QUANTUM FIELD THEORY in Visual Studio .NET

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32 QUANTUM FIELD THEORY
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1-3 PROPAGATION AND RADIATION 1-3-1 Green Functions
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The dynamical equations of field theory are typically of the Klein-Gordon form: (1-164) 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 (1-164) 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 second-order 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 space-like 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 space-like surfaces widely separated by a time-like interval. It is then useful to construct standard solutions to (1-164) where the right-hand side is replaced by a distribution concentrated around a point x'. We shall generically denote G(x, x') the solution of (1-165) 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 (1-164) will be generated by
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cfJ(x) = cfJ(O)(x) +
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f d4x ' G(x -
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XI)j(X ' )
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(1-166)
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where cfJ(O)(x) obeys the homogeneous equation and is chosen in such a way that cfJ satisfies the boundary conditions. Making further use of translation invariance, (1-165) is solved through a Fourier transformation which replaces it by an algebraic equation. Setting
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G(x - x') = _1_ fd4pe-iP'(X-X')(;(P)
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(2n)4
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(1-167)
we get (1-168) To divide both sides by - p2 + m 2 we have to cope with the zero of this expression on the two-sheeted hyperboloid p2 - m 2 = 0 (or the cone p2 = 0 if m 2 = 0). This in turn is equivalent to prescribing in (1-167) a slightly deformed contour of
CLASSICAL THEORY
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
(1-169)
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 )
(1-170)
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 (1-170), a reflection of the fact that the small x 2 behavior is entirely dictated by -the differential operator in (1-165). The mass term is then responsible for the fact that the support is not concentrated on the light cone as in (1-170), but also involves signals propagating at a speed smaller than one. Returning to the integral representation (1-169) let us perform the Po integral using the decomposition
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