Single-Particle Electrodynamics - Assassination Science

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that fall off as R −1 ), then we are ignoring the mechanical energy contained in the near fields (those that fall off as R −2 )—and, indeed, the interference between the near and far fields. Of course, if the system is accelerated for a certain finite period of time, and then left alone forever after, the near fields will eventually stabilise themselves, and only the mechanical energy pilfered by the radiation fields will come off the bottom line of the balance sheet for the mechanical energy. But this small amount of information is rarely acceptable to us: we would generally like to know what is happening now to the electric charge—as it is radiating—not what it will be doing next week. Thus, we clearly need to expand our considerations, to also take into account the near fields of the charge. But now we hit another problem: as we approach the position of the point charge, we find that the field (and hence mechanical energy density) expressions grow divergently. We must therefore be extremely careful how we evaluate integrals of the mechanical field energy density: we need to “regularise” them in such a way that we avoid any danger of manipulating infinite expressions meaninglessly. One way we can do this is to let the point particle have a finite size to start off with, and then shrink it to a point at the end of the calculations. Another way is to exclude a small region surrounding the charge, and take into account the energy and momentum crossing the boundary of this region when computing our balance sheets. Whichever way we proceed, we find that we can, in fact, obtain meaningful answers. But there is another, more deeply philosophical question that one may ask: who is responsible for this radiation reaction force? We have figured out what its effects are—by some nifty detective work,—but we have not really pinpointed its fundamental source. The Lorentz force law—which we can derive, most elegantly, from the interaction Lagrangian—does not give us any clue whatsoever that such a force is lurking on our doorsteps. Where else, then, could it come from? Maybe we have to add extra terms to our fundamental Lagrangian, by hand, that yield the correct force? Unfortunately—or, 221
perhaps, fortunately,—such a “fudge factor” approach cannot even be made to work. We are thus left with the Lagrangian on the one hand, describing all of the evolution of the system except for the effects of radiation reaction; and the radiation reaction terms on the other, sticking out like the canine’s proverbials. And then Lorentz makes the statement quoted at the head of this chapter. When first heard, it invariably sounds absurd. Imagine, for example, a small sphere, uniformly charged throughout its volume. The Coulomb field of each infinitesimal volume of charge imparts an electric force on every other infinitesimal volume of charge in the sphere. Indeed, the like charges all repel, and the sphere explodes. This is not our intention; let us, therefore, add other, non-electromagnetic forces that balance the Coulomb forces, and hence bind the infinitesimal constituent charges together. These rigid body forces are completely internal, so they do not contribute to the motion of the centre of mass of the body. Lorentz’s statement then amounts to saying: let’s add up all the Coulomb forces, and see what the resultant net force on the body as a whole is. But of course it is zero: the force of the Coulomb field of constituent charge A on constituent charge B is, quite trivially, equal and opposite to that of constituent B on constituent A. But still Lorentz insists that we should add up the interactions between the constituents of the charge. To extract any physics from such an intuitively useless pastime is the mark of sheer brilliance. One is therefore completely humbled when one realises that one has, in fact, forgotten Maxwell’s words, quoted at the head of Chapter 5: “. . . without assuming the existence of forces capable of acting directly at sensible distances.” The Coulomb force is not instantaneously propagated from sender to receiver; it propagates at the speed of light. The Coulomb force felt now by constituent A is determined by the motion of constituent B as it was some time ago; the Coulomb force felt now by constituent B is likewise determined by the motion of constituent A as it was some time ago. We may still fail to see, through this observation, 222
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Single-Particle Electrodynamics Mr.
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Copyright Declaration I understand
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I apologise to Dr. Geoff Taylor for
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Contents 1 Overview of this Thesis
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3.2.2 Einsteinian rigidity . . . .
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6 Radiation Reaction 220 6.1 Introd
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A Notation and Conventions 314 A.1
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A.8.18 MCLF and CACS components . .
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E.2 Quantum field theory . . . . .
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G.7 test3int: Testing of 3-d integr
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lood pressure: such inane things ar
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1.2 What do I need to read? If the
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Section 2.7: Classical limit of qua
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Section 5.2: History of the retarde
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Section 6.7: Inverse-cube integrals
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Section A.7: Matrices Notation and
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Appendix C: Supplementary Proofs Se
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Section E.2: Quantum field theory T
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esults, not the establishment of ne
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e forgiven by their respective auth
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tron. The proton and neutron clearl
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most appropriate in the task of rem
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(see Section A.8): f ≡ d τ p. (2
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T 0i ≡ T i0 = E×B, T ij = 1 2 (E
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densities and the conservation laws
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2.4 Lagrangian mechanics An alterna
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canonical momentum: b a ≡ ∂ ˙q
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to be the appropriate degrees of fr
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task. Firstly, we need to select ap
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canonical energy, canonical momentu
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Let us return to the electric charg
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should also be carefully noted agai
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For an elementary particle, the mas
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2.6.7 Unit four-spin It is often ne
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the case. The reason, pointed out b
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period it appeared that the magnitu
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tionally reflect the difference bet
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has precessed by an amount ˙σ dt.
