Università degli Studi di Perugia

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Unit QUANTUM FIELD THEORY

Course
Physics
Study-unit Code
GP005520
Curriculum
Fisica delle particelle elementari
Teacher
Marta Orselli
Teachers
  • Marta Orselli - Didattica Ufficiale
Hours
  • 42 ore - Didattica Ufficiale - Marta Orselli
CFU
6
Course Regulation
Coorte 2017
Offered
2018/19
Learning activities
Affine/integrativa
Area
Attività formative affini o integrative
Sector
FIS/02
Type of study-unit
Opzionale (Optional)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
Klein-Gordon field. Dirac field. Interacting Fields and Feynman diagrams. Elementary processes in quantum electrodynamics. Spontaneous symmetry breaking. Goldstone theorem and Higgs mechanism. Path integrals for quantum field theories. Radiative corrections. Renormalization and renormalization group. Quantization of non-Abelian gauge theories. Beta function of non-Abelian gauge theories. Standard Model.
Reference texts
M. E. Peskin, D. V. Shroeder, An Introduction to Quantum Field Theory, Perseus books.
Educational objectives
This course represents the first advance course in Quantum Field Theory (QFT). The main aim of this teaching is to provide students with the bases needed to address and solve the most important problems in QFT.
Main knowledge acquired will be:
Knowledge of the Feynman rules and Feynman diagrams from a lagrangian.
Knowledge of regularization and renormalization.
Knowledge of running coupling constants.
Quantization of non-abelian gauge theories.
Standard Model.
The main competence (i. e. the ability to apply the acquired knowledge) will be:
Ability to compute Feynman rules given a Lagrangian.
Ability to renormalize a QFT and understand the consequences of the renormalization process.
Capacity of computing the cross sections of the main processes in the electroweak theory.
Prerequisites
In order to be able to understand and apply the majority of the techniques described within the Course, it is necessary to master the most important topics of the Quantum Mechanics and Theoretical Physics Courses.
Teaching methods
face to face lectures, seminars, theoretical lessons and practical training
Other information
none
Learning verification modality
The exam consists of an oral test. The oral test consists on an interview of about 45 minutes long aiming to ascertain the knowledge level and the understanding capability acquired by the student on theoretical and methodological contents as indicated on the program. The oral exam will also test the student communication skills and his autonomy in the organization and exposure of the theoretical topics. During the oral text the student will be asked to solve one of the problems given during the course.
Extended program
Canonical quantization of the scalar field real. Commutators. Feynman propagators. Wick rotation. Complex scalar field and phase invariance. Non-relativistic limit of phi ^ 4 theory. Noether's theorem. Conserved charges for quantum fields. Stress-energy tensor of the scalar field. Electromagnetic current and Gauss theorem. Spontaneous breaking of a global symmetry. Effects of tunneling and recovery of the symmetry. Some hints of group theory. Goldstone theorem. Abelian Higgs model and spontaneous simmetry breaking. Stueckelberg local Lagrangian for the electromagnetic field massive. Polarizations for the electromagnetic field.
Lorentz algebra and relativistic fields. Weyl fermions. Dirac and Majorana masses. Weyl kinetic terms. Spinors and Dirac gamma matrices. Lorentz transformations. Dirac equation. Non-relativistic base. Lagrangian of QED. Non-relativistic limit of the Dirac theory: Pauli equation. Discrete symmetries, C, P and T transformations. Fierz transformations. Currents associated to the phase and chiral symmetries. Quantization of the Dirac field. Plane waves. Anticommutators. Feynman propagator.
Kernel of evolution: free particle. Lagrangian and Hamiltonian formulation of the functional integral. Euclidean kernel. Harmonic oscillator. T-ordered products: interaction representation. Adiabatic shut-down of the interaction. Feynman propagator. Wick theorem. Determinant of an operator. Gelfand and Yaglom equation. Generating functional of the Green's functions: Z [J]. Generating functional of the Connected Green's functions: W [J]. Effective action Gamma[phi]. Perturbation theory and Feynman diagrams. Feynman propagator in the coordinate space in generic dimensions. Feynman diagrams.
Ultraviolet divergences. Dimensional regularization. S matrix and forced oscillator. Functional integral and double well. Instanton classical action and zero modes. Instantons: collective coordinates. Separation of levels. Similar problems: periodic potential and decay of a metastable state. Theory phi ^ 4 Wick rotation and Euclidean Feynman rules. Examples of one-loop diagrams. Effective potential. Vacuum energy. Regularization and renormalization: theory phi ^ 4. Ultraviolet divergences and superficial degree of divergence. Dimensional regularization and minimal subtraction scheme. Overlapping divergences. Examples of two loop diagrams. Subtracted diagrams and polynomial algorithm for subtraction.
Renormalization group. Beta function. Landau pole. Anomalous dimenssions gamma_d and gamma_m. Recursive relations for higher order poles. Effective coupling constant: behavior above and below the threshold. Effective action. Callan-Symanzink or Gellman-Low equation. Beta function and asymptotic behaviors. Fixed points. Relevant and irrelevant operators. Epsilon expansion. Critical exponents. Wilson's method for phi ^ 4 theory. Representation Kallen-Lehman representation. Reduction formula. Probability decay per unit time.
Cross sections. Conserved Currents and Ward identities: global symmetry case. Quantization of a massive vector field. Massless vector field, gauge invariance and gauge fixing. Ghost of Faddeev-Popov. Ward identities for QED. BRST (Becchi-Rouet-Stora-Tyutin) symmetry. Slavnov Taylor identity for the Yang-Mills theory. Standard Model.
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