## Unit QUANTUM FIELD THEORY

- Marta Orselli - Didattica Ufficiale

- 42 ore - Didattica Ufficiale - Marta Orselli

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.

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.