expressions reflecting the phenomenon of “finite sig-nal speed” (e.g. [42]). 12 .1 Retarded potentials
The retarded electromagnetic potentials are the solutions of the inhomogeneous Bopp-Podolsky equations (18) and (19) and for zero initial conditions they are given as convo-lution of the (retarded) Green function G BP and the given charge and current densities( ρ , J ) φ = 1 ε G BP ∗ ρ , (58) A = µ G BP ∗ J . (59)Explicitly, the convolution integrals (58) and (59) read as φ ( n ) ( r , t ) = 1 ε Z t −∞ d t ′ Z R n d r ′ G BP( n ) ( r − r ′ , t − t ′ ) ρ ( r ′ , t ′ ) , (60) A ( n ) ( r , t ) = µ Z t −∞ d t ′ Z R n d r ′ G BP( n ) ( r − r ′ , t − t ′ ) J ( r ′ , t ′ ) , (61)where r ′ is the source point and r is the field point. Here n denotes the spatial dimension.Substituting Eqs. (58) and (59) into the generalized Lorentz gauge condition (17) andusing Eqs. (22) and (10), it can be seen that the generalized Lorentz gauge condition issatisfied (cid:2) ℓ (cid:3) (cid:3) (cid:18) c ∂ t φ + ∇ · A (cid:19) = µ (cid:2) ℓ (cid:3) (cid:3) G BP ∗ (cid:0) ∂ t ρ + ∇ · J (cid:1) = µ (cid:0) ∂ t ρ + ∇ · J (cid:1) ∗ G (cid:3) = 0 . (62) Substituting the Bopp-Podolsky Green function (32) into Eqs. (60) and (61), the three-dimensional retarded electromagnetic potentials read as φ (3) ( r , t ) = c πε ℓ Z t −∞ d t ′ Z R d r ′ H (cid:0) cτ − R (cid:1) √ c τ − R J (cid:18) √ c τ − R ℓ (cid:19) ρ ( r ′ , t ′ )= c πε ℓ Z t − R/c −∞ d t ′ Z R d r ′ ρ ( r ′ , t ′ ) √ c τ − R J (cid:18) √ c τ − R ℓ (cid:19) (63)and A (3) ( r , t ) = µ c πℓ Z t −∞ d t ′ Z R d r ′ H (cid:0) cτ − R (cid:1) √ c τ − R J (cid:18) √ c τ − R ℓ (cid:19) J ( r ′ , t ′ )= µ c πℓ Z t − R/c −∞ d t ′ Z R d r ′ J ( r ′ , t ′ ) √ c τ − R J (cid:18) √ c τ − R ℓ (cid:19) (64)since H ( cτ − R ) = 0 for t ′ > t − R/c . In the Bopp-Podolsky electrodynamics, thethree-dimensional retarded potentials (63) and (64) possess an afterglow, since they drawcontribution emitted at all times t ′ from −∞ up to t − R/c . The retarded time is a resultof the finite speed of propagation for electromagnetic signals.13 .1.2 2D
Substituting the Bopp-Podolsky Green function (37) into Eqs. (60) and (61), the two-dimensional retarded electromagnetic potentials become φ (2) ( r , t ) = c πε Z t −∞ d t ′ Z R d r ′ H (cid:0) cτ − R (cid:1) √ c τ − R (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19)(cid:21) ρ ( r ′ , t ′ )= c πε Z t − R/c −∞ d t ′ Z R d r ′ ρ ( r ′ , t ′ ) √ c τ − R (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19)(cid:21) (65)and A (2) ( r , t ) = µ c π Z t −∞ d t ′ Z R d r ′ H (cid:0) cτ − R (cid:1) √ c τ − R (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19)(cid:21) J ( r ′ , t ′ )= µ c π Z t − R/c −∞ d t ′ Z R d r ′ J ( r ′ , t ′ ) √ c τ − R (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19)(cid:21) (66)since H ( cτ − R ) = 0 for t ′ > t − R/c . Thus, the two-dimensional retarded potentials (65)and (66) show an afterglow, since they draw contribution emitted at all times t ′ from −∞ up to t − R/c . In the version of the Bopp-Podolsky electrodynamics in one spatial dimension, the po-tentials φ (1) and A (1) are both a scalar field, and the current density J is also a scalarfield.Substituting the Bopp-Podolsky Green function (41) into Eqs. (60) and (61), the one-dimensional retarded electromagnetic potentials read φ (1) ( x, t ) = c ε Z t −∞ d t ′ Z ∞−∞ d x ′ H (cid:0) cτ − | X | ) (cid:20) − J (cid:18) √ c τ − X ℓ (cid:19)(cid:21) ρ ( x ′ , t ′ )= c ε Z t −| X | /c −∞ d t ′ Z ∞−∞ d x ′ ρ ( x ′ , t ′ ) (cid:20) − J (cid:18) √ c τ − X ℓ (cid:19)(cid:21) (67)and A (1) ( x, t ) = µ c Z t −∞ d t ′ Z ∞−∞ d x ′ H (cid:0) cτ − | X | ) (cid:20) − J (cid:18) √ c τ − X ℓ (cid:19)(cid:21) J ( x ′ , t ′ )= µ c Z t −| X | /c −∞ d t ′ Z ∞−∞ d x ′ J ( x ′ , t ′ ) (cid:20) − J (cid:18) √ c τ − X ℓ (cid:19)(cid:21) (68)since H ( cτ − | X | ) = 0 for t ′ > t − | X | /c . It can be seen that the one-dimensional retardedpotentials (67) and (68) draw contribution emitted at all times t ′ from −∞ up to t −| X | /c .In the Bopp-Podolsky electrodynamics, the retarded potentials possess an afterglowin 1D, 2D and 3D since they draw contribution emitted at all times t ′ from −∞ up to t − R/c unlike in the classical Maxwell electrodynamics where only the retarded potentialspossess an afterglow in 1D and 2D (see, e.g., [36, 38]).14 .2 Retarded electromagnetic field strengths
Substituting Eqs. (58) and (59) into the electromagnetic fields (3) and (4) or solvingEqs. (15) and (16), the electromagnetic fields ( E , B ) are given by the convolution of theGreen function G BP and the given charge and current densities ( ρ , J ) and read as E ( n ) ( r , t ) = − ε Z t −∞ d t ′ Z R n d r ′ (cid:16) ∇ G BP( n ) ( r − r ′ , t − t ′ ) ρ ( r ′ , t ′ )+ 1 c ∂ t G BP( n ) ( r − r ′ , t − t ′ ) J ( r ′ , t ′ ) (cid:17) , (69) B ( n ) ( r , t ) = µ Z t −∞ d t ′ Z R n d r ′ ∇ G BP( n ) ( r − r ′ , t − t ′ ) × J ( r ′ , t ′ ) . (70) Substituting the derivatives of the Bopp-Podolsky Green function (43) and (44) intoEqs. (69) and (70), the three-dimensional retarded electromagnetic field strengths read as E (3) ( r , t ) = 18 πε ℓ Z R d r ′ (cid:18) R R ρ (cid:0) r ′ , t − R/c (cid:1) − c J (cid:0) r ′ , t − R/c (cid:1)(cid:19) − c πε ℓ Z t − R/c −∞ d t ′ Z R d r ′ (cid:2) R ρ ( r ′ , t ′ ) − τ J ( r ′ , t ′ ) (cid:3) ( c τ − R ) J (cid:18) √ c τ − R ℓ (cid:19) (71)and B (3) ( r , t ) = − µ πℓ Z R d r ′ R R × J (cid:0) r ′ , t − R/c (cid:1) + µ c πℓ Z t − R/c −∞ d t ′ Z R d r ′ R × J ( r ′ , t ′ )( c τ − R ) J (cid:18) √ c τ − R ℓ (cid:19) . (72)In the first part of Eqs. (71) and (72), the δ -function in Eqs. (43) and (44) picked outthe value of ρ and J at the retarded time, t − R/c , which is earlier than t by as longas it takes a signal with speed c to travel from the source point r ′ to the field point r .From each point r ′ , the first part of Eqs. (71) and (72) draws contributions emitted at theretarded time t − R/c . The second part of Eqs. (71) and (72) is due to the discontinuouspart of Eqs. (43) and (44) and they draw contribution emitted at all times t ′ from −∞ up to t − R/c .It can be seen that Eqs. (71) and (72) have some similarities but also differencesto the so-called Jefimenko equations in Maxwell’s electrodynamics [42] (see also [43]).The differences are based on the appearance of the Bopp-Podolsky Green function (32)in the Bopp-Podolsky electrodynamics instead of the Green function of the d’Alembertoperator (30) in the Maxwell electrodynamics.
