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

• Grant number: 15/10813-4
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Duration: July 13, 2015 - July 24, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Pedro Lauridsen Ribeiro
• Grantee: Pedro Lauridsen Ribeiro
• Visiting researcher: Romeo Brunetti
• Visiting researcher institution: Universitá degli Studi di Trento, Italy
• Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil

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)