First cycle
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Second cycle
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degree courses
School of Science
Course unit
SCP7081702, A.A. 2017/18

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course Second cycle degree in
SC2382, Degree course structure A.Y. 2017/18, A.Y. 2017/18
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Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination QUANTUM FIELD THEORY
Website of the academic structure
Department of reference Department of Physics and Astronomy
Mandatory attendance No
Language of instruction English

Teacher in charge MARCO MATONE FIS/02

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Educational activities in elective or integrative disciplines FIS/02 Theoretical Physics, Mathematical Models and Methods 6.0

Mode of delivery (when and how)
Period Second semester
Year 1st Year
Teaching method frontal

Organisation of didactics
Type of hours Credits Hours of
Hours of
Individual study
Lecture 6.0 48 102.0 No turn

Start of activities 26/02/2018
End of activities 01/06/2018

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.
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: The course's characteristic is to provide, as far as possible and without inessential formalisms, a step-by-step derivation of the path-integral formulation of quantum field theories. In this respect the course includes several details on the most subtle points including the proof of basic theorems usually not considered in the literature. To this end, in addition to the mentioned texts, on the site


there are informal and far from complete notes, 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
  • S. Weinberg, The Quantum Theory of Fields. Vol I.. --: Cambridge University Press, 2005. Cerca nel catalogo
  • Peskin, Michael E.; Schroeder, Daniel V., <<An >>introduction to quantum field theoryMichael E. Peskin, Daniel V. Schroeder. Reading: Mass. [etc.], Addison-Wesley, --. Errata corrige available at
  • Pierre Ramond, Field Theory: A Modern Primer, 2nd Edition. --: Addison-Wesley, 1989. Errata corrige available at
  • M. Matone & Studenti, Note su alcune parti del corso di Teoria dei Campi 1. --, 2017.