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APPLICATIONS OF GROUP THEORY

Sergei Borisenok

1. Introduction. Symmetry

The model of molecule HNO3. Main oscillations. Fundamental frequencies. Multiplicity. Symmetry (definition). Symmetries for HNO3. Group of symmetries. The group C3v. Cayley table. Linearity. The main oscillations of HNO3: the cases of multiplicity 1,2.

2. General scheme of group theory application

Linear space. Linear operator. Group. Abelian group. Group representation. The first main problem of group theory application. Equivalent representations. Addition of representations. Reducible and irreducible representations. The second main problem of group theory application. Canonical solution.

3. Continuous Lie groups. Rotation group and its application

Rotations in 3-dimensional space. Euler angles. Group of rotations O+(3). 1st main problem of group theory application: irreducible representation of rotation group. 2nd main problem: examples.
Lie group. Compact group. General properties of Lie groups. Infinitesimal matrices (generators). Structural constants.
The group of 3-dimensional rotations O+(3) as Lie group. Infinitesimal matrices of representations for O+(3). Irreducible representation of group O+(3). Double-valued representation. Spherical harmonic functions as basis for irreducible representation of group O+(3). Homomorphism of group O+(3) with SU(2).
Schoedinger equation for the particle in central field. Orbital angular momentum. Direct product of irreducible representations. Tensor representations.
The classification of physical fields and the group of rotations. The symmetry of equations for physical field.

4. Lorentz and Poincare groups

Poincare group. Minkowski space. Interval. Lorentz group. Special theory of relativity. Galilei transformation. 2-dimensional unimodular group and Lorentz group:
homomorphism. Spinors and spinor representation of Lorentz group.

5. Space groups

Crystals. Crystal lattices. Group of translations. Bravais lattice. Syngony. General form for element of space group. Irreducible representation of translation group. Inverse lattice. Brillouin zone. Group classification of normal oscillations in crystal.

6. Permutation group

The systems of identical quantum particles. Permutation group of n symbols.
Irreducible representation of group Sn. Symmetrization operator (symmetrizator).
Young tableau. Combinatorial lemma.

7. The theory of finite group representation and its application

The unitary theorem for representations and its consequences. Projection operator and orthogonality relations. Schur’s lemma. General scheme for the 2nd main problem of finite group theory applications. The character of representation. Completeness theorem and Fourier coefficients. Complex conjugate representation.
Small oscillations of symmetrical mechanical systems. Energy in mechanics. Symmetric coordinates. Potential energy in symmetric coordinates. Potential energy in real coordinates. Multiplicity of fundamental frequencies and the forms of main oscillations. Example of theory application: eight Na-Cl point oscillations.

Literature:
1. M. Ayub. Group Theory for Physical Application. Islamabad: 1996.
2. G. G. Hall. Applied Group Theory. London: Longmans, 1967.
3. M. Hamermesh. Group Theory and Its Application to Physical Problems. New York: Dover Publications, 1989.
4. J. W. