Single-Particle Electrodynamics - Assassination Science

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A.5.3 Heaviside step function . . . . . . . . . . . . . . . . . . 339 A.5.4 Alternating function . . . . . . . . . . . . . . . . . . . 339 A.6 Associativity and commutativity . . . . . . . . . . . . . . . . . 340 A.6.1 C-numbers and q-numbers . . . . . . . . . . . . . . . . 341 A.6.2 Commutators and anticommutators . . . . . . . . . . . 341 A.6.3 Symmetrisors . . . . . . . . . . . . . . . . . . . . . . . 342 A.7 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 A.7.1 Rows and columns . . . . . . . . . . . . . . . . . . . . 343 A.7.2 Square matrices . . . . . . . . . . . . . . . . . . . . . . 344 A.7.3 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 A.7.4 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 A.7.5 Explicit listing of matrix elements . . . . . . . . . . . . 345 A.8 Relativistic mechanics . . . . . . . . . . . . . . . . . . . . . . 345 A.8.1 Lab frames . . . . . . . . . . . . . . . . . . . . . . . . 345 A.8.2 Lorentz tensors . . . . . . . . . . . . . . . . . . . . . . 346 A.8.3 Notation for Lorentz quantities . . . . . . . . . . . . . 346 A.8.4 Raising and lowering indices . . . . . . . . . . . . . . . 347 A.8.5 Explicit listing of components . . . . . . . . . . . . . . 347 A.8.6 Outer-products . . . . . . . . . . . . . . . . . . . . . . 347 A.8.7 Inner-products, dot-products . . . . . . . . . . . . . . . 348 A.8.8 Chained dot-product rings . . . . . . . . . . . . . . . . 349 A.8.9 Epsilon products . . . . . . . . . . . . . . . . . . . . . 349 A.8.10 Cross-products . . . . . . . . . . . . . . . . . . . . . . 350 A.8.11 Wedge-products . . . . . . . . . . . . . . . . . . . . . . 351 A.8.12 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . 351 A.8.13 Timelike, spacelike, lightlike . . . . . . . . . . . . . . . 352 A.8.14 Four-position of a particle . . . . . . . . . . . . . . . . 352 A.8.15 Three-velocity of a particle . . . . . . . . . . . . . . . . 353 A.8.16 MCLF of a particle . . . . . . . . . . . . . . . . . . . . 353 A.8.17 CACS of a particle . . . . . . . . . . . . . . . . . . . . 353 13
A.8.18 MCLF and CACS components . . . . . . . . . . . . . . 353 A.8.19 Proper time of a particle . . . . . . . . . . . . . . . . . 355 A.8.20 Overdots . . . . . . . . . . . . . . . . . . . . . . . . . . 355 A.8.21 Convective derivative . . . . . . . . . . . . . . . . . . . 355 A.8.22 Kinematical proper-time derivatives . . . . . . . . . . . 355 A.8.23 Four-velocity of a particle . . . . . . . . . . . . . . . . 356 A.8.24 Mechanical four-momentum . . . . . . . . . . . . . . . 356 A.8.25 Nonrelativistic and pre-relativistic limits . . . . . . . . 357 A.8.26 C, P, T . . . . . . . . . . . . . . . . . . . . . . . . . . 357 A.9 Euclidean three-space . . . . . . . . . . . . . . . . . . . . . . . 358 A.9.1 Three-tensors . . . . . . . . . . . . . . . . . . . . . . . 358 A.9.2 Notation for three-space quantities . . . . . . . . . . . 359 A.9.3 The null three-vector . . . . . . . . . . . . . . . . . . . 359 A.9.4 The three-gradient operator . . . . . . . . . . . . . . . 359 A.9.5 Explicit listing of components . . . . . . . . . . . . . . 360 A.9.6 Mixed notation . . . . . . . . . . . . . . . . . . . . . . 360 A.9.7 Uniqueness of symbols . . . . . . . . . . . . . . . . . . 361 A.9.8 Parentheses and brackets . . . . . . . . . . . . . . . . . 361 A.9.9 Magnitude of a three-vector . . . . . . . . . . . . . . . 362 A.9.10 Outer-products . . . . . . . . . . . . . . . . . . . . . . 362 A.9.11 Dot-products . . . . . . . . . . . . . . . . . . . . . . . 362 A.9.12 Epsilon products . . . . . . . . . . . . . . . . . . . . . 363 A.9.13 Cross-products . . . . . . . . . . . . . . . . . . . . . . 364 B Supplementary Identities 365 B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 B.2 The electromagnetic field . . . . . . . . . . . . . . . . . . . . . 365 B.2.1 The four-potential . . . . . . . . . . . . . . . . . . . . 365 B.2.2 The field strength tensor . . . . . . . . . . . . . . . . . 366 B.2.3 The homogeneous Maxwell equations . . . . . . . . . . 366 14
Page 1 and 2: Single-Particle Electrodynamics Mr.
