Quantum Field Theory 1
A.Y. 2025/2026
Learning objectives
The course provides an introduction to relativistic quantum
field theory, its theoretical foundations, and its application to the
perturbative computation of scattering processes.
field theory, its theoretical foundations, and its application to the
perturbative computation of scattering processes.
Expected learning outcomes
The course provides an introduction to relativistic quantum
field theory, its theoretical foundations, and its application to the
perturbative computation of scattering processes.
Risultati di apprendimento attesi (inglese )
At the end of this course the student will know how to
Decouple the dynamics of coupled finite-and infinite-dimensional
- 33 -
system in terms of normal coordinates
Obtain a classical field as the continuum limit of a system of coupled
harmonic oscillators
Construct a relativistic classical field theory for scalar, vector and
spin 1/2 fields
Determine the conserved currents in the presence of both internal and
space-time symmetry, specifically the enrrgy-momentum tensor
Quantize a free scalar field and construct its Fock space
Quantize a Fermi field
Obtain the time evolution of a quantum field theory from its path
integral
Compute the path integral and propagator for a free field theory of
Bosons or Fermions
Write down the path integral for an interacting field theory and use
it to calculate Green functions
Relkate aplitudes to Green functions through the reduction formula
Determine the Feynman rules for a given theory from the path integral
Compute amplitudes and cross-sections for simple processes
Understand the origin of divergences in perturbative computations, and
how to tame them through regularization and renormalization
Determine the Feynman rules for a renormalized field theory
Determinare le regole di Feynman per una teoria rinormalizzata
Understand under which conditions a theory is renormalizable or not,
and what it means
field theory, its theoretical foundations, and its application to the
perturbative computation of scattering processes.
Risultati di apprendimento attesi (inglese )
At the end of this course the student will know how to
Decouple the dynamics of coupled finite-and infinite-dimensional
- 33 -
system in terms of normal coordinates
Obtain a classical field as the continuum limit of a system of coupled
harmonic oscillators
Construct a relativistic classical field theory for scalar, vector and
spin 1/2 fields
Determine the conserved currents in the presence of both internal and
space-time symmetry, specifically the enrrgy-momentum tensor
Quantize a free scalar field and construct its Fock space
Quantize a Fermi field
Obtain the time evolution of a quantum field theory from its path
integral
Compute the path integral and propagator for a free field theory of
Bosons or Fermions
Write down the path integral for an interacting field theory and use
it to calculate Green functions
Relkate aplitudes to Green functions through the reduction formula
Determine the Feynman rules for a given theory from the path integral
Compute amplitudes and cross-sections for simple processes
Understand the origin of divergences in perturbative computations, and
how to tame them through regularization and renormalization
Determine the Feynman rules for a renormalized field theory
Determinare le regole di Feynman per una teoria rinormalizzata
Understand under which conditions a theory is renormalizable or not,
and what it means
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1. Introduction: normal co-ordinates and continuum limit of classical field theory,
2. Lorentz and Poincaire symmetry: Lorentz group and algebra, spinors, field representations, Poncaire group.
3. Classical field theory: principle of least action, Noether theorem, Klein-Gordon equation.
4. Canonical quantization of free fields: real and complex scalar fields.
5. Canonical quantization of spin-1 field: radiation gauge, covariant quantization (Gupta-Bleuler)
6. Spinor fields: classical theory of spinor fields, canonical quantization of free Dirac field
7. Interacting fields: time evolution, Lehmann-Symanzik-Zimmermann reduction formula, perturbative expansion of correlators and interaction picture.
8. Time-ordered correlators and Feynman diagrams: Feynman propagator for scalar field, Wick's theorem, Feynman diagrams, Feynman rules for fermions and QED
9. Computation of e+e- → mu+ mu-
10. Path Integral Quantization: path integral formalism in quantum mechanics, path integral formalism for scalar fields, Feynman rules from path integral formalism, Grassmann variables and path integrals for fermion fields
11. Loop amplitudes and UV divergences
12.Loop amplitudes, including Wick rotation, in phi-4 theory: counting UV divergences, renormalization of UV divergences in phi-4 theory
· Dimensional regularization in phi-4 theory
◦ Feynman parametrization
◦ UV counterterms
2. Lorentz and Poincaire symmetry: Lorentz group and algebra, spinors, field representations, Poncaire group.
3. Classical field theory: principle of least action, Noether theorem, Klein-Gordon equation.
4. Canonical quantization of free fields: real and complex scalar fields.
5. Canonical quantization of spin-1 field: radiation gauge, covariant quantization (Gupta-Bleuler)
6. Spinor fields: classical theory of spinor fields, canonical quantization of free Dirac field
7. Interacting fields: time evolution, Lehmann-Symanzik-Zimmermann reduction formula, perturbative expansion of correlators and interaction picture.
8. Time-ordered correlators and Feynman diagrams: Feynman propagator for scalar field, Wick's theorem, Feynman diagrams, Feynman rules for fermions and QED
9. Computation of e+e- → mu+ mu-
10. Path Integral Quantization: path integral formalism in quantum mechanics, path integral formalism for scalar fields, Feynman rules from path integral formalism, Grassmann variables and path integrals for fermion fields
11. Loop amplitudes and UV divergences
12.Loop amplitudes, including Wick rotation, in phi-4 theory: counting UV divergences, renormalization of UV divergences in phi-4 theory
· Dimensional regularization in phi-4 theory
◦ Feynman parametrization
◦ UV counterterms
Prerequisites for admission
Knowledge of Lagrangian classical mechanics, non-relativistic quantum mechanics and special relativity.
Teaching methods
The course consists of blackboard lectures in which the topics included in the syllabus will be presented. Interaction with the students in class is very much encouraged, through questions and discussions. A tutor will be assigned to the course to assist students and follow the exercises.
Teaching Resources
Reference textbook:
M. Maggiore: A Modern Introduction to Quantum Field Theory; Oxford
University Press, 2005
Further reading:
M.E. Peskin, D.V. Schroeder: An introduction to Quantum Field Theory; Addison-Wesley, 1995
S. Weinberg: The Quantum Theory of Fields: Vol. I (foundations); Cambridge University Press, 1995
A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010
Exercises:
V. Radovanovic: Problem Book in Quantum Field Theory; Springer, 2007
M. Maggiore: A Modern Introduction to Quantum Field Theory; Oxford
University Press, 2005
Further reading:
M.E. Peskin, D.V. Schroeder: An introduction to Quantum Field Theory; Addison-Wesley, 1995
S. Weinberg: The Quantum Theory of Fields: Vol. I (foundations); Cambridge University Press, 1995
A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010
Exercises:
V. Radovanovic: Problem Book in Quantum Field Theory; Springer, 2007
Assessment methods and Criteria
The final exam consists of a a written exam of 2.5 hours and an oral exam of approximately 30 minutes. During the written exam, the student is required to determine the conserved charges and Feynman rules for a given Lagrangian, as well as compute simple matrix elements for a process in this theory and discuss its ultraviolet behavior. During the oral exam, the student should discuss a topic in the course, chosen by the examiner at the time of the exam.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 42 hours
Professor:
Röntsch Raoul Horst
Professor(s)