Single-Particle Electrodynamics - Assassination Science

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A description is given of the files included in digital copies of this thesis, and instructions are given for the digital processing of the whole thesis, or any single chapter or appendix. The computer algebra programs included in digital copies of this thesis are described in more detail in Appendix G. Section A.3: Language and typography Descriptions are given of the document preparation system used to prepare this thesis, the spelling and punctuation choices made, the system of units used, and the symbols used to represent quantities. Special or ambiguous notation or nomenclature, that is either of a universal nature or is not specifically mentioned elsewhere, is defined here. Section A.4: Enumeration sets The terminology surrounding enumerations is borrowed from the field of computer programming. The extra nomenclature serves to simplify the description of other mathematical operations in the following sections. Section A.5: Special functions Notation and physical interpretation is supplied for the Kronecker and Dirac delta functions, and the Heaviside step function. The sign convention used in this thesis for the alternating function is specified. Section A.6: Associativity and commutativity Various functions and pieces of nomenclature are established to deal with the associativity and commutativity of quantities. 29
Section A.7: Matrices Notation and nomenclature are defined for matrix quantities. Section A.8: Relativistic mechanics Various issues surrounding the use of relativistic mechanics are addressed. A substantial amount of nomenclature is given a precise definition for the purposes of this thesis. Notation for Lorentz-covariant quantities is described in detail, as are the operations between them, and the notation used to denote such operations. Notation for the kinematical quantities of a point particle’s motion is established. Particular attention is given to the notation for the proper-time derivatives of kinematical quantities, for which there are two subtly different definitions (see Chapter 2). Section A.9: Euclidean three-space As with the previous section, a substantial amount of nomenclature is given a precise definition, and notations are specified for various operations on three-vectors. Appendix B: Supplementary Identities Section B.1: Introduction Appendix B is a convenient collection of various mathematical identities that are used throughout this thesis, that are either of common knowledge or can be derived simply. Identities that require substantial argument on the part of the author to establish their veracity are derived elsewhere in this thesis. 30
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: Section 6.7: Inverse-cube integrals
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
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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; (
Page 91 and 92:
wish of them—not least of which b
Page 93 and 94:
The author will, in the remaining c
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(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
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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 [
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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, −(
Page 139 and 140:
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
Page 161 and 162:
invariance requirements. They sough
Page 163 and 164:
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 (
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
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moments of a particle. There are a
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It can be noted that equation (5.51
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the overall Lorentz-gauge four-pote
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Applying (5.61) to the delta functi
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The results above are for a particl
Page 199 and 200:
(One should carefully note that the
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under differentiation, the latter e
Page 203 and 204:
Because the field expressions diver
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expression (5.81) in the direction
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We now note that the non-zero elect
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clearly, for any ε, this parametri
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For a point electric dipole moment,
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where the convenient constant η 1
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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
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Chapter 6 Radiation Reaction Accord
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perhaps, fortunately,—such a “f
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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
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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 ∫
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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
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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
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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
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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
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6.7.1 The bare inverse-cube integra
Page 279 and 280:
6.7.2 Inverse-cube integral with tw
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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
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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
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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
Page 303 and 304:
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,
Page 309 and 310:
Substituting (6.151) into (6.152),
Page 311 and 312:
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
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
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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
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+ 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
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− 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
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− 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
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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
Page 481 and 482:
+ 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)
Page 491 and 492:
− 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
Page 497 and 498:
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 +
Page 501 and 502:
+ 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·
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·
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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,
Page 547 and 548:
N F (1) dd = 1 10 η 1( ˙v·σ) ˙
Page 549 and 550:
% 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
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
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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
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It doesn’t matter whether you suc
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