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div E = P oE. cur1Bat=J in .NET framework
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Create EAN 128 In Java Using Barcode drawer for Java Control to generate, create GS1128 image in Java applications. Data Matrix 2d Barcode Creation In ObjectiveC Using Barcode creation for iPhone Control to generate, create DataMatrix image in iPhone applications. pd 3 x; the flux ofE through a closed surface equals the enclosed
charge (Gauss' law). B dx =
Is (j + ~~) dS; the circulation of B along a closed curve C
0; the flux of B through a closed surface vanishesmagnetic
bordering a surface S equals the flux through S of the sum of the usual current plus Maxwell's displacement current (oE/ot). B dS
charges (monopoles) are absent.
(d)  E . dx
dS .
~~ ; a varying magnetic flux generates an electromotive
force (Faraday's induction law). It is convenient to use a compact notation for vector, tensor, etc., fields which exhibits relativistic covariance. With greek indices running from zero to three: jll = (p, j) Xll = (t, x) we write
E1 FIlV = _FVIl = E2 E3
while derivatives are abbreviated as a/axil == Ow We also define LeviCivita's antisymmetric symbol ellvpa to be equal to 1 or 1 according to whether (/lvpa) is an even or odd permutation of (0, 1, 2, 3) and zero otherwise. Note that ellvpa =  ellvpa . This symbol is used to transform an antisymmetric tensor into its dual, as for instance _B1 _B2 _ _ B1 E3 _E2 0 Fllv  _Fvll 2" 1ellVpaFpa  B2 (134) E1 E 3 0 B3 E2 _E1 0 B3 _B2
_E2 _B3
B2 _B1
(133) E') In words, P is obtained from F by substituting E + B, B +
iellvpappa = _FIlV
E, and
CLASSICAL THEORY
Under Lorentz transformations F and F transform as antisymmetric tensors. In particular, if we perform the boost (123) we obtain (c = 1) ' (E) n n E = + n x (E x n) + v x B ',====~
(135) + ;===~
n x (B x n)  v x E
where These explicit forms are not always useful. For instance, to find the relativistic invariants constructed from E and B it is easier to use tensor notations and observe that they can only be combinations of F IlVpv =  F IlvFIlV =  2(E2  B2) FllvFIlV
Note the following identities: 4E B
(136) FIlAFJ..v FIlVF vp
FIlJ..FJ..v
+ ig/FapFaP
gllpEoB
With these notations and Einstein's summation convention, Maxwell's equations take the compact form ( 137) while current conservation appears as a natural compatibility condition alljll=o
(138) To proceed, we have to identify the space coordinate x with the index i of the various degrees of freedom, this correspondence being such that Li + Sd 3 x. However, we are at first slightly embarrassed since only firstorder derivatives with respect to time occur in Maxwell's equations if we are to identify E(x, t) and B(x, t) with the configuration variables. Furthermore, Lorentz in variance does not appear naturally in this formulation. To overcome these difficulties let us first transform the equations into equivalent secondorder ones. The trick is to introduce the fourpotential All by recognizing that the homogeneous set of equations (137b) is precisely the condition enabling one to write FIlV = allAv _ aVAil
that is, E= VA
aA at
(139) B = curl A
This is in general a local statement with a particular solution, in the vicinity of a
10 QUANTUM FIELD THEORY
point taken as origin, of the form
AIl(X) =  fl dA APV(h)xv
(140) Such a potential is, however, not uniquely defined by (139). It can be modified by the addition of a fourgradient. This is called a gauge transformation For regular fields satisfying (137) AIl(X) can in fact be defined over all spacetime, leading to the equivalent form of Maxwell's equation (141) where D stands for the d'alembertian: D == 0IlOIl == (a/at  Ll. Equation (141) is clearly not affected by a gauge transformation. Gauge arbitrariness of electrodynamics may appear sometimes annoying and sometimes a deep and farreaching principle. Clever choices of gauge lead to interesting simplifications but can also destroy manifest covariance. Our first goal has 'been reached in the sense that we now have secondorder equations. Compact notations may, however, be misleading. Let us therefore have a closer look at these equations, which read explicitly a at p = D A o  at (OA
+ d'IV A) = + div A) LlA  at d'IV A a
V(:t
For some gaugefixing conditions, such as div A = 0 (Coulomb gauge), no time derivatives appear in the first of these relations (Poisson's equation), which plays the role of a constraint. One may then solve for AO: AO(t, x) f 4n Ix  xii fd4n (op/ot)(t, x') Ix  xii
d 3X
p(t, x') The vector potential is now given by
DA = j  V
The divergence of the righthand side vanishes, of course, due to current conservation. The above choice is sometimes used in order to get a lagrangian formulation as a first step toward quantization. It has the obvious drawback of breaking manifest covariance even though the underlying physical picture, corresponding to the elimination of some irrelevant degrees of freedom, may be appealing in specific instances such as boundstate problems. We want, however, to write an action using a Lagrange function preserving

