Michael MOVSHEV

MAT 570: Concepts of Quantum Mechanics

Syllabus:

Day of:		Material Covered	Suggested Reading
Week 1:	Jan. 28	Lagrangian Mechanics:The principle of the least action, Euler-Lagrange equation.	[Bryant] Lect4
	Jan. 30	Symmetries and Noether theorem,Legendre transform	[Bryant] Lect4
	Feb. 1	One-dimensional motion. The motion in a central field and the Kepler problem.	[Takhtajan] Sec.1.5
Week 2:	Feb. 4	Hamiltonian Mechanics. Hamilton’s equations. The action functional in the phase space.	[Bryant] Lect5, Lect6 [Takhtajan] Sec.2.2
	Feb. 6	Classical observables and Poisson bracket. Symplectic and Poisson manifolds.	[Takhtajan] Sec2.4-2.7
	Feb. 8	Hamilton’s and Liouville’s representations.	[Takhtajan] Sec2.8
Week 3:	Feb. 11	Observables, states and dynamics in Quantum Mechanics.	[Takhtajan] Sec2.1
	Feb. 13	Procedure of Quantization. Heisenberg commutation relations. Coordinate and momentum representations. Weyl relations.	[Takhtajan] Sec2.2 Sec 2.3
	Feb. 15	Schroedinger Equation. Harmonic oscillator. Hydrogen atom	[Takhtajan] Sec3.1-Sec3.5
Week 4:	Feb. 18	Geometric asymptotics.	[Guillemin, Sternberg] Ch 2 [Takhtajan] Sec3.6
	Feb. 20	Geometric asymptotics.	[Guillemin, Sternberg] Ch 2 [Takhtajan] Sec 3.6
	Feb. 22	Geometric quantization.	[Woodhouse] Sec 8
Week 5:	Feb. 25	Geometric quantization.	[Woodhouse] Sec 9
	Feb. 27	Geometric quantization.	[Woodhouse] Sec 9
	Feb. 29	Fedosov's quantization	[Fedosov]
Week 6:	March 3	Kontsevich's quantization	[Kontsevich]
	March 5	Kontsevich's quantization	[Kontsevich]
	March 7	Spin and representation theory of SU(2). Charged spin particle in the magnetic field	[Takhtajan] Ch4.1-Sec 4.2
Week 7:	March 10	Feynman path integral Symbols of the evolution operator and path integrals.	[Grosche] SecII [Rabin] Lec 6
	March 12	Feynman path integral Symbols of the evolution operator and path integrals. Continued	[Grosche] SecII [Rabin] Lec6
	March14	Gaussian path integrals. Regularized determinants of differential operators	[Takhtajan] Sec 5.3Sec 5.5
Week 8:	March 17,19,21	Spring Recess
Week 9:	March 24	The Harmonic Oscillator via Path Integral	[Rabin] Lect. 7 [Takhtajan] Sec5.4
	March 26	Semi-classical asymptotics . Diagrammatic	[Itzykson_Zuber] Sec6.2 [Kazhdan]
	March 28	Schwinger-Dyson equation. Semi-classical asymptotics continued	[Takhtajan] Sec5.6
Week 10:	March 31	Introduction to super algebra	[Manin] [Berezin(ru)]
	Apr. 2	Introduction to super geometry	[Manin] [Berezin(ru)]
	Apr. 4	Classical mechanics of a super particle, Supersymmetry.	[Deligne Freed] [Guillemin_V._Sternberg_S._Supersymmetry]
Week 11:	Apr. 7	Quantization of a super particle. Atiyah-Singer index theorem: Topological prerequisites	[Palais(ru)]
	Apr. 9	Localization and Duistermaat-Heckman formula,	[Guillemin V.."Moment_maps..."] Sec2 [Atiyah, M. F.,Bott R.] Sec3
	Apr. 11	One-loop computations	[Alvarez]
Week 12:	Apr. 14	Classical systems with constraints. First and second order constraints.	[Stasheff] [Henneaux_Teitelboim] Ch1
	Apr. 16	Quantization of systems with constraints. BRST and BV	[Stasheff] [Schwarz1] [Schwarz2]
	Apr. 18	Symplectic and quantum reductions. Toric manifolds.	[Bryant] Lec7 [Guillemin V.."Moment_maps..."]

