New Equation of Motion of an Electron: the Covariance of Self-action

Transcription

1 New Equation of Motion of an Eletron: the Covariane of Self-ation Xiaowen Tong Sihuan University Abstrat It is well known that our knowlege about the raiation reation of an eletron in lassial eletroynamis is unambiguous, but the self-ation is not. The latter orrespons to an eletromagneti mass whih is not relativistially ovariant. In this paper we first erive a new formula for energy ensity of eletrostati fiels. By establishing a elay oorinate system, a lassial equation of motion of an eletron is then obtaine base on the onservation of energy an momentum. Finally we alulate the self-energy of an eletron in quantum eletroynamis an fin that it merely leas to an aitional mass of the eletron. Thus the ovariane of the self-ation is prove without altering lassial eletroynamis but with a iret ut-off impose on the integral of the self-energy. The etail that the self-ation beomes ovariant in quantum eletroynamis is unknown. However, the interation energy of an eletron interating with vauum flutuations an be easily alulate by assuming that every moe of the raiation fiels is oupie by one real photon. Making use of all the results we obtain a semi-lassial an ovariant equation of motion of an eletron. Energy of Eletrostati Fiels In many books about lassial eletroynamis[], the eletrostati energy ensity is erive from W = 2 u = 8π E2 () n q i V (r i ) (2) i= () an not be orret sine one will get an infinite result after integrating it while (2) is always finite. Obviously () ontains the potential energy share Corresponing author: yztsztxw@63.om

2 by harges as well as the fiel energy owe by harges themselves whih remains onstant in eletrostati. (2) is just the former. The latter shall not be ounte in the eution of (). Hene we will give a new erivation about the eletrostati energy ensity. We rewrite (2) as W = 2 n q i V /ii (3) i= where V /ii is the total potential at r i proue by all the harges exept the harge i. The seon subsript of V /ii means at r i. In orer to use integral afterwars we write q i as q i = ρ i V i = q i δ(r i ) V i where V i is a small volume at point r i. Substitute this equation into (3) we obtain W = 2 n q i δ(r i ) V i V /ii (4) i= Aoring to Gauss s law we have at r i E = 4πρ i = 4πq i δ(r i ) We use E ij to enote the fiel strength at r j proue by q i. Then the eletri fiel strength E at r i an be split into two parts E = E ii + E /ii Where the two parts satisfy respetively an E ii = 4πq i δ(r i ) (5) E /ii = E i + E 2i + + E i,i + E i+,i + E ni = (6) The reason for (6) is that exept the harge q i all the other harges are not at r i, i.e. E ki =, for k i Thus we an retain the term E /ii or abanon it. Retaining it means we are alulating the ation V /ii of on E /ii, whih is exatly the self-ation. Therefore we abanon it. Plugging (5) into (4) we get W = 8π n E ii V /ii V i (7) i= The meaning of the subsripts of E /ii is the same as that of V /ii. Similarly, E ii enotes the fiel strength at r i proue only by q i. 2

3 As mentione previously, the ivergene of E ij vanishes everywhere exept at r i. We an a the term below to every term of the sum on the rhs of (7) Consequently W = 8π = 8π = 8π j=,j i E ij V /ij V j n E ii V /ii V i + i= n i= j= n i= E ij V /ij V j V E i V /i V j=,j i E ij V /ij V j In the last step we have thrown away the seon inex of E ij an V /ij sine we are integrating them in a volume V. V shall be big enough to inlue all the harges. After applying an integration by parts we obtain W = 8π n i= ( ) E i V /i V + V /i E i S V S If we hoose the volume V to be the whole spae, the seon integral in the braket of (8) woul vanish. Due to (8) beomes W = 8π V /i = E /i n i= Consier a system of two point harges, we have (8) E i E /i V (9) (9) gives E / = E 2, E /2 = E W = 4π E E 2 V () (3) being always finite guarantees the same hols for (9). The form of (9) implies that the energy of eletrostati fiels origins from interation between harge partiles. One an verify easily () for two point harges[2]. 3

