Single-Particle Electrodynamics - Assassination Science

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where φ(t) ≡ Ω P t. We then find Ω ′ RR = Ω 3 P γ 2 σ×i − 8Ω 2 MΩ P γ 4 σ×i + 2Ω 3 Mγ 3 (γ + 1) −2 σ×i − 4Ω 3 Mγ 5 (γ + 1) −2 σ×i + 2Ω 3 Mγ 7 (γ + 1) −2 σ×i − 3Ω M Ω 2 P γ 2 (γ + 1) −2 σ×i − 3Ω M Ω 2 P γ 3 (γ + 1) −3 σ×i + 6Ω M Ω 2 P γ 5 (γ + 1) −2 σ×i − 3Ω M Ω 2 P γ 6 (γ + 1) −3 σ×i + 2Ω 2 MΩ P γ 2 (γ + 1) −3 σ×i + 20Ω 2 MΩ P γ 5 (γ + 1) −1 σ×i − 10Ω 2 MΩ P γ 6 (γ + 1) −2 σ×i + 2Ω 2 MΩ P γ 7 (γ + 1) −3 σ×i . Substituting the Lorentz–Thomas result of Ω P = (1 + aγ)Ω M , where a ≡ g − 2 2 is the magnetic anomaly, we find Ω ′ RR = −Ω 3 Mγ 3 σ×i + 2Ω 3 Mγ 5 σ×i − Ω 3 Maγ 3 σ×i + 4Ω 3 Maγ 5 σ×i + 3Ω 3 Ma 2 γ 5 σ×i + Ω 3 Ma 3 γ 5 σ×i . This result is discussed in Chapter 6. G.5 retfield: Retarded fields G.5.1 Introduction This program verifies that the explicit expressions obtained by the author for the retarded fields agree with those extractable directly from the manifestlycovariant expressions. 439
G.5.2 Covariant field expressions Since a significant amount of intelligence is required to convert the manifestly-covariant retarded field expressions to their final simplified non-covariant form, we will here only verify that the initial and final expressions agree, when all convenient quantities are expanded out explicitly. We start with the manifestly covariant field expressions which, in the case of the dipole fields, have been explicitly verified against the Cohn–Wiebe expressions: F q 2 F q 1 F3 d F2 d = ϕ 3 ζ ∧U, = ϕ 2 ζ ∧ ˙U − ϕ 3 ˙χζ ∧U, = ϕ 3 U ∧Σ − 3ϕ 5 ψζ ∧U, = ϕ 2 ˙U ∧Σ + ϕ 3 [ζ ∧ ˙ Σ ] + ϕ 3 ψU ∧ ˙U − ϕ 3 ˙χU ∧Σ + 6ϕ 5 ˙χψζ ∧U − 3ϕ 4 ˙ψζ ∧U − 3ϕ 4 ψζ ∧ ˙U + ϕ 3 ˙ϑζ ∧U, F d 1 = ϕ 2 [ζ ∧ ¨Σ ] − ϕ 3 ψ[ζ ∧Ü] + ϕ4 ψ ¨χζ ∧U − 2ϕ 3 ˙ψζ ∧ ˙U + 3ϕ 4 ˙χ ˙ψζ ∧U − ϕ 3 ˙χ[ζ ∧ ˙ Σ ] − ϕ 3¨ηζ ∧U − 3ϕ 5 ˙χ 2 ψζ ∧U + 3ϕ 4 ψ ˙χζ ∧ ˙U, where ζ ≡ x − z, ϕ ≡ (ζ ·U) −1 , ˙χ ≡ (ζ · ˙U), ¨χ ≡ [ζ ·Ü], ψ ≡ (ζ ·Σ ), ˙ψ ≡ [ζ · ˙ Σ ], ˙ϑ ≡ [U · ˙ Σ ], ¨η ≡ [ζ · ¨Σ ]. 440
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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
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lead to delta-function contribution
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(6.99) then gives { (a·b) − 5(n
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6.7.1 The bare inverse-cube integra
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6.7.2 Inverse-cube integral with tw
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we immediately find that the r d -i
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6.8.4 Duality symmetry consideratio
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in order that Maxwell’s equations
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This asserted result of the author
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obtain a non-vanishing integral ove
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− 1 2 ˜µ2 η 1 ( ˙v· ¨σ)σ
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If one examines the author’s calc
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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
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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,
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Substituting (6.151) into (6.152),
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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
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Filename algebra.h algebra1.c algeb
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costella, british or american must
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fonts of other resolutions, or font
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typographically, the mathematical n
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quantity which has been designated
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noted that the inline fraction symb
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The d’Alembertian operator is den
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It should be noted that quantum mec
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A.3.15 Pointlike objects If an obje
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as a mathematically G-covariant qua
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The adjective enumerated will be us
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Each of the i m take values from th
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distributed factor; for example, A
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A.7.2 Square matrices Matrices with
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A.8.2 Lorentz tensors Rank-zero, ra
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AB denotes the four-tensor that is
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and G are four-tensors, then ε(A,
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The use of this signature of metric
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ackets; e.g., C α | MCLF ≡ [C α
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The first way is to simply measure
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is referred to as the parity operat
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three-vector conversion. In any cas
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A.9.9 Magnitude of a three-vector T
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A.9.13 Cross-products An alternativ
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B.2.2 The field strength tensor The
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B.2.9 Explicit electromagnetic comp
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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
Page 389 and 390: Appendix E The Interaction Lagrangi
Page 391 and 392: G µ ≡ G 1 k/k µ + G 2 k/B µ +
Page 393 and 394: u 2 σ µν u 1 −→ ˜Σ µν ,
Page 395 and 396: powers of ∂ 2 acting on the field
Page 397 and 398: International Journal of Modern Phy
Page 399 and 400: we shall not claim any predictive c
Page 401 and 402: least, if one wants to derive the e
Page 403 and 404: and electric charge interaction ter
Page 405 and 406: way that the “Lagrangian-based”
Page 407 and 408: Θ ≡ ζ 2 (E·B) , ∂ ′ ≡
Page 409 and 410: prospective solution of the combine
Page 411 and 412: Appendix G Computer Algebra Mrs. Du
Page 413 and 414: ) dependence to extract the three-g
Page 415 and 416: mately chosen, both for simplicity,
Page 417 and 418: appendical reference. In following
Page 419 and 420: Program Filename Description kinema
Page 421 and 422: G.4 kinemats: Kinematical quantitie
Page 423 and 424: The reason for us naming it after F
Page 425 and 426: [ ˙ Σ 0 ] = γ 2 (v· ˙σ) + γ
Page 427 and 428: − 25γ 10 (γ + 1) −2 ˙v 2 (v
Page 429 and 430: + 124γ 11 (v· ˙v) 2 (v·¨v), (
Page 431 and 432: + 2γ 5 (γ + 1) −2 ( ˙v·σ)¨v
Page 433 and 434: ( Σ ... ) 0∣ ∣ ∣v=0 = 0, (
Page 435 and 436: + 4γ 8 (v· ˙v) 2 (v·σ ′ ), (
Page 437 and 438: fields. These products are labelled
Page 439: + 3γ 4 (γ + 1) −1 (v· ˙σ)¨v
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)
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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}{
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
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[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]. [
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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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