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is in terms of the quantities of Sp
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2.6.10 The FitzGerald three-spin In
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such systems. On the other hand, fo
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equations, one then finds, for eith
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i.e., the torque. Clearly, the corr
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The component ∆L 0 is our warning
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vanishes, for v = 0: P | v=0 = 0; (
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wish of them—not least of which b
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The author will, in the remaining c
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(see equation (A.58) of Section A.8
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6(c 1·c 3 ) + 4c 2 2 = 0, 8(c 1·c
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and reverting t(τ) from (2.91), on
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Chapter 3 Relativistically Rigid Bo
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than considering the rest frame of
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We also note Pearle’s proof of hi
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properties of the constituent r are
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where we set u(τ) = r (3.7) becaus
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+ 1 { [1 ] .... + (r· ˙v) v + 2 [
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applied than the former.) We shall
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which varies with time. We deem tha
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desirable properties—but unfortun
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the magnitude d of d describes the
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shall compute the net force on the
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we shall make the analysis relativi
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Now, let us consider such a spin-ha
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adreact of Appendix G; the expressi
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4.2.2 The magnetic-charge dipole Ma
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ut its employment of moving constit
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point of the loop at angle θ to th
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Now, the power into each positive c
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or, on reärranging the terms, −(
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Again noting the relations (4.23) a
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its discovery. Mathematically, the
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direction: E ≡ jE y . Now, let us
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(i.e., 2πε/v orb with v = 1, the
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a sufficient amount of inertia, the
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The fact that the Penfield-Haus eff
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tum of ±µ×E. Firstly, one can sh
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concepts precisely and correctly, a
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on the structure of spacetime, one
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4.3.4 Relativistic Lagrangian deriv
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we know that the operator equations
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invariance requirements. They sough
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That (4.73) is an amazing result is
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is an important example. Clearly, t
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let us therefore add a Pauli term (
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On the other hand, for a Pauli mome
Page 171 and 172: moment actually cancels with the Di
Page 173 and 174: Chapter 5 The Retarded Fields I hav
Page 175 and 176: to particles carrying dipole moment
Page 177 and 178: workers. In 1969, Cohn [53] unfortu
Page 179 and 180: manifestly-covariant field expressi
Page 181 and 182: as a mathematical aid—it not bein
Page 183 and 184: integrate over the proper time τ,
Page 185 and 186: Equation (5.28) is, in the above no
Page 187 and 188: this is Jackson’s equation (14.13
Page 189 and 190: moments of a particle. There are a
Page 191 and 192: It can be noted that equation (5.51
Page 193 and 194: the overall Lorentz-gauge four-pote
Page 195 and 196: Applying (5.61) to the delta functi
Page 197 and 198: The results above are for a particl
Page 199 and 200: (One should carefully note that the
Page 201 and 202: under differentiation, the latter e
Page 203 and 204: Because the field expressions diver
Page 205 and 206: expression (5.81) in the direction
Page 207 and 208: We now note that the non-zero elect
Page 209 and 210: clearly, for any ε, this parametri
Page 211 and 212: For a point electric dipole moment,
Page 213 and 214: where the convenient constant η 1
Page 215 and 216: the former generates solely an elec
Page 217 and 218: The position of the mechanical cent
Page 219 and 220: when the particle is in arbitrary m
Page 221: Chapter 6 Radiation Reaction Accord
Page 225 and 226: Lorentz performed this calculation
Page 227 and 228: 6.2 Previous analyses To the author
Page 229 and 230: other than pedagogy, in the past fi
Page 231 and 232: immediate practicality, since then
Page 233 and 234: 6.3 Infinitesimal rigid bodies We n
Page 235 and 236: enefit that, by Gauss’s law, ther
Page 237 and 238: particular times depend on the acce
Page 239 and 240: ground our understanding of the qua
Page 241 and 242: whence the reverse transformation i
Page 243 and 244: We define η 0 ≡ 1 ( ) 4 −2 ∫
Page 245 and 246: and ending values of each loop are
Page 247 and 248: = 1 8( 4 3 π(2ε)3 ) 2 , (6.23) wh
Page 249 and 250: = 4 3 π(2ε)5 { 1 5 − 3 4 α +
Page 251 and 252: where by the notation ∫ d 2 n d w
Page 253 and 254: 6.4.6 Angular outer integrals We no
Page 255 and 256: 6.4.7 Is there an easier way? It mi
Page 257 and 258: 6.5.3 Final expression for the reta
Page 259 and 260: Using the fact that t ret ≡ −R,
Page 261 and 262: 6.5.9 Modified retarded normal vect
Page 263 and 264: 6.6 Divergence of the point particl
Page 265 and 266: “godlike” parameter, i.e., one
Page 267 and 268: Using the chain rule on (6.74), we
Page 269 and 270: Our first use of (6.84) is to compu
Page 271 and 272: The significance of this result is
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lead to delta-function contribution
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(6.99) then gives { (a·b) − 5(n
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6.7.1 The bare inverse-cube integra
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6.7.2 Inverse-cube integral with tw
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we immediately find that the r d -i
