Single-Particle Electrodynamics - Assassination Science

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in this chapter has been published [65]; this paper is included verbatim in Appendix F. However, it was found by the author, shortly before the printing of this thesis, that an extra term needs to be added to the results of the published paper; this is discussed in detail in Section 4.2.1. 4.2 Newtonian mechanics In this section, we analyse various classical dipoles from the viewpoint of Newtonian mechanics. 4.2.1 The electric dipole The simplest dipole to consider is an electric dipole: a separation of positive and negative electric charge. Its behaviour is easily derivable, and is not subject to any controversy whatsoever. Let us choose to analyse what is arguably the simplest model of a fixed electric dipole: a positive charge glued to one end of a rigid stick, with a negative charge of the same magnitude glued to the other end of the stick. (We only consider fixed moments in this thesis, as are applicable for the intrinsic moments of spin-half particles; “induced” moments are not, in general, considered.) The author shall describe how the following analysis can be made absolutely rigorous, according to the relativistically rigid body formalism described in Chapter 3, at the end of this section. We shall, as with all considerations of this thesis, consider only pointlike particles in this chapter; but we shall of course start off with an object that is of finite size, and then take the point limit at the end of the calculations. Now, an electric dipole moment is fundamentally described by a threevector, d, in the rest frame of the particle. The direction ˆd of d describes the direction in which the positive-charge end of our rigid stick is pointing; 117
the magnitude d of d describes the strength of the electric dipole moment. We shall make use of the unit three-vector σ of Section 2.6.7, via σ ≡ ˆd , to simplify notationally the analysis of the electric dipole. Let us call the positive charge on our stick q 1 , and the negative charge q 2 , and likewise label the kinematical quantities of these charges by the subscripts 1 and 2 . For simplicity, let us place the origin of our spatial coördinates at the midpoint of the rigid stick, at the instant t = 0 at which we wish to analyse the particle. Let us say that the stick is of length ε, which we shall shrink to zero at the end of the analysis. according to these definitions, located at the position The positive charge q 1 is thus, z 1 = 1 εσ, (4.1) 2 at t = 0; and the negative charge q 2 is located at the position z 2 = − 1 εσ. (4.2) 2 By convention, the definition of the magnitude d of the dipole moment is such that our charges q 1 and q 2 take the values q 1 ≡ −q 2 ≡ d ε . (4.3) Let us now consider the motion of the charges q 1 and q 2 , in the nonrelativistic limit. Each charge will obviously partake in the motion, v, of the centre of the dipole (the mid-point of the rigid stick); but the charges will also have equal and opposite contributions to their velocity if the stick is rotating (or “precessing”, as it is generally referred to). This precession is described by the three-vector ˙σ, the rate of change of the direction of the dipole. Clearly, since σ is of constant magnitude (i.e., the dipole moment is fixed in magnitude), we have the identity (σ· ˙σ) ≡ 0. 118
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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
Page 67 and 68: 2.6.7 Unit four-spin It is often ne
Page 69 and 70: the case. The reason, pointed out b
Page 71 and 72: period it appeared that the magnitu
Page 73 and 74: tionally reflect the difference bet
Page 75 and 76: has precessed by an amount ˙σ dt.
Page 77 and 78: is in terms of the quantities of Sp
Page 79 and 80: 2.6.10 The FitzGerald three-spin In
Page 81 and 82: such systems. On the other hand, fo
Page 83 and 84: equations, one then finds, for eith
Page 85 and 86: i.e., the torque. Clearly, the corr
Page 87 and 88: The component ∆L 0 is our warning
Page 89 and 90: vanishes, for v = 0: P | v=0 = 0; (
Page 91 and 92: wish of them—not least of which b
Page 93 and 94: The author will, in the remaining c
Page 95 and 96: (see equation (A.58) of Section A.8
Page 97 and 98: 6(c 1·c 3 ) + 4c 2 2 = 0, 8(c 1·c
Page 99 and 100: and reverting t(τ) from (2.91), on
Page 101 and 102: Chapter 3 Relativistically Rigid Bo
Page 103 and 104: than considering the rest frame of
Page 105 and 106: We also note Pearle’s proof of hi
Page 107 and 108: properties of the constituent r are
Page 109 and 110: where we set u(τ) = r (3.7) becaus
Page 111 and 112: + 1 { [1 ] .... + (r· ˙v) v + 2 [
Page 113 and 114: applied than the former.) We shall
Page 115 and 116: which varies with time. We deem tha
Page 117: desirable properties—but unfortun
Page 121 and 122: shall compute the net force on the
Page 123 and 124: we shall make the analysis relativi
Page 125 and 126: Now, let us consider such a spin-ha
Page 127 and 128: adreact of Appendix G; the expressi
Page 129 and 130: 4.2.2 The magnetic-charge dipole Ma
Page 131 and 132: ut its employment of moving constit
Page 133 and 134: point of the loop at angle θ to th
Page 135 and 136: Now, the power into each positive c
Page 137 and 138: or, on reärranging the terms, −(
Page 139 and 140: Again noting the relations (4.23) a