4 2 A Classial Equation of Motion of an Eletron The suess of eliminating the self-ation of a point harge in last setion motivates us to go further. If we ontinue to apply the metho to general eletromagneti phenomena in whih eletrons are usually in motion, however, we woul still obtain only the energy of interation. As it is known, when an eletron is aelerate its own fiel will twist, whih implies the fiel has inee hange. In aition, eletromagneti raiation always takes away some energy an momentum from the eletron. Therefore reation of the fiel exerting on the eletron shall exist. It is expete then we woul enounter some infinity in view of (). But we shall keep in min that only the hange of the energy an the momentum of the fiel, whih orrespons to the reation, is meaningful to us. In other wors, the motion of the eletron interating with the eletromagneti fiel may be learne by alulating the hange of the latter. We want to obtain an equation of motion with manifest Lorentz ovariane. To this en, it is suitable to aopt the ovariant form of eletroynamis. The equations for the eletron interating with eletromagneti fiels are p µ meh t = F µν totj ν 3 x () µ F µν tot = 4π jν (2) where p µ µν meh is the mehanial energy-momentum vetor of the eletron an F tot is the total fiel strength tensor. Aoring to the priniple of superposition, the latter an be split into two terms, namely, those of external fiels an that of the fiel of the eletron itself: F µν tot = F µν sel + F µν ext (3) There aren t any other harge partiles exept the eletron in the spae we onsier. Therefore, µ F µν ext = In view of (2) we obtain then a equation Using the four-potential we write F µν sel as In Lorentz gauge (4) now beomes We use g µν = iag{,,, } in this setion. µ F µν sel = 4π jν (4) F µν sel = Aν sel Aµ sel (5) x µ x ν A µ sel = 4π jµ (6) 4

5 Let χ µ (τ) be the worl-line funtion of the eletron whih we regar it as a point harge. The urrent ensity is then j µ (x) = e δ(x χ(τ))v µ (τ)τ In orer to use some elay quantities to express the solution of (6), we introue a vetor u µ whih satisfies u µ u µ =, u µ v µ = The potential to be solve at x µ is proue by the eletron when it was at χ µ. If we let The elay relation is then We also introue an invariant (7) now an be written R µ = x µ χ µ (7) R µ R µ = ρ = Rµ v µ We fin that the solution of (6) is[3] R µ = ρ(u µ + vµ ) (8) A µ sel (x) = e v µ ρ The fiel strength tensor an be alulate from (5). The result is[4] F µν sel = e ρ 2 (vµ u ν v ν u µ )+ e 2 ρ [ (aµ v ν a ν v µ )+u ν (a µ + a uv µ ) u µ (a ν + a uv ν )] (9) where a u is efine by a u = a µ u µ We are intereste in the symmetri tensor whih is given by Θ µν tot = 6π gµν (F tot ) αβ (F tot ) αβ + 4π (F tot) µα (F tot ) α ν Substituting (3) into it yiels where Θ µν tot = Θ µν sel + Θµν ext + Θ µν rs 5

6 Θ µν rs = 8π gµν (F sel ) αβ (F ext ) αβ + 4π [(F ext) µα (F sel ) α ν +(F ext ) να (F sel ) α µ ] (2) We all Θ µν rs the ross term. It is ompose of the fiel variables of the eletron an the external fiel variables while Θ µν sel ontains only the former an Θµν ext the latter. From (9) it an be shown that Θ µν sel = e2 4πρ 4 (uµ u ν vµ v ν 2 2 gµν ) }{{} Θ µν v + e2 4π 4 ρ 2 (a u 2 a λ a λ ) Rµ R ν ρ }{{ 2 } Θ µν a =Θ µν v + Θ µν va + Θ µν a 2π 2 ρ 3 (a R µ R ν v (µ R ν) u ρ 2 a u a(µ R ν) ) ρ ρ }{{} Θ µν va + e2 The ivergene of the total symmetri tensor is µ Θ µν tot = F µν totj ν (2) We integrate the two sies of the above equation in a four-imensional volume V of whih we hoose the time oorinate varies from to t. Due to () it follows that t 4 x µ Θ µν tot = t pν meh (22) V t Aoring to Gauss s theorem, the lhs of the above equation an be rewritten as 4 x µ Θ µν tot = Θ µν tot V Σ where n µ is the tangent vetor of the three-imensional area element S. ε(n) is + when S is spae-like an when S is time-like. Suppose that the relation between the oorinate time t an the proper time τ of the eletron is We set t = t(τ) t() =, t(τ ) = t Differentiate both sies of (22) with respet to τ we obtain τ Sε(n)n Θ µν tot µ = pv meh (23) τ Σ 6

