Subjects

Your browser does not support JavaScript, or its support is disabled. Some features may not be available.

Geometrical Methods of Theoretical Physics II - NTMF060

Title:	Geometrické metody teoretické fyziky II
Guaranteed by:	Institute of Theoretical Physics (32-UTF)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2012 to 2018
Semester:	summer
E-Credits:	4
Hours per week, examination:	summer s.:3/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech
Teaching methods:	full-time
Teaching methods:	full-time
Additional information:	http://utf.mff.cuni.cz/vyuka/TMF060

Guarantor:	prof. RNDr. Jiří Bičák, DrSc., dr. h. c. prof. RNDr. Pavel Krtouš, Ph.D.
Comes under:	Doporučené přednášky 1/2

Annotation -

Last update: doc. RNDr. Karel Houfek, Ph.D. (04.02.2021)

Riemann geometry in terms of forms, Hodge theory, Lie groups and algebras, fibre bundles, geometry of gauge fields, two-component spinors.

Literature - Czech

Last update: prof. RNDr. Pavel Krtouš, Ph.D. (30.04.2020)

M. Fecko: Diferenciálna geometria a Lieove grupy pre fyzikov, IRIS, Bratislava 2004.

O. Kowalski: Základy Riemannovy geometrie, skripta, Karolinum, Praha 1995.

S. Kobayashi a K. Nomizu: Foundations of Differential Geomatry I, Interscience Publishers, New York 1963.

M. Spivak: A Comprehensive Introduction to Differential Geometry, Publish or Perish Press, New York 1970-1979.

T. Frankel: The Geometry of Physics - An Introduction, Cambridge Univ. Press, Cambridge 1999.

M. Nakahara: Geometry, Topology and Physics, Taylor & Francis, London 2003.

J. A. de Azcárraga, J. M. Izquierdo: Lie Groups, Lie Algebras, Cohomology and some Applications in Physics, Cambridge Univ. Press, Cambridge 1995.

Ch. J. Isham: Moddern Differential Geometry For Physicists, World Scientific, Singapore 1989.

C. von Westenholz: Differential Forms in Mathematical Physics, North-Holland, Amsterdam 1978.

R. Penrose, W. Rindler: Spinors and space-time, Volume 1, Cambridge Univ. Press, Cambridge 1984.

P. O'Donnell: Introduction to 2-Spinors in General Relativity, World Scientific, Singapore 2003.

C. W. Misner, K. S. Thorne a J. A. Wheeler: Gravitation, Freedman, San Francisco 1973.

S. W. Hawking a G. F. R. Ellis: The Large Scale Structure of Space-Time, Cambridge Univ. Press, Cambridge 1973.

R. Wald: General Relativity, Univ. of Chicago Press, Chicago 1984.

R. A. Bertlmann: Anomalies in Quantum Field Theory, Oxford Univ. Press, Oxford 1996.

Syllabus -

Last update: prof. RNDr. Pavel Krtouš, Ph.D. (28.01.2021)

Riemann geometry in terms of forms: Exterior calculus (overview). Maxwell theory. Othonormal frames, Cartan structure equations, Ricci coefficients. Bianchi identities. Calculation of the curvature, example - Vaidya metric.
Integration on manifolds: Divergence and rotation, integration on submanifolds, Stokes and Gauss theorems.
Hodge theory: Scalar product on forms, Hodge dual, coderivative, de Rham-Laplace and Beltrami-Laplace operators. Hodge decomposition, potential and copotential, harmonics, cohomology.
Lie groups and algebras: Lie groups, construction of Lie algebra, exponential mapping, Killing metric and structure constants. Bi-invariant metric, measure and covariant derivative. Adjoint representations. Action of Lie group on a manifold, flows and their generators. Representations on vector spaces.
Fibre bundles: Abstract fibre bundles. Vector bundles and their geometry, covariant derivative, vector potential and curvature. Objects on gauge algebra bundle. Introduction to characteristic classes, Chern characters and Chern-Simons forms.
Geometry of gauge fields: Inner degrees of freedom and their description in terms of vector bundles. Gauge and Yang-Mills fields, action and field equations. Gauge symmetry. Electromagnetic and charged fields.
Two-component spinors: Space of spinors, antisymmetric metric and soldering form. Relation between spinors and vectors. Geometric quantities and physical fields in term of spinors, Newman-Penrose formalism.