In two-dimensional electrodynamics, the magnetic field strength is a scalar field B (2) = ∇ × A (2) = ǫ ij ∂ i A j , where ǫ ij is the two-dimensional Levi-Civita tensor, and the electricfield strength E (2) = ( E x , E y ) is a two-dimensional vector field (see, e.g., [44]).15ubstituting the derivatives of the Bopp-Podolsky Green function (48) and (49) intoEqs. (69) and (70), the two-dimensional retarded electromagnetic field strengths become E (2) ( r , t ) = − c πε Z t − R/c −∞ d t ′ Z R d r ′ (cid:2) R ρ ( r ′ , t ′ ) − τ J ( r ′ , t ′ ) (cid:3) ( c τ − R ) (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19) − √ c τ − R ℓ sin (cid:18) √ c τ − R ℓ (cid:19)(cid:21) (73)and B (2) ( r , t ) = µ c π Z t − R/c −∞ d t ′ Z R d r ′ R × J ( r ′ , t ′ )( c τ − R ) (cid:20) − cos (cid:18) √ c τ − R ℓ (cid:19) − √ c τ − R ℓ sin (cid:18) √ c τ − R ℓ (cid:19)(cid:21) , (74)where R × J = ǫ ij R i J j . The two-dimensional retarded electromagnetic field strengths (73)and (74) show an afterglow, since they draw contribution emitted at all times t ′ from −∞ up to t − R/c . This version of the Bopp-Podolsky electrodynamics in one spatial dimension has a scalarelectric field and no magnetic field (see, e.g., [45] for classical electrodynamics in onespatial dimension).Substituting the derivatives of the Bopp-Podolsky Green function (54) and (55) intoEqs. (69) and (70), the one-dimensional retarded electromagnetic field strengths read as E (1) ( x, t ) = c ε ℓ Z t −| X | /c −∞ d t ′ Z ∞−∞ d x ′ (cid:2) Xρ ( x ′ , t ′ ) − τ J ( x ′ , t ′ ) (cid:3) √ c τ − X J (cid:18) √ c τ − X ℓ (cid:19) , (75) B (1) ( x, t ) = 0 . (76)The one-dimensional retarded electric field strength (75) possesses an afterglow, becauseit draws contribution emitted at all times t ′ from −∞ up to t − | X | /c . We consider a non-uniformly moving point charge carrying the charge q at the position s ( t ). The electric charge density and the electric current density vector are given by ρ ( r , t ) = q δ ( r − s ( t )) , J ( r , t ) = q V ( t ) δ ( r − s ( t )) , (77)where V ( t ) = ∂ t s ( t ) = ˙ s ( t ) is the arbitrary velocity of the non-uniformly moving pointcharge. We consider the case that the velocity of the point charge is less than the speed oflight: | V | < c . Therefore, the retarded potentials for non-uniformly moving point chargeslead to the generalized Li´enard-Wiechert potentials in the framework of Bopp-Podolskyelectrodynamics. 16 .1 Generalized Li´enard-Wiechert potentials Substituting Eq. (77) into Eqs. (63) and (64) and performing the spatial integration, thethree-dimensional generalized Li´enard-Wiechert potentials read as φ (3) ( r , t ) = qc πε ℓ Z t R −∞ d t ′ p c τ − R ( t ′ ) J (cid:18) p c τ − R ( t ′ ) ℓ (cid:19) (78)and A (3) ( r , t ) = µ qc πℓ Z t R −∞ d t ′ V ( t ′ ) p c τ − R ( t ′ ) J (cid:18) p c τ − R ( t ′ ) ℓ (cid:19) , (79)where R ( t ′ ) = r − s ( t ′ ) and the retarded time t R being the root of the equation (cid:2) x − s x ( t R ) (cid:3) + (cid:2) y − s y ( t R ) (cid:3) + (cid:2) z − s z ( t R ) (cid:3) − c ( t − t R ) = 0 . (80)Due to the condition | V | < c , there is only one solution of Eq. (80) which is the retardedtime t R . The three-dimensional generalized Li´enard-Wiechert potentials (78) and (79)draw contributions emitted at all times t ′ from −∞ up to t R . The generalized Li´enard-Wiechert potentials (78) and (79) are in agreement with the expressions