Groups. Role of symmetry in physics; examples.

Definition of group, group properties.

Examples: discrete groups (point groups, C2, C3, D3, D3h; permutation group, S3; continuous groups, R2, R3). Isomorphism, subgroup, direct product; conjugate elements, classes.

Representations of finite groups. Invariant subspaces, irreducibility, equivalent representations, Maschke theorem.

Orthogonality properties of irreducible

representations, Schur lemmas, orthogonality

relations, examples. Characters of representations, orthogonality relations for characters of irreducible representations, reduction of representations, irreducibility criterion. Regular representation. Construction of character table. Subduction. Projection operators. Wigner-Eckart theorem.

Symmetry in quantum mechanics. Labelling of degenerate electron states, selection rules, variational solution of quantum-mechanical problems, perturbation theory.

Molecular vibrations. Classical vibration modes,

classification of normal modes, vibration

patterns, vibrational levels and wave functions

(ground state, fundamental states, combined

states).

Crystallographic point groups. Stereogram,

proper groups, improper groups, class

structure of point groups. Point and

translational symmetry. Crystal systems.

Irreducible representations.

Atom in crystal field. Spin and double group. Splitting of atomic levels in weak and moderate crystal field.

Continuous groups. Infinitesimal operators, Lie algebras, Unitary representations, Wigner-Eckart theorem for continuous groups, irreducible representations, adjoint representation, simple algebras and groups. Examples: R2, R3, SU(2), SU(3).

Construction of irreducible representations of continuous groups. Casimir operators, Cartan subalgebra, roots and weights, positive weights, simple roots, fundamental representations, Dynkin indices and diagrams. Particle physics examples: SU(3) in QCD.

Tensor methods in SU(3). Invariant tensors, decomposition of product representations, Young tableaux and SU(3): dimension of representation, product representations. Particle-physics examples: Flavor symmetry of light quarks and spectra of hadronic states.

Space and time. Lorentz transformations, Lorentz group, generators of SO(3,1). Examples of representations of Lorentz group: Dirac representation. Translations, Poincare group and its representations for massive and massless particles. Coleman-Mandula theorem and supersymmetry.

# Symmetry in physics

W. Ludwig in C. Falter, Symmetries in Physics, Springer, Berlin, 1996,

J. P. Elliott in P. G. Dawber, Symmetry in Physics, MacMillan, Houndmills, 1979,

M. Hamermesh, Group Theory and Its Application to Physical Problems, Dover, New York, 1989,

M. Tinkham, Group Theory and Quantum Mechanics, McGraw Hill, New York, 1964,

S. K. Kim, Group Theoretical Methods and Applications to Molecules and Crystals, Cambridge University Press, Cambridge, 1999,

W.-K. Tung, Group Theory in Physics, World Scientific, Singapore, 1985,

F. Stancu, Group Theory in Subnuclear Physics, Oxford Science Publications, Oxford, 1996,

H. Georgi, Lie algebras in particle physics, Westview Press, Boulder, CO, 1999.

To provide a balanced overview of the role of symmetry in physical systems using a selection of examples from physics of elementary particles, atomic and molecular physics as well as condensed-matter physics, and to introduce the group theory and the theory of representations as tools for the description of symmetry.

Knowledge and understanding

To master the methods of group theory and to understand the role of symmetry in physics. To gain an overview of various physical systems that are as illustrative as possible from the group-theory perspective, thereby complementing the phenomenological insight into the physics of elementary particles, atoms, molecules, and condensed matter provided by the respective specialized courses.

Application

Students learn to use the methods of group theory, especially the theory of representations, to study symmetry in various physical systems.

Reflection

The course helps to appreciate the meaning of symmetry as a factor that constrains and systemizes the set of possible states of physical systems. As it builds on examples from a range of branches of physics, it offers an overview of formally similar yet distinct phenomena.

Transferable skills

Identification of aspects of physical problems that can be studied using group theory.

lectures, tutorials, seminars, homework assignments, consultations

completed homework assignment (written report, presentation) counts as problem-solving examination

oral examination

grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

doc. dr. J. Fesel Kamenik

1. HURTH, Tobias, ISIDORI, Gino, KAMENIK, Jernej, MESCIA, Federico. Constraints on new physics in MFV models : a model-independent analysis of [Delta]F=1 processes. Nuclear physics. Section B, ISSN 0550-3213. [Print ed.], 2009, vol. 808, no. 1/2, str. 326-346. [COBISS-SI-ID 22772007].

2. DORŠNER, Ilja, FAJFER, Svjetlana, KAMENIK, Jernej, KOŠNIK, Nejc. Light colored scalars from grand unification and the forward-backward asymmetry in t[bar]t production. Physical review. D, Particles, fields, gravitation, and cosmology, ISSN 1550-7998, 2010, vol. 81, no. 5, str. 055009-1-055009-11. [COBISS-SI-ID 23517735].

3. DROBNAK, Jure, FAJFER, Svjetlana, KAMENIK, Jernej. Probing anomalous t W b interactions with rare B decays. Nuclear physics. Section B, ISSN 0550-3213. [Print ed.], 2011, vol. 855, no. 1, str. 82-99, doi: 10.1016/j.nuclphysb.2011.10.004. [COBISS-SI-ID 25202215].

4. GEDALIA, Oram, KAMENIK, Jernej, LIGETI, Zoltan, PEREZ, Gilad. On the universality of CP violation in [delta]F = 1 processes. Physics letters. Section B, ISSN 0370-2693. [Print ed.], 2012, vol. 714, no. 1, str. 55-61, doi: 10.1016/j.physletb.2012.06.050. [COBISS-SI-ID 25960999].

5. FAJFER, Svjetlana, KAMENIK, Jernej, NIŠANDŽIĆ, Ivan, ZUPAN, Jure. Implications of lepton flavor universality violations in B decays. Physical review letters, ISSN 0031-9007. [Print ed.], 2012, vol. 109, issue 16, str. 161801-1-161801-5, doi: 10.1103/PhysRevLett.109.161801. [COBISS-SI-ID 26186535].

izr. prof. dr. P. Ziherl

1. DOTERA, T., OSHIRO, T, ZIHERL, Primož. Mosaic two-lengthscale quasicrystals. Nature, ISSN 0028-0836, 2014, vol. 506, no. 7487, str. 208-211, doi: 10.1038/nature12938. [COBISS-SI-ID 27499815].

2. GLASER, M. A., GRASON, G. M., KAMIEN, Randall D., KOŠMRLJ, Andrej, SANTANGELO, C. D., ZIHERL, Primož. Soft spheres make more mesophases. Europhysics letters, ISSN 0295-5075, 2007, vol. 78, str. 46004-1-46004-5. [COBISS-SI-ID 20719143].

3. ZIHERL, Primož, KAMIEN, Randall D. Soap froths and crystal structures. Physical review letters, ISSN 0031-9007. [Print ed.], 2000, vol. 85, str. 3528-3531. [COBISS-SI-ID 15476775].

4. HOČEVAR BREZAVŠČEK, Ana, EL SHAWISH, Samir, ZIHERL, Primož. Morphometry and structure of natural random tilings. The European physical journal. E, Soft matter, ISSN 1292-8941, 2010, vol. 33, no. 4, str. 369-375. [COBISS-SI-ID 24277287].

5. HOČEVAR BREZAVŠČEK, Ana, ZIHERL, Primož. Periodic three-dimensional assemblies of polyhedral lipid vesicles. Physical review. E, Statistical, nonlinear, and soft matter physics, ISSN 1539-3755, 2011, vol. 83, no. 4, str. 041917-1-041917-10. [COBISS-SI-ID 24719655].