7 We hoose that the volume V ontains only the eletron being stuie. Hene, µ Θ µν ext = Thus we o not nee to onsier the external term in (22). Consequnetly, (23) beomes τ (Θµν sel + Θµν rs) = pv meh (24) τ Σ We assume that the eletron was moving uniformly with veloity v until it begins to aelerate at t = uner the ation of the external fiel, as shown below. The shape of volume V an be hosen arbitrarily without affeting our result of alulation as long as the eletron is loate in V. We hoose it to be a retangle so that the alulation woul be simple. As exhibite in the figure, the furthest points at whih the eletromagneti signals arrive at t are labele by P an P. With the exeption of the segment P P, we enote with Σ the remainer of the bounary Σ. The eletromagneti fiels on Σ are all proue by the uniform motion of the eletron, of whih we use Θ µν v to represent the symmetri tensor. When evaluating the integral on the lhs of (24), we on t Figure : the worl-line of the eletron in two imensional spae-time. nee to alulate iretly that involving Θ µν v. If the eletron ontinues to move uniformly after t = with the absene of external fiels, we have from (23) or τ Σ Θµν v = P ( τ P Θµν v + Σ Θµν v ) = τ Σ Θµν v = P τ P Θµν v For the sake of brevity, we rew the iagram in only two imensions. (25) 7

8 is Aoring to (2) an the above equation, the first term on the lhs of (24) τ = τ [ = τ Σ P Θµν sel P P (24) beomes then (Θµν a + Θ µν va + Θ µν v ) + P (Θµν a + Θ µν va + Θ µν v Θ µν v ) Σ Θµν v ] τ Σ Θµν rs+ P τ P (Θµν a +Θ µν va +Θ µν v Θ µν v ) = pv meh τ (26) To get an equation of motion of the eletron, what we nee to o next is to alulate the integrals on the lhs of the above equation. The alulation has been exhibite in Appenix A. The result is(see (A.)) p meh τ E meh τ + ( 2 τ 3 ( + τ e 2 ) ( γβ = γ ee + v B a ) 2 e 2 γ ( + 3 β2) a = γ ev E ) 2 e aλ a λ γv 2 e aλ a λ γ (27a) (27b) where a is the raius of ut-off. Note that this equation is exat without any approximation. 3 Moifiation from Quantum Eletroynamis We immeiately reognize the terms in the brakets on the LHS of (27) if we write own the energy an the momentum aoring to (A.5) an (A.6) of the fiel of a point harge moving uniformly p fiel = 2 e 2 γβ, E fiel = e 2 γ( + 3 β2 ) (28) 3 a 2 a For omparison, the eletrostati energy of a point harge is e 2 /2a. Therefore, the terms in the brakets on the LHS of (27), whih o not form a 4-vetor, orrespon to the eletromagneti mass of an eletron. The LHS of (22) is generally a 4-vetor unless the symmetri tensor Θ µν tot has singularities, whih is our ase in virtue of (2). An interesting aspet of our result (27) is that the raiation reation 2 e a λa λ v µ 8

