QUANTUM FIELD THEORY

Second cycle degree in PHYSICS

Language: English

Teaching period: Second Semester

Lecturer: MARCO MATONE

Number of ECTS credits allocated: 6

Syllabus
 Prerequisites: Relativistic quantum mechanics. Klein-Gordon equation. Dirac equation. Canonical quantization of the scalar and fermionic fields. Examination methods: The examination is oral and concerns the full programm. It starts with the explicit calculation of a Feynman diagram (phi^4 or QED) to be chosen by the student. Course unit contents: INTRODUCTION. General aspects of Quantum Field Theories. Perturbative and non-perturbative formulations. Wigner and von Neumann theorems. Spontaneous symmetry breaking. Elitzur theorem. Minkowskian and euclidean formulations. Overview of the axiomatic formulation: Wightman axioms, Wightman functions, Wightman reconstruction theorem. Schwinger functions and the Osterwalder-Schroeder reconstruction theorem. OPERATOR FORMALISM. Covariance of the Dirac equation. Spin statistics theorem. PCT theorem. The Lehman, Symanzik and Zimmerman theorem. PATH-INTEGRAL IN QUANTUM MECHANICS. Dirac paper at the basis of the Feynman idea. Forced harmonic oscillator. The vacuum-vacuum amplitude. Wick rotation. Quadratic lagrangians. Bohm-Aharonov effect. PATH-INTEGRAL FOR SCALAR THEORIES. Functional derivative. General properties of the path-integral for scalar theories. Convergence methods Feynman propagator. Green functions. Effective action. Schwinger-Dyson equation. The case of phi^4. Linked-cluster theorem. Euclidean formulation. Computational techniques of functional determinants, the heat equation. Scaling properties of the coupling constant, determinants and anomaly under dilatation. Feynman rules. Computation of some Feynman diagrams for phi^4. Vertex functions and Jona-Lasinio theorem. RENORMALIZATION. Ultraviolet and infrared divergences. Dimensional regularization. Super-renormalizable, renormalizable and non-renormalizable theories. Counterterms. Relation between renormalized and bare vertex functions. Beta function. Landau pole. Ultraviolet and infrared fixed points. Asymptotic freedom and confinement. FERMIONIC PATH-INTEGRAL. Integration over Grassmann numbers. Path integral for the free fermion fields. Feynman rules for spinor fields. Fermion determinants. QUANTUM ELECTRODYNAMICS (QED): Gauge symmetries. Feynman rules for the gauge fields. Gauge fixing. Evaluation of 1-loop Feynman diagrams of QED. Ward identities. Anomalous magnetic moment of the electron. Renormalization of the QED.