Introduction to Hamiltonian formulation of QFT 1100-4IHFQFT
The purpose of the course is to discuss the advanced methods of constructing relativistic Hamiltonians for basic theories of particles and fields, including regularization, renormalization and description of bound states, for students who think about applying such methods in physics of the standard model as well as further development of the general quantum theory. Such Hamiltonians include interactions that involve extraordinarily large range of scales, such as between the size of an electron and the size of a macroscopic chunk of matter, or literally infinity when one thinks about a point particle in an infinite space. Therefore, to manage the great number of variables in a computationally feasible way, trying to describe observables at the experimentally accessible scales, one is forced to compute equivalent effective Hamiltonians. The course aims at presenting methods of the required constructions using the renormalization group procedure for effective particles, which in the non-relativistic limit provides a conceptual way for relating the standard model with atomic and quantum condensed matter physics via a derivation of the Schroedinger equation for a fixed number of particles from quantum field theory. The lecture and homework exercises will provide participants with hands-on experience with practical application of the general principles to simple models. The course intends to cover:
0. Foreword - from Schroedinger equation to theory of universe;
1. Dirac's classification of forms of relativistic dynamics;
2. Gell-Mann-Goldberger theory of scattering;
3. Canonical Hamiltonians in QFT;
4. The problem of ground state, or vacuum;
5. Renormalizability and renormalization groups;
6. Wilsonian renormalization group equations for Hamiltonians;
7. Model examples of triviality, asymptotic freedom, limit cycles and chaos;
8. The concept of universality on the example of a quartic oscillator;
9. Theory of effective particles in application to massive QED;
10. Concept of quantum potentials at a distance;
11. Description of hadrons using Hamiltonian of QCD;
12. Symmetry breaking, mass generation and neutrino oscillations;
and may evolve as a result of questions and discussions in class.
Students may work in small teams and become familiar with the subject matter by discussing and solving problems. Such work may eventually lead to publications, e.g. see
S.Dawid, R.Gonsior, J.Kwapisz, K.Serafin, M.Tobolski,
Phys. Lett. B 777, 260-264 (2017) or
J. Dereziński, O. Grocholski,
J. Math. Phys. 63 (2022) 1, 013504.
The lecturer would welcome collaboration with students interested in contributing to the subject, or preparing a script for the course.
Description by Stanisław Głazek, September 2023.
Time estimate:
Lecture = 45 hours (15 x 3) x 2 semesters
Homework = 30 hours x 2 semesters
Exam preparation = 30 hours in summer semester
Total of about 180 hours
Research on issues of interest to students = unlimited
Kierunek podstawowy MISMaP
fizyka
astronomia
Tryb prowadzenia
Założenia (opisowo)
Koordynatorzy przedmiotu
Efekty kształcenia
1. Student writes Hamiltonian operators for particles of the standard model
2. Student describes the concepts of renormalized energy and charge
3. Student describes the connection between fundamental and effective theories
4. Student describes the concepts of triviality, asymptotic freedom, fixed points and limit cycles
6. Student derives effective Hamiltonians for bound states in simple models
7. Student applies the relativistic concept of effective particle in perturbation theory
Kryteria oceniania
Assessment methods and assessment criteria:
Written report on the work carried out during the course and oral exam at the end of each semester.
Literatura
Original articles cited during the lecture, including:
P. A. M. Dirac, Forms of Relativistic Dynamics, Rev. Mod. Phys. 21, 392 (1949);
M. Gell-Mann, M. L. Goldberger, The Formal Theory of Scattering, Phys. Rev. 91, 398 (1953);
P. A. M. Dirac, Quantum Electrodynamics without Dead Wood, Phys. Rev. 139, B684 (1965);
K. G. Wilson, Model of Coupling-Constant Renormalization, Phys. Rev D 2, 1438 (1970);
S. D. Glazek, K. G. Wilson, Renormalization of Hamiltonians, Phys. Rev. D 48, 5863 (1993);
F. Wegner, Flow equations for Hamiltonians, Ann. Physik 506, 77 (1994),
and textbooks such as:
E. M. Henley and W. Thirring, Elementary quantum field theory (McGraw-Hill, 1962);
J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields (McGraw-Hill, 1965);
C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill,1980);
J. Collins, Renormalization (Cambridge University Press, 1984);
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Perseus, 1995);
S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, 1995);
S. Coleman, Quantum Field Theory Lectures of (World Scientific,2019).
Więcej informacji
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