9 exatly onstitutes a 4-vetor, although Θ µν a has a quarati singularity ρ 2. This is ue to the quarati zero of the volume element of integral (See (A.4)). It has been a problem up to now[5] that the eletromagneti mass being not ovariant. It is very interesting that if the eletron moves with veloity, the energy an the momentum of the fiel woul form a 4-vetor in view of (28). But this zero an not anel the ivergent in Θ µν v an Θ µν va. To solve the problem many attempts have be mae[2]. One of them, for example, was brought forwar by Born an Infel. They moifie the Maxwell equations in the region of small istane away from an eletron to reue the infinity of the fiels to a finite value. But we are not in the situation yet in whih we have to propose hypotheses. Beause it has been prove in quantum eletroynamis that the effet of the self-ation of an eletron an be inlue in the observe mass[6]. However, it seems that the meaning of this proeurerenormalization, hasn t been larifie thoroughly, espeially the link between it an the lassial theory. The self-energy of a free eletron an be alulate by using the perturbation theory. What the hange of the wave-funtion the self-ation leas to is just a fator e i ET. The first-orer of the hange is given by[7](ūu =, g µν = {,,, }) lim i ET = T N e2 4 x 4 y ψ(x)( iγ µ )is F (x y)( iγ ν )id µν F (x y)ψ(y) (29) where N is a normalization fator. It equals to (2π) 3 δ 3 () if ontinuum normalization is applie to ψ(x). After integrating over one of the oorinate variables, we get E = e 2 i It reas in momentum spae E = 4πe 2 i m E 4 x ψ(x)γ µ S F (x y)γ ν D µν F (x y)ψ(y) 4 k (2π) 4 ūγµ /p /k m γ µu k 2 (3) This integral ivergenes. One an regulate it using Pauli-Villars proeure or Feynman s metho. Instea of using the usual regularization, we apply iretly a ut-off to the integral with the help of Wik rotation[9]. This proeure, whih uts off integrals at a large momentum, is essentially equivalent to the metho we aopte in Appenix A, i.e. igging out a sphere from the volume we onsier in the instantaneously rest frame, while in Feynman s metho eletroynamis have been moifie in the short-range aroun an eletron[7]. To illustrate our metho, let us alulate first 4 k (2π) 4 k 2 + iε L we make a substitution: k ik, so that 9

10 k E is obviously a Lorentz invariant. Therefore, k 2 = (k 2 + k 2 ) = k 2 E (3) 4 k (2π) 4 k 2 + iε L = i (2π) 4 = i (2π) 4 4 k E k 2 E + L ke 3 Ω 4 k E ke 2 + L = i 8π 2 (λ2 2 L 2 log λ2 + L L ) Here in the last step we have ut off the integral at λ. Replaing the integration variable k with k p gives 4 k (2π) 4 k 2 2p k = i 8π 2 (λ2 2 + p2 log λ2 + + p p 2 ) Differentiating both sies of the above equation with respet to an p α respetively yiels 4 k (2π) 4 (k 2 2p k ) 2 = i [ 8π 2 2 log λ2 + + p 2 λ 2 ] + p 2 2(λ p 2 ) 4 k (2π) 4 k α (k 2 2p k ) 2 = i [p 6π 2 α log λ2 + + p 2 + p 2 λ 2 ] p α λ p 2 With the help of these integrals an Feynman parameters, we are now able to alulate (3): Or E = m E e 2 2π 3m log λ m m = e2 2π 3m log λ m This aitional mass woul lea to aitional momentum an energy whih are of ourse relativistially ovariant. This is what we faile to ahieve in the regime of lassial theory. Thus we on t nee to propose any assumption about the struture of eletrons. The self-ation of an eletron in quantum Note that the way we prove the ovariane is not the same as that we use in previous isussion about lassial eletroynamis. What the perturbation theory of quantum mehanis onerns is the hange of the energies of free eigenstates when their momentums are fixe. At least it is so when one only nee to explain the ovariane of the self-ation.