---

DJVU plug-in available here

Books:

• [Alvarez]Alvarez, Orlando Lectures on quantum mechanics and the index theorem. Geometry and quantum field theory (Park City, UT, 1991), 271--322, IAS/Park City Math. Ser., 1, Amer. Math. Soc., Providence, RI, 1995.

• [Atiyah, M. F.,Bott R.]Atiyah, M. F.; Bott, R.The moment map and equivariant cohomology. Topology 23 (1984), no. 1, 1--28.

• [Berezin(ru)]Berezin, Felix Alexandrovich Introduction to superanalysis. Edited and with a foreword by A. A. Kirillov. With an appendix by V. I. Ogievetsky. Translated from the Russian by J. Niederle and R. Koteck\'y. Translation edited by Dimitri Le\u\i tes. Mathematical Physics and Applied Math

• [Bryant]Bryant, Robert L.An introduction to Lie groups and symplectic geometry. Geometry and quantum field theory (Park City, UT, 1991), 5--181, IAS/Park City Math. Ser., 1, Amer. Math. Soc., Providence, RI, 1995.

• [Deligne Freed] Deligne, Pierre; Freed, Daniel S. Classical field theory. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 137--225, Amer. Math. Soc., Providence, RI, 1999.

• [Grosche] Christian Grosche An Introduction into the Feynman Path Integral arXiv:hep-th/9302097v1

• [Guillemin V.."Moment_maps..."] Guillemin, Victor Moment maps and combinatorial invariants of Hamiltonian $T\sp n$-spaces. Progress in Mathematics, 122. Birkhäuser Boston, Inc., Boston, MA, 1994. viii+150 pp. ISBN: 0-8176-3770-2

• [Guillemin, Sternberg] Guillemin, Victor; Sternberg, Shlomo Geometric asymptotics. Mathematical Surveys, No. 14. American Mathematical Society, Providence, R.I., 1977. xviii+474 pp. (one plate).

• [Henneaux_Teitelboim] Henneaux, Marc; Teitelboim, Claudio Quantization of gauge systems. Princeton University Press, Princeton, NJ, 1992. xxviii+520 pp. ISBN: 0-691-08775-X; 0-691-03769-8

• [Kazhdan]Kazhdan, David Introduction to QFT. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 377--418, Amer. Math. Soc., Providence, RI, 1999.

• [Kontsevich]Kontsevich, Maxim Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (2003), no. 3, 157--216.

• [Manin]Manin, Yuri I. Gauge field theory and complex geometry. Translated from the Russian by N. Koblitz and J. R. King. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 289. Springer-Verlag, Berlin, 1988. x+297 pp. ISBN: 3-540-1

• [Palais(ru)]Palais, Richard S. Seminar on the Atiyah-Singer index theorem. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Annals of Mathematics Studies, No. 57 Princeton University Press, Princeton, N.J. 1965 x+366 pp.

• [Stasheff]Stasheff, Jim Homological reduction of constrained Poisson algebras. J. Differential Geom. 45 (1997), no. 1, 221--240.

• [Schwarz1] Schwarz, Albert Topological quantum field theories. XIIIth International Congress on Mathematical Physics (London, 2000), 123--142, Int. Press, Boston, MA, 2001.

• [Schwarz2]Schwarz, Albert Semiclassical approximation in Batalin-Vilkovisky formalism. Comm. Math. Phys. 158 (1993), no. 2, 373--396.

• [Takhtajan] Takhtajan L.A. A Neoclassic Introduction to Quantum Mechanics.(DRAFT)

• [Woodhouse] Woodhouse, Nicholas Geometric quantization. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1980. xi+316 pp. ISBN: 0-19-853528-7