Page 3 and 4: Copyright Declaration I understand
Page 5 and 6: I apologise to Dr. Geoff Taylor for
Page 7 and 8: Contents 1 Overview of this Thesis
Page 9 and 10: 3.2.2 Einsteinian rigidity . . . .
Page 11 and 12: 6 Radiation Reaction 220 6.1 Introd
Page 13: A Notation and Conventions 314 A.1
Page 17 and 18: E.2 Quantum field theory . . . . .
Page 19 and 20: G.7 test3int: Testing of 3-d integr
Page 21 and 22: lood pressure: such inane things ar
Page 23 and 24: 1.2 What do I need to read? If the
Page 25 and 26: Section 2.7: Classical limit of qua
Page 27 and 28: Section 5.2: History of the retarde
Page 29 and 30: Section 6.7: Inverse-cube integrals
Page 31 and 32: Section A.7: Matrices Notation and
Page 33 and 34: Appendix C: Supplementary Proofs Se
Page 35 and 36: Section E.2: Quantum field theory T
Page 37 and 38: esults, not the establishment of ne
Page 39 and 40: e forgiven by their respective auth
Page 41 and 42: tron. The proton and neutron clearl
Page 43 and 44: most appropriate in the task of rem
Page 45 and 46: (see Section A.8): f ≡ d τ p. (2
Page 47 and 48: T 0i ≡ T i0 = E×B, T ij = 1 2 (E
Page 49 and 50: densities and the conservation laws
Page 51 and 52: 2.4 Lagrangian mechanics An alterna
Page 53 and 54: canonical momentum: b a ≡ ∂ ˙q
Page 55 and 56: to be the appropriate degrees of fr
Page 57 and 58: task. Firstly, we need to select ap
Page 59 and 60: canonical energy, canonical momentu
Page 61 and 62: Let us return to the electric charg
Page 63 and 64: should also be carefully noted agai
Page 65 and 66:
For an elementary particle, the mas
Page 67 and 68:
2.6.7 Unit four-spin It is often ne
Page 69 and 70:
the case. The reason, pointed out b
Page 71 and 72:
period it appeared that the magnitu
Page 73 and 74:
tionally reflect the difference bet
Page 75 and 76:
has precessed by an amount ˙σ dt.
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is in terms of the quantities of Sp
Page 79 and 80:
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
Page 87 and 88:
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
Page 93 and 94:
The author will, in the remaining c
Page 95 and 96:
(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
Page 101 and 102:
Chapter 3 Relativistically Rigid Bo
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than considering the rest frame of
Page 105 and 106:
We also note Pearle’s proof of hi
Page 107 and 108:
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 [
Page 113 and 114:
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
Page 125 and 126:
Now, let us consider such a spin-ha
Page 127 and 128:
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
Page 135 and 136:
Now, the power into each positive c
Page 137 and 138:
or, on reärranging the terms, −(
Page 139 and 140:
Again noting the relations (4.23) a
Page 141 and 142:
its discovery. Mathematically, the
Page 143 and 144:
direction: E ≡ jE y . Now, let us
Page 145 and 146:
(i.e., 2πε/v orb with v = 1, the
Page 147 and 148:
a sufficient amount of inertia, the
Page 149 and 150:
The fact that the Penfield-Haus eff
Page 151 and 152:
tum of ±µ×E. Firstly, one can sh
Page 153 and 154:
concepts precisely and correctly, a
Page 155 and 156:
on the structure of spacetime, one
Page 157 and 158:
4.3.4 Relativistic Lagrangian deriv
Page 159 and 160:
we know that the operator equations
Page 161 and 162:
invariance requirements. They sough
Page 163 and 164:
That (4.73) is an amazing result is
Page 165 and 166:
is an important example. Clearly, t
Page 167 and 168:
let us therefore add a Pauli term (
Page 169 and 170:
On the other hand, for a Pauli mome
Page 171 and 172:
moment actually cancels with the Di
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Chapter 5 The Retarded Fields I hav
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to particles carrying dipole moment
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workers. In 1969, Cohn [53] unfortu
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manifestly-covariant field expressi