11 eletroynamis iffers greatly with that in lassial theory. The ifferene an be lassifie into three aspets: spin, vauum flutuations an the eletrons in negative energy states in vauum[8]. It is possible to extrat out separately their ontributions from the total m so that the reason will be reveale why the self-ation in quantum theory is ovariant. For example, we an alulate the ontribution of vauum flutuations here. The vauum flutuations are the outome of not being null of the zeropoint energy of the quantize raiation fiel. Unlike the energies of the states in whih there are real photons being present, every moe of the zero-point energy posseesses a half of one quanta. It an be seen, however, from the spontaneous emission that effet of a half of one quanta belonging to the zeropoint osillations to ause transitions is the same as one real photon. Therefore, we infer that interation between the eletron an the states of whih every moe is oupie by one real photon woul lea to the same energy as that ause by the zero-point osillations. Analogous to the self-ation, the moifiation ause by the former to the wave funtion of the free eletron is to multiply it a phase fator e i E vf T. The first-orer of the moifiation orrespons to the proess where only one photon is absorbe or emitte. The interation energy E vf, therefore, an be alulate as follows: The S-matrix element of the Compton sattering in whih the eletron is sattere into the same eigenstate as its initial state is S(k, λ) =( ie) 2 4 x 4 m 4π x 2 E 2ω ūγ µis F (x x 2 )γ ν ue ip x e ip x2 [ε µ (k, λ)ε ν (k, λ)e ik x e ik x2 + ε µ (k, λ)ε ν (k, λ)e ik x e ik x2] Sine there is one photon in every moe, we nee to sum all the moes to get the moifiation of the wave funtion: In momentum spae, lim T i E vf T = N 2 λ= 3 k 3 S(k, λ) (32) (2π) E vf =4πe 2 m 2 3 k E (2π) 3 2ω k λ= [ ] (33) ū ε(k, λ) p + k m ε(k, λ)u + ū ε(k, λ) p k m ε(k, λ)u Using we obtain E vf = 4πe2 E a b = b a + 2a b, a2 = a 2 3 k (2π) 3 = 4πe2 ω k E 4 k (2π) 4 2i k 2 + iε = m E e 2 2πm λ2

12 or m vf = e2 2πm λ2 where the ientity k 2 = has been applie sine the photons are all real. This result is in aorane with previous treatments[8]. While the ontribution of the vauum flutuations an be easily alulate, the other terms of the self-energy are not. Nevertheless, it is enough for us to be able to moify our equation of motion in the last setion sine we have prove the ovariane of the self-ation of a free eletron without moifying the lassial eletroynamis(with only a ut-off). When an eletron is not free, the ivergene part of the effet of the self-ation an also be inlue in the observe mass(vertex orretion). Another ivergene in quantum eletroynamis is the vauum polarization. It results in the harge renormalization. Besies these ivergenes, we an neglet the other quantum effets when onsiering the semi-lassial equation of motion of an eletron. Therefore, instea of (27) we obtain p meh τ E meh τ or in the ovariant form + ( τ ( mγv) = γ ee + v B ) 2 e aλ a λ γv + ( mγ 2 ) = γ ev E 2 e 2 τ 3 4 aλ a λ γ p µ τ = e F extv µα α 2 e a λa λ v µ where the mass an the harge are of observe values. Unlike Abraham-Lorentz equation[2] or Lorentz-Dira equation[4], it oes not ontain a thir orer erivative of the position of the eletron. 4 Conlusion To erive the eletrostati energy ensity, we nee to eompose the total fiels into self-fiels an external-fiels. The perspetive enable us to obtain an lassial equation of motion of an eletron without any approxymation. As is expete, the eletromagneti mass ontaine in the equation is not relativistially ovariant but in quantum eletroynamis. To reveal the unerlying ause of this situation, further stuy is neee. As a first step, we extrat out the ontribution of vauum flutuations to the total self-energy by proposing a moel for them. The moel may be useful in other problems involving vauum flutuations. Our result is half of Weisskopf s. This is ue to (3), i.e. the square of the ut-off momentum is ouble that Weisskopf use. This obviously oesn t affet the logarithmi epenene of the self-energy on λ. 2

13 Appenix A We first establish a oorinate system for the four-imensional spae-time. It is onvenient to employ the elay quantities as the oorinates. From (7) an (8), we obtain x µ = χ µ (τ) + ρ(u µ + vµ ) (A.) Consier the ase where the eletron is moving along the x axis of our referene frame. We set χ µ (τ) = (t(τ), x(τ),, ) (A.2) v µ = (γ, γv,, ) (A.3) Note that the spee v is a unknown funtion of τ an so is γ. ρ an u µ is R an (, ˆR) respetively in the frame where the eletron is instantaneously rest. ˆR is the unit elay vetor. More preisely, (u µ ) Rest = (, os θ, sin θ os ϕ, sin θsinϕ) We an infer then u µ is in our referene system u µ = (γβ os θ, γ os θ, sin θ os ϕ, sin θsinϕ) Aoring to (A.), (A.2), (A.3)an the above equation, every point x µ in the spae-time now an be etermine by using ρ, τ, θ an ϕ. Thus the oorinate system has been establishe as we want. The segment P P in Figure is atually a three-imensional sphere in our spae-time. To label the points of this part, one nees only 3 oorinates. In fat, the th omponent of (A.) is t = t(τ) + ργβ os θ + ργ where t is fixe an equal to t on P P. Therefore we an eliminate from the above equation a oorinate suh as ρ whih woul be given by ρ = t t(τ) γ( + β os θ) The points of P P are now labele by τ, θ an ϕ. After some umbersome alulation we fin the Jaob s eterminant from the spae omponents of (A.) (x, y, z) (τ, θ, ϕ) = ρ2 sin θ γ( + β os θ) (A.4) where we have retaine the appearane of ρ. With the above equation we are able to ompute the integrals on the lhs of (26). The normal vetor of the surfae represente by P P is n µ = (,,, ) 3

