nLab renormalization scheme

Idea

In perturbative quantum field theory, for a suitable free field theory to perturb around, given an interaction Lagrangian density then the renormalization of the corresponding Lagrangian field theory is in each order $\omega \in \mathbb{Z}$ a choice from an affine space of renormalization parameters. A renormalization scheme is a way to pick for all interaction Lagrangians at once an origin in these affine spaces, making them finite dimensional vector spaces relative to this choice.

This means that, given a free field theory $\mathbf{L}_{kin}$, then a renormalization scheme makes the construction of the perturbative S-matrix become a function

$$S \;\colon\; LocObs(E)[ [\hbar] ] \longrightarrow PolyObs(E,\mathbf{L}_{kin})_{mc}$$

which assigns to each local observable $\mathbf{L}_{int} \in LocObs(E)$, regarded as an adiabatically switched interaction Lagrangian density, the corresponding perturbative scattering matrix $S(\mathbf{L}_{int}) \in PolyObs(E,\mathbf{L}_{kin})$ (in the form of an element of the Wick algebra $(PolyObs(E,\mathbf{L})_{mc}, \star_{H})$) satisfying some conditions (notably causal additivity).

Concretely, in causal perturbation theory renormalization is understood as the choice of extension of distributions of the Feynman amplitudes (which are products of distributions, along edges of Feynman diagrams, of Feynman propagators with the interaction Lagrangian densities) to the diagonal where the interaction points coincide. For translation-invariant theories this amounts, locally, to extension of distributions from the complement $\mathbb{R}^n \setminus \{0\}$ of the origin in some (high-dimensional) Cartesian space to the full space. For given scaling degree of a distribution $\omega$, the space of such extensions is an affine space modeled on the finite dimensional vector space of point-supported distributions at the origin of degree $\leq \omega$.

A choice of such extensions for all distributions at once is naturally fixed by a choice of projection from the space of all bump functions on $\mathbb{R}^n$ to the subspace of those all whose partial derivatives to order $\omega$ vanish at the origin (Brunetti-Fredenhagen 00, p. 24-25, following Epstein-Glaser 73, p. 27-28 (236-237)).

Such a choice hence amounts to a renormalization scheme.

The main theorem of perturbative renormalization states that any two choices of renormalization schemes differ by a unique re-definition of the interaction Lagrangian density by an element in the Stückelberg-Petermann renormalization group.

For more see at geometry of physics – A first idea of quantum field theory.

References

Discussion in relativistic field theory via causal perturbation theory is due to

• Henri Epstein and Vladimir Glaser, section 5 of The role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)

• Romeo Brunetti, Klaus Fredenhagen, section 5.2 of Microlocal analysis and interacting quantum field theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

Review is in

• Frédéric Paugam, section 20.6 of Towards the mathematics of quantum field theory, Springer 2014 (pdf)

Discussion in Euclidean field theory is in

• Kevin Costello, def. 1.4.1, def. 8.0.2 Renormalisation and the Batalin-Vilkovisky formalism (arXiv:0706.1533)

Further discussion of dependence of quantities on choice of renormalization scheme includes

• Renormalization scheme dependence, lecture notes (pdf)

• Matin Mojaza, Stanley J. Brodsky, Xing-Gang Wu, A Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in PQCD (arXiv:1212.0049)