Advanced Quantum Field Theory:
Modern Applications in High Energy, Astro-, and Condensed Matter Physics

Лектор — ассистент О. Г. Харланов
(магистратура/аспирантура, 68 часов)

Course topics

    I. Quantum effects in finite volume

  1. The Casimir effect: different approaches and their equivalence (mode summation, dyadic Green's function, zeta function regularization & explicit renormalization). Scalar and electromagnetic Casimir effects in different geometries
  2. Fundamentals of the Dynamical Casimir effect: two uniformly moving mirrors and one accelerated mirror, the role of conformal invariance
  3. Quantization of classical solutions: the kink in D=1+1 theories. Topological sectors, spectra of excitations, and renormalization of the kink mass as a self-consistent version of the Casimir effect.
  4. II. Quantum effects in strong external fields

  5. Photon splitting in strong magnetic field: soft photon limit and the role of gauge invariance, reduction to the Heisenberg-Euler effective action
  6. Collective neutrino oscillations in dense media: finite-density field theory, forward scattering, effective Hamiltonian for neutrino oscillations, flavor polarization vector
  7. Radiative Lorentz violation in Axion-Wess-Zumino extension of QED: an example of Coleman-Weinberg radiative symmetry violation
  8. Hawking radiation as a tunneling process: enter Gravity
  9. III. There and back again: From QFT to the lattice field theory

  10. Tight-binding model and its roots: Hartree-Fock approximation for a many-body system; periodic potential well and lattice wave function; overlap & transfer integrals
  11. The solid ground: Hohenberg-Kohn theorems, the Thomas-Fermi model
  12. A deeper insight into Density Functional Theory (DFT): Kohn-Sham equations, exchange-correlation functional, local density approximation
  13. Lattice field theory: "field" operators, "wave" functions, the Fermi-Dirac sea vs. the QFT vacuum, (quasi)particles and holes, lattice Green's function; symmetries of the tight-binding model, Peierls substitution
  14. "Interaction": the Coulomb interaction (brief consideration), the Hubbard model and its weak- and strong-coupling regimes
  15. Fermi surface instabilities: susceptibilities and nesting vectors; mean-field theory for Hubbard model, ferromagnetic and anti-ferromagnetic (Neél) orders; Cooper pairing and BCS superconductivity
  16. IV. There and back again: QFT as a continuum limit of the lattice theory

  17. Graphene, the honey from the honeycomb: band theory for graphene, the effective Dirac wave equation and massless non-relativistic fermion quasiparticles; graphene in external fields
  18. A complete field theory for interacting graphene: the Lagrangian, self-energy; effective "amplification" of the Coulomb interaction in graphene

Suggested readings, incl. original papers (Ref. numbers with dots indicate further reading):

I. Quantum effects in finite volume

[1] K. A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, Singapore, 2001); e-Print arXiv:hep-th/9901011.
[2] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in the Casimir Effect (Oxford Science Publications, Oxford, 2009).
[3] K. A. Milton, L. L. DeRaad, Jr., and J. Schwinger, Casimir Self-Stress on a Perfectly Conducting Spherical Shell, Ann. Phys. 115, 388 (1978).
[4] M. Bordag, G. Petrov, and D. Robaschik, Calculation of the Casimir effect for a scalar field with the simplest non-stationary boundary conditions, Sov. J. Nucl. Phys. 39, 828 (available in Russian as Yad. Fiz. 39, 1315 and from the KEK server
[5] E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, Lecture Notes In Physics series, volume 855 (Springer, Berlin, 2012).
[6] S. Hawking, Zeta Function Regularization of Path Integrals in Curved Spacetime, Commun. Math. Phys. 55, 133 (1977).
[7] S. A. Fulling and P. C. W. Davies, Radiation from a Moving Mirror in Two Dimensional Space Time: Conformal Anomaly, Proc. Roy. Soc. Lond. A 348, 393 (1976).
[7.1] H. Epstein, V. Glaser, A. Jaffe, Nonpositivity of the energy density in quantized field theories, Nuovo Cimento 36, 1016 (1965).
[8] R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, (Elsevier, Amsterdam, 1982). Available in Russian as Р. Раджараман, Солитоны и инстантоны в квантовой теории поля (Москва, "Мир", 1985).
[8.1] A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, The soliton: A new concept in applied science, IEEE Proc. 61, 1443 (1973).

