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.

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.

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.

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.

Exercises: Exercise Sheet 6 on the Exercise web page.

**Lecture November 26, 2015:**

Introduction to the Grassman algebra; Gaussian integrals over anticommuting variables.

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.

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);

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.

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.

Exercises: Exercise Sheet 1 on the Exercise web page.

**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