Blog of the Lectures "Quantum Field Theory" Prof. Nora Brambilla

Welcome to the "Quantum Field Theory tutorial and blog web page". Here you find the detailed subjects discussed during the lectures, indication on books chapters where these subjects are clearly discussed (with reference to Suggested Books). Additional material, link to research articles and notes will be also posted here.

Here you find the Lectures Notes of the course of "Relativity, Particles and Fields" of last semester: you may find useful material also in these notes.

Lecture January 28, February 2, 4 2016:
Gauge fields; interaction of spin one-half particles and spin one gauge fields; Noether current associated to charge conservation; local gauge symmetry principle, covariant derivative and the form of the interaction; abelian gauge theory; abelian gauge transformation of the matter fields, the photon field and the field strength; QED Lagrangian; discussion about the explicit breaking of the gauge symmetry by vector boson mass terms; Lie group and Lie algebra; representation of Lie group; non-abelian gauge symmetry global and local; non abelian gauge transformations of the matter fields, the non-abelian vector fields and non-abelian field strengths; representation of SU(N); fundamental, adjoint and conjugate representations; normalization of the generators; Casimir operator and its value in the fundamental and adjoint representations; Yang-Mills Lagrangian (pure gauge); lagrangian for the interaction of matter fields with non-abelian gauge fields; equations of motion; Bianchi identity; discussion about the dynamics of a non-abelian pure gauge theory. Path integral quantization for gauge fields; discussion on the difficulties related to a naive path integral quantization, redundancy of the gauge degrees of freedom and correspondent infinity in the functional generator; definition of gauge equivalent configurations: orbits of the gauge group; detailed discussion of the Faddeev-Popov procedure for the path integral quantization of abelian and non-abelian gauge field theories; gauge fixing term and related effective Lagrangian; Faddeev-Popov determinant and ghost degrees of freedom; final form of the functional generator for gauge fields; the case of QED: the ghosts can be eliminated; the case of QCD: the ghosts can be eliminated in axial gauges; discussion on the gauge dependence of the gauge boson Green's functions and the gauge invariance of the scattering matrix; example of gauge fixing and vector boson propagators: Landau gauge, general covariant gauges, non covariant gauges; the full action and the full functional generator for the theory of non-abelian gauge fields interaction with fermions in the fundamental representation; Feynman rules for all the propagators; free functional generator; perturbation theory. Derivation of the Feynman rules for abelian and nonbelian gauge field theories: interaction vertices; discussion on the self-interaction terms for gauge vector bosons; polarization vectors for gauge vector boson: the case with mass and the massless case. Sum over the polarization vectors and gauge choice.

Here you find my Lectures Notes

Books for gauge field theories: Bailin, Love Chap.9; Srednicki 69, 70; Peskin, Schroeder Chap. 15. Books for path integral quantization of gauge fields: Pokorski Chap. 3, Bailin, Love Chap.10; Ryder Chap. 7. Books for Feynman rules: Pokorski Chap. 3, Bailin, Love Chap.10; Ryder Chap. 7.

Exercises: Exercise Sheet 11 on the Exercise web page.

