Educational offer - University of Padova

Syllabus

Prerequisites:	Relativistic quantum mechanics. Klein-Gordon equation. Dirac equation. Canonical quantization of the scalar and fermionic fields.
Target skills and knowledge:	The course is focused on the formulation of perturbative quantum field theory. In particular, the expertise and skills to be acquired concerns a good knowledge of the path-integral formulation of bosonic and fermionic quantum field theories. Part of the course covers the path-integral formulation of quantum electrodynamics and renormalization theory. In addition to these skills the student will be able to calculate the contributions up to 2-loops in the scalar case (phi^4) and at 1-loop in the case of quantum electrodynamics.
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. Then the knowledge and skills of the student will be verified with questions on the various topics of the course. However, the details of the proofs of the theorems introduced in the course are not required.
Assessment criteria:	The student should demonstrate that she/he acquired a good knowledge of the path integral formulation of quantum field theory. This concerns the general logical structure, the mathematical aspects and the physical motivations.
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.
Planned learning activities and teaching methods:	The teaching method is based on an introductory "ab-initio" presentation of the path-integral formulation of quantum field theory.
Additional notes about suggested reading:	One aim of the course is to provide a step-by-step derivation of the path integral formulation of quantum field theories. In this respect the course includes several details on some subtle points, including the proof of basic theorems usually not considered in the literature. To this end the references include the lecture notes https://www2.pd.infn.it/~matone/QFTCourseNotes.pdf to which the students contributed. Students are encouraged to provide additional contributions.
Textbooks (and optional supplementary readings)	Itzykson, Claude; Zuber, Jean-Bernard, Quantum field theoryClaude Itzykson and Jean-Bernard Zuber. Mineola: Dover, 2005. Errata corrige available at http://www.lpthe.jussieu.fr/~zuber/ZE_Errata.html S. Weinberg, The Quantum Theory of Fields. Vol I.. --: Cambridge University Press, 2005. Peskin, Michael E.; Schroeder, Daniel V., An introduction to quantum field theoryMichael E. Peskin, Daniel V. Schroeder. --: Westview Press, 1995. Errata corrige available at http://www.slac.stanford.edu/~mpeskin/ Pierre Ramond, Field Theory: A Modern Primer, 2nd Edition. --: Addison-Wesley, 1989. Errata corrige available at https://www2.pd.infn.it/~matone/ M. Matone & Students, QFT. --: --, 2018. Notes written by master students, available at https://www2.pd.infn.it/~matone/QFTCourseNotes.pdf