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By Machal A.D.

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Outgoing) soft photon of momentum k and polarization r is described by the à (r) wave function e(r)  (k) (resp. e (k) ), where the (r) polarization vectors e (k), r = 1, 2, 3 solve the vector field wave equation (& þ 2 )A = @ A = 0 and obey the orthonormality and completeness conditions eðrÞ ðkÞà Á eðsÞ ðkÞ ¼ Àrs Gauge Fields The issues we have dealt with thus far have interesting difficulties when dealing with gauge fields. We will now discuss some general aspects of the perturbation expansion of gauge theories using as prototypical examples quantum electrodynamics (QED) and quantum chromodynamics (QCD) in four spacetime dimensions.

I1 2i ðk2 Þlþz where the contour of integration is chosen to lie to the right of the poles of the Euler function À(l þ z) and to the left of the poles of À(Àz) in the complex z-plane. Alternatively, one may apply the negative-dimension method in which D is regarded as a negative integer in intermediate calculations and the problem of loop integration is replaced with that of handling infinite series. When combined with the above methods, it may be used to derive powerful recursion relations among scattering amplitudes.

63: 399–409. Weinberg S (1995) The Quantum Theory of Fields, Volume 1: Foundations. Cambridge: Cambridge University Press. Perturbative Renormalization Theory and BRST K Fredenhagen, Universita¨t Hamburg, Hamburg, Germany M Du¨tsch, Universita¨t Zu¨rich, Zu¨rich, Switzerland ª 2006 Elsevier Ltd. All rights reserved. Main Problems in the Perturbative Quantization of Gauge Theories Gauge theories are field theories in which the basic fields are not directly observable. Field configurations yielding the same observables are connected by a gauge transformation.

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Concerning Certain Solvable Equations with Functional Derivatives by Machal A.D.

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