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QUANTUM FIELD THEORY in Visual Studio .NET
32 QUANTUM FIELD THEORY PDF417 2d Barcode Recognizer In .NET Framework Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in VS .NET applications. PDF417 2d Barcode Encoder In .NET Framework Using Barcode creation for VS .NET Control to generate, create PDF417 image in .NET applications. 13 PROPAGATION AND RADIATION 131 Green Functions
PDF417 2d Barcode Scanner In VS .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Creator In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create barcode image in .NET framework applications. The dynamical equations of field theory are typically of the KleinGordon form: (1164) where j may depend on the fields cfJ and extra indices have been omitted. We had already an example with Maxwell's equations for the potential in the Lorentz gauge, where the mass term of (1164) was absent. For the time being, let us assume the source j(x) to be given and m 2 ; : O. We are thus dealing with an hyperbolic secondorder partial differential equation, which determines cfJ in the neighborhood of a point x in terms of its values together with those of its normal derivative on a spacelike surface element passing through x. Characteristic elements are tangent to the light cone, showing that causality is locally obeyed. In scattering theory one seldom has to tackle the problem in the way just mentioned. Boundary conditions on cfJ are rather imposed along spacelike surfaces widely separated by a timelike interval. It is then useful to construct standard solutions to (1164) where the righthand side is replaced by a distribution concentrated around a point x'. We shall generically denote G(x, x') the solution of (1165) with an appended suffix to characterize the boundary conditions imposed on G. The latter will most frequently be translationally invariant in such a way that the corresponding Green functions (or propagators) will only depend on the argument x  x'. From the superposition principle, solutions to (1164) will be generated by Barcode Decoder In .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. Encoding PDF 417 In Visual C# Using Barcode encoder for VS .NET Control to generate, create PDF417 2d barcode image in .NET applications. cfJ(x) = cfJ(O)(x) +
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integration. We note that all these choices differ at most by a contribution g(p, Poll Po I) c5(p2  m2 ) to G(p), an expression corresponding to the solution of the homogeneous equation. This choice is, of course, related to boundary conditions at infinity. Let us first define the retarded and advanced Green functions: Gret (p) adv
+ ie)' p2 Po  _ m2
(1169) The e prescription is equivalent to a slight contour deformation in the integration over Po and shows that G(x) is to be interpreted as a distribution. If we first perform the integral over Po, in the case of Gret(x), say, we can close the integration contour in the upper complex plane if X O < 0, without encountering any singularity. From Cauchy's theorem we conclude that Gret(x) vanishes for Xo < O. The opposite conclusion applies to Gadv(x). It can be checked that these distributions are Lorentz invariant so that Gret(x) vanishes outside the forward light cone while Gadv(x) vanishes outside the backward one. These properties are in agreement with causal propagation. We also note that both Green functions are real, with Gadv(x) = Gret (  x). When m 2 = 0 we recover Gret (x) I 2 =O m adv
1 2 2n e( xo)c5(x ) (1170) while for m 2 > 0 the explicit expressions involving Bessel functions are not too illuminating. However, no matter what m 2 is, the singularity of these Green functions on the light cone remains given by (1170), a reflection of the fact that the small x 2 behavior is entirely dictated by the differential operator in (1165). The mass term is then responsible for the fact that the support is not concentrated on the light cone as in (1170), but also involves signals propagating at a speed smaller than one. Returning to the integral representation (1169) let us perform the Po integral using the decomposition

