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Course, academic year 2018/2019
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Geometrical Methods of Theoretical Physics II - NTMF060
Title in English: 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 [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
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.
Annotation -
Last update: T_UTF (10.05.2012)
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. (17.05.2012)

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

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

P. Krtouš: Geometrické metody ve fyzice, http://utf.mff.cuni.cz/vyuka/%TMF060/GeometrickeMetody/, 2008.

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. (17.05.2012)
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.
 
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