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as a mathematical aid—it not bein
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integrate over the proper time τ,
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Equation (5.28) is, in the above no
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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
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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
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when the particle is in arbitrary m
Page 221 and 222:
Chapter 6 Radiation Reaction Accord
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perhaps, fortunately,—such a “f
Page 225 and 226:
Lorentz performed this calculation
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6.2 Previous analyses To the author
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other than pedagogy, in the past fi
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immediate practicality, since then
Page 233 and 234:
6.3 Infinitesimal rigid bodies We n
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enefit that, by Gauss’s law, ther
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particular times depend on the acce
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ground our understanding of the qua
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whence the reverse transformation i
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We define η 0 ≡ 1 ( ) 4 −2 ∫
Page 245 and 246:
and ending values of each loop are
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= 1 8( 4 3 π(2ε)3 ) 2 , (6.23) wh
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= 4 3 π(2ε)5 { 1 5 − 3 4 α +
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where by the notation ∫ d 2 n d w
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6.4.6 Angular outer integrals We no
Page 255 and 256:
6.4.7 Is there an easier way? It mi
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6.5.3 Final expression for the reta
Page 259 and 260:
Using the fact that t ret ≡ −R,
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6.5.9 Modified retarded normal vect
Page 263 and 264:
6.6 Divergence of the point particl
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“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
Page 273 and 274:
lead to delta-function contribution
Page 275 and 276:
(6.99) then gives { (a·b) − 5(n
Page 277 and 278:
6.7.1 The bare inverse-cube integra
Page 279 and 280:
6.7.2 Inverse-cube integral with tw
Page 281 and 282:
we immediately find that the r d -i
Page 283 and 284:
6.8.4 Duality symmetry consideratio
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in order that Maxwell’s equations
Page 287 and 288:
This asserted result of the author
Page 289 and 290:
obtain a non-vanishing integral ove
Page 291 and 292:
− 1 2 ˜µ2 η 1 ( ˙v· ¨σ)σ
Page 293 and 294:
If one examines the author’s calc
Page 295 and 296:
performed the electric charge radia
Page 297 and 298:
and the spin renormalisation term f
Page 299 and 300:
motion of the particle. This has, i
Page 301 and 302:
chrotron” radiation (i.e., the ra
Page 303 and 304:
6.10.2 The Ternov-Bagrov-Khapaev ef
Page 305 and 306:
where the characteristic polarisati
Page 307 and 308:
detail, for arbitrary values of g,
Page 309 and 310:
Substituting (6.151) into (6.152),
Page 311 and 312:
importantly, the hyperbolic tangent
Page 313 and 314:
twice, then it clearly will not do
Page 315 and 316:
Appendix A Notation and Conventions
Page 317 and 318:
Filename bibliog.tex claspcle.tex c
Page 319 and 320:
Filename algebra.h algebra1.c algeb
Page 321 and 322:
costella, british or american must
Page 323 and 324:
fonts of other resolutions, or font
Page 325 and 326:
typographically, the mathematical n
Page 327 and 328:
quantity which has been designated
Page 329 and 330:
noted that the inline fraction symb
Page 331 and 332:
The d’Alembertian operator is den
Page 333 and 334:
It should be noted that quantum mec
Page 335 and 336:
A.3.15 Pointlike objects If an obje
Page 337 and 338:
