Teaching methods (Werkvormen)
Lecture hours and exercise class. (2+2 or 3+1).
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Pre-requisites (Vereiste voorkennis)
Some knowledge of quantum field theory (as taught by physicists) would be helpful, but is not required. Some inclination for rigorous mathematics. Analysis I+II and some basic knowledge of ordinary differential equations are helpful.
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Objectives (Leerdoelen)
- The student will have some understanding of the 'philosophy' of the renormalization group,
- The student has some understanding of in particular its incarnation as the flow equation as applied to perturbative QFT.
- He/she will understand that perturbative renormalization gives rise to a well-defined formal power series.
- The student will have seen at least a glimpse of other applications of the renormalization group philosophy.
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Contents (Beschrijving)
We will begin with a quick introduction to (perturbative) quantum field theory, in particular the (modest) mathematical tools that will be used. As is well known, quantum field theory (QFT) is plagued by convergence problems even in perturbation theory. This led to renormalization theory, a subject that is notorious for its difficulty and its supposed but untrue lack of mathematical rigor. We will discuss the approach to renormalization via flow equations, which is the simplest of all, rather painless and entirely rigorous. This approach is an adaptation of Wilson's renormalization group philosophy to perturbative QFT and avoids almost all the difficult combinatorics of other approaches. Depending on the interests of the audience, we can then turn to the perturbative renormalization of gauge theories, but other possible continuations might involve non-perturbative uses of the renormalization group in QFT, statistical physics, Hamiltonian dynamics (KAM theory) or pure mathematics.
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Subjects (Onderwerpen)
- Foundations of perturbative QFT
- Generating functionals, Gaussian measures
- The flow equation
- Proof of renormalizabilty of scalar theories in 2-4 dimensions
- (Non-)Abelian gauge theories
- Other subjects related to rigorous renormalization group
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Examination (Tentaminering)
Probably oral, taking homework into account.
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Literature (Literatuur)
- V.F. Mueller: Perturbative renormalization by flow equations. Review in Mathematical Physics 15, 491-558 (2003).
- M. Salmhofer: Renormalization. An Introduction. Springer, 1999. (Texts and monographs in physics)
- K. Costello: Renormalization and effective field theory. American Mathematical Society, 2011. (Mathematical surveys and monographs)
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Extra information (Bijzonderheden)
This course should be of interest to master's students and PhD students in theoretical physics and to master's students in mathematics with a minor in physics or a double bachelor in mathematics and physics.
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