Lecture January 7, 12, 14, 19, 21 2016:
Minimal subtraction (MS) scheme; calculation of the counterterms and of the two point and four point renormalized Green's functions in MS scheme; MS scheme is a mass independent scheme, general discussion. Discussion on the characteristics of each such scheme; how to perform a change of scheme: example, relation between the renormalized mass and coupling constant in MOM and MSbar scheme. Discussion about lambdaphi4 renormalization at two loops and in general at all loops; contribution to the field counterterm at two loops in lambda phi4; overlapping divergences; renormalization group (RG); renormalization group transformations; invariance of the bare Green's function and renormalization equation for the renormalized Green's function; definition of the beta functions and of the anomalous dimensions; formal solution of the RG equation; explicit calculation of the solution of the RG equation without masses; behaviour of the renormalized Green's functions under rescaling of the momenta; calculation of the explicit solution of the corresponding renormalization group equation (in a mass independent scheme) and its physical significance; discussion on the anomalous dimension and the canonical dimension; breaking of classical scale invariance via quantum corrections; behaviour of the QFT at large and small momenta in terms of the behaviour of the running coupling constant and running masses; beta function and running coupling constant of lambda phi4 at one loop; Landau pole and related physical discussion in lambda phi4 and QED; the beta function, fixed points and renormalization group flux; different scenarios for different form and behaviour of the beta functions. Renormalization group, renormalization group transformations; detailed discussion about zeroes (fixed points) of the beta function; behaviour of the coupling constant close to the zeros of the beta functions; infrared stable fixed points and ultraviolet fixed stable points; discussion on the form of the beta function in QED and in lambda phi^4; discussion on the form of the beta function in non abelian filed theories in four dimension, and in particular in QCD; asymptotic freedom and infrared slavery in QCD; example of beta functions with several fixed points; explicit calculation of the beta function at one loop in lambda phi^4; explicit calculation of the mass anomalous dimension at one loop in lambda phi^4.

Here you find my Lectures Notes

Books: Bailin, Love Chap. 7, 4-5; Ramond Chap. IV, 5-6; Ryder Chap. 9.3-9.4 ; Chen and Li Chap.3; for the trace anomaly you can see Peskin Chap. 19.5 and the references quoted there. G. Leibbrandt, Rev. of Mod. Phys. and at J. Collins, Chap.4. Books for the beta function: Ramond Chap. IV, 5-6; Pascual and Tarrach Chap.4; Books for gauge field theories: Bailin, Love Chap.9; Srednicki 69, 70; Peskin, Schroeder Chap. 15. Books for path integral quantization of gauge fields: Pokorski Chap. 3, Bailin, Love Chap.10; Ryder Chap. 7.

Exercises: Exercise Sheet 10 on the Exercise web page.

Lecture December 17, 22 2015:
Regularization in field theory: cutoff regularization, Pauli-Villars regularization; dimensional regularization; calculation of primitive divergence at one loop in lambda phi^4; regularization of ultraviolet and infrared divergences. Properties of Dimensional Regularization (DR): it preserve gauge-invariance, it does not preserve dilatation invariance; problems with chiral invariance; Definition and properties of the gamma matrices in D-dimensions; trace of the identity, contractions of gamma matrices and trace of gamma matrices in D-dimension; problems with the definition of the gamma5 in D-dimension; different implementations of the gamma5 in D-dimension; t'Hooft-Veltman implementation of the gamma5 in D-dimensions; renormalization in general and in concrete for lambda phi^4; field strength renormalization; definition of renormalized parameters and counterterms; renormalized Lagrangian, counterterms Lagrangian and corresponding Feynman rules; one loop calculation of the two point and four point renormalized proper Green's functions as a function of the external moment, the counterterms and the renormalized parameters; discussion about how the counterterms cancel the divergences in the proper Green's functions; general relation between the bare and renormalized Green's functions and proper Green's functions; check of the general relation with the one loop calculation of the two point renormalized Green's function; renormalization schemes and general definitions of the counterterms valid in all renormalization schemes.

Here you find my Lectures Notes

Books: Bailin, Love Chap. 7, 1-3, 4-5; Ramond Chap. IV, 1-6; Peskin and Schroeder Chap. 10, 7.1, 7.5 If you want to know more details about dimensional regularization you can have a look at G. Leibbrandt, Rev. of Mod. Phys. and at J. Collins, Chap.4.

Exercises: Exercise Sheet 9 on the Exercise web page.

Lecture December 8, 10, 15 2015:
Superficial degree of divergence in lambda phi^4; primitive divergences in lambda phi^4; Weinberg theorem; superficial degrees of in lambda phi^n and in generic d-dimensions; UV behaviour of Quantum Field Theories; definition of renormalizable, superenormalizable and non-renormalizable theories; relation to the dimension of the coupling constant; renormalized mass, coupling constant and field strength; field strength renormalization; The Kallen-Lehmann representation of two point Green's function; analytic structure of the two point Green's function in an interacting theory. Kallen-Lehmann representation of two point Green's function for fermions; analytic structure of the Green's function in an interacting theory; comments on the case of particles of mass zero and unstable particles; handling of loop diagrams: Feyman parameterization and Wick rotation; properties of the Gamma function; regularization in field theory.

