Quantum Field Theory Proceedings of the Ringberg Workshop Held at Te
On the occasion of W. Zimmermann's 70th birthday some eminent scientists gave review talks in honor of one of the great masters of quantum field theory. It was decided to write them up and publish them in this book, together with reprints of some seminal
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Introduction
The relation between symmetries and quantum theory is an important and fundamental issue. For instance, symmetry relations among correlation functions (Ward identities) are often used in order to prove that a quantum field theory is unitary and renormalizable. Conversely, the violation of a classical symmetry at the quantum level (anomalies) often indicates that the theory is inconsistent. Furthermore, in recent years symmetries (such as supersymmetry) have been instrumental in uncovering non-perturbative aspects of quantum theories (see, for example, [1]). It is, thus, desirable to understand the interplay between symmetries and quantization in a manner which is free of the technicalities inherent in the conventional Lagrangiaa approach (regularization/renormalization) and in a way which is model independent as much as possible. In a recent paper [2] we have presented a general method, the Quantum Noether Method, for constructing perturbative quantum field theories with global symmetries. Gauge theories are within this class of theories, the global symmetry being the BRST symmetry [3]. The method is established in the causal approach to quantum field theory introduced by Bogoliubov and Shirkov [4] and developed by Epstein and Glaser [5,6]. This explicit construction method rests directly on the axioms of relativistic quantum field theory. The infinities encountered in the conventional approach are avoided by a proper handling of the correlation functions as operator-valued distributions. In particular, the wellknown problem of ultraviolet (UV) divergences is reduced to the mathematically well-defined problem of splitting an operator-valued distribution with causal support into a distribution with retarded and a distribution with advanced support or, alternatively [6,7], to the continuation of time-ordered products to coincident points. Implicitly, every consistent renormalization scheme solves this problem. P. Breitenlohner and D. Maison (Eds.): Proceedings 1998, LNP 558, pp. 86 105, 2000. 9 Springer-Verlag Berlin Heidelberg 2000
The Quantum Noether Conditionin Terms of Interacting Fields
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Thus, the explicit Epstein-Glaser (EG) construction should not be regarded as a special renormalization scheme but as a general framework in which the conditions posed by the fundamental axioms of quantum field theory (QFT) on any renormalization scheme are built in by construction. In this sense our method is independent from the causal framework. Any renormalization scheme can be used to work out the consequences of the general symmetry conditions proposed in [2]. In the EG approach the S-matrix is directly constructed in the Fock space of free asymptotic fields in a form of formal power series. The coupling constant is replaced by a tempered test function g(x) (i.e. a smooth function rapidly decreasing at infinity) which switches on the interaction. Instead of evaluating the S-matrix by first computing off-shell Greens functions by means of Feynman rules and then applying the LSZ formalism, the S-matrix
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