Single-Particle Electrodynamics - Assassination Science

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Lorentz force equation of motion for charged particles works well in a wide variety of applications; no one would deny its applicability to the real world. Further than this, one must tread carefully. In the early days of quantum mechanics, it was commonly stated that the newly-invented spin degrees of freedom had no classical limit whatsoever. But accelerator physicists know this to be an exaggeration: the Thomas– Bargmann–Michel–Telegdi equation of motion [214, 25]—which is a completely classical equation—describes the precession of spins to a high degree of accuracy. (Indeed, in the interpretation of the extremely precise measurements of the magnetic moments of fundamental particles, the Thomas–Bargmann–Michel–Telegdi equation is generally used as an analytical tool [19].) Of course, a classical object may indeed possess “spin” angular momentum: it is simply the angular momentum of the object in its rest frame, about its centre of energy; it therefore comes as no surprise that a classical limit should exist. What is surprising, to some, is that this limit exists, and is entirely valid, even for spin-half particles—the lowest non-zero quantum of spin possible; in other words, it is not necessary to invoke the “large quantum number” arguments of the Old Quantum Theory to make contact with the classical formalism. In fact, one finds that, for a single particle, the classical spin precession equation describes accurately the evolution of the three-vector characterising the expectation value of the spin: its “latitude” describes the ratio of the amplitudes in the “up” and “down” states; and its “longitude” describes the relative phase between these two amplitudes. The overall phase of the wavefunction is—as with any reduction to the classical limit—lost. (See, e.g., [155, App. 1.3] for a thorough and rigorous derivation of this correspondence.) But the successes of the Lorentz force law, and the Thomas–Bargmann– Michel–Telegdi spin precession equation, do not mean that classical equations of motion may be used carte blanche. One must always recall that there will be situations in which the wave-packets assumptions fail. 91
The author will, in the remaining chapters of this thesis, obtain numerous consequences arising from the careful consideration of classical electrodynamics; some of these results are new. The reader may validly ask: In which physical situations may these results be used? The author is not ashamed to admit that he does not precisely know; it is, arguably, a question to which a hasty answer would almost definitely be wrong. The author therefore suggests a “Suck It and See” approach, as has been used, to great advantage, with the Lorentz force and the Thomas–Bargmann–Michel–Telegdi equations. If the equations derived herein by the author describe your physical application well, then count your blessings: you have a new analytical tool available to you. 2.8 Pointlike trajectory parametrisation Consider a pointlike particle. We shall, in later chapters, have need to parametrise its (relativistic) path, z(τ), around some particular event, which we shall refer to as the zero event. The following question then arises: What is the best way to perform this parametrisation? Consider, first, Lorentz invariance: no matter which Lorentz frame we use to perform our computations, the final, Lorentz-invariant results must be the same. Choice of an arbitrary Lorentz frame gives us ten degrees of freedom that we may choose at will: the origin of the four spacetime coördinates; the three-velocity of the frame relative to that of the particle, at the zero event; and the three Euler angles describing the spatial orientation of the frame. Clearly, the kinematical motion of the particle provides “natural” choices for seven of these degrees of freedom, that will clearly simplify the mathematics considerably: namely, setting the origin of spacetime to be at the zero event (we also set the origin of τ to be at this same event); and setting the relative three-velocity of the Lorentz frame and the particle’s motion to be zero at this same event. We shall leave the spatial orientation of the Lorentz 92
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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
Page 41 and 42: tron. The proton and neutron clearl
Page 43 and 44: most appropriate in the task of rem
Page 45 and 46: (see Section A.8): f ≡ d τ p. (2
Page 47 and 48: T 0i ≡ T i0 = E×B, T ij = 1 2 (E
Page 49 and 50: densities and the conservation laws
Page 51 and 52: 2.4 Lagrangian mechanics An alterna
Page 53 and 54: canonical momentum: b a ≡ ∂ ˙q
Page 55 and 56: to be the appropriate degrees of fr
Page 57 and 58: task. Firstly, we need to select ap
Page 59 and 60: canonical energy, canonical momentu
Page 61 and 62: Let us return to the electric charg
Page 63 and 64: should also be carefully noted agai
Page 65 and 66: 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: wish of them—not least of which b
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 and 118: desirable properties—but unfortun
Page 119 and 120: the magnitude d of d describes the
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
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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
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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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