**Suggested course**: Theoretical Physics

**Target skills and knowledge**: This course will give an introduction to the path integral formulation of quantum field theories to discuss their structural aspects.
At the end of the course students should be able to master basic techniques to compute relevant quantities for quantum field theories using the path integral.

**Planned learning activities and teaching methods:** Lectures. Weekly assignments.

**Examination method**: Questions on the topics presented during the course and solution of a problem,

**Textbooks**:

- M. Srednicki, "Quantum Field Theory", CUP
- L. Brown, "Quantum Field Theory", CUP
- M. E. Peskin and D. V. Schroeder, "An Introduction to Quantum Field Theory", 1995, Addison Wesley
- S. Weinberg, "The Quantum Theory of Fields", vol I and II, 1995 CUP

**Lectures**: Lectures will be delivered at the Physics and Astronomy department:

Day |
Time |
Room |

MON |
14.30 - 16.15 |
Aula C |

TUE |
14.30 - 16.15 |
Aula C |

**Office hour**: Fridays from 15:00 to 16:00, preferably by appointment (also by e-mail or telephone).

**1. The LSZ Reduction Formula**.

1.1 A new approach to Quantum Field Theory. 1.2 Correlators and the LSZ reduction formula.

**2. The Path integral in Quantum Mechanics**.

2.1 Intuitive introduction to path integrals. 2.2 From Schroedinger equation to the path integral. 2.3 The partition function. 2.4 Operators and time ordering. 2.5 The continuum limit and non-commutativity.

**3. Perturbation Theory**.

3.1 Correlators and scattering amplitudes. 3.2 Free field theory. 3.3 Perturbation theory. 3.4 Feynman Diagrams. 3.5 Borel resummation*. 3.6 Exact results - localization*.

**4. Effective and quantum action**.

4.1 Wilsonian effective action. Integrating out fields. 4.2 The 1pI effective action.

**5. Path integral quantization of λ φ**^{4}.

5.1 Dimensional analysis. 5.2 The free theory. 5.3 The interacting theory. 5.4 The Coleman-Weinberg potential.

**6. Quantising spin 1/2 and spin 1 fields.**

6.1 Path integral for Dirac fermions. 6.2 Path integral for photons.

**7. Perturbative renormalization**.

7.1 Divergences. 7.2 Superficial degree of divergence and BPHZ theorem. 7.3 1-loop propagator in λ φ^{4}. 7.4 On-shell renormalisation. 7.5 Dimensional regularization. 7.6 λ φ^{4} at two loops. 7.7 QED Renormalization.

**8. The Renormalization Group**.

8.1 Renormalization and integrating out degrees of freedom. 8.2 The Callan-Symanzik equations. 8.3 Anomalous dimensions. 8.4 Renormalization group flow. 8.5 Countertems and the continuum limit. 8.6 Polchinski equations. 8.7 The local potential approximation. 8.8 Dimensional regularization. 8.9 The Gaussian Critical point and Landau poles. 8.10 The Wilson-Fishler critical point. 8.11 Zamolodchikov's C-theorem.

**9. Symmetries**.

9.1 Slavnov-Taylor identities. 9.2 Schwinger-Dyson equation and Ward-Takahashi identities. 9.3 Ward identities for QED.

**10. Quantization of non-abelian gauge theories**.

10.1 Classical Yang-Mills theories. 10.2 Quantization issues. 10.3 Faddeev-Popov determinant. 10.4 Ghosts. Nakanishi-Lautrup fields. 10.5 BRST symmetry 10.5 BRST charge and physical Hilbert space.

* Not part of the examination.