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Energy Momentum Tensor in .NET
122 Energy Momentum Tensor Decode PDF417 2d Barcode In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications. PDF 417 Generation In .NET Using Barcode printer for VS .NET Control to generate, create PDF417 2d barcode image in VS .NET applications. For an infinite system we assume a lagrangian depending on the spacetime coordinates x only through fields and their gradients. Under a translation we therefore have Read PDF417 2d Barcode In VS .NET Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Bar Code Encoder In Visual Studio .NET Using Barcode creator for VS .NET Control to generate, create bar code image in .NET applications. 2'(x
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d X{Ov2'  O/l[O(~~i) OVCPi(X)]} E>aV(x) (196) CLASSICAL THEORY
We obtain in this way a generalization of our previous discussion of energy momentum in a completely local form. From the vanishing of M for arbitrary baV(x), we deduce that the energy momentum flow described by the canonical tensor (197) satisfies the conservation law (198) (From this derivation the two indices of llv playa dissimilar role.) It follows that the four quantities Pv corresponding to total energy (v = 0) and threemomentum (v = 1,2,3): (199) are time independent, since ftv =
d 3 x ooeOV(x, t) =  d 3 x itl Oieiv(X, t) = 0 provided that the fields vanish sufficiently rapidly for large arguments, i.e., no energy or momentum escapes at infinity. This result is a typical case of Noether's theorem. The latter states that to any continuous oneparameter set of invariances of the lagrangian is associated a local conserved current. Integrating the fourth component of this current over threespace generates a conserved "charge." Invariance of the lagrangian means in this geometrical context that we also allow the possible transformation of the spacetime argument as appears, for instance, in (194). Furthermore, the space integral defining the charge can be performed on a spacelike surface <J with a surface element d<J Il without affecting the result: Pv =
d<J1l
(1100) It can, of course, happen that the lagrangian also depends explicitly on the coordinates x, in which case (194) is no longer valid and Eq. (198) is replaced by (1101) Such is the case if f is a sum of an invariant part f plus a coupling term to external sources ji(X) of the form f 1 = Li (MX)ji(X). The energy momentum v tensor gets a contribution from f plus an added term from f 1 and reads (1102) with
<PiOVji
QUANTUM FIELD THEORY
according to (1101). This last equation can be rewritten as 0lleg v = 'Ij;(X)OvCPi
(1103) These two ways of expressing the local variation of energy and momentum differ according to whether the interaction energy  Sd 3 x 2 1 is included or not in the system. Let us apply this to the electromagnetic field coupled to an external conserved current j where the lagrangian is given by 2 = _!F2  j
(1104) (1105) + gllVjo A egv =  pIlP OV Ap + !gIlVF 2
e llV =
The tensor e is not gauge invariant even if j is zero! Under a transformation A+A+ocp: (1106) In the absence of external sources the added term is a divergence and will not contribute to the value of the total energy momentum if the fields vanish at infinity. The energy momentum density is in principle measurable and, moreover, is coupled to the gravitational field. It is therefore very unpleasant to find a gaugedependent expression. In addition, the antisymmetric part of e llv is nonvanishing: (1107) We know that the lagrangian is not entirely determined by the equations of motion. Correspondingly, the energy momentum tensor admits some arbitrariness. Generally speaking, let kll(x) denote a conserved current. Then the charge K = Sd 3x kO(x, t) and the local conservation law are unchanged if kll goes into (1108) provided that 0lll1k ll = 0
(1109) d 3x I1kO(x, t) (1110) A solution to these constraints is given by (1111) where k llP is antisymmetric and depends locally on the fields. Indeed, (1109) is verified and so is (1110), since Jd 3XO p k OP = Jd3xtOikOi=0
(1112) CLASSICAL THEORY
Returning to the electromagnetic energy momentum tensor, we see that we can v add to the canonical a piece (1113) where e"P,v is antisymmetric in (j1, p) and local in the fields. The discussion of the gauge dependence where (1114)

