Single-Particle Electrodynamics - Assassination Science

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american.tex or british.tex. (The exception is the appendix compalg.- tex, which also requires the presence of the files kmoutth.tex, rfoutth.- tex, rroutth.tex, t3outth.txt and crsoutth.txt.) These files must reside in a single directory of the computer system. One then simply needs to LaT E X the file corresponding to the chapter or appendix in question (several times, to link the cross-references correctly). The files have been constructed so that no errors or warnings are generated on the final pass of LaT E X. Cross-references to sections, subsections or equations in other chapters or appendices, or to entries in the Bibliography, are replaced with harmless ‘x’ characters, rather than generating an error. Cross-references to sections, subsections and equations within the chapter or appendix in question are processed normally. Page numbering is initialised at the beginning of the chapter or appendix; however, the Chapter and Section numbers take the values that they do in the complete thesis. For convenience, a Table of Contents, covering only the chapter or appendix in question, is added to the output. (In the complete thesis, only a single complete Table of Contents is included, before the first chapter.) A.3 Language and typography A.3.1 Document preparation system This thesis has been prepared using the LaT E X document preparation system [129]. The default text type is 12-point Computer Modern, 1.5 spaced. All fonts, styles, commands and macro constructions used in this thesis follow the guidelines given in the LaT E X manual [129]. This thesis has been designed to be viewed or printed using standard LaT E X fonts of a resolution of precisely 300 dots per inch; reproductions using 321
fonts of other resolutions, or fonts that are not those standard to LaT E X, do not represent authorised copies of this thesis. A.3.2 Dialect-independent English This thesis is written in dialect-independent English: it may be printed with either British English, American English or the author’s English conventions; the printing you are reading uses the author’s conventions. (All official copies submitted for examination use the author’s conventions.) Diæreses are used to separate nondiphthonged adjacent vowels. Nouns collected together into the one entity are separated by en dashes. Some words of foreign origin have their original accenting retained. Hyphens whose omission would not lead to ambiguity, ugliness or a violation of an accepted custom are omitted. Foreign phrases not naturalised into the English language are generally italicised. Latin abbreviations are italicised and punctuated. Terms are generally italicised when they are introduced or defined. Italicisation is also used for emphasis in the text. Foreign names are reproduced as faithfully as possible, following the printed spelling in the source literature. Some of the above specifications do not apply to the published paper included verbatim in Appendix F. Words and phrases used by the author of arguably questionable spelling, grammar or semantics have been checked by the author with the full Oxford English Dictionary [161]. Typographical and other errors may of course still be present in the text of this thesis. A.3.3 Political incorrection The author does not belong to the Political Incorrection movement. Readers of that persuasion are warned that phrases of an offensive nature may be found within this thesis. For example, quantities are referred to as 322
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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
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: costella, british or american must
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·
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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
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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
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
Page 559 and 560:
+\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
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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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