Page 141 and 142: its discovery. Mathematically, the
Page 143 and 144: direction: E ≡ jE y . Now, let us
Page 145 and 146: (i.e., 2πε/v orb with v = 1, the
Page 147 and 148: a sufficient amount of inertia, the
Page 149 and 150: The fact that the Penfield-Haus eff
Page 151 and 152: tum of ±µ×E. Firstly, one can sh
Page 153 and 154: concepts precisely and correctly, a
Page 155 and 156: on the structure of spacetime, one
Page 157 and 158: 4.3.4 Relativistic Lagrangian deriv
Page 159 and 160: we know that the operator equations
Page 161 and 162: invariance requirements. They sough
Page 163 and 164: That (4.73) is an amazing result is
Page 165 and 166: is an important example. Clearly, t
Page 167 and 168: 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,
Page 353 and 354:
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, (
Page 435 and 436:
+ 4γ 8 (v· ˙v) 2 (v·σ ′ ), (
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fields. These products are labelled
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+ 3γ 4 (γ + 1) −1 (v· ˙σ)¨v
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G.5.2 Covariant field expressions S
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B ′d 1 = κ 2 ¨σ ′ ×n + κ 3
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Expanding out these expressions exp
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G.6 radreact: Radiation reaction G.
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Firstly, iterating the differential
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we define the redshift factor λ
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G.6.8 Sum and difference variables
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− 1 16 r d(r d· ˙v) 3 − 1 8 r
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− 1 120 r4 d (n d·.... v)n d + 1
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+ 1 4 r2 d (n d· ˙v) 2 ˙v − 1
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+ 1 8 r2 d r s ˙v 2 (n s· ˙v)n d
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− 1 48 r3 d r s (n d· ˙v)(n s·
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Using (G.28) and (G.55) in (G.56),
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+ 1 8 r4 d (n d· ˙v) 2 ¨σ − 1
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+ 3 2 r dr s ˙v 2 (n s· ˙v) ˙σ
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E q 2 = r −2 d n d + 1 2 r−1 d
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− r d (n d· ˙v)( ˙v· ¨σ)n d
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+ 39 4 r s(n d· ˙v)(n d·¨v)(n d
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+ 3rd −1 r s(n d· ˙v)(n d·σ)(
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+ 1 4 r s(n s·... v) ˙σ + 15 8 r
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+ 1 8 r−2 d r3 s (n d· ˙v)(n s
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− 3 4 r−2 d r2 s (n d· ˙σ)(n
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− 9 16 r s(n d·σ)(n s·¨v)¨v
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B q 1 = −rd −1 n d× ˙v + n d
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− 39 8 r d(n d· ˙v) 2 (n d·σ)
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− 1 2 r−1 d r s(n s·¨v)n d ×
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+ 3 4 r s(n d·σ)(n s· ˙v) ˙v×
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− 5 12 r d(n d·¨v)¨v×σ − 1
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If we add the various E a n and B a
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+ 15 16 r d ˙v 2 (n d· ¨σ)n d +
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+ 3 16 r−2 d r3 s (n d· ˙v)(n d
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+ 3 8 r s(n d·¨v)(n d·σ)(n s·
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+ 1 24 r(n·¨v)(¨v·σ)n − 5 24
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λE q 2 = r −2 d n d + 1 2 r−1
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+ 27 4 (n d· ˙v) 3 (n d·σ)n d
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+ 1 2 r−1 d r s(n d·σ)(n s·¨v
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− 1 2 r−1 d r s(n s· ˙v)n d
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+ 1 8 r−1 d r s(n s· ˙v)( ˙v·
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λ(σ·∇)E d 1 = −rd −2 ¨v
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− 11rd −1 (n d· ˙v)(n d·σ)(
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+ 1 6 (¨v· ¨σ)σ − 1 2 (... v
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− 3rd −2 r2 s (n d· ˙v)(n d·
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d·... − 3rd −1 (n d·σ)(n σ)
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− 12rd −2 r s(n d·¨v)(n d·σ
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+ 101 4 (n d·¨v)(n d·σ)( ˙v·
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+ 7rd −2 (n d·σ)(¨v·σ)n d +
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− 57 2 r−1 d (n d· ˙v) 2 (n d
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+ (n d· ˙v)(n d·... σ)σ + 3(n
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− rd −2 r2 s (n s· ˙v)(n s·
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+ 1 8 r−1 d (n d· ˙v) 2 ( ˙v·
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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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