Translation Group and Particle Representations in Quantum Field Theory

1. Front Matter

   Pages I-VII

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2. The Axioms of the Theory of Local Observables

   Pages 1-27

3. Translations and the Spectrum Condition

   Pages 28-60

4. The Opposite Edge of the Wedge Problem

   Pages 61-88

5. Locality Condition and the Spectrum of Translations

   Pages 89-119

6. Back Matter

   Pages 120-131

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At the time I learned quantum field theory it was considered a folk theo­ rem that it is easy to construct field theories fulfilling either the locality or the spectrum condition. The construction of an example for the latter case is particularly easy. Take for instance an irreducible representation of the Poincare group with positive energy, and as an algebra of observables all compact operators in that representation space. This algebra of observables is even an asymptotically Abelian algebra. Since it has only a single repre­ sentation - except for multiples of this one - it is hardly possible to replace locality in order to obtain a theory with a reasonable physical structure. This example shows that it is not sufficient to replace locality by asymptotic Abelian-ness. The construction of a theory fulfilling locality without a pos­ itive energy representation was first done by Doplicher, Regge, and Singer [DRS]. However, modern investigations on the locality ideal in the algebra oftest functions, started by Alcantara and Yngvason [AY], seem to indicate that this is a general feature; this means that most of the algebras of ob­ servables fulfilling the locality condition will not have representations that also fulfil the spectrum condition. This discussion shows that quantum field theory becomes a subject of interest only if both conditions are satisfied at the same time.

• C*- and W*-Algebras
• C*- und W*-Algebren
• Lokale Quantenphysik
• Mehrere Komplexe Veränderliche
• Quantenfeldtheorie
• quantum field theory
• quantum physics

• Institut für Theoretische Physik, Universität Göttingen, Göttingen, Germany

  Hans-Jürgen Borchers

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