TL;DR
This paper generalizes the Laplace Hopf algebra to create an algebraic framework for quantum field theory, unifying various products and renormalization as deformations within a group-structured setting.
Contribution
It introduces a unified algebraic approach to quantum field theory products and renormalization, extending the Laplace Hopf algebra to encompass operator and time-ordered products.
Findings
Provides an algebraic interpretation of Wick's theorem
Shows the renormalized time-ordered product as an associative deformation
Identifies the renormalization group as a group acting on the product
Abstract
The Laplace Hopf algebra created by Rota and coll. is generalized to provide an algebraic tool for combinatorial problems of quantum field theory. This framework encompasses commutation relations, normal products, time-ordered products and renormalisation. It considers the operator product and the time-ordered product as deformations of the normal product. In particular, it gives an algebraic meaning to Wick's theorem and it extends the concept of Laplace pairing to prove that the renormalised time-ordered product is an associative deformation of the normal product involving an infinite number of parameters. The parameters themselves form a group: the renormalisation group, which acts on the product instead of on the algebra.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Noncommutative and Quantum Gravity Theories