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6.8.4 Duality symmetry consideratio
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in order that Maxwell’s equations
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This asserted result of the author
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obtain a non-vanishing integral ove
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− 1 2 ˜µ2 η 1 ( ˙v· ¨σ)σ
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If one examines the author’s calc
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performed the electric charge radia
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and the spin renormalisation term f
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motion of the particle. This has, i
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chrotron” radiation (i.e., the ra
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6.10.2 The Ternov-Bagrov-Khapaev ef
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where the characteristic polarisati
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detail, for arbitrary values of g,
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Substituting (6.151) into (6.152),
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importantly, the hyperbolic tangent
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twice, then it clearly will not do
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Appendix A Notation and Conventions
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Filename bibliog.tex claspcle.tex c
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Filename algebra.h algebra1.c algeb
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costella, british or american must
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fonts of other resolutions, or font
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typographically, the mathematical n
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quantity which has been designated
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noted that the inline fraction symb
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The d’Alembertian operator is den
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It should be noted that quantum mec
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A.3.15 Pointlike objects If an obje
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as a mathematically G-covariant qua
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The adjective enumerated will be us
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Each of the i m take values from th
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distributed factor; for example, A
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A.7.2 Square matrices Matrices with
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A.8.2 Lorentz tensors Rank-zero, ra
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AB denotes the four-tensor that is
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and G are four-tensors, then ε(A,
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The use of this signature of metric
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ackets; e.g., C α | MCLF ≡ [C α
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The first way is to simply measure
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is referred to as the parity operat
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three-vector conversion. In any cas
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A.9.9 Magnitude of a three-vector T
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A.9.13 Cross-products An alternativ
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B.2.2 The field strength tensor The
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B.2.9 Explicit electromagnetic comp
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Also, (a×b)·(c×d) ≡ (a·c)(b·
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{ (σ·∇)rd m r s (n s·A)n d = r
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For two electric charges q 1 and q
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where z is the position of the char
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Appendix D Retarded Fields Verifica
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(−2) ≡ − e { 3 a U mρ 2 2 ρ
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F µν + 1 2ȧUM [να U α R µ]
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Comparing (D.11) and (D.12), we see
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efore neutral particles were discov
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Appendix E The Interaction Lagrangi
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G µ ≡ G 1 k/k µ + G 2 k/B µ +
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u 2 σ µν u 1 −→ ˜Σ µν ,
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powers of ∂ 2 acting on the field
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International Journal of Modern Phy
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we shall not claim any predictive c
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least, if one wants to derive the e
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and electric charge interaction ter
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way that the “Lagrangian-based”
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Θ ≡ ζ 2 (E·B) , ∂ ′ ≡
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prospective solution of the combine
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Appendix G Computer Algebra Mrs. Du
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) dependence to extract the three-g
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mately chosen, both for simplicity,
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appendical reference. In following
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Program Filename Description kinema
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G.4 kinemats: Kinematical quantitie
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The reason for us naming it after F
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[ ˙ Σ 0 ] = γ 2 (v· ˙σ) + γ
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− 25γ 10 (γ + 1) −2 ˙v 2 (v
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+ 124γ 11 (v· ˙v) 2 (v·¨v), (
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+ 2γ 5 (γ + 1) −2 ( ˙v·σ)¨v
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( Σ ... ) 0∣ ∣ ∣v=0 = 0, (
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+ 4γ 8 (v· ˙v) 2 (v·σ ′ ), (
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fields. These products are labelled
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+ 3γ 4 (γ + 1) −1 (v· ˙σ)¨v
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G.5.2 Covariant field expressions S
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B ′d 1 = κ 2 ¨σ ′ ×n + κ 3
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Expanding out these expressions exp
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G.6 radreact: Radiation reaction G.
