Communications in Mathematical Physics

, Volume 105, Issue 1, pp 49–84 | Cite as

On the connection between quantum fields and von Neumann algebras of local operators

  • Wulf Driessler
  • Stephen J. Summers
  • Eyvind H. Wichmann


The relationship between a standard local quantum field and a net of local von Neumann algebras is discussed. Two natural possibilities for such an association are identified, and conditions for these to obtain are found. It is shown that the local net can naturally be so chosen that it satisfies the Special Condition of Duality. The notion of an intrinsically local field operator is introduced, and it is shown that such an operator defines a local net with which the field is locally associated. A regularity condition on the field is formulated, and it is shown that if this condition holds, then there exists a unique local net with which the field is locally associated if and only if the field algebra contains at least one intrinsically local operator. Conditions under which a field and other fields in its Borchers class are associated with the same local net are found, in terms of the regularity condition mentioned.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Field Operator 
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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Wulf Driessler
    • 1
  • Stephen J. Summers
    • 1
  • Eyvind H. Wichmann
    • 2
  1. 1.Fachbereich PhysikUniversität OsnabrückOsnabrückFederal Republic of Germany
  2. 2.Department of PhysicsUniversity of CaliforniaBerkeleyUSA

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