# QFT_S3

Quantum Field Theory

Contact: Federico Piazza & Laurent Lellouch

Quantum Field Theory (QFT) provides a theoretical framework for the description of systems with an infinite number of degrees of freedom, as opposed to Quantum Mechanics (relativistic or not) which deals with systems possessing only a finite number of degrees of freedom. Accordingly, the framework of QFT is relevant to many areas of theoretical physics, e.g. statistical physics or condensed matter physics. When QFT is elaborated in the context of systems invariant under special relativity transformations, it provides an appropriate framework for the description of the fundamental interactions occurring in particle physics. Lectures will provide a broad introduction to the inner workings of relativistic QFTs, but also a concrete working tool to derive predictions from these theories.

Outline of the course

Chap. 1: The problems of relativistic, single-particle quantum mechanics

• a. Construction of a relativistic single-particle quantum mechanics
• b. Violations of causality
• c. The need for a multiparticle theory

Chap. 2: From Fock space to quantum field theory and back

• a. Construction of Fock space and occupation number representation
• b. Harmonic oscillator formalism
• c. Observables and the need for a quantum theory of fields
• d. From quantum field theory to Fock space

Chap. 3: Ingredients for constructing a relativistic quantum field theory

• a. Review of lagrangian and hamiltonian mechanics
• b. Canonical quantization
• c. The Lorentz group and its representations
• d. The Poincaré group, its representations and the notion of a particle

Chap. 4: Action functionals for relativistic quantum field theory

• a. Behavior of local fields under the Poincaré group: from scalar to spin-2 fields
• b. General properties of the action for relativistic fields
• c. Continuous symmetries, Noether’s theorem and conserved charges
• d. The action for scalar fields

Chap. 5: The quantum field theory of spinors

• a. The action for spinor fields
• b. Generators of Poincaré transformations
• c. Solutions of the free Dirac equation
• d. Canonical quantization of the Dirac field
• e. Fock space for fermions
• f. The fermionic propagator

Chap. 6: Building spinor QED

• a. Electromagnetic interactions of the Dirac field
• b. Covariant theory of the photon
• c. Canonical quantization of the gauge field
• d. The photon propagator

Chap. 7: S-matrix expansion, Wick’s theorem and QED at tree level

• a. The S-matrix expansion
• b. Wick’s theorem
• c. Derivation of QED’s Feynman rules
• d. The cross section and Compton scattering
• e. Bremsstrahlung, infrared divergences and Bloch-Nordsieck theorem

Chap. 8: Radiative corrections and renormalization

• a. $O(\alpha)$ corrections to tree-level processes
• b. Vacuum polarization and renormalization of the charge
• c. Electron self energy and charge and mass renormalization
• d. Renormalization of external lines
• e. Vertex correction and Ward identity
• f. Counter terms and bare lagrangian g. Applications: anomalous magnetic moments and Lamb shift

Chap. 9: Higher order radiative corrections and renormalizability (heuristic)

• a. One-particle irreducible diagrams at higher orders
• b. Skeleton diagrams and systematic calculations at higher orders
• c. Primitive divergences and renormalizability

Chap. 10: Dimensional regularization

• a. Different kinds of regularization
• b. Mathematical preliminaries: analytic continuations, integrals in d-dimensions and Feynman parametrization
• c. Vacuum polarization revisited

Chap. 11: Functional methods and path integrals

• a. From quantum mechanics to the Feynman path integral
• b. Path integrals and scattering
• c. Gaussian integrals in many bosonic and fermionic dimensions

Chap. 12: Non-abelian gauge theories

• a. Invariance under an SU(N) gauge symmetry
• b. Yang-Mills theory
• c. Path integral quantization of Yang-Mills theory: Faddeev-Popov approach
• d. SU(N) quantum chromodynamics and its Feynman rules

Prerequisites (though some reminders will be given):

- Special relativity: Lorentz transformations, covariant notation, scalars, four-vectors, higher rank tensors,...

- Quantum Mechanics at M1 level: Hilbert space, Dirac notation, angular momentum...

- Elements of group theory: continuous groups, (projective) representations,...

- Classical field theories: Euler-Lagrange equations, symmetries, Noether’s theorem(s)...

References:

- Peskin and Schroeder: An Introduction to Quantum Field Theory

- Schwartz: Quantum Field Theory and the Standard Model - Weinberg: The Quantum Theory of Fields, vols. 1 & 2

- Ryder: Quantum Field Theory - Ramond: Field Theory - A Modern Primer

- Itzykson and Zuber: Quantum Field Theory