Here you find my Lectures Notes

Books: Bailin, Love Chap. 7, 1-3; Ramond Chap. IV, 1-6; Peskin and Schroeder Chap. 10, 7.1, 7.5. You can compare the field strength renormalization that we obtained in field theory with the wave function renormalization that arises in quantum mechanics in perturbation theory, see e.g. J. Sakurai. Modern Quantum Mechanics Chapter 5.1.

Exercises: Exercise Sheet 7 and 8 on the Exercise web page.

Lecture December 1, 2015:
Path integral for fermions, Fermion propagators and fermion two-points, four-points and n-points Green's functions in the path integral approach; calculation of fermion loops; introduction to renormalization in classical theory and in quantum field theory; bare and renormalized parameters and bare and renormalized Lagrangian; UV divergences and momentum power counting.

Here you find my Lectures Notes

Books: Bailin, Love Chap. 7; Peskin and Schroeder Chap. 10.1, 7.1.

Exercises: Exercise Sheet 6 on the Exercise web page.

Lecture November 26, 2015:
Introduction to the Grassman algebra; Gaussian integrals over anticommuting variables.

Here you find my Lectures Notes

Books: An introduction to the path integral over fermions is given in all field theory books addressing path integral quantization, see e.g. Bailin, Love Chap. 8, for more details you can see F.A. Berezin, The Method of Second Quantization, Academic Press, (1966).

Exercises: Exercise Sheet 5 on the Exercise web page.

Lectures November 19, 24 2015:
Spontaneous symmetry breaking; example of ferromagnetic/paramagnetic transition and analogy between the magnetization seen as an order parameter of the phase transition and a nonzero vacuum expectation value of the scalar field; effective action in the case of spontaneous symmetry breaking; effective potential; interpretation of the effective potential as vacuum energy density; calculation of the effective potential at order lambda. Spontaneous symmetry breaking (SSB) in classical systems; the example of the lambda phi4 with a discrete symmetry; SSB, vacuum degeneracy and vacuum instability. SSB in quantum systems: effective action and effective potential; definition of the effective potential in terms of the sum of proper vertex functions at zero momentum; information contained in the effective potential: vacuum structure, renormalized mass, renormalized coupling constant; divergences in a theory with SSB versus a theory without SSB; loop expansion of the effective potential; calculation at one loop in lambda phi4 of the effective potential: saddle point approximation calculation and diagrammatic one loop calculation; Definition of the scattering operator ad the scattering amplitudes in the path integralism formalism; calculation of the equation that gives the scattering amplitudes in terms of amputated on shell Green's functions.

Here you find my Lectures Notes

Books: for the effective action: S. Pokorski Chap. 2; Le Bellac (Chap. 1-intro to SSB in statistical mechanics) Chap. 5.3, 5.4, ; Bailin, Love Chap. 4.4 and 6.3. You find an extended treatment of the effective potential for example in Hatfield Chap. 18; for the spontaneous symmetry breaking (SSB) Books: K. Huang Chap. 10.1-10.5; Bailin, Love Chap. 5.2, 6.4, 6.5 You find a beautiful treatment of SSB in S. Coleman, "Aspects of Symmetry" (Selected Erice lectures), Cambridge University Press 1985 (reprinted 1995). For the Scattering Matrix: Bailin, Love Chap. 6.4-6.5.

Exercises: Exercise Sheet 4 on the Exercise web page.

