Background field method

15A4 Damiano Anselmi
Background field method and the cohomology of renormalization

November 5, 2015

Using the background field method and the Batalin-Vilkovisky formalism, we prove a key theorem on the cohomology of perturbatively local functionals of arbitrary ghost numbers, in renormalizable and nonrenormalizable quantum field theories whose gauge symmetries are general covariance, local Lorentz symmetry, non-Abelian Yang-Mills symmetries and Abelian gauge symmetries. Interpolating between the background field approach and the usual, nonbackground approach by means of a canonical transformation, we take advantage of the properties of both approaches and prove that a closed functional is the sum of an exact functional plus a functional that depends only on the physical fields and possibly the ghosts. The assumptions of the theorem are the mathematical versions of general properties that characterize the counterterms and the local contributions to the potential anomalies. This makes the outcome a theorem on the cohomology of renormalization, rather than the whole local cohomology. The result supersedes numerous involved arguments that are available in the literature.

PDF

Phys. Rev. D 93 (2016) 065034 | DOI: 10.1103/PhysRevD.93.065034

arXiv: 1511.01244 [hep-th]

Posted in Papers, Background field method | Tags: Batalin-Vilkovisky formalism, Renormalization, Adler-Bardeen theorem, Background field method, Cohomology, General gauge theories

14B1 Damiano Anselmi
Renormalization

August 3, 2014

Course on renormalization, taught in 2015.

Last update: September 15th 20123, 242 pages

Buy hard copy of 2023 Edition (final one) on
amazon.com
amazon.it
amazon.de
amazon.fr
amazon.es
amazon.co.uk
amazon.jp
Contents:

Preface

1. Functional integral

1.1 Path integral
- Schroedinger equation
- Free particle
1.2 Free field theory
1.3 Perturbative expansion
- Feynman rules
1.4 Generating functionals, Schwinger-Dyson equations
1.5 Advanced generating functionals
1.6 Massive vector fields
1.7 Fermions

2. Renormalization

2.1 Dimensional regularization
- 2.1.1 Limits and other operations in $D$ dimensions
- 2.1.2 Functional integration measure
- 2.1.3 Dimensional regularization for vectors and fermions
2.2 Divergences and counterterms
2.3 Renormalization to all orders
2.4 Locality of counterterms
2.5 Power counting
2.6 Renormalizable theories
2.7 Composite fields
2.8 Maximum poles of diagrams
2.9 Subtraction prescription
2.10 Regularization prescription
2.11 Comments about the dimensional regularization
2.12 About the series resummation

3. Renormalization group

3.1 The Callan-Symanzik equation
3.2 Finiteness of the beta function and the anomalous dimensions
3.3 Fixed points of the RG flow
3.4 Scheme (in)dependence
3.5 A deeper look into the renormalization group

4. Gauge symmetry

4.1 Abelian gauge symmetry
4.2 Gauge fixing
4.3 Non-Abelian global symmetry
4.4 Non-Abelian gauge symmetry

5. Canonical gauge formalism

5.1 General idea behind the canonical gauge formalism
5.2 Systematics of the canonical gauge formalism
5.3 Canonical transformations
5.4 Gauge fixing
5.5 Generating functionals
5.6 Ward identities

6. Quantum electrodynamics

6.1 Ward identities
6.2 Renormalizability of QED to all orders

7 Non-Abelian gauge field theories

7.1 Renormalizability of non-Abelian gauge theories to all orders
- Raw subtraction

A. Notation and useful formulas

The PDF of the 2015 edition is available here below.

PDF

Posted in Books, Quantum gravity, Adler-Bardeen theorem, Background field method, Renormalization of general gauge theories, Renormalization group, Conformal field theory, Dimensional regularization | Tags: Gauge theories, Quantum field theory, Renormalization, Quantum gravity, Renormalization-group flow, Adler-Bardeen theorem, Background field method

13A3 D. Anselmi
Background field method, Batalin-Vilkovisky formalism and parametric completeness of renormalization

November 13, 2013

We investigate the background field method with the Batalin-Vilkovisky formalism, to generalize known results, study parametric completeness and achieve a better understanding of several properties. In particular, we study renormalization and gauge dependence to all orders. Switching between the background field approach and the usual approach by means of canonical transformations, we prove parametric completeness without making use of cohomological theorems, namely show that if the starting classical action is sufficiently general all divergences can be subtracted by means of parameter redefinitions and canonical transformations. Our approach applies to renormalizable and non-renormalizable theories that are manifestly free of gauge anomalies and satisfy the following assumptions: the gauge algebra is irreducible and closes off shell, the gauge transformations are linear functions of the fields, and closure is field-independent. Yang-Mills theories and quantum gravity in arbitrary dimensions are included, as well as effective and higher-derivative versions of them, but several other theories, such as supergravity, are left out.

PDF

Phys. Rev. D 89 (2014) 045004 | DOI: 10.1103/PhysRevD.89.045004

arXiv: 1311.2704 [hep-th]

Posted in Papers, Background field method, Renormalization of general gauge theories | Tags: Gauge theories, Batalin-Vilkovisky formalism, Quantum field theory, Renormalization, Nonrenormalizable theories

RENORMALIZATION

Quantum Field Theory and Quantum Gravity

Course

Book

Recent Papers