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suggests the compensating term
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e" + op(PPAV) = g"V(!F + j. A) v
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Using Maxwell's equations, we get apF"P = for e" v is
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(1-117)
In the absence of sources this energy momentum density enjoys the following properties: it is gauge invariant, conserved, symmetric, and traceless. According to the general framework, if j is nonvanishing then
(1-118)
This can be also stated as
olle~v
= OA!g"VF 2 + PPF /) = jpFPV
(1-119)
The new tensor eo can thus be identified with the pure electromagnetic contribution. The energy density ego = i(E 2 + B2) is positive and the momentum density eW = (E x B)i is the familiar Poynting vector. We recover the known result that
After these manipulations 0"' is stilI gauge dependent in the presence of sources. This should not be too surprising because the system is an open one. To see what happens in a closed system, taking the sources into account, consider a set of charged point particles interacting electromagnetically. The total action reads
(1-120)
It is in variant under a space-time translation
A(x) -> A(x
+ a)
(1-121)
If we consider infinitesimal variations of the form:
DA"(x)
<5a'(x) iJ,A"(x)
(1-122)
QUANTUM FIELD THEORY
we find
(1-123)
where
0"'(x) = !g"'p 2 - F"P 8'Ap
+ ~ d,.e.x"(,.) ,54[X x.('.)]
x.(,.)]A'[x.(,.)]
~ m. d,.x~(,.)x~(,.) ,54[X !g"'p 2 + F"PP -'
(1-124)
Repeating the steps leading to a gauge-invariant tensor, i.e., adding 8 p(P"P A'), we get 0"'
~ m. d,.x~x;; ,54[X -
x.('.)]
(1-125)
All unpleasant terms have disappeared leaving a gauge-invariant, symmetric, and conserved total energy momentum tensor. Observe that even though the dynamical equations couple the material points to the fields, their contributions appear additively in (1-125).
Let us now study the consequences of Lorentz invariance about a fixed point to generalize the conservation of angular momentum. In infinitesimal form (1-126) with bol antisymmetric. We observe that the lagrangian is not, in general, invariant under such transformation on the arguments alone, since they should be accompanied by corresponding transformations on the fields. In other words, the fields carry a representation of the homogeneous Lorentz group. The elements A of the latter transform them according to
A<p(X) = S(A)<p(A -lX)
(1-127)
We have collected the fields <p in a column vector and S(A) is a matrix representation of the group. For A close to unity, A -1 X will be equal to x - bx with bx given by (1-126), and S(A) will assume a corresponding form. We leave it as an exercise for the reader to study the case of the electromagnetic field as an example of such a circumstance, and we will limit ourselves to the simpler situation where the fields are scalars [S(A) = 1J. To apply Noether's theorem we have to consider bro llv in (1-126) as functions of x and to identify their coefficient in the variation of the action, with due care paid to the fact that bro llv = - bro vil . If e llv stands for the symmetric energy momentum tensor, one finds the conservation of the generalized angular momentum:
(1-128)
0IlJIl,VP
This represents only the orbital part in the case where the fields transform according to nontrivial representations of the Lorentz group. Additional parts corresponding to the internal contributions appear in jIl'vp to build the conserved quantity. In
CLASSICAL THEORY
Chaps. 2 and 3 we shall deal explicitly with such instances. It is crucial that e llv be symmetric for (1-128) to yield such conserved densities and time-independent associated charges (six in number):
J"P =
d 3x JO,VP(x, t)
(1-129)
Note that J"P is not translationally invariant. Under a displacement all of the origin of coordinates, the orbital part varies by an amount aVpP - apr. To obtain what really deserves to be called the intrinsic angular momentum we construct the Pauli-Lubanski vector
iGapyo
JPYpO 1D2 v' p2
(1-130)
which reduces in a rest frame (P = 0) to ordinary three-dimensional angular momentum. Up to now in this relativistic context we have avoided appealing to the hamiltonian formalism or introducing Poisson brackets. The reason is that we did not want time to playa special role. However, nothing prevents us from doing so. At a given time t and for a generic cp(x, t) we can associate the conjugate field
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