14 From (2), we get p a = P P Θµ a = 4π 2π [ γ( + β os θ)] γ(β os θ + ) τ = τ 2 e a λa λ γv where we have use: τ ϕ τ γ( + β os θ) τ p a = τ 2 e γa λa λ a λ a λ a 2 u = a λ a λ ( os 2 θ) os θ ( ) e2 4 (a u 2 a λ a λ ) The remaining omponents of p µ a are equal to. The veloity term is p v = P The term Θ µν va gives p va = P Θµ v P τ p v = τ = P Θµ va = τ p va = τ 2 3 τ e 2 γ 3 2 (t t(τ)) 2 ( + 3 β2 ) τ 2 e 2 γ 3 v 3 (t t(τ)) 2 τ τ e2 γ(2a 2 3 βa ) 2 (t t(τ)) e 2 γ(βa + a ) 2 (t t(τ)) We may expet the term p µ va an be anele out. This is true. We apply an integral by parts to p v an p v: p v = e 2 γ 2 ( + 3 β2 τ ) τ 2 (t t(τ)) + τ e2 γ(2a 2 3 βa ) 2 (t t(τ)) p v = 2 e 2 γ 2 τ β τ 3 (t t(τ)) + τ 2 e 2 γ(βa + a ) 3 2 (t t(τ)) The result is what we expete. The term orresponing to Θ µν v is p v τ = τ e 2 γ 3 ( 2 t t(τ) ) 2 ( + 3 β 2 ) = 2 p v = 2 e 2 γβ 2 3 (t γ τ) 4 τ e 2 γ 2 ( + 3 β 2 ) (t γ τ) τ (A.5) (A.6)

15 where γ = / v 2 /2 an t(τ) = γ τ, t = γ τ (A.7) is the relation between the oorinate t an the proper time τ for uniform motion of the eletron. When substituting the boun τ (or τ ) of the integrals we woul get infinities. These infinities will not bother us, as we will see in setion 3. Hene we retain the zeros in the enominators but in the limit form. For instane, t t(τ ) = lim a t t(τ a ) = lim a γ(τ ) a γ(τ ) a where we have use t(τ a ) t(τ ) γ(τ ) a a is an invariant an has the imension of length. The treating for the substituting of τ is analogous. Summing up p µ v, p µ va an p µ v together gives P τ P (Θµ va + Θ µ v Θ µ v ) = ( e 2 γ( + ) 3 β2 ) τ 2 a τ P P (Θµ va + Θ µ v Θ µ v ) = ( 2 e 2 ) γβ τ 3 a When alulating the integral of the term Θ µν rs we regar the external fiels as onstants. This an be justifie by two reasons. One is that we an make the volume V very small so that the external fiels beomes homogeneous. The seon is the eletron is subjete only to the fiel at the loation where it is. Thus we an just hoose the external fiel as a uniform eletri fiel E without loss of generality. Analogous to our previous isussion, it is not neessary to alulate iretly the part on Σ. Instea, it an be evaluate in the following way: Suppose that a onstant fore f = ee begins to at on the eletron at t = so that it keeps moving uniformly. Thus (24) beomes τ Σ (Θµν v +Θ µν rs(v )+f ν ) = τ Σ Θµν rs(v ) +f ν = pv meh τ = in virtue of (25). Here Θ µν rs(v ) is the Θµν rs when the eletron is moving uniformly with veloity v an f ν is the ovariant form of f: Hene, f ν = γ( f v, f) We an t treat the fiels of the eletron in this way beause they have a soure-the eletron, i.e. a singularity. 5

