Poslední úprava: doc. Mgr. Milan Krtička, Ph.D. (30.04.2019)
Rozdíly mezi časoprostorovými a vnitřními
symetriemi (na úrovni teoremu Noetherové pro klasická pole
ve čtyřech dimenzích) a dvěma přístupy (obecná relativita ve
třech dimenzích a supersymetrie)
v nichž byly tyto symetrie sjednoceny.
Supersymetrie bude zkoumána způsobem, který se nevyskytuje v tradiční
literatuře.
Vytvoření superalgebry a jejích reprezentací skýtá možnost podívat se
na další projevy symetrií jako jsou centrální náboje, monopóly a
obecně BPS stavy.
Po diskusi rozšíření Lorentzovy invariance na supersymetrii se budeme
zabývat možnými projevy narušení této invariance v rámci nekomutativní
teorie pole.
[1] A. Iorio, Lecture notes, 2009.
[2] L. H. Ryder, Quantum Field Theory, Cambridge Univ. Press, 1985.
[3] S. Weinberg, The Quantum Theory of Fields, Cambridge Univ. Press,
1995 (Vols. 2 and 3).
[4] J. D. Walecka, Advanced Modern Physics, World Scientific, 2010.
[5] A. Altland, B. Simons, Condensed Matter Field Theory, Cambridge
Univ. Press, 2006.
[6] S. Coleman, Aspects of Symmetry, Cambridge Univ. Press, 1985.
[7] R. Jackiw, Diverse topics in Theoretical and Mathematical Physics,
World Scientific, 1995.
[8] C. Nash, S. Sen, Topology for physicists, Academic Press,1988.
[9] M. Nakahara, Geometry, Topology and Physics, IOP Publ., 1990.
[10] J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton
Univ. Press, 1992.
[11] S. Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge Univ.
Press, 2003.
Poslední úprava: doc. Mgr. Milan Krtička, Ph.D. (30.04.2019)
Important concepts and mathematical structures - such as the conformal and the super symmetry algebras, central charges and topological objects - and their role in classical and quantum field theories and solid state physics will be introduced.
Special emphasis will be given to topological objects in the framework of gauge theories of the Yang-Mills type and of the Chern-Simons type, and in the framework of gravity theories in three and lower dimensions.
The course is divided in three parts:
PART I: Noether charges and Supersymmetry.
The Noether theorem for classical field theories will be presented and discussed in general. Spatiotemporal symmetries from the conformal symmetry to supersymmetry will be introduced. The latter will be achieved via the Haag-Lopuszanski-Sohnius construction of the SUSY algebra.
PART II: Topological objects in gauge theories.
Conservation laws not descending from the Noether theorem will be presented. Important topological objects such as Dirac and 't Hooft-Polyakov monopoles and topological gauge theories, such as the Chern-Simons theory, will be discussed.
PART III: Topological objects in gravity theories.
Three dimensional Einstein gravity will be shown to be equivalent to a gauge theory of the Chern-Simons type. The Chern-Simons gravitational term (conformal gravity) will be presented.
Applications of all the above to condensed matter systems could be outlined.
Bibliography:
[1] A. Iorio, Lecture notes, 2009.
[2] L. H. Ryder, Quantum Field Theory, Cambridge Univ. Press, 1985.
[3] S. Weinberg, The Quantum Theory of Fields, Cambridge Univ. Press,
1995 (Vols. 2 and 3).
[4] J. D. Walecka, Advanced Modern Physics, World Scientific, 2010.
[5] A. Altland, B. Simons, Condensed Matter Field Theory, Cambridge
Univ. Press, 2006.
[6] S. Coleman, Aspects of Symmetry, Cambridge Univ. Press, 1985.
[7] R. Jackiw, Diverse topics in Theoretical and Mathematical Physics,
World Scientific, 1995.
[8] C. Nash, S. Sen, Topology for physicists, Academic Press,1988.
[9] M. Nakahara, Geometry, Topology and Physics, IOP Publ., 1990.
[10] J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton
Univ. Press, 1992.
[11] S. Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge Univ.
Press, 2003.
