Info about the course and the exam.
General introduction.
Non relativistic Quantum Mechanics and Schoedinger's equation.
Wave
function and probability density. Description of a single
particle
state at low energies. The need to include Special Relativity.
The
Klein-Gordon equation as a relativistic wave equation.
Probability
density. Negative energy solutions. Attempts to get a correct
relativistic wave equation. Dirac's equation. Probability
density.
Negative energy solutions. Hole theory and the positron. Multi
particle formalism. Energy and mass. Classical Electrodynamics.
Wave
equations as the correct classical field equations. “Second”
quantization.
Galilean relativity and composition of velocities
in Newtonian mechanics. Maxwell's equations and Lorentz
transformations. Constance of the speed of light. Attempts to
reformulate a relativistically covariant version of Mechanics.
(2h)
Einstein's postulates of relativity. Criticism of the "absolute time". Simultaneous events. Light front propagation. Invariant interval. Time-like, light-like and space-like intervals and the causality structure of Space-Time. Light cone. Lorentz transformations: Boost in the x direction. (1h)
Lorentz
transformations: general case. Limit of small velocity and the
recovery of Galilean transformations. Transformation of
the
3-velocity under Lorentz transformations and the relativistic
composition of velocities.
Boosts and the tensorial
relation that defines a boost. Lorentz transformations form a
Group.
Structure of the group. Poincare' Group. Infinitesimal
transformations. (2h)
Vectors. Contravariant and covariant components. Transformation of contravariant and covariant components of a vector under a basis change. Scalar product. Metric tensor and its transformation under a basis change. Tensors and properties. Necessity to express Physics in terms of tensorial relations. Minkowski space, metric tensor. Contravariant and covariant vectors. (2h)
Dynamics of a classical free particle and covariant
quantities. Four-velocity and proper time, four-acceleration.
Four-momentum. Lagrangian and Hamiltonian. Mass-shell
relation.
Lagrangian and Hamiltonian mechanics. The free particle.
(1h)
Least Action Principle.
Lagrangian and Relativity. Relativistic free particle.
Euler-Lagrange
equations. Conservation laws. Lagrangian invariant under
Poincare'
transformation and conservation of the four-momentum and angular
momentum. (2h)
Fields. Lagrangian density. Least Action Principle in the case
of fields and Euler-Lagrange Equations. (1h)
Symmetries
and Noether's Theorem. Space-time symmetries and Internal
symmetries.
Continuity equation. Conserved current. Conservation of the
charge.
Space-Time symmetries. Invariance under Lorentz transformations.
(2h)
Energy-momentum Tensor. Angular momentum. Scalar fields and
pure inhomogeneous Poincare' transformations, conservation
of
four-momentum. Proper Lorentz transformation and conservation of
angular momentum. Symmetries and unitary transformations. Lie
groups and representations. (2h)
The Lie Algebra of the group. Generators.
(1h)
Casimirs.
Unitary representations. Example: SO(2). Poincare' algebra.
Finite
dimensional representations of the Lorentz
group. SU(2)xSU(2) and (1/2,0)+(0,1/2). (2h)
Scalar field. Tensorial representations,
vector representations and examples of the generators in this
representation. SO(3,1). (2h)
Rotations,
SO(3) and SU(2), spinors, Pauli matrices, boosts in the
2-dimensional
representation. L- and R-spinors. Parity and Dirac spinors.
(1h)
Recap of
finite-dimensional representation of the Lorentz group.
One-particle
states. Pauli-Lubanski vector. Massive and massless particles.
Helicity. (2h)
Natural Units.
Klein-Gordon (KG) Equation. Motivations. Recapitulation of
Schroedinger's Equation and non relativistic QM point of view.
The
scalar product in L2 and its time independence. Covariance of
the KG
equation. Scalar product for the KG field and non positive
definite
probability density. Lagrangian and Hamiltonian of the KG field.
Charged scalar field and global U(1) symmetry. Conserved charge
and
scalar product. (2h)
Potential
in the Lagrangian and mass term. Plane wave solutions of the KG
equation. Positive and negative energy solutions. Normalization
of
the solutions. General solution as superposition of positive and
negative energy solutions. Canonical Quantization. Conjugated
momenta, Hamiltonian density, probability density. Change of
perspective and quantization. Field in terms of creation and
annihilation operators. (1h)
Heisenberg
picture, time-dependent operators, time-independent states.