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Firstly, iterating the differential
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we define the redshift factor λ
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G.6.8 Sum and difference variables
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− 1 16 r d(r d· ˙v) 3 − 1 8 r
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− 1 120 r4 d (n d·.... v)n d + 1
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+ 1 4 r2 d (n d· ˙v) 2 ˙v − 1
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+ 1 8 r2 d r s ˙v 2 (n s· ˙v)n d
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− 1 48 r3 d r s (n d· ˙v)(n s·
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Using (G.28) and (G.55) in (G.56),
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+ 1 8 r4 d (n d· ˙v) 2 ¨σ − 1
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+ 3 2 r dr s ˙v 2 (n s· ˙v) ˙σ
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E q 2 = r −2 d n d + 1 2 r−1 d
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− r d (n d· ˙v)( ˙v· ¨σ)n d
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+ 39 4 r s(n d· ˙v)(n d·¨v)(n d
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+ 3rd −1 r s(n d· ˙v)(n d·σ)(
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+ 1 4 r s(n s·... v) ˙σ + 15 8 r
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+ 1 8 r−2 d r3 s (n d· ˙v)(n s
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− 3 4 r−2 d r2 s (n d· ˙σ)(n
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− 9 16 r s(n d·σ)(n s·¨v)¨v
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B q 1 = −rd −1 n d× ˙v + n d
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− 39 8 r d(n d· ˙v) 2 (n d·σ)
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− 1 2 r−1 d r s(n s·¨v)n d ×
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+ 3 4 r s(n d·σ)(n s· ˙v) ˙v×
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− 5 12 r d(n d·¨v)¨v×σ − 1
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If we add the various E a n and B a
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+ 15 16 r d ˙v 2 (n d· ¨σ)n d +
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+ 3 16 r−2 d r3 s (n d· ˙v)(n d
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+ 3 8 r s(n d·¨v)(n d·σ)(n s·
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+ 1 24 r(n·¨v)(¨v·σ)n − 5 24
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λE q 2 = r −2 d n d + 1 2 r−1
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+ 27 4 (n d· ˙v) 3 (n d·σ)n d
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+ 1 2 r−1 d r s(n d·σ)(n s·¨v
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− 1 2 r−1 d r s(n s· ˙v)n d
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+ 1 8 r−1 d r s(n s· ˙v)( ˙v·
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λ(σ·∇)E d 1 = −rd −2 ¨v
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− 11rd −1 (n d· ˙v)(n d·σ)(
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+ 1 6 (¨v· ¨σ)σ − 1 2 (... v
Page 523 and 524:
− 3rd −2 r2 s (n d· ˙v)(n d·
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d·... − 3rd −1 (n d·σ)(n σ)
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− 12rd −2 r s(n d·¨v)(n d·σ
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+ 101 4 (n d·¨v)(n d·σ)( ˙v·
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+ 7rd −2 (n d·σ)(¨v·σ)n d +
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− 57 2 r−1 d (n d· ˙v) 2 (n d
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+ (n d· ˙v)(n d·... σ)σ + 3(n
Page 537 and 538:
− rd −2 r2 s (n s· ˙v)(n s·
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+ 1 8 r−1 d (n d· ˙v) 2 ( ˙v·
Page 541 and 542:
P (n) ad (r d, r s ) = ˙σ·E a n(
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F (2) dq = 1 15 η 1 ˙v 2 σ − 1
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F (M) µd = −3η ′ 3σ× ˙σ,
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N F (1) dd = 1 10 η 1( ˙v·σ) ˙
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% ID string: Test 3-d integrations.
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+rd^{-3}(n.a)(n.b) +rd^{-3}(n.c)(n.
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+(n.q)(n.r)(ns.s)(ns.t)ns +(ns.u)v
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+\f{1}{105}(i.o)(j.m)(k.l)a +\f{2}{
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+rd^{-2}rs^2(n.u)(n.v)(ns.w)(ns.x)y
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+\f{1}{30}(o.p)a*m +\fourthirds a
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+\f{1}{105}(w.y)(x.z)a’*a +\f{1}{
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Bibliography [1] M. Abraham, Ann. P
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[39] H. J. Bhabha, Proc. Roy. Soc.
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[80] J. R. Ellis, J. Math. Phys. 7
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[124] M. Kolsrud and E. Leer, Phys.
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[167] M. Pavsic (1987): see [34]. [
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[212] I. M. Ternov, V. G. Bagrov an
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It doesn’t matter whether you suc
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