Single-Particle Electrodynamics - Assassination Science

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Hamiltonian H DP in the Dirac–Pauli representation. (An unstated assumption, crucial to the validity of the procedure, is that the “odd” part of the Hamiltonian is in fact of no higher order in m than m 0 . This is usually the case, but the assumption has the latent ability to trip one up. For example, if one tries to Foldy–Wouthuysen-transform a Hamiltonian in which the mass term βm has been multiplied by e iθγ 5 (say, by a canonical transformation of the representation), then one can be led to quite erroneous conclusions if one assumes that the terms omitted in the subsequent Foldy–Wouthuysen process are of high order in m; in fact, the omitted terms are of exactly the same order as the terms that are retained; the Foldy–Wouthuysen transformation is, if applied in this way, completely useless. In such cases, the correct procedure is to first perform a simple canonical transformation to remove the order m +1 terms from the “odd” parts of the Hamiltonian; the resulting representation may then be fruitfully subjected to the Foldy–Wouthuysen transformation.) It may be wondered, after hearing of all of the wonderful properties of the Newton–Wigner representation, why one should bother with any other representation at all. In particular, why do we usually only concentrate on the Dirac–Pauli representation of the Dirac equation? (Or representations “trivially” related to it; we shall define this term with more precision shortly.) The answer is subtle, but beautiful. The massive leptons in Nature are excellently described by a minimal coupling of their Dirac fields to the electromagnetic field, in the Dirac–Pauli representation only. It is not often stressed that minimal coupling—the use of the prescription b −→ b − qA in the corresponding non-interacting formalism—is not a universal, representation-independent transformation. The reason is that, in general, a canonical transformation used to effect a change in representation may be momentum-dependent; indeed, the Foldy–Wouthuysen transformation itself 163
is an important example. Clearly, the processes of using minimal coupling, and then performing a momentum-dependent transformation, on the one hand; and that of performing the momentum-dependent transformation first, and then using minimal coupling, on the other; will lead to completely different relativistic wave equations, in general. A priori, one cannot know which representation one should use the minimal coupling prescription on. (Clearly, “trivial” changes of representation, in the sense used above, are therefore those in which the canonical transformation does not involve the canonical momentum operator.) Dirac was therefore not just brilliant, but also somewhat lucky: he invented a representation, in which to use minimal coupling, that just happened to be the right representation for the electron (and, as we now know, the muon and tauon also). But what distinguishes the Dirac representation of spin-half particles from all others? It gives a free-particle Hamiltonian that is linear in the components of the canonical momentum operator. This property is, of course, precisely what Dirac was striving for in the first place; even though his reasoning for its necessity was, as we now know, flawed, his intuition nevertheless led him along the right path. We therefore come to recognise that, in reality, there are two representations of the Dirac equation that are singled out above all others,—each having qualities unique to itself,—that have a truly direct correspondence with Nature: The Dirac–Pauli representation is unique due to its linearity; it is the representation in which the massive leptons are minimally coupled. The Newton–Wigner representation is unique due to its decoupling of positiveand negative-energy states; it is the representation in which the operators of the free theory correspond to their classical counterparts. We may go even further, conceptually speaking, in our description of the massive leptons: they are, in effect, two types of particle in the one being: on the one hand, they are four-component animals, in the Dirac– Pauli representation, in which all four components are inextricably coupled, 164
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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 [
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
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: That (4.73) is an amazing result is
Page 167 and 168: let us therefore add a Pauli term (
Page 169 and 170: On the other hand, for a Pauli mome
Page 171 and 172: moment actually cancels with the Di
Page 173 and 174: Chapter 5 The Retarded Fields I hav
Page 175 and 176: to particles carrying dipole moment
Page 177 and 178: workers. In 1969, Cohn [53] unfortu
Page 179 and 180: manifestly-covariant field expressi
Page 181 and 182: as a mathematical aid—it not bein
Page 183 and 184: integrate over the proper time τ,
Page 185 and 186: Equation (5.28) is, in the above no
Page 187 and 188: this is Jackson’s equation (14.13
Page 189 and 190: moments of a particle. There are a
Page 191 and 192: It can be noted that equation (5.51
Page 193 and 194: the overall Lorentz-gauge four-pote
Page 195 and 196: Applying (5.61) to the delta functi
Page 197 and 198: The results above are for a particl
Page 199 and 200: (One should carefully note that the
Page 201 and 202: under differentiation, the latter e
Page 203 and 204: Because the field expressions diver
Page 205 and 206: expression (5.81) in the direction
Page 207 and 208: We now note that the non-zero elect
Page 209 and 210: clearly, for any ε, this parametri
Page 211 and 212: For a point electric dipole moment,
Page 213 and 214: 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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