Single-Particle Electrodynamics - Assassination Science

More documents

Recommendations

Info

putations: they are simply algebraic manipulations (with some—but not too much—subtlety required to handle the three-vector gradients and integrations), more suited to a computer than a human. (This is to be contrasted with the the calculation of the Taylor series coëfficients, which is almost indecipherable if done the naïve way, but most manageable if iterated order by order.) Thus, it was suggested by the author’s supervisor, Professor McKellar, that a computer algebra program such as Mathematica be used to perform these calculations. Investigations were made into this possibility. Two major obstacles were found. Firstly, it did not appear that Mathematica possessed sufficient inherent intelligence to handle three-vectors in the abstract way required for these calculations. This could, of course, be programmed in, but that would require a substantial programming investment, in a language the author was totally unfamiliar with, at a very late stage of the game. Secondly, it was not clear how the output of Mathematica’s manipulations could be compared with the existing hand-computed results, on the one hand; or integrated into the LaT E Xable body of this thesis, on the other; without a further substantial programming (or painful transcription) process. Investigation into other available computer algebra systems showed possibility for alleviating the first problem, but the second remained. There was also the logistical problem that, while Mathematica is widely used throughout the University, and hence help and support is readily available, the use of other packages would be a major task, undertaken essentially alone, and from scratch. For these reasons, the author decided to write a simplified computer algebra system, that would be powerful enough to perform the computations actually required for this thesis, without further bells or whistles; and, equally important, the system would output the results using exactly the same macros that the author has used to prepare the remainder of this thesis. Familiarity with the two obvious language contenders, ANSI C and C++, yielded a difficult choice; despite the elegance of C++ for such tasks, ANSI C was ulti- 413
mately chosen, both for simplicity, and to ensure a standard result that could be run on any machine—the author already having been burnt previously by the implementation-dependencies of C++, which has not, unfortunately, been standardised as yet. Of course, the output of computer programs must always be eyed suspiciously, being only as reliable as the programmer’s concentration level at the most fatigued point of his work. It is for this reason that the fully checked, manually-obtained results for the electric charge field contributions to the radiation reaction terms—computed from beginning to end—were crucial, for the purpose of debugging the program at each stage of its development. The computations up to n of the final-order-in-ε results were also important, in order to verify that the algorithms were correctly handling the extra order of complexity involved. Once the integrity of the system had been exhaustively verified, the remaining computations were performing exclusively by computer—being, of course, too lengthy to continue further on paper. At a late stage, it was recognised that the definition of the relativistically rigid body originally used by the author did not correctly handle the subtle Thomas precession of the constituents, as the body made the transition to a finite velocity and acceleration (see Section 3.3). This rendered almost all of the author’s manually-computed results erroneous, in the lower orders. Fortunately, the program radreact was completed before this oversight was noted, and it was quite a minor task adding the extra precession expression to the program. When the program was then run anew, it was found that, indeed, practically all expressions which contained sufficient orders in τ or ε (specifically, those for which terms involving both ˙v and ¨v were present) were changed by the redefinition of the rigid body. However,—and most surprisingly,—the final radiation reaction equations of motion were not changed at all, despite the fact that even the penultimate equations had significant modifications. Although the deep physical meaning behind such an “invariance” of the re- 414