given by Land´eand Thomas [29]. Substituting Eq. (77) into Eqs. (65) and (66) and performing the spatial integration, thetwo-dimensional generalized Li´enard-Wiechert potentials become φ (2) ( r , t ) = qc πε Z t R −∞ d t ′ p c τ − R ( t ′ ) (cid:20) − cos (cid:18) p c τ − R ( t ′ ) ℓ (cid:19)(cid:21) (81)and A (2) ( r , t ) = µ qc π Z t R −∞ d t ′ V ( t ′ ) p c τ − R ( t ′ ) (cid:20) − cos (cid:18) p c τ − R ( t ′ ) ℓ (cid:19)(cid:21) , (82)where R ( t ′ ) = r − s ( t ′ ) and the retarded time t R being the root of the equation (cid:2) x − s x ( t R ) (cid:3) + (cid:2) y − s y ( t R ) (cid:3) − c ( t − t R ) = 0 . (83)It can be seen that the two-dimensional generalized Li´enard-Wiechert potentials (81) and(82) draw contributions emitted at all times t ′ from −∞ up to t R .17 .1.3 1D Substituting Eq. (77) into Eqs. (67) and (68), the spatial integration can be performed togive the one-dimensional generalized Li´enard-Wiechert potentials φ (1) ( x, t ) = qc ε Z t R −∞ d t ′ (cid:20) − J (cid:18) p c τ − X ( t ′ ) ℓ (cid:19)(cid:21) (84)and A (1) ( x, t ) = µ qc Z t R −∞ d t ′ V ( t ′ ) (cid:20) − J (cid:18) p c τ − X ( t ′ ) ℓ (cid:19)(cid:21) , (85)where X ( t ′ ) = x − s ( t ′ ) and t R is the retarded time, which is the root of the equation (cid:2) x − s ( t R ) (cid:3) − c ( t − t R ) = 0 . (86)Also the one-dimensional generalized Li´enard-Wiechert potentials (84) and (85) drawcontributions emitted at all times t ′ from −∞ up to t R .Like the retarded potentials in the Bopp-Podolsky electrodynamics, the generalizedLi´enard-Wiechert potentials in 3D, 2D and 1D, Eqs. (78), (79), (81), (82) and (84), (85)depend on the entire history of the point charge up to the retarded time t R and contain“tail terms”. Substituting Eq. (77) into Eqs. (71) and (72) and performing the spatial integration (see,e.g., [46, 47, 43]), the three-dimensional electromagnetic fields in the generalized Li´enard-Wiechert form read as E (3) ( r , t ) = q πε ℓ (cid:20) R ( t ′ ) P ( t ′ ) − V ( t ′ ) R ( t ′ ) cP ( t ′ ) (cid:21) t ′ = t R − qc πε ℓ Z t R −∞ d t ′ R ( t ′ ) − τ V ( t ′ )( c τ − R ( t ′ )) J (cid:18) p c τ − R ( t ′ ) ℓ (cid:19) (87)and B (3) ( r , t ) = − µ q πℓ (cid:20) R ( t ′ ) × V ( t ′ ) P ( t ′ ) (cid:21) t ′ = t R + µ qc πℓ Z t R −∞ d t ′ R ( t ′ ) × V ( t ′ )( c τ − R ( t ′ )) J (cid:18) p c τ − R ( t ′ ) ℓ (cid:19) , (88)where P ( t ′ ) = R ( t ′ ) − V ( t ′ ) · R ( t ′ ) /c . (89)18n the first part of Eqs. (87) and (88), the expression inside the brackets has to be takenat the retarded time t ′ = t R , which is the unique solution of Eq. (80). The second part ofEqs. (87) and (88) draws contribution emitted at all times t ′ from −∞ up to the retardedtime t R . Note that the term R ( t ′ ) /P ( t ′ ) in the first part of Eqs. (87) and (88) possessesa (directional) discontinuity (see also [14]). Substituting Eq. (77) into Eqs. (73) and (74) and performing the spatial integration, thetwo-dimensional electromagnetic fields in the generalized Li´enard-Wiechert form become E (2) ( r , t ) = − qc πε Z t R −∞ d t ′ R ( t ′ ) − τ V ( t ′ )( c τ − R ( t ′ )) (cid:20) − cos (cid:18) p c τ − R ( t ′ ) ℓ (cid:19) − p c τ − R ( t ′ ) ℓ sin (cid:18) p c τ − R ( t ′ ) ℓ (cid:19)(cid:21) (90)and B (2) ( r , t ) = µ qc π Z t R −∞ d t ′ R ( t ′ ) × V ( t ′ )( c τ − R ( t ′ )) (cid:20) − cos (cid:18) p c τ − R ( t ′ ) ℓ (cid:19) − p c τ − R ( t ′ ) ℓ sin (cid:18) p c τ − R ( t ′ ) ℓ (cid:19)(cid:21) . (91)It can be seen that the two-dimensional electromagnetic fields (90) and (91) draw contri-butions emitted at all times t ′ from −∞ up to t R , being the unique solution of Eq. (83). Substituting Eq. (77) into Eqs. (75) and (76), the spatial integration can be performed togive the one-dimensional electromagnetic fields in the generalized Li´enard-Wiechert form E (1) ( x, t ) = qc ε ℓ Z t R −∞ d t ′ X ( t ′ ) − τ V ( t ′ ) p c τ − X ( t ′ ) J (cid:18) p c τ − X ( t ′ ) ℓ (cid:19) , (92) B (1) ( x, t ) = 0 . (93)Thus, the one-dimensional electric field (92) draws contributions emitted at all times t ′ from −∞ up to t R , which is the unique solution of Eq. (86). We have investigated the Bopp-Podolsky electrodynamics as prototype of a dynamical gra-dient theory with weak nonlocality in space and time. The retarded potentials, retardedelectromagnetic field strengths, generalized Li´enard-Wiechert potentials and electromag-netic field strengths in generalized Li´enard-Wiechert form have been calculated for 3D, 2Dand 1D and they depend on the entire history from −∞ up to the retarded time t R . The19able 1: Behaviour of the Green function of the Bopp-Podolsky electrodynamics and itsfirst derivatives on the light cone.Spatial dimension Green function G BP First derivatives of G BP
3D finite and discontinuous singular and discontinuous2D approaching zero singular and discontinuous1D approaching zero finite and discontinuousBopp-Podolsky field is a superposition of the Maxwell field describing a massless photonand the Klein-Gordon field describing a massive one. In particular, the Klein-Gordonpart of the Bopp-Podolsky field gives rise to a decreasing oscillation around the classicalMaxwell field. The Green function of the Bopp-Podolsky electrodynamics and its firstderivatives have been calculated and studied in the neighbourhood of the light cone (seetable 1). It turned out that the Bopp-Podolsky Green function is the regularization ofthe Green function of the d’Alembert operator: G BP = reg (cid:2) G (cid:3) (cid:3) , (94)corresponding to the simplest case of the Pauli-Villars regularization with a single “aux-iliary mass” proportional to 1 /ℓ . The Green function of the Klein-Gordon operator playsthe mathematical role of the regularization function in the Bopp-Podolsky electrodynam-ics. Moreover, the retarded Bopp-Podolsky Green function and its first derivatives showdecreasing oscillations inside the forward light cone. The behaviour of the electromag-netic potentials and electromagnetic field strengths on the light cone is obtained fromthe behaviour of the Green function and its first derivatives in the neighbourhood of thelight cone. Only in 1D the electric field strength of the Bopp-Podolsky electrodynamicsis singularity-free on the light cone. In 2D and 3D, the electromagnetic field strengths inthe Bopp-Podolsky electrodynamics possess weaker singularities than the classical singu-larities of the electromagnetic field strengths in the Maxwell electrodynamics. In orderto regularize the 2D and 3D electromagnetic field strengths in the Bopp-Podolsky elec-trodynamics towards singular-free fields on the light cone, generalized electrodynamics ofhigher order might be used. Acknowledgement
The author gratefully acknowledges the grant from the Deutsche Forschungsgemeinschaft(Grant No. La1974/4-1).
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