16 Now we have Σ Σ Θµν rs(v ) = f ν τ Θµν rs = Σ = f ν τ Θµν P P P P P rs(v + ) P Θµν rs(v ) Θµν rs P Θµν rs(v + ) Let the eletri fiel be along the x axis, we have then P Θµν rs (A.8) (F ext ) = E, (F ext ) = E All the other omponemts of (F ext ) µν are equal to. We only nee to onsier Θ ν rs ue to n µ. Let us alulate first its omponent. From (2) it an be shown that Aoring to (9), Θ rs = E 4π (F sel) (F sel ) = e ρ 2 (v u v u ) + = e ea os θ ρ2 2 ρ os2 θ γβ The following equations are helpful for us: a = a β, The veloity term of Θ rs gives e 2 ρ [ (a v a v ) + u (a + a uv γ v = γ3 β, a = γ τ = γ3 β v τ ) u (a + a uv )] P P S( ) E e 4π ρ 2 os θ = E 4π = ee 2 = ee 2 2π τ τ Then we an immeiately write own τ τ ϕ τ τ x γ os θ x x γ γ 2 ( + βx) [ β x + β ( + βx) e ρ 2 os θ ρ 2 γ( + β os θ) ] 6

17 P P Θµν rs(v = ee ) 2 τ = ee 2 t [ x γ β x + β ( + β x) [ x β x + β ( + β x) where (A.7) has been use in the last step. The aeleration term gives ] ] P P S( ) E ea os 2 θ 4π 2 = ee τ ρ γβ 2 = ee 2 τ = ee t 2 [t t(τ)] v τ τ [ [t t(τ)] Summing up these results aoring to (A.8) yiels τ rs = ee γβτ Σ Θµ x x + β ( + β x) + ee 2 x x2 ( + βx) 2 x x + β ] ( + βx) The treatment for the omponent similar. Thus it follows that τ Σ Θµν rs = e F extv νλ λ After olleting all the results of our alulation, we finally obtain(replae τ with τ) p meh τ E meh τ + ( 2 e 2 ) γβ τ 3 a ( + τ 2 ( = γ ee + v B e 2 γ ( + 3 β2) a ) = γ ev E ) 2 e aλ a λ γv 2 e aλ a λ γ Now we apply a ut-off on τ, i.e. we replae a with a onstant a : p meh + ( 2 e 2 ) ( γβ = γ ee + v B ) 2 e 2 τ τ 3 a 3 5 aλ a λ γv ( E meh + e 2 γ ( + 3 β2) ) = γ ev E 2 e 2 τ τ 2 a 3 4 aλ a λ γ τ γτ (A.9a) (A.9b) (A.a) (A.b) This proeure is equivalent to igging out a sphere of raius a (in the instantaneously rest referene frames) from the volume in whih we alulate the energy an momentum of the eletromagneti fiels. Although they are erive for the ase of retilinear motion, they an apply to the ases of urvilinear motion sine there is no limit on the uration of the motion of the eletron in our alulation. x x + β ( + βx) 7

18 Referenes [] D. J. Griffiths, Introution to Eletroynamis, Aison-Wesley, 999 [2] J. D. Jakson, Classial Eletroynamis, John Wiley an Sons, 998 [3] J. L. Synge, Relativity: The Speial Theory, North-Hollan, 956 [4] F. Rohrlih, Classial Charge Partiles, Aison-Wesley, 965 [5] Julian Shwinger, Founations of Physis, Vol.3, No.3, 983 [6] Julian Shwinger, Phys. Rev. 75, 65, 949 [7] R. P. Feynman, Quantum Eletroynamis, W.A. Benjamin, INC. 96 [8] V. F. Weisskopf, Phys. Rev. 56, 72, 939 [9] M. E. Peskin an D. V. Shroeer, An Introution to Quantum Fiel Theory, Perseus Books Publishing, 995: p.93 8