Regensburg 1998 – scientific programme

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 7: Strukturelle Fragen der Quantenfeldtheorie

MP 7.2: Fachvortrag

Thursday, March 26, 1998, 14:20–14:40, H45

From euclidean field theory to quantum field theory. — •Dirk Schlingemann — The Erwin Schr"odinger International Institute for Mathematical Physics, Vienna

Starting from an appropriate set of euclidean n-point functions (Schwinger distributions), a Wightman theory can be reconstructed by an application of the famous Osterwalder-Schrader reconstruction theorem. This procedure (Wick rotation), which relates models for classical statistical mechanics to quantum field theory models, is, however, somewhat subtle. It essentially relies on the analytic properties of the euclidean n-point functions.

We present a C*-algebraic version of the Osterwalder-Scharader reconstruction theorem. Starting from an euclidean C*-algebra A, which can be regarded as the euclidean analogue of quasi-local algebra of a Haag-Kastler net, and an appropriate linear functional on A, which plays the role of the functional integral, we reconstruct directly a Haag-Kastler net of bounded operators in a vacuum representation.

Our reconstruction theorem also applies to extended objects, like Wilson loop variables. In this sense we archive a generalization of previous schemes. However, this is not a total generalization since we have to assume the existence of (non-trivial) operators which can be localized at sharp euclidean times. On the other hand, the known interacting models like the P(φ)₂, the Yukawa₂ as well as the φ₃⁴ model fulfill this assumption and, therefore, we think that this is a sensible requirement.