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Perturbative Construction of Models of Algebraic Quantum Field Theory

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Advances in Algebraic Quantum Field Theory

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

The construction of models of algebraic quantum field theory by renormalized perturbation theory is reviewed.

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Notes

  1. 1.

    There is another natural way to introduce a smooth manifold structure on \(\mathcal {E}_S\). We define the atlas where charts are given by maps \(\varphi +\mathcal {D}\rightarrow \mathcal {E}_{S,sc}\), \(\varphi +\overrightarrow{\varphi }\mapsto \varDelta _S\overrightarrow{\varphi }\), with \(\varphi \in \mathcal {E}_S\), where \(\mathcal {E}_{S,sc}\) is the space of solutions with compactly supported Cauchy data. We have \(\varDelta _S:\mathcal {D}\rightarrow \mathcal {E}_{S,sc}\) and we equip \(\mathcal {E}_{S,sc}\) with the final topology with respect to all curves of the form \(\lambda \mapsto \varphi + \varDelta _S(\overrightarrow{\varphi }(\lambda ))\), where \(\lambda \mapsto \overrightarrow{\varphi }(\lambda )\) is a smooth curve in \(\mathcal {D}\). This gives \(\mathcal {E}_{S}\) the structure of an affine manifold in the sense of convenient calculus [35].

  2. 2.

    Since \(\varDelta _S\) is a bi-distribution rather than a smooth function, the map \(\varphi \mapsto \varDelta _S\) doesn’t induce an actual bi-vector field on \(\mathcal {E}\), but belongs to a suitable completion of \(\varGamma (\varLambda ^2T\mathcal {E})\).

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Author information

Authors and Affiliations

  1. II. Institut für Theoretische Physik, 22761, Hamburg, Germany

    Klaus Fredenhagen

  2. University of York, Heslington, York, YO10 5DD, UK

    Katarzyna Rejzner

Authors
  1. Klaus Fredenhagen
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  2. Katarzyna Rejzner
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Correspondence to Klaus Fredenhagen .

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Editors and Affiliations

  1. University of Trento, Trento, Italy

    Romeo Brunetti

  2. University of Pavia, Pavia, Italy

    Claudio Dappiaggi

  3. University of Hamburg, Hamburg, Germany

    Klaus Fredenhagen

  4. University of Vienna, Vienna, Austria

    Jakob Yngvason

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Fredenhagen, K., Rejzner, K. (2015). Perturbative Construction of Models of Algebraic Quantum Field Theory. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds) Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-21353-8_2

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