Lectures November 10, 12, 17 2015:
Feynman rules in momentum space. One particle irreducible diagrams; proper vertex functions; the effective action: the functional generator of the proper vertex functions; diagrammatics of the two points connected Green's function up to order lambda^3 included; discussion on the resummation of one particle irreducible amputated diagrams for the two points connected Green's function: definition of the self-energy. Self energy; relation between the bare mass, the physical mass and the self energy; renormalized running mass; two point vertex function as inverse of the two points connected Green's function; introduction of the classical field and its relation to the vacuum expectation value of the scalar field; definition of the classical field in terms of the functional derivative of the generator of the connected Green's functions with respect to the external current; introduction of the effective action via the Legendre transformation; discussion about the thermodinamical equivalent of this Legendre transformation in terms of internal energy, free energy and entropy; relation between the external current and the classical field; calculation of the classical field and of the effective action for the free scalar theory and for the lambda phi^4 at order lambda. general relation between the connected Green's function of order n and the proper vertex functions of order n and lower (analytic and diagrammatic calculation);

Here you find my Lectures Notes

Books: S. Pokorski Chap. 2; Bailin & Love Chap. 4 and 6, L. Ryder Chap. 6. Le Bellac Chap. 5.3, 5.4, ; Bailin, Love Chap. 4.4 and 6.3. For the effective action: S. Pokorski Chap. 2; Le Bellac (Chap. 1-intro to SSB in statistical mechanics) Chap. 5.3, 5.4, ; Bailin, Love Chap. 4.4 and 6.3.

Exercises: Exercise Sheet 3 on the Exercise web page.

Lectures October 27, October 29, November 3, November 5 2015:
Functional determinant; functional generator of connected Green's functions; Feynman propagator and pole prescription. Calculation of the Green's function from the functional generator in the free case; Wick theorem; momentum space Feynman rules for the functional generator. Perturbation theory within the path integral formulation of quantum field theory: general method; Real scalar self-interaction theory: the perturbative treatment of the lambda phi^4 interaction term; calculation of the order lambda contribution in the functional generator of the scalar self interacting theory; calculation of the two point Green's function at order lambda; example of divergence at order lambda, power counting; mass renormalization. Calculation of the four points Green's functions at order lambda; combinatorics and symmetry factors; vacuum graphs, their significance and their treatment. Feyman rules in position space.

Here you find my Lectures Notes

Books: S. Pokorski Chap. 2; Bailin & Love Chap. 4 and 6, L. Ryder Chap. 6. Le Bellac Chap. 5.3, 5.4, ; Bailin, Love Chap. 4.4 and 6.3.

Exercises: Exercise Sheet 2 on the Exercise web page.

Lectures October 20, October 22 2015:
Matrix elements of time ordered quantum mechanical position operators in path integral representation; transition amplitude in presence of an external source; generating functional; functional derivatives. Path integral formulation of quantum field theory; Green's functions and functional generator; Euclidean formulation and Wick rotation; Euclidean Green's functions; calculation of the functional generator for the free real scalar field theory; Gaussian integration in real and complex vector spaces; Gaussian integrals in the field theory path integral formulation; determinant of an operator; functional generator of connected Green's functions; Feynman propagator and pole prescription.

Here you find my Lectures Notes

Books: S. Pokorski Chap. 2; Bailin & Love Chap. 2, L. Ryder Chap. 6.

Exercises: Exercise Sheet 1 on the Exercise web page.

See also exercise 1.3 on the functional derivatives in sheet 1 of the Course of "Relativity, Particles and Fields of last semester"

Lectures October 13, October 15 2015:
Quantum mechanical description of nonrelativistic system with N degrees of freedom; first quantization, observables; Schroedinger equation and vector states; Schroedinger and Heisenberg picture. Path integral quantization in Quantum Mechanics; transition amplitudes as path integrals in quantum mechanics; classical limit and classical equations of motion; matrix elements of time ordered quantum mechanical position operators in path integral representation; transition amplitude in presence of an external source; generating functional; functional derivatives

Books: for the path integral in Quantum Mechanics see J. Sakurai, J. Napolitano Modern Quantum Mechanics (Pearson eds) Chap. 2.6; For the rest: S. Pokorski Chap. 2; Bailin & Love Chap. 2.