[Paper Review] On the mathematical sense of renormalization

Alexander Roi Stoyanovsky|arXiv (Cornell University)|Sep 30, 2011

Homotopy and Cohomology in Algebraic Topology3 references3 citations

TL;DR

This paper frames renormalization in quantum field theory as a natural consequence of quantizing a Poisson algebra of classical field theory Hamiltonians. By constructing a Poisson algebra of regular and singular Hamiltonians and showing that deformation quantization cannot extend the standard Moyal product to singular functionals, the authors derive renormalization as a non-canonical map from classical non-local Hamiltonians to themselves, arising from the failure of quantum embeddings to commute with classical symbols.

ABSTRACT

We place the renormalization procedure in quantum field theory into the familiar mathematical context of quantization of Poisson algebras. The Poisson algebra in question is the algebra of classical field theory Hamiltonians constructed in a previous paper (arXiv:1008.3333). Its quantum deformations presumably contain (non-canonically) the algebra of functional differential operators. We explain that this picture contains renormalization as a natural ingredient.

Motivation & Objective

• To provide a rigorous mathematical foundation for renormalization in quantum field theory.
• To place the renormalization procedure within the framework of deformation quantization of Poisson algebras.
• To show that renormalization emerges naturally when quantizing singular Hamiltonians in field theory.
• To clarify the role of non-canonical embeddings of functional differential operators into the quantum algebra.
• To establish a map from classical non-local Hamiltonians to themselves as the mathematical essence of renormalization.

Proposed method

• Construct a topological Poisson algebra Symb of classical field theory Hamiltonians using functionals on Schwartz space with first-order functional derivatives in the dual space of tempered distributions.
• Define regular Hamiltonians as those with smooth, rapidly decreasing kernels in their functional derivative representation, forming a dense Poisson subalgebra Symb^reg.
• Show that the standard Moyal or differential operator product quantizes Symb^reg but fails to extend to Symb due to distributional pairings of higher-order derivatives.
• Propose that a full quantization of Symb must be constructed via QFT methods, leading to a non-canonical embedding R: (Symb^reg, *_{Diff}) ↪ (Symb, *).
• Derive the renormalization map as the inverse of this embedding, H_Λ ↦ R^{-1}(H_Λ), which diverges as Λ → ∞ for families approaching singular limits.
• Identify the failure of the quantum embedding to commute with the classical symbol map as the source of the renormalization map.

Experimental results

Research questions

• RQ1 How can renormalization in quantum field theory be understood as a mathematical operation within a consistent algebraic framework?
• RQ2 What is the role of singular Hamiltonians in the deformation quantization of classical field theory?
• RQ3 Why does the standard Moyal product fail to extend to the full space of classical Hamiltonians, and what replaces it?
• RQ4 How does the non-canonical embedding of regular Hamiltonians into the quantum algebra give rise to a renormalization map?
• RQ5 What is the precise mathematical definition of the renormalization map in terms of classical and quantum Hamiltonian algebras?

Key findings

• Renormalization arises naturally as a map from classical non-local Hamiltonians to themselves, induced by the failure of quantum embeddings to preserve classical symbols.
• The Poisson algebra Symb of classical Hamiltonians includes both regular and singular functionals, with regular ones forming a dense subalgebra.
• The standard Moyal product quantizes only the regular subalgebra Symb^reg, but cannot be extended to Symb due to distributional pairings in higher derivatives.
• A full quantization of Symb requires new methods beyond standard deformation quantization, possibly involving QFT techniques.
• The renormalization map is defined as R^{-1}(H_Λ) for a family H_Λ of regular Hamiltonians tending to a singular limit, and diverges as Λ → ∞.
• The non-uniqueness of the quantum embedding leads to a non-canonical, yet mathematically well-defined, renormalization procedure.