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Mathematics of Linear Quantum Field Theory
Mathematics of Linear Quantum Field Theory
ABSTRACT
Linear quantum field theory (QFT) has a wide range of applications in physics. It can be used to model quantum fields in the presence of external sources or on a curved spacetime.
The common wisdom says that linear QFT is quite straightforward and not very interesting. In my lectures I will try to convince the audience that this is not true.
The mathematical status of nonlinear QFT, such as QED or the Standard model, is (and, probably, will always be) quite problematic, because one is forced to use perturbative techniques, which are useful only in the presence of a small parameter.
Linear QFT can be well understood beyond perturbation theory. It often has surprising properties. It sheds light onto various aspects of full nonlinear QFT. It shows the power of advanced mathematics. Various relatively sophisticated mathematical concepts, such as of tempered distributions, boundary conditions for differential operators, spectral shift function, Krein spaces, scales of Hilbertizable spaces, wave front sets, are very useful in linear QFT.
SCHEDULE
Monday, April 4
meeting room (7th floor), 10:00 - 12:00Tuesday, April 5
meeting room (7th floor), 10:00 - 12:00Wednesday, April 6
meeting room (7th floor), 10:00 - 12:00Thursday, April 7
meeting room (7th floor), 14:30 - 16:30Monday, April 11
meeting room (7th floor), 10:00 - 12:00Tuesday, April 12
meeting room (7th floor), 10:00 - 12:00Wednesday, April 13
meeting room (7th floor), 10:00 - 12:00Tuesday, April 19
meeting room (7th floor), 10:00 - 12:00Wednesday, April 20
meeting room (7th floor), 10:00 - 12:00REFERENCES
Preliminary lecture notes 1
Preliminary lecture notes 2
Prerequisites: basic theory of operators on Hilbert spaces, tempered distributions, Fourier transformation, basic Quantum and Classical Mechanics, Special Relativity and Differential Geometry. Prior exposition to Quantum Field Theory is welcome, since it provides an additional motivation and openness to the physical jargon, but is not necessary.
LECTURE PLAN
1) Lorentz and Poincare groups.
2) Laplace and Helmholtz operators and their Green's functions.
3) Wave and Klein-Gordon equation and their propagators.
4) Second quantization and Fock spaces.
5) Hamiltonian and Lagrangian formalism of classical field theory.
6) Canonical Commutation Relations.
7) Free scalar bosons.
8) Generating functions of symplectic transformations and metaplectic group.
9) Time-dependent Hamiltonians.
10) Gaussian integrals and path integrals.
11) Scalar fields with masslike perturbations and their renormalization.
12) Euclidean fields and the Wick rotation.
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