Higher Currents for the sine-Gordon Model in perturbative Algebraic Quantum Field Theory

by Fabrizio Zanello
Doctoral thesis
Date of Examination:2023-09-28
Date of issue:2024-09-19
Advisor:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Chenchang Zhu
Referee:Prof. Dr. Ralf Meyer
Persistent Address: http://dx.doi.org/10.53846/goediss-10736

 

 

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Abstract

English

The main results of this thesis are the establishment of the super-renormalizability by power counting and of the summability of an infinite number of (classically conserved) higher currents for the sine-Gordon model in the framework of perturbative Algebraic Quantum Field Theory (pAQFT). In order to achieve this, we first consider the classical theory. Combining the notion of Bäcklund transformations with Noether’s Theorem, we obtain recursive formulas for the components of the higher currents and also characterize them introducing a suitable notion of degree. We then move to the pAQFT setting and, by means of some technical results, we compute explicit formulas for the unrenormalized interacting components of the currents. In the context of the Epstein and Glaser approach to renormalization, we prove a uniform bound on the scaling degree of the interacting components given by the notion of degree introduced previously, which directly implies superrenormalizability. Subsequently, we describe the concrete renormalization of the interacting components by a procedure which we call piecewise renormalization. Finally, we show that the formal power series arising as expectation values of the renormalized interacting components in a generic Gaussian state are, under suitable conditions, summable.
Keywords: Perturbative Algebraic Quantum Field Theory; Renormalization; Integrable systems
 
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