Single-Particle Electrodynamics - Assassination Science

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− 3 4 r−1 d r s(n s· ˙v)( ˙v·σ)n d × ˙v + 3rd −1 r s(n d· ˙v)(n d·σ)(n s· ˙v)n d × ˙v + O(ε). Adding these results together, we find 4πλE q = r −2 d n d − 1 2 r−1 d ˙v + 1 2 r−2 d r s(n s· ˙v)n d + 2 3 ¨v − 3 8 ˙v2 n d + 1 4 (n d· ˙v) ˙v − 1 8 (n d· ˙v) 2 n d + O(ε), 4πλE d = −r −3 d − 1 2 r−3 d + 3 8 r−1 d − 3 8 r−1 d σ + 3r−3(n d·σ)n d − 1 2 r−2(n d·σ) ˙v + 1 2 r−2( ˙v·σ)n d d d r s(n s· ˙v)σ + 3 2 r−3r s(n d·σ)(n s· ˙v)n d − 1 2 r−1 ¨σ ˙v 2 σ + 1 8 r−1 d d (n d· ˙v) 2 σ − 1 2 r−1 d ˙v 2 (n d·σ)n d + 1 4 r−1 d (n d· ˙v)(n d·σ) ˙v d d (n d· ¨σ)n d − 1 4 r−1 ( ˙v·σ) ˙v + 1 4 r−1 d (n d· ˙v)( ˙v·σ)n d − 3 8 r−1 d (n d· ˙v) 2 (n d·σ)n d + 2 ... σ 3 − 1 4 (n d·¨v) ˙σ − 1 (nd·... v)σ + 3 8 8 (n d·σ) v ... + (n d· ˙σ)¨v d + 3 4 (n d· ¨σ) ˙v − 2 3 ( ˙v·¨v)σ + 1 3 ( ˙v·σ)¨v − 3 4 ( ˙v· ¨σ)n d − 1 (¨v·σ) ˙v 3 − 1 2 (¨v· ˙σ)n d − 1 8 (... v·σ)n d − 3 8 ˙v2 (n d· ˙v)σ + 15 16 ˙v2 (n d·σ) ˙v − 3 16 ˙v2 ( ˙v·σ)n d − 1 16 (n d· ˙v) 2 (n d·σ) ˙v + 1 16 (n d· ˙v) 2 ( ˙v·σ)n d + 1 4 (n d·¨v)(n d· ˙σ)n d + 1 (nd·... v)(n d·σ)n d 8 + 3 8 ˙v2 (n d· ˙v)(n d·σ)n d + 1 4 r−1 d − 3 16 r−1 d + 1 4 r−1 d r s(n s· ˙v)¨σ + 1 4 r−1r s(n s·¨v) ˙σ r s ˙v 2 (n s· ˙v)σ − 1 16 r−1r s(n d· ˙v) 2 (n s· ˙v)σ d r s(n d· ˙σ)(n s·¨v)n d + 1 4 r−1r s(n d· ¨σ)(n s· ˙v)n d d d 513
+ 1 8 r−1 d r s(n s· ˙v)( ˙v·σ) ˙v + 3 − 1 8 r−1 d 16 r−1 d r s ˙v 2 (n d·σ)(n s· ˙v)n d r s(n d· ˙v)(n d·σ)(n s· ˙v) ˙v − 1 8 r−1r s(n d· ˙v)(n s· ˙v)( ˙v·σ)n d + 3 16 r−1 d r s(n d· ˙v) 2 (n d·σ)(n s· ˙v)n d + O(ε), d 4πλB q = 1 2 n d×¨v, 4πλB d = −r −2 d n d× ˙σ + 1 2 r−1 d ˙v× ˙σ + 1 2 r−1 d ¨v×σ − 1 2 r−1 d (n d· ˙v)n d × ˙σ + 1 2 r−1 d (n d·σ)n d ×¨v + rd −1 (n d· ˙σ)n d × ˙v + 1 2 n d× σ ... − 2 ¨v× ˙σ 3 − 1 ... v×σ + 3 3 8 ˙v2 n d × ˙σ − ˙v 2 ˙v×σ − 1 8 (n d· ˙v) 2 n d × ˙σ − 1 4 (n d·¨v) ˙v×σ + 1 4 (n d·σ) ˙v×¨v − 1 4 (¨v·σ)n d× ˙v − 1 4 (n d·¨v)(n d·σ)n d × ˙v − 1 4 r−1 d r s(n s· ˙v) ˙v× ˙σ − 1 4 r−1 d + 1 4 r−1 d − 1 4 r−1 d + O(ε). r s(n s· ˙v)¨v×σ − 1 4 r−1r s(n s·¨v) ˙v×σ d r s(n d· ˙v)(n s· ˙v)n d × ˙σ − 1 4 r−1r s(n d·σ)(n s· ˙v)n d ×¨v r s(n d·σ)(n s·¨v)n d × ˙v − 1 2 r−1r s(n d· ˙σ)(n s· ˙v)n d × ˙v d d G.6.22 Gradients of the self-fields We now compute the spatial gradients of the self-fields of Section G.6.19 in the direction of σ, and multiplied by λ. (Only those expressions actually required for the radiation reaction calculations are computed.) Because of the excessive length of the resulting expressions (on the order of 14 gpages each), we omit terms of order ε 0 that are odd in either n d or n s , since they 514