Hamiltonian density in normal modes. Creation and annihilation
operators. Canonical quatization and commutation relations on
the
fields. Commutation relations for the creation and annihilation
operators. Normal ordering. (2h)
Noether's theorem and conserved quantities. Momentum in terms
of creation and annihilation operators. Fock space. One-particle
state. Two-particle state and Bose symmetry.
Charged
scalar field. Energy, momentum and charge. Fock space. Particles
and
anti-particles. (2h)
Locality
and causality in Quantum Field Theory. Dirac equation.
Properties of
alpha and beta matrices. Covariance of Dirac's
equation. (1h)
Unitarity
in gamma_0 metric and Dirac conjugated. Adjoint Dirac equation.
Norether's theorem. Current and Probability density. Lagrangian
density and conjugated momenta. Energy momentum tensor. Energy
density. (2h)
Angular
Momentum. Spin. Internal transformations and invariance of
Dirac's
equation under U(1) global phase transformations. Charge.
gamma_5.
Bilinear covariants and their behaviour under Lorentz
transformations. (2h)
Algebra
of gamma matrices. Traces of gamma matrices. (1h)
Plane
wave solutions of Dirac's equation. Spinors. Negative energy
solutions. (2h)
Energy
projectors. Polarization sum. Spin projectors. Non-relativistic
limit
and giromagnetic factor of the electron. (2h)
Discrete
symmetries: Parity, Charge conjugation. (1h)
Charge
conjugation. Time reversal. CPT invariance. Quantization of
Dirac's
field. Energy density and momentum in terms of creation and
annihilation operators. (2h)
Anti-commutation
relations and positive definite energy density. Fock space,
antisymmetric two-particle state and Fermi statistics. Locality
and
Causality. Massless fermions and Weyl neutrinos.
Electromagnetic
field. Maxwell's equations. Scalar and vector potentials. Gauge
invariance. Covariant form of Maxwell's equations. (2h)
Lorentz gauge. Electromagnetic tensor and field equations in covariant form. Larangian density. Counting of the actual degrees of freedom in covariant form (and Coulomb gauge). (2h)
Canonical (covariant) Quantization of the field A^mu. Problems
arising from the vanishing of the momentum conjugated to A^0.
Gauge
fixing. Gupta-Bleuler quantization. Plane wave solutions.
Physical
states. Temporal, longitudinal and transverse modes. Conditions
fulfilled by the physical states. Energy and momentum densities
and
contribution of the physical states. The Photon.
(2h)
Non-homogeneous
differential equations and Green's function. The case of the
scalar
field. Fourier transform and integration in the complex plain.
Closed
paths. Open paths. Retarded and Advanced Green's functions. (1h)
The
Feynman propagator for the scalar field. Covariant form.
Propagator
as the vacuum expectation value of T-ordered product of two
fields.
Feynman's propagator for the spinor field and for the
electromagnetic
field. (2h)
Electromagnetic interaction of fields and Lagrangian density.
Gauge transformation and Gauge Principle. Quantization of the
interaction Lagrangian. Non derivative interactions. (2h)
Scattering
matrix. Interaction representation. Time-dependent perturbation
theory and Dyson's formula. Unitarity and Lorentz invariance of
the
scattering matrix. Properties of the time-ordered exponential.
Wick's
theorem. Time ordered product of 2, 3, 4 … fields. Bosonic and
fermionic fields. (2h)
Time-ordered product of normal ordered fields evaluated
in
the same space-time point. Evaluation of the scattering
matrix at the first order in QED. Building blocks, vertex
interactions. Evaluation of the scattering matrix at the
second
order in QED. One-contraction contributions: electron Compton
scattering, positron Compton scattering, pair creation and pair
annihilation, Moller scattering. (2h)
Bhabha
scattering. Two-contraction contributions: electron and positron
self-energy, vacuum polarization of the photon. (1h)
Feynman's
rules in momentum space. Propagators, spinors ... (2h)
The
production Cross Section. Calculation of the cross section for
the
scattering process e+e-→μ+μ- (2h)