Quantum Field Theory

Program for exam 2005

 

  1. Electromagnetic field in the vacuum. Casimir effect.

                                                    i.     Contribution of each photon mode to the energy of the vacuum.

                                                  ii.     Total energy of the photon vacuum:

o      Why is it infinite ?

o      Which effect this infinity bears onto quantum gravity ?

                                                iii.     The idea of the renormalization:

o      Calculate those quantities, which are directly measured in the experiment

o      Verify that all infinities are cancelled out in these calculations

                                                iv.     The energy of the vacuum of the electromagnetic field between two metal plates.

                                                  v.     Casimir effect

o      What the Casimir effect is

o      Calculations, which lead to the Casimir effect (definitely the starting point)

  1. Self-interaction of charged particles; Lamb shift

                                                    i.     What is the Lamb shift.

                                                  ii.     Physical ideas, which qualitatively explain the Lamb shit:

o      Electron surrounded by the electromagnetic field, “dressed” electron

o      Difference between the location of the center of mass of  and the location of charge in the “dressed” electron

o      Self-interaction of the electron

o      Vacuum polarization

                                                iii.     Calculation of the Lamb shift in the logarithmic approximation; Bethe approach

o      The second order of the perturbation theory

o      The dipole matrix element responsible for absorbtion and radiation of dipole quanta.

  1. Dirac equation

o      Dirac equation for free electrons

o      Gauge invariance and electromagnetic field

o      Presentation of the Dirac equation as a

a.      first-order differential equation

b.     second-order differential equation

o      Perturabation theory for the second-order differential equation

o      Nonrelativistic approximation, Pauli equation, magnetic moment of the electron, hyromagnetic ration

  1. Vacuum polarization, Uehling potential, running coupling constant, Landau pole.

                                                    i.     Physical ideas, which explain the vacuum polarization by external electromagnetic field

o      Dirac sea

o      Virtual excitation of electron-positron pairs

                                                  ii.     Dirac equation:

                                                iii.     Vacuum polarization

o      How to organize the calculations:

a.      Calculate the correction to the wave-function of electrons in the Dirac sea.

b.     From the correction to the wave function find correction to the charge denity

c.      And finally from charge find polarization potential

o      The role of renormalization

a.      Renormalization of the electron charge

b.     Renormalization which keeps the photon mass zero

                                                iv.     Physical consequencies of the vacuum polarization:

o      Increase of the charge at small distances (large momenta) in the first order of the perturbation theory over

o      Summation of higher orders of the expansion; running coupling constant; Landau pole.

o      The role, which the electron charge and spin play in phenomena leading to the Landau pole (qualitatively).

                                                  v.     Uehling potential:

o      Behaviour at large and small distances.

o      Contribution to the Lamb shift.

o      Contribution to the wave function of the electron at small distances.

  1. Anomalous magnetic moment of the electron

                                                    i.     Qualitative physical picture, which explains the Schwinger correction  to the electron magnetic moment.

  1. Non-Abelian gauge theories

                                                    i.     Definition of the SU(2) gauge potentials and fields

                                                  ii.     Qualitative physical explanation of asymptotic freedom; the roles played by the charge, spin and statistics. Comparison with the vacuum polarization for spinor and scalar particles.