as a mathematically G-covariant qua
Page 339 and 340:
The adjective enumerated will be us
Page 341 and 342:
Each of the i m take values from th
Page 343 and 344:
distributed factor; for example, A
Page 345 and 346:
A.7.2 Square matrices Matrices with
Page 347 and 348:
A.8.2 Lorentz tensors Rank-zero, ra
Page 349 and 350:
AB denotes the four-tensor that is
Page 351 and 352:
and G are four-tensors, then ε(A,
Page 353 and 354:
The use of this signature of metric
Page 355 and 356:
ackets; e.g., C α | MCLF ≡ [C α
Page 357 and 358:
The first way is to simply measure
Page 359 and 360:
is referred to as the parity operat
Page 361 and 362:
three-vector conversion. In any cas
Page 363 and 364:
A.9.9 Magnitude of a three-vector T
Page 365 and 366:
A.9.13 Cross-products An alternativ
Page 367 and 368:
B.2.2 The field strength tensor The
Page 369 and 370:
B.2.9 Explicit electromagnetic comp
Page 371 and 372:
Also, (a×b)·(c×d) ≡ (a·c)(b·
Page 373 and 374:
{ (σ·∇)rd m r s (n s·A)n d = r
Page 375 and 376:
For two electric charges q 1 and q
Page 377 and 378:
where z is the position of the char
Page 379 and 380:
Appendix D Retarded Fields Verifica
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(−2) ≡ − e { 3 a U mρ 2 2 ρ
Page 383 and 384:
F µν + 1 2ȧUM [να U α R µ]
Page 385 and 386:
Comparing (D.11) and (D.12), we see
Page 387 and 388:
efore neutral particles were discov
Page 389 and 390:
Appendix E The Interaction Lagrangi
Page 391 and 392:
G µ ≡ G 1 k/k µ + G 2 k/B µ +
Page 393 and 394:
u 2 σ µν u 1 −→ ˜Σ µν ,
Page 395 and 396:
powers of ∂ 2 acting on the field
Page 397 and 398:
International Journal of Modern Phy
Page 399 and 400:
we shall not claim any predictive c
Page 401 and 402:
least, if one wants to derive the e
Page 403 and 404:
and electric charge interaction ter
Page 405 and 406:
way that the “Lagrangian-based”
Page 407 and 408:
Θ ≡ ζ 2 (E·B) , ∂ ′ ≡
Page 409 and 410:
prospective solution of the combine
Page 411 and 412:
Appendix G Computer Algebra Mrs. Du
Page 413 and 414:
) dependence to extract the three-g
Page 415 and 416:
mately chosen, both for simplicity,
Page 417 and 418:
appendical reference. In following
Page 419 and 420:
Program Filename Description kinema
Page 421 and 422:
G.4 kinemats: Kinematical quantitie
Page 423 and 424:
The reason for us naming it after F
Page 425 and 426:
[ ˙ Σ 0 ] = γ 2 (v· ˙σ) + γ
Page 427 and 428:
− 25γ 10 (γ + 1) −2 ˙v 2 (v
Page 429 and 430:
+ 124γ 11 (v· ˙v) 2 (v·¨v), (
Page 431 and 432:
+ 2γ 5 (γ + 1) −2 ( ˙v·σ)¨v
Page 433 and 434:
( Σ ... ) 0∣ ∣ ∣v=0 = 0, (
Page 435 and 436:
+ 4γ 8 (v· ˙v) 2 (v·σ ′ ), (
Page 437 and 438:
fields. These products are labelled
Page 439 and 440:
+ 3γ 4 (γ + 1) −1 (v· ˙σ)¨v
Page 441 and 442:
G.5.2 Covariant field expressions S
Page 443 and 444:
B ′d 1 = κ 2 ¨σ ′ ×n + κ 3
Page 445 and 446:
Expanding out these expressions exp
Page 447 and 448:
G.6 radreact: Radiation reaction G.
Page 449 and 450:
Firstly, iterating the differential
Page 451 and 452:
we define the redshift factor λ
Page 453 and 454:
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)(
Page 479 and 480:
+ 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
Page 499 and 500:
+ 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
Page 519 and 520:
− 11rd −1 (n d· ˙v)(n d·σ)(
Page 521 and 522:
+ 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 +
Page 533 and 534:
− 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(
Page 543 and 544:
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.
Page 553 and 554:
+(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}{
Page 557 and 558:
+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}{
Page 563 and 564:
Bibliography [1] M. Abraham, Ann. P
Page 565 and 566:
[39] H. J. Bhabha, Proc. Roy. Soc.
Page 567 and 568:
[80] J. R. Ellis, J. Math. Phys. 7
Page 569 and 570:
[124] M. Kolsrud and E. Leer, Phys.
Page 571 and 572:
[167] M. Pavsic (1987): see [34]. [
Page 573 and 574:
[212] I. M. Ternov, V. G. Bagrov an
Page 575:
It doesn’t matter whether you suc
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