Modular structure of chiral Fermi fields in conformal quantum field theory

by Gennaro Tedesco
Doctoral thesis
Date of Examination:2014-09-05
Date of issue:2014-09-19
Advisor:Prof. Dr. Karl-Henning Rehren
Referee:Prof. Dr. Karl-Henning Rehren
Referee:Prof. Dr. Laura Covi
Persistent Address: http://dx.doi.org/10.53846/goediss-4686

 

 

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Abstract

English

The following thesis deals with the modular theory of Fermi fields in low dimensions; in particular, making use of the algebraic approach to quantum field theory, we have investigated the behaviour of two- dimensional theories which split into two separate copies of chiral fields, each one of them depending on one lightray variable at a time only. The remarkable result we have found is the existence of a vacuum preserving isomorphism β connecting the vacuum states between the algebra of N Fermi fields localised in one single interval and the algebra of one Fermi field localised in N disjoint intervals. Since this map preserves the vacuum states, it therefore intertwines the respective modular groups; as a result, the modular automorphism flow for a Fermi field localised in several intervals turns out to mix the field among different points, with the mixing itself being described through suitable differential equations. Moreover, using the fact that Wick products are as well preserved, one can even embed via β the sub-theories of local observables, as currents and the stress-energy tensor. Consequently, since the isomorphism β is multi-local, a new class of multi-local gauge transformations and diffeomorphisms arise. Interestingly enough, such characterisation of the modular group for multi-local algebras was already presented by [Casini and Huerta, 2009] using different techniques, and so far it is a special feature of free Fermi fields only (although outlooks of generality are fascinating to investigate). The isomorphism that we have found is deeply related to the split property and the way fields transform under diffeomorphism covariance. In particular, it only differs from the action of diffeomorphisms by a gauge transformation, whose features we have characterised in the cases at hand, namely for the local algebras of Fermi fields, currents and stress-energy tensor.
Keywords: Conformal field theory; Modular theory; Algebraic quantum field theory; Tomita-Takesaki theory
 
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