II. Quantum effects in strong external fields

[1] S. L. Adler, Photon Splitting and Photon Dispersion in a Strong Magnetic Field, Ann. Phys. 67, 599 (1971).
[1.1] S. L. Adler and C. Schubert, Photon Splitting in a Strong Magnetic Field: Recalculation and Comparison with Previous Calculations, Phys. Rev. Lett. 77, 1695 (1996).
[1.2] C. Schubert, QED in the worldline representation, AIP Conf. Proc. 917, 178 (2007).
[2] J. Schwinger, On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664 (1951).
[3] H. Nunokawa, V. B. Semikoz, A. Yu. Smirnov, and J. W. F. Valle, Neutrino conversions in polarized medium, Nucl. Phys. B 501, 17 (1997).
[4] S. Esposito and G. Capone, Neutrino propagation in a medium with a magnetic field, Z. Phys. C 70, 55 (1996).
[4.1] S. J. Hardy and D. B. Melrose, Ponderomotive force due to neutrinos, Phys. Rev. D 54, 6491 (1996).
[5] A. A. Andrianov and R. Soldati, Lorentz symmetry breaking in Abelian vector-field models with Wess-Zumino interaction, Phys. Rev. D 51, 5961 (1995).
[5.1] A. A. Andrianov, R. Soldati, and L. Sorbo, Dynamical Lorentz symmetry breaking from a (3+1)-dimensional axion-Wess-Zumino model, Phys. Rev. D 59, 025002 (1998).
[5.2] A. A. Andrianov, P. Giacconi, and R. Soldati, Lorentz and CPT violations from Chern-Simons modifications of QED, JHEP 02, 030 (2002).
[5.3] D. Colladay and V. A. Kostelecky, Lorentz-violating extension of the standard model, Phys. Rev. D 58, 116002 (1998).
[6] M. K. Parikh and F. Wilczek, Hawking radiation as tunneling, Phys. Rev. Lett. 85, 5042 (2000).
[7.1] Lectures of prof. Scrucca in Advanced quantum field theory (e.g., for the first study of the worldline formalism):

III. There and back again: From QFT to the lattice field theory

[1] W. A. Harrison, Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond (Dover, 1989).
[2] E. Fradkin, Field Theories of Condensed Matter Physics, (2nd ed., CUP, 2013).
[3] P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Phys. Rev. 136, B864 (1964).
[4] W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Phys. Rev. 140, 1133 (1965).
[5] DFT lecture notes by Roi Baer, including a nice tutorial on the Thomas-Fermi model (chapter II), are available at
[6] A very handy Hubbard model introduction: R. T. Scalettar, Elementary Introduction to the Hubbard Model, lecture notes, UC Davis,
[7] E. H. Lieb and F. Y. Wu, Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension, Phys. Rev. Lett. 20, 1445 (1968).

IV. There and back again: QFT as a continuum limit of the lattice theory

[1] A. H. Castro-Neto et al., The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
[2] M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea, Gauge fields in graphene, Phys. Rept. 496, 109 (2010).
[3] G. Semenoff, Condensed-Matter Simulation of a Three-Dimensional Anomaly, Phys. Rev. Lett. 53, 2449 (1984).
[3.1] P. R. Wallace, The Band Theory of Graphite, Phys. Rev. 71, 622 (1947).
[4] K. Wakabayashi et al., Electronic and magnetic properties of nanographite ribbons, Phys. Rev. B 59, 8271 (1999).
[5.1] D. Gunlycke and C. T. White, Valley and spin polarization from graphene line defect scattering, J. Vac. Sci. Technol. B 30, 03D112 (2012).
[6] O. V. Gamayun, E. V. Gorbar, and V. P. Gusynin, Gap generation and semimetal-insulator phase transition in graphene, Phys. Rev. B 81, 075429 (2010).
[7] J. Gonzalez, F. Guinea, and M. A. H. Vozmediano, Non-Fermi liquid behavior of electrons in the half-filled honeycomb lattice (A renormalization group approach), Nucl. Phys. B 424, 595 (1994).

Handouts, examination tickets, etc.:

Handout 1/2 (Spring term) (pdf)

Handout 2/2 (Spring term) (pdf)

Examination syllabus (Spring term) (pdf)

Handout 1/1 (Fall term) (pdf)

Examination syllabus (Fall term) (pdf)

Checkpoints leaderboard (Fall 2020, restricted to course attendees) (pdf)

Selected lecture notes & presentations (access granted to students attending the course):

I. Quantum effects in finite volume

Lecture 1, The Outline & Before Casimir: the van der Waals forces (presentation, pdf)

Lecture 2, The Casimir effect: different approaches and their equivalence (presentation, pdf)

Lecture 3, The Casimir effect: explicit renormalization, zeta function regularization (presentation, pdf)

Lecture 4, The Casimir effect: the Green function technique (presentation, pdf)

Lecture 5, The Casimir effect: the dyadic Green function technique, spherical Casimir effect (presentation, pdf)

Lecture 6, Fundamentals of the Dynamical Casimir effect (presentation, pdf)

Lecture 7, The Dynamical Casimir Effect: arbitrarily moving ‘plate’ in D = 1 + 1 (presentation, pdf)

Lecture 8, Quantization of classical solutions: the kink (or quantized fields in a self-consistent finite volume) (presentation, pdf)

II. Quantum effects in strong external fields

Lecture 9, Photon splitting in strong magnetic field: Heisenberg, Euler, and Adler (presentation, pdf)

Lecture 10, Neutrino in dense media and fundamentals of collective neutrino oscillations (presentation, pdf)

Lecture 11, Radiative Lorentz violation: the Axion-Wess-Zumino model (presentation, pdf)

III. There and back again: From QFT to the lattice field theory

Lectures 1-2, Tight-binding model and its origins (presentation, pdf)

Lectures 3-4, Many-particle to one-particle: some recipes. Hartree--Fock & DFT (presentation, pdf)

Lecture 5, Lattice field theory: ‘free fields’ (presentation, pdf)

Lectures 6-8, Lattice field theory: ‘interacting fields’. The Hubbard model & Fermi surface instabilities; BCS superconductivity (presentation, pdf)

IV. There and back again: QFT as a continuum limit of the lattice theory

Lecture 9, Graphene: honey from the honeycomb. Gauge fields, strains, EM interaction (presentation, pdf)