Single-Particle Electrodynamics - Assassination Science

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dimensionless, rather than dimensionally challenged; erroneous results are referred to as incorrect, rather than fictionally gifted; and so on. Readers for whom such terminology would be psychologically challenging may request a digital copy of this thesis, and digitally censor the offending phrases themselves. A.3.4 Units The base system of units for this thesis is the SI system. Theoretical investigations are carried out in the system of units derived from the SI system when they are “naturalised” according to c = ε 0 = µ 0 = 1. (A.1) A.3.5 Decimal separator The decimal separator is the full stop, as in the value 137.036. A.3.6 Standard symbol set The standard symbol set is the set of typographical symbols that are used in this thesis to represent general mathematical quantities. Any symbol from the standard symbol set is referred to as a standard symbol. Some classes of mathematical quantities are, as described in the following sections, denoted in a unique, identifiable way. This may be done by typographically modifying a standard symbol’s typeface, or attaching an extra mark to it (collectively referred to as adorning the symbol); or it may be performed by using symbols not in the standard symbol set. All of the above specialised forms of notation are referred to as special symbols. Special symbols are not a part of the standard symbol set. All mathematical quantities not denoted by special symbols are denoted by symbols from the standard symbol set. There is no way to distinguish, 323
typographically, the mathematical nature of such quantities. The nature of the quantity will either be explicitly spelt out in the text, or it will be implied by context. The standard symbol set consists of the following symbols: a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F , G, H, I, J, K, L, M, N, P , Q, R, S, T , U, V , W , X, Y , Z, α, β, γ, δ, ε, ζ, η, θ, ϑ, κ, λ, µ, ν, ξ, π, ρ, σ, τ, φ, ϕ, χ, ψ, ω, Γ , ∆, Θ , Λ, Ξ , Π , Σ , Υ , Φ, Ψ and Ω . It will be noted that all standard symbols are italicised. All typographical modifications to the standard symbol set used in this thesis retain the italicisation of the standard symbol part of the modified symbol. A symbol is referred to as a collection symbol if it represents a collection of quantities rather than a single quantity; these quantities are referred to as being housed in the collection symbol. The act of typographically modifying a collection symbol to denote the extraction of a quantity housed within it is referred to as dereferencing the collection symbol. A.3.7 Decommissioning and recommissioning If a symbol, word or phrase is explicitly specified as decommissioned by the author, then it is thenceforth deemed to be an invalid symbol, word or phrase for the purposes of the remainder of this thesis, even if it would otherwise be a valid construct according to the rules laid down in this appendix. Decommissioned symbols, words or phrases may only be used for the purpose of recommissioning. If a symbol, word or phrase is recommissioned by the author, after having been previously decommissioned, then it is thenceforth deemed to again be a valid construct. If a phrase is recommissioned as nonmodifiable, then no typographical symbols may appear in the interior of the phrase, apart from line-breaking hyphenation marks. In particular, additional words may not be added be- 324
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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
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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
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(One should carefully note that the
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under differentiation, the latter e
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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
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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
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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
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Using the chain rule on (6.74), we
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Our first use of (6.84) is to compu
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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
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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: fonts of other resolutions, or font
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
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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
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− 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
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− 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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