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Taking the divergence of both sides we see that, for A #- 0,
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Hence if 0IlAIl vanishes for large It I it will be guaranteed to vanish at all times, and (1-49) will therefore be equivalent to Maxwell's equations in the Lorentz gauge. Let us summarize the logic:
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= - iF2 - j . A. Gauge invariance <-+ current conservation. Least-action principle -+ Maxwell's equations.
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CLASSICAL THEORY
2. 2' = -!F2 + H(o A)2 - j A and 0 j = O. Least-action principle DOJlAJl = O. Boundary conditions -+ Lorentz gauge -+ Maxwell's equations.
Simple remarks enable one to remember the signs in (l-4S) irrespective of the conventions. The "kinetic" terms in (8A/8tf should appear with a factor +l Moreover, since pO A O is a potential energy it has to occur with a minus sign. The rest follows from Lorentz invariance. Note also that in rationalized units Jd 3 x(E2/2) is an energy and therefore Jd4 x(E2/2) has indeed the dimension of an action: energy x time.
1-1-3 Electromagnetic Interaction of a Point Particle
The four-vector current of a charged point particle is localized on its space-time trajectory xJl(r). Therefore Jl Jl dx (1-51) r(y, t) = dx J3[y - x(r)] [t=x O ) = f dr d; J4[y - x(r)]
with e denoting the conserved charged J d 3 y /(t, y) = e. The current jJl as expressed by (1-51) fulfills the continuity equation oJljll = O. For the combined system particle + field we simply add the actions 2 4 4 I=-fd X(:2 +joA)-m fds=-fd x: -e fdXJlAJl[X(r)]-m fdS
= f d4 x 2'em(X)
+ f dt{ -mJ1 -
v2 + eA v - eAO)
(1-52)
Any charged particle acts as a source, thus modifying the surrounding field. As a first approximation we shall neglect this effect and assume AJl fixed by external conditions. This allows us to drop the contribution of 2'em in (1-52). Observe that we take in this case an opposite view to the one followed at the beginning of this discussion. Genuinely coupled systems, whether classical or quantum mechanical, are obviously difficult to study. It is a fair practice to first investigate simplified models before looking at the more realistic ones. In Sec. 1-3-2, we shall briefly investigate a classical coupled system. We are thus led to study the motion of a particle under the action of a given external field. Its Lagrange function is
-mJ1=V2 + e(A
p=-=
v - AO)
(1-53)
We see that its conjugate momentum p differs from the free value mvlJ 1 - v , since
oL mv ov J1-v
+ eAo =
(1-54)
Hamilton's function is expressed as
H = v
oL ov
J1=7
[m 2 + (p - eA)2] 1/2
+ eAo
(1-55)
QUANTUM FIELD THEORY
while the equation of motion (d/dt)(oI/ov) = aI/ax takes the form d dt
}1 -
- e - - eVA at
+ ev x curl A = e(E + v x B)
(1-56)
which is nothing but the Lorentz force law. Finally the variation of energy g is dg d m d mv at =dt ( y'T=7 ) =v dt J1=V2= eE' v (1-57)
a well-known result expressing the fact that only electric fields do work. Equations (1-56) and (1-57) can be written compactly as dUll m-=epvu v dr
(1-58)
with u ll = dxll/dr being the four-velocity. The reader can verify that the variational principle applied to the action I = - Jdr m(u 2 /2) - JeA . dx leads directly to (1-58).
Let us now study three simple examples:
1. Motion in a constant uniform field Let F ~v be independent of x. Equations (1-56) and (1-57) lead at once to
- - - - ---= =
Jl- v6
e(Et
x B)
,g - ,go = eE x
(1-59)
We can also solve directly the covariant Eq. (1-58) in matrix form:
u~(r) =
(ex p : F }vUV(O)
Finally it is instructive to use a spinorial representation. For a four-vector u we introduce the 2 x 2 matrix:
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