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 and 14:
A Notation and Conventions 314 A.1
Page 15 and 16:
A.8.18 MCLF and CACS components . .
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.
Page 77 and 78:
is in terms of the quantities of Sp
Page 79 and 80:
2.6.10 The FitzGerald three-spin In
Page 81 and 82:
such systems. On the other hand, fo
Page 83 and 84:
equations, one then finds, for eith
Page 85 and 86:
i.e., the torque. Clearly, the corr
Page 87 and 88:
The component ∆L 0 is our warning
Page 89 and 90:
vanishes, for v = 0: P | v=0 = 0; (
Page 91 and 92:
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
Page 97 and 98:
6(c 1·c 3 ) + 4c 2 2 = 0, 8(c 1·c
Page 99 and 100:
and reverting t(τ) from (2.91), on
Page 101 and 102:
Chapter 3 Relativistically Rigid Bo
Page 103 and 104:
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
Page 109 and 110:
where we set u(τ) = r (3.7) becaus
Page 111 and 112:
+ 1 { [1 ] .... + (r· ˙v) v + 2 [
Page 113 and 114:
applied than the former.) We shall
Page 115 and 116:
which varies with time. We deem tha
Page 117 and 118:
desirable properties—but unfortun
Page 119 and 120:
the magnitude d of d describes the
Page 121 and 122:
shall compute the net force on the
Page 123 and 124:
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
Page 129 and 130:
4.2.2 The magnetic-charge dipole Ma
Page 131 and 132:
ut its employment of moving constit
Page 133 and 134:
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
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 and 222:
Chapter 6 Radiation Reaction Accord
Page 223 and 224:
perhaps, fortunately,—such a “f
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
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
Page 285 and 286:
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
Page 381 and 382: (−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: ) dependence to extract the three-g
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
Page 455 and 456: − 1 16 r d(r d· ˙v) 3 − 1 8 r
Page 457 and 458: − 1 120 r4 d (n d·.... v)n d + 1
Page 459 and 460: + 1 4 r2 d (n d· ˙v) 2 ˙v − 1
Page 461 and 462: + 1 8 r2 d r s ˙v 2 (n s· ˙v)n d
Page 463 and 464: − 1 48 r3 d r s (n d· ˙v)(n s·
Page 465 and 466:
Using (G.28) and (G.55) in (G.56),
Page 467 and 468:
+ 1 8 r4 d (n d· ˙v) 2 ¨σ − 1
Page 469 and 470:
+ 3 2 r dr s ˙v 2 (n s· ˙v) ˙σ
Page 471 and 472:
E q 2 = r −2 d n d + 1 2 r−1 d
Page 473 and 474:
− r d (n d· ˙v)( ˙v· ¨σ)n d
Page 475 and 476:
+ 39 4 r s(n d· ˙v)(n d·¨v)(n d
Page 477 and 478:
+ 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
Page 481 and 482:
+ 1 8 r−2 d r3 s (n d· ˙v)(n s
Page 483 and 484:
− 3 4 r−2 d r2 s (n d· ˙σ)(n
Page 485 and 486:
− 9 16 r s(n d·σ)(n s·¨v)¨v
Page 487 and 488:
B q 1 = −rd −1 n d× ˙v + n d
Page 489 and 490:
− 39 8 r d(n d· ˙v) 2 (n d·σ)
Page 491 and 492:
− 1 2 r−1 d r s(n s·¨v)n d ×
Page 493 and 494:
+ 3 4 r s(n d·σ)(n s· ˙v) ˙v×
Page 495 and 496:
− 5 12 r d(n d·¨v)¨v×σ − 1
Page 497 and 498:
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 +
Page 501 and 502:
+ 3 16 r−2 d r3 s (n d· ˙v)(n d
Page 503 and 504:
+ 3 8 r s(n d·¨v)(n d·σ)(n s·
Page 505 and 506:
+ 1 24 r(n·¨v)(¨v·σ)n − 5 24
Page 507 and 508:
λE q 2 = r −2 d n d + 1 2 r−1
Page 509 and 510:
+ 27 4 (n d· ˙v) 3 (n d·σ)n d
Page 511 and 512:
+ 1 2 r−1 d r s(n d·σ)(n s·¨v
Page 513 and 514:
− 1 2 r−1 d r s(n s· ˙v)n d
Page 515 and 516:
+ 1 8 r−1 d r s(n s· ˙v)( ˙v·
Page 517 and 518:
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·
Page 525 and 526:
d·... − 3rd −1 (n d·σ)(n σ)
Page 527 and 528:
− 12rd −2 r s(n d·¨v)(n d·σ
Page 529 and 530:
+ 101 4 (n d·¨v)(n d·σ)( ˙v·
Page 531 and 532:
+ 7rd −2 (n d·σ)(¨v·σ)n d +
Page 533 and 534:
− 57 2 r−1 d (n d· ˙v) 2 (n d
Page 535 and 536:
+ (n d· ˙v)(n d·... σ)σ + 3(n
Page 537 and 538:
− rd −2 r2 s (n s· ˙v)(n s·
Page 539 and 540:
+ 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
Page 545 and 546:
F (M) µd = −3η ′ 3σ× ˙σ,
Page 547 and 548:
N F (1) dd = 1 10 η 1( ˙v·σ) ˙
Page 549 and 550:
% ID string: Test 3-d integrations.
Page 551 and 552:
+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
Page 555 and 556:
+\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
Page 559 and 560:
+\f{1}{30}(o.p)a*m +\fourthirds a
Page 561 and 562:
+\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
show all

Single-Particle Electrodynamics - Assassination Science

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?