Introduction to Hamiltonian formulation of QFT

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fizyka
matematyka

Familiarity with elements of QFT.

w sali

The summer semester 2024 starts with the explanation of WIlsonian concept of renormalization procedure for Hamiltonians, including simple, illustrative examples of its application, before it is generalized to enable one to handle bound states and proceed to a front form of Hamiltonian approach to the Standard Model with a host of new projects to tackle.

The whole yearly course introduces students to modern methods of constructing renormalized Hamiltonian operators of relativistic quantum field theory in application to particle and nuclear physics. The stress is put on addressing the conceptual issues of quantization, regularization, renormalization and evaluation of effective Hamiltonians, including implications of the discussed methods concerning computational strategies for solving dynamical problems, especially the relativistic bound state eigenvalue problems. In the non-relativistic limit, the methods a priori also apply to atomic and condensed matter physics.

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

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).

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

Assessment methods and assessment criteria:

Written report on the work carried out during the course and oral exam at the end of each semester.