nLab
differential renormalization

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

In perturbative quantum field theory, differential renormalization (Freedman-Johnson-Latorre 92) is a method of ("re"-)normalization which makes choices for defining time-ordered products/Feynman amplitudes at coincident interaction vertices.

Differential normalization follows directly (Prange 97) from Epstein-Glaser renormalization in the guise of extensions of distributions of time-ordered products/Feynman amplitude to coinciding interaction points (this prop.):

In the proof of that construction (this prop) over Minkowski spacetime p,1\mathbb{R}^{p,1}, a choice of ("re"-)normalization is embodied at each order kk \in \mathbb{N} (number of vertices in Feynman diagrams) by a choice of projection operator

p ρ k:𝒟( (p+1)(k1))𝒟 ρ k( (p+1)(k1)) p_{\rho_k} \;\colon\; \mathcal{D}(\mathbb{R}^{(p+1)(k-1)}) \longrightarrow \mathcal{D}_{\rho_k}(\mathbb{R}^{(p+1)(k-1)})

(this equation) from all test functions to those that vanish to order ρ k\rho_k at the origin. (This projction is often denoted “WW”, see this remark).

Restricted to the image of this projector the relevant time-ordered product T kT_k has a unique extension of distributions to the origin, and this defines the ("re"-)normalization.

This defines a new distribution T k,RT_{k,R} by

T k,R,bT k,p ρ k(b). \left\langle T_{k,R}, b \right\rangle \;\coloneqq\; \left\langle T_k, p_{\rho_k}(b)\right\rangle \,.

Differential renormalization focuses on manipulating theses expressions T k,RT_{k,R} (Prange 97, section 1.1).

References

The concept is due to

  • D. Z. Freedman, K. Johnson, J. I. Latorre, Differential regularization and renormalization: a new method of calculation in quantum field theory, Nucl. Phys. B 371 (1992) 353-414

  • J. I. Latorre, X. Vilasis-Cardona, Systematic Differential Renormalization to All Orders, Ann. Phys. (N.Y.) 231 (1994) 149

Discussion in causal perturbation theory/relation to Epstein-Glaser renormalization is due to

Discussion of application to anomalous magnetic moment and supergravity:

  • F. del Aguila, A. Culatti, R. Munoz-Tapia, M. Perez-Victoria, Supergravity corrections to (g2) l(g-2)_l in differential renormalization, Nuclear Physics B 504 (1997) 532-550 (arXiv:hep-ph/9702342)

Last revised on February 6, 2018 at 07:48:11. See the history of this page for a list of all contributions to it.