nLab differential renormalization



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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).


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 12:48:11. See the history of this page for a list of all contributions to it.