## PHY 411: Quantum Field Theory II

3 credits | Prerequisites: PHY 410

### Course rationale

This course is a continuation of Quantum Field Theory (QFT) 1. In QFT 1, the focus was on second quantization and the computation of some elementary processes. In the end, we faced the diverging integrals which require renormalization. In QFT 2, the focus will be on the Feynman path integral formulation of QFT. By this formulation, many of the formulae learned from QFT 1 emerge naturally. Renormalization is done by adding counter-terms, and an alternate view is provided by Wilsonian renormalization. In the last part, this course discusses perturbation theory anomalies

### Course content

Path Integral: path integral formulation of quantum mechanics; Path Integral Quantization: scalar field, spinor field, gauge field, correlation functions, 2-point functions, 4-point functions, Feynman rules, generating functional. Functional derivatives, QED, functional determinants; Renormalization of φ4 Theory: power counting of ultraviolet divergences, one-loop structure if φ4 theory, dimensional regularization, Pauli-Villar regularization; Renormalization of QED: one-loop structure of QED, renormalization beyond the leading order; Spontaneous Symmetry Breaking: renormalization and symmetry, explicit and spontaneous symmetry breaking, Nambu-Goldstone theorem, example; Effective Action: idea of the effective action, effective action for the linear sigma model, effective action as the generating function; Wilsonian Renormalization: Wilson’s approach to renormalization, renormalization condition, the Callan-Symanzik equation, beta and gamma function in renormalization group; Evolution of Parameters: solution of Callan-Symanzik equation, application in QED, concept of running coupling constant, evolution of parameters and operators; Statistical Mechanics and Field Theory: quantum field theory and statistical mechanics, partition function in statistical mechanics, the critical phenomena; Critical Exponents: theory of critical exponents, spin correlation function, exponents of thermodynamic functions, values of the critical exponents, critical behavior in 4-dimensions; Perturbation Theory Anomaly: definition, axial anomaly, toy example: axial current in 2-dimensions, vacuum polarization operator, current operator, fermion number non-conservation, axial current in 4-dimensions, triangle diagrams, spontaneous breaking of chiral symmetry, anomalies of chiral current, chiral gauge theories, anomalous breaking of scale invariance.

### Course objectives

- Understand the concept of path integral.
- Use path integral to quantize quantum fields.
- Understand the concept of renormalization.
- Observe the connection between QFT and statistical mechanics.
- Familiarize with spontaneous symmetry breaking and the Nambu-Goldstone theorem.
- Understand the quantum anomalies.

### References

- An Introduction to Quantum Field Theory (3rd edition) by Michael E. Peskin, Daniel V. Schroeder
- Quantum Field Theory (2nd edition) by Lewis H. Ryder
- The Quantum theory of fields. Vol. 1: Foundations by Steven Weinberg
- The Quantum theory of fields. Vol. 2: Modern Applications by Steven Weinberg
- Quantum Field Theory and the Standard Model (1st edition) by Matthew D. Schwartz
- Quantum Field Theory and Critical Phenomena (5th edition) by Jean Zinn-Justin