Properties of the S-Matrix
The inductive construction of causal perturbation theory is very well suited to derive properties of the S-matrix that are valid in all orders. One has only to show that a property is true in first order and that it is preserved in the inductive step. In this way we get comparably simple proofs of normalizability, various symmetries, gauge invariance and unitarity. Before coming to these themes we will analyse vacuum graphs and show that the perturbative S-matrix S(g) is a well-defined operator in Fock space. The chapter closes with some more sophisticated techniques, namely the renormalization group and interacting fields. The latter are widely used in Lagrangian field theory. However, we shall obtain different results: the interacting fields must depend not only on the space-time argument, but also on the switching function g(x). The limit g → 1 is impossible in general. Nevertheless, these fields fulfill suitably defined field equations which are similar to the basic equations of classical Lagrangian field theory.
KeywordsGauge Invariance Ward Identity Vacuum Polarization Interact Field Adiabatic Limit
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