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n(x,t)=aa ( )=5::1 ( ) oCP x, t uuoCP x, t in VS .NET
n(x,t)=aa ( )=5::1 ( ) oCP x, t uuoCP x, t Scanning PDF417 In .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. PDF417 2d Barcode Creation In .NET Framework Using Barcode maker for .NET Control to generate, create PDF417 image in .NET framework applications. dy2'(y,t) PDF417 2d Barcode Decoder In .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications. Create Barcode In .NET Using Barcode encoder for Visual Studio .NET Control to generate, create barcode image in .NET framework applications. (1131) Bar Code Scanner In VS .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Making PDF417 2d Barcode In Visual C# Using Barcode drawer for .NET framework Control to generate, create PDF417 2d barcode image in .NET applications. Similarly, we define the Poisson bracket of two functionals Ll and L2 of the fields cP and n at time t as follows: {Ll L2} = {L2 Lt} ' , Make PDF417 2d Barcode In .NET Framework Using Barcode maker for ASP.NET Control to generate, create PDF 417 image in ASP.NET applications. Painting PDF417 2d Barcode In Visual Basic .NET Using Barcode maker for VS .NET Control to generate, create PDF417 2d barcode image in .NET applications. fd3X[~ bcp(x,2t)  bcp(x, t) bn(x, t) bL ~~J bn(x, t) Painting Universal Product Code Version A In Visual Studio .NET Using Barcode creator for Visual Studio .NET Control to generate, create UPCA Supplement 2 image in .NET framework applications. Generate Barcode In Visual Studio .NET Using Barcode maker for VS .NET Control to generate, create barcode image in .NET applications. (1132) Matrix Barcode Drawer In .NET Using Barcode generation for VS .NET Control to generate, create 2D Barcode image in VS .NET applications. GS1  8 Drawer In .NET Framework Using Barcode generator for .NET Control to generate, create EAN8 image in Visual Studio .NET applications. These functional derivatives are to be taken with a grain of salt. For instance, if L is expressed as a space integral over a density involving gradients of the fields, a suitable integration by parts is always understood. In particular, Decoding UPC Code In .NET Framework Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. EAN13 Decoder In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. {n(x, t), cp(y, t)} {n(x, t), n(y, t)} Create UPCA In Java Using Barcode creator for Android Control to generate, create UPC Code image in Android applications. Matrix Barcode Creator In VS .NET Using Barcode generator for ASP.NET Control to generate, create 2D Barcode image in ASP.NET applications. b3(X  y) Create UCC  12 In Java Using Barcode drawer for Java Control to generate, create EAN / UCC  14 image in Java applications. Code 128 Code Set A Drawer In ObjectiveC Using Barcode generator for iPad Control to generate, create Code128 image in iPad applications. {cp(x, t), cp(y, t)} Painting Code 128 Code Set A In Visual Basic .NET Using Barcode drawer for VS .NET Control to generate, create Code128 image in VS .NET applications. Code 39 Extended Drawer In Java Using Barcode drawer for Android Control to generate, create Code 39 image in Android applications. (1133) and, for instance, It is easy to see that the field equations (144) take the form: 8o<p(x, t) = {H, <p(x, t)} = <  H un(x, t) (1134) 8on(x, t) {H, n(x, t)} =  <  u<p(x, t) 28 QUANTUM FIELD THEORY
in complete analogy with nonrelativistic dynamics, when we have identified Hamilton's function H as the integral of the energy density: d 3 x E)00(x, t) (1135) expressed in terms of <p(x, t), V<p(x, t), and n(x, t). Equation (1134) expresses the fact that H generates time translations of the system. The reader should find it easy to generalize it to space translations and infinitesimal Lorentz transformations. 123 Internal Symmetries
Systems such as those of particle physics are endowed with internal symmetries which playa prominent part in analyzing their spectrum and interactions. As far as we know most ofthese symmetries are, however, only approximate, being broken by forces weaker than the ones under consideration. It is nevertheless useful to pretend at first that these are exact invariances and to study the pattern of violations as a second approximation. Let us briefly derive Noether's theorem in this classical context. Let 4h ... CPN stand for N interacting fields. Some of them can be real or complex and carry Lorentz indices. For each value of x we consider them as a vector in an Ndimensional space where we are given a representation of a group G. The latter can leave certain subspaces invariant (such will necessarily be the case if we have fields having inequivalent transformation properties under the Lorentz group). If G is a compact Lie group we shall write T'(s = 1, ... , r), the corresponding generators with T' antihermitian, and (1136) Recall that the structure constants C S ,S2S3 can be chosen totally antisymmetric for a compact group. For brevity we do not distinguish the group from its linear realization on cp. Let us consider spacetimedependent variations of the fields bcp(x) = bCl.s(x)T'cp(x) bcpt(x) = bCl.s(x)cpt(x)T' (1137) under which we have Sf(cp) 4 Sf(cp + bcp) = il!, where il! is a function of cp, OIlCP (and their complex conjugates), bCl.., and 0llbCl.s. A variation around the stationary solution will yield 0= M f d x bCl..(x) ail! ail!} obCl..(x)  all o[ollbCl..(x)] (1138) We define the corresponding currents
ail! j:(x) = o[ollbCl.s(x)] (1139) Only the part of the lagrangian containing field derivatives will contribute to
CLASSICAL THEORY
jf. Noether's theorem follows from (1138) and gives the divergence of these currents as
.Jl _ 02(x) oJlJs(x)  Obr:J. s
(1140) where we have noted that 02(x)/Obr:J.s(X) can be identified with the same derivative for constant variations br:J.s. A current jf will be conserved if the corresponding term on the righthand side of (1140) vanishes, which means that the original lagrangian is invariant under the onedimensional subgroup of G generated by 7,. Stated differently, to each such subgroup is associated a conserved charge Qs =
d3 x
j~(x, t) dd;s =
d3 x
o~f(x, t) = 0 (1141) Let us return to the general case without assuming the conservation of the currents. This does not prevent us from defining the charges Qs(t) at time t. In what follows, t will be fixed and we shall not write it explicitly. If g; stands for a functional of the fields and conjugate momenta at time t, we want to compute the Poisson bracket (1142) The dependence of 2 on Oobr:J.s(x) arises only from the dependence of 2 on OOcPa. To simplify, let us assume that the fields are real (in which case 7, is real and antisymmetric). If this were not the case, we would consider separately the real and imaginary parts. Then it follows that .0 02 Js (x) = O[Oobr:J.s(X)] QS(X) = We have, therefore, (1143) d 3 x n(x)T'cP(x) (1144) In particular, {Qs, cP(x)} = T'cP(x)

