Renormalization Group - 1º Semester 2017/2018

Program

Introduction: the Wilsonian approach to quantum field theory. Renormalization in quantum mechanics. The Casimir effect.

Path integrals: path integral in quantum mechanics. Path integral in quantum field theories. Generating functionals. Partition functions and correlation functions. 1PI effective actions.

Fermionic path integrals.

Effective actions and Schwinger proper time. QED: Euler-Heisenberg Lagrangian. One-loop beta function. Schwinger pair production.

Relating Schwinger pair production in scalar QED to topological string theory and Gopakumar-Vafa invariants.

Scalar field theory: regularization and renormalization. 1PI effective action and the Coleman-Weinberg potential.

Renormalization group: fixed points, anomalous dimensions, critical exponents.

QCD: renormalization, beta functions.

Bibliography

Quantum mechanics for mathematicians, Leon A. Takhtajan, Graduate Studies in Mathematics Vol. 95, American Mathematical Society.

Quantum theory for mathematicians, Brian C. Hall, Graduate Texts in Mathematics 267, Springer.

Quantum field theory and the Standard Model, Matthew D. Schwartz, Cambridge University Press, 2014.

Field theory: a modern primer, Pierre Ramond.

Quantum field theory, Lowell S. Brown, Cambridge University Press.

Quantum field theory and critical phenomena, Jean Zinn-Justin.

Evaluation

Homework and exam.

Weekly assignments

Homework assignment 1 (due October 06).

Homework assignment 2 (due October 20).

Homework assignment 3 (due November 3).

Homework assignment 4 (due November 30).

Summary

Introduction to renormalization and effective field theory: The Wilsonian approach to QFT.

Irrelevant, relevant, marginal couplings.

Renormalization in quantum mechanics.

Casimir effect.

Path integrals in quantum mechanics.

Path integral in quantum field theories. Generating functionals. Partition functions and correlation functions. 1PI effective actions.

Effective actions and Schwinger proper time. QED: Euler-Heisenberg Lagrangian. One-loop beta function.

Schwinger pair production. Relating Schwinger pair production in scalar QED to topological string theory and Gopakumar-Vafa invariants.

Scalar field theory: regularization and renormalization. 1PI effective action and the Coleman-Weinberg potential.