Quantum Field Theory
This is an introductory course on Quantum Field Theory where quantization
is described in terms of path integrals.
Although most of the subject is worked out in detail, I assume that students
already know the properties of relativistic equations for particles
of spin 0, 1/2 and 1.
The goal is understanding the properties of Quantum Electrodynamics
beyond the tree level.
Risultati compito scritto 11.4.2014
Recommended Textbooks
- An Introduction to Quantum Field Theory
by M. E. Peskin and D. V. Schroeder
1995, Addison Wesley
- The Quantum Theory of Fields, vol I and II,
by S. Weinberg,
1995 Cambridge University Press
Reference Program (A.Y. 2009-2010)
Path Integrals and Perturbation Theory
Beyond the Tree Approximation
Renormalization Theory for QED
Effective Program (A.Y. 2009-2010)
[12/4/2010:] General discussion about the plan of the lectures
Path Integrals in Quantum Mechanics
- [12/4/2010:] 1 T
Path integral in Quantum Mechanics in phase space for one degree of freedom [MK 2.1].
- [13/4/2010:] 2 T
Path integral in configuration space for Hamiltonians quadratic in momentum [MK 2.1];
classical limit: the classical trajectory [MK 3];
matrix element of a time-ordered product of Heisenberg operators.
- [14/4/2010:] 1 E
Path integral for the free particle [MK 2.2.1]; integral of an exponential of a quadratic form.
- [19/4/2010:] 2 T
Aharonov-Bohm effect [MK 4.1]; Green's functions in quantum mechanics [MK 7]; Green's functions of the scalar field theory [MK 8].
- [20/4/2010] 2 T
Path Integrals in Quantum Field Theory
Generating functional Z[J] of the scalar Green's functions; Z[J] in the free case; 2 and 4 point functions [MK 8.1]; Wick theorem.
- [21/4/2010] 1 E
Perturbative expansion of Z[J]; 2 and 4 point functions at order lambda [MK 8.2].
- [26/4/2010] 2 T
Green's functions for the scalar throry from Feynman rules in configuration space;
the LSZ reduction formula [IZ/5-1-2,5-1-3]; Feynman rules in momentum space for the scattering amplitudes in the scalar theory.
- [27/4/2010] 1 E + 1 T
Scattering amplitude for the process (1 2 -> 1' 2') in the scalar theory at the second order in the coupling constant;
Grassmann variables; path integral quantization of the free Dirac field [PS 9.5].
- [28/4/2010] 1 E
Path integral for the free electromagnetic field: problems related to the gauge invariance.
- [3/5/2010] 2 T
The Fadeev Popov procedure for the quantization of the free electromagnetic field [PS 9.4]; quantum electrodynamics (QED).
Quantum Electrodynamics
- [4/5/2010] 1 T + 1 E
Feynman rules in QED [PS 9.5]; cross section and decay width [PS 4.5]; phase space for two particles in the final state.
- [5/5/2010] 1 E
Tree-level cross section for electron positron scattering into a muon antimuon pair in QED [PS 5.1].
Higher Orders
- [17/5/2010] 1 E + 1 T
Tree-level cross section for elastic electron muon scattering; Feynman diagrams for electron-muon scattering at one-loop;
Loop integrals in d dimensions [PS/7.5].
- [18/5/2010] 2 T
QED in d dimensions: gamma matrices and field dimensions. The t'Hooft parameter;
one-loop vacuum polarization [PS/7.5].
- [19/5/2010]1 E
One-loop vacuum polarization: the final result [PS/7.5].
- [24/5/2010]2 T
One-loop renormalization in QED: general ideas;
wave function renormalization of the electromagnetic field at one-loop [PS/7.5];
renormalization schemes and renormalized parameters, on-shell scheme [PS/10.3].
- [25/5/2010]2 T
Consequences of vacuum polarization: Uehling potential, running gauge coupling constant;
generating functionals of connected and amputated, one-particle-irreducible Green functions [PS/11.3,PS/11.5].
- [26/5/2010]1 T
Gauge invariance of the generating functional Gamma_0; Ward identities.
- [31/5/2010]2 T
General parametrization of electron-positron-photon vertex for on-shell fermions; form factors [W/10.6,PS/6.2];
anomalous magnetic moment of the electron.
- [1/6/2010]2 E
Electron-positron-photon vertex in the one-loop approximation [W/11.3,PS/6.3];
anomalous electron magnetic moment at one-loop; renormalization of the vertex.
- [3/6/2010]1 T
Renormalization conditions for QED in the on-shell scheme;
contribution of the electron wave function renormalization to the form factor F [PS/7.1,W/11.4];
Ward identity between the vertex and the electron self-energy [PS/7.4,PS/10.3].
- [7/6/2010]2 E
Differential cross section for electrom-muon elastic scattering at one-loops [PS/6.1];
[infrared divergences, soft photons and cancellation of infrared divergences; Sudakov double logarithm [PS/6.4]]*;
On-shell renormalization conditions in QED [PS/10.3].
- [8/6/2010]2 T
General properties of renormalization; superficial degree of divergence; renormalizable and non-renormalizable theories [W/12.1,PS/10.1];
polynomial character of divergences in one-loop vertex Green functions; local counterterms as redefinitions
of fields and parameters [W/12.2,PS/10.1].
- [9/6/2010]1 T
Divergent vertex functions in QED [W/12.2,PS/10.1]; divergences beyond one-loop; removal of non-local divergences [W/12.1,12.2,PS/10.4].
[T=theory; E=exercise] The topics marked with * are optional.
References
- [MK]
Path Integral Methods and Applications
by Richard MacKenzie
available at arXiv:quant-ph/0004090v1
- [IZ] Quantum Field Theory
by C. Itzykson and J.-B. Zuber
1980, McGraw-Hill
- [LL] Fisica Teorica, Teoria dei Campi
L. D. Landau and E. M. Lifsits
Ed. Riuniti
- [PS] An Introduction to Quantum Field Theory
by M. E. Peskin and D. V. Schroeder
1995, Addison Wesley
- [W] The Quantum Theory of Fields, vol I and II
by S. Weinberg
1995 Cambridge University Press
- [FLS] Electromagnetism, vol I
by R. Feynman, Leighton and Sands
- [R]
The Path Integral approach to Quantum Mechanics Lecture Notes for Quantum Mechanics IV
by Riccardo Rattazzi
available here
Exercises
To absorbe the subject of this course you also need to practice it.
You are strongly invited to solve the exercises proposed below.
Path Integrals and Perturbation Theory
Beyond the Tree-Level Approximation
How to prepare the examination
How can you judge that you have reached a
good level of understanding and that you are "ready to go"? I tried to figure out a number
of questions that you should be able to answer. If you have doubts on any of these points,
please ask me to clarify them during the class. Do not esitate. To begin with, have a look
to what you should know about the chapter
Beyond the tree level approximation.
If you have been able to answer to all questions, you can move to
General renormalization theory
. Are you reasonably happy with your answers to the
previous questions? If this is the case, you are ready for the examination.
Last modified: June 9 2010 - (feruglio@padova.infn.it)
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