Single-Particle Electrodynamics - Assassination Science

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where F (−n) are the expressions (D.7), by using the relations of Section D.4 to convert the remaining notation. Simplifying the algebra generated by the product of the two alternating functions from (D.21) and (D.20) requires the use of identities (B.44). When this is done, and the various terms collected together, one obtains F d 3 = ϕ 3 U ∧Σ − 3ϕ 5 ψζ ∧U, (D.22) F d 2 F d 1 = 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, (D.23) = 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. (D.24) It may be verified that equations (D.22), (D.23) and (D.24) agree precisely with their counterparts (5.70), (5.71) and (5.72) of Section 5.4.5. Thus, the analysis of Chapter 5 agrees completely with that of Cohn and Wiebe [54]; the two quite separate paths taken to obtain the end results mean that one can have confidence in their veracity. 387
Appendix E The Interaction Lagrangian E.1 Introduction This thesis is concerned with the motion of a single particle, in the classical limit, under the influence of a classical electromagnetic field. In Nature, the only particles for which we expect this to be a very good approximation are the massive leptons: the bare electron, muon and tauon. We also expect that the stable fermionic hadrons, such as the proton and neutron, will also be conveniently described by a single-particle description, as long as the electromagnetic fields involved are sufficiently well-behaved so as not to excite the internal states of these particles. Likewise, the anomalous contributions to the magnetic moments of the massive leptons should be describable quite well by an equivalent Pauli interaction in the single-particle equation, while again keeping in mind that the QED and other processes leading to these contributions are in reality multi-particle effects. Thus, in the real world, we will generally apply the results of this thesis to spin-half particles. It is therefore of interest to note that, in full generality, there are only four possible electromagnetic properties that a spin-half particle may possess: electric charge, electric dipole moment, magnetic dipole moment and anapole moment; each may in general have a structure func- 388
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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
Page 337 and 338: as a mathematically G-covariant qua
Page 339 and 340: The adjective enumerated will be us
Page 341 and 342: Each of the i m take values from th
Page 343 and 344: distributed factor; for example, A
Page 345 and 346: A.7.2 Square matrices Matrices with
Page 347 and 348: A.8.2 Lorentz tensors Rank-zero, ra
Page 349 and 350: AB denotes the four-tensor that is
Page 351 and 352: and G are four-tensors, then ε(A,
Page 353 and 354: The use of this signature of metric
Page 355 and 356: ackets; e.g., C α | MCLF ≡ [C α
Page 357 and 358: The first way is to simply measure
Page 359 and 360: is referred to as the parity operat
Page 361 and 362: three-vector conversion. In any cas
Page 363 and 364: A.9.9 Magnitude of a three-vector T
Page 365 and 366: A.9.13 Cross-products An alternativ
Page 367 and 368: B.2.2 The field strength tensor The
Page 369 and 370: B.2.9 Explicit electromagnetic comp
Page 371 and 372: Also, (a×b)·(c×d) ≡ (a·c)(b·
Page 373 and 374: { (σ·∇)rd m r s (n s·A)n d = r
Page 375 and 376: For two electric charges q 1 and q
Page 377 and 378: where z is the position of the char
Page 379 and 380: Appendix D Retarded Fields Verifica
Page 381 and 382: (−2) ≡ − e { 3 a U mρ 2 2 ρ
Page 383 and 384: F µν + 1 2ȧUM [να U α R µ]
Page 385 and 386: Comparing (D.11) and (D.12), we see
Page 387: efore neutral particles were discov
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
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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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