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r::J in VS .NET
r::J PDF417 2d Barcode Scanner In .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. Making PDF 417 In VS .NET Using Barcode printer for Visual Studio .NET Control to generate, create PDF417 image in .NET framework applications. 1. Draw all distinct diagrams with 2n external points Xl, ... , X2n and p vertices Yl, ... , YP' and sum all the contributions according to the following rules. Scan PDF417 2d Barcode In VS .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications. Bar Code Creation In .NET Using Barcode creator for Visual Studio .NET Control to generate, create bar code image in Visual Studio .NET applications. Figure 61 Examples of diagrams with a symmetry factor: in case (a), S = 2!; in case (b), S = 3!. Recognizing Barcode In VS .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in VS .NET applications. PDF417 Creator In C#.NET Using Barcode printer for VS .NET Control to generate, create PDF417 2d barcode image in .NET applications. QUANTUM FIELD THEORY
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USS Code 128 Maker In .NET Using Barcode generation for ASP.NET Control to generate, create USS Code 128 image in ASP.NET applications. Encoding Barcode In Java Using Barcode creation for Java Control to generate, create barcode image in Java applications. 2. To each vertex attach a factor  iA. 3. To each line between Zi and Zj attach a factor cP(Zi)cP(Zj) given by (616). 4. Each diagram has to be divided by a symmetry factor S. As an example, let us examine the twopoint function ( iA)P <01 Tcp(Xl)CP(X2) : cp4(Yl) :/4! : cp4(yp) :/4! 10) to low orders. For p = 0, we get cP(Xl)<P(X2), while there is no p = 1 contribution, since we forbid contractions cP(y) <b(y). For p = 2, there are three contributions, depicted in Fig. 62 UPCA Creator In ObjectiveC Using Barcode maker for iPhone Control to generate, create Universal Product Code version A image in iPhone applications. Scanning Code 3 Of 9 In VS .NET Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications. (a) (b) (c) Barcode Creator In None Using Barcode generator for Software Control to generate, create barcode image in Software applications. UCC128 Generator In Java Using Barcode generation for BIRT Control to generate, create UCC  12 image in BIRT applications. (  iA.f r;;, r;;, 4 4! CP(Xl)CP(X2) [CP(Yl)CP(Y2)] The same with Yl ++ Y2
Let us emphasize the importance and the meaning of the word "distinct" in rule 1 above. In configuration space, diagrams are considered as distinct when they are topologically different, all the points Xi and Yj being fixed. For instance, diagrams (b) and (c) of Fig. 62 are distinct, whereas diagrams (a) and (b) of Fig. 63 are not; in the latter case only one of them has to be taken into account. Let us divide the diagrams into two classes. The first class contains no vacuumvacuum subdiagram, i.e., subdiagrams (like the one of Fig. 62a) which are not connected to the external points; the second class includes such subdiagrams. To each diagram of the first class, we may associate a set of secondclass diagrams, by addition of vacuumvacuum subdiagrams. Denoting by the Figure 63 Example of indistinguishable diagrams; only one of them has to be taken into account.
PERTURBATION THEORY
superscript (1) the firstclass contribution to the Green function, we may write
G(x1o ... ,X2n)<01 Texp
{i fd4x 2'int[<P(X)]} 10) = p~op! fd4Yl"'d4YPkf:O (p) <01 T<p(Xl)'" <p(X2n)2'int(Yl)"'2'int(Yk) 10)(1) k x <OIT2'int(Yk+l)"'2'int(Yp)IO) (618) since the (p  k)thorder vacuumvacuum subdiagram corresponds to the vacuum expectation value of the timeordered product of P  k interaction lagrangians. Moreover, the combinatorial factor p) p! ( k = k!(p  k)! results from the fact that, after integration over Y 10 .. , YP' all possible permutations of the Y give rise to equal contributions. It is easy to see that the vacuum amplitudes <01 Texp [if d4x 2'int(X)] 10) factorizes on both sides of Eq. (618) and G(X1o"" X2n) = p=o p.
d4Yl'" d4yp <01 T<p(Xl)'" <p(X2n)2'int(yd'" 2'int(Yp) 1 )(1) (619) Only diagrams without vacuum subdiagrams contribute to the righthand side ofEq. (619). It may be simpler to give the Feynman rules in momentum space, i.e., to consider the Fourier transform G: G(P1o ',P2n) = d4Xl" d4x 2n exp
(i j~l
(620) Pj" Xj) G(X1o" "X2n) With our conventions, all the P are incoming momenta. Translation invariance entails momentum conservation Iin Pj = 0, which implies G(Pl"",P2n)= (2n)4t5 4(pl + ... + P2n)G(P1o"',P2n) (621) For simplicity we use the same symbol in x or P space. We have to integrate over all spacetime points x and Y to get G from (619) and (620). Using the Fourier representation (616) of the elementary contraction, the integration over Xi produces a contribution i(Pf  m 2 + is)l for each external line. We are left with the integrations over the Yj and the k/, which are the momenta running through the internal lines. Let us concentrate on the contribution attached to a given configuration of lines and vertices for fixed internal momenta and given external configuration arguments x 10 , X2n' Any permutation of the labels Y1o"" YP of the internal vertices yields the same contribution. However, not all QUANTUM FIELD THEORY
of the p! permutations of the y need generate topologically distinct diagrams in the original configuration space. If this were the case we would get a factor p! compensating the factor (p!)  1 appearing in Eq. (619). It may happen for some diagrams that two (or more) vertices play identical roles (this is the case in the example of Fig. 63a where Y3 and Y4 are such vertices). If the p vertices are divided into groups of Vb V2,"" Vs vertices playing the same role (Vl + ... + Vs = p), the integration over the y variables yields a degeneracy factor p !/Vl ! ... vs!. This compensates only partially the (p !)1 factor of Eq. (619) and leaves a vertex symmetry factor I/Vl!'" vs!, which multiplies the line symmetry factor discussed previously. For instance, the diagram of Fig. 63a has, in momentum space, a global symmetry factor i xi. Finally, every y integral is of the form d4Yj e iYj ' qj = (2n)4J4(qj) where qj denotes the sum of all external or internal momenta flowing into the vertex Yj. Momentum is conserved at each vertex, and is thus globally conserved. Feynman rules in momentum space may now be written, for the Green function G(Pb ... , P2n), as follows: 1. Draw all topologically distinct diagrams with 2n external lines of incoming momertta Pl," ',P2n and without vacuum subdiagrams. For each diagram, denote by k b . .. , kJ the momenta of internal lines. In a scalar theory without derivative coupling, the choice of an orientation ofthe internal lines is irrelevant. 2. To the jth external line, assign the factor i/(pJ  m 2 + is). 3. To the lth internal line, assign [d 4k//(2n)4] [i/(kf  m 2 + is)]. 4. To each vertex, assign ( iA)(2n)4 J4(qj) where qj is the sum of all incoming momenta at vertex j. 5. Integrate over the k variables the product of all these contributions and divide by the symmetry factor of internal lines and vertices of the diagram. 6. Sum the contributions of all topologically distinct diagrams. Generally, if V is the number of vertices (called p previously) and I the number of internal lines, there are VI conservation rules among the k after separation of the global conservation (621), and thus at most I  V + 1 nontrivial integrations. The Feynman rules must be slightly modified when some external lines are connected to no vertex at all. For instance, the single propagator of Fig. 64 gives in configuration space the contribution