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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
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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
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)
Page 491 and 492: − 1 2 r−1 d r s(n s·¨v)n d ×
Page 493 and 494: + 3 4 r s(n d·σ)(n s· ˙v) ˙v×
Page 495 and 496: − 5 12 r d(n d·¨v)¨v×σ − 1
Page 497 and 498: If we add the various E a n and B a
Page 499 and 500: + 15 16 r d ˙v 2 (n d· ¨σ)n d +
Page 501 and 502: + 3 16 r−2 d r3 s (n d· ˙v)(n d
Page 503 and 504: + 3 8 r s(n d·¨v)(n d·σ)(n s·
Page 505 and 506: + 1 24 r(n·¨v)(¨v·σ)n − 5 24
Page 507 and 508: λE q 2 = r −2 d n d + 1 2 r−1
Page 509 and 510: + 27 4 (n d· ˙v) 3 (n d·σ)n d
Page 511 and 512: + 1 2 r−1 d r s(n d·σ)(n s·¨v
Page 513: − 1 2 r−1 d r s(n s· ˙v)n d
Page 517 and 518: λ(σ·∇)E d 1 = −rd −2 ¨v
Page 519 and 520: − 11rd −1 (n d· ˙v)(n d·σ)(
Page 521 and 522: + 1 6 (¨v· ¨σ)σ − 1 2 (... v
Page 523 and 524: − 3rd −2 r2 s (n d· ˙v)(n d·
Page 525 and 526: d·... − 3rd −1 (n d·σ)(n σ)
Page 527 and 528: − 12rd −2 r s(n d·¨v)(n d·σ
Page 529 and 530: + 101 4 (n d·¨v)(n d·σ)( ˙v·
Page 531 and 532: + 7rd −2 (n d·σ)(¨v·σ)n d +
Page 533 and 534: − 57 2 r−1 d (n d· ˙v) 2 (n d
Page 535 and 536: + (n d· ˙v)(n d·... σ)σ + 3(n
Page 537 and 538: − rd −2 r2 s (n s· ˙v)(n s·
Page 539 and 540: + 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
Page 545 and 546: F (M) µd = −3η ′ 3σ× ˙σ,
Page 547 and 548: N F (1) dd = 1 10 η 1( ˙v·σ) ˙
Page 549 and 550: % ID string: Test 3-d integrations.
Page 551 and 552: +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
Page 555 and 556: +\f{1}{105}(i.o)(j.m)(k.l)a +\f{2}{
Page 557 and 558: +rd^{-2}rs^2(n.u)(n.v)(ns.w)(ns.x)y
Page 559 and 560: +\f{1}{30}(o.p)a*m +\fourthirds a
Page 561 and 562: +\f{1}{105}(w.y)(x.z)a’*a +\f{1}{
Page 563 and 564: Bibliography [1] M. Abraham, Ann. P
Page 565 and 566:
[39] H. J. Bhabha, Proc. Roy. Soc.
Page 567 and 568:
[80] J. R. Ellis, J. Math. Phys. 7
Page 569 and 570:
[124] M. Kolsrud and E. Leer, Phys.
Page 571 and 572:
[167] M. Pavsic (1987): see [34]. [
Page 573 and 574:
[212] I. M. Ternov, V. G. Bagrov an
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It doesn’t matter whether you suc
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