Algebraic approach to classical field theory and its perturbative quantization around arbitrary Background Fields

Abstract

We seek to address the problem of constructing realistic quantum field theoretical models as a formal deformation quantization of the corresponding classical field theories, based on the algebraic formulation for classical field theory put forward in the ongoing series of papers by the visitor, Prof. Klaus Fredenhagen (University of Hamburg, Germany) and the host in the case of real scalar fields, partially extended by several authors to spinor, gauge, gravitational and membrane fields. The framework is based on singling out a suitable algebra $A_{cl}$ of functionals on the space of classical field configurations, which play the role of observables. The dynamics is imposed on this algebra by quotienting out the ideal generated by a hyperbolic Euler-Lagrange operator, which in turn induces a(n almost) Poisson structure on $A_{cl}$ by means of its Peierls bracket. This structure will then be used to deform the algebraic structure of the algebra $A_{cl}[[\hbar]]$ of formal power series in Planck constant $\hbar$ with values in $A_{cl}$, extending the setup of perturbative algebraic quantum field theory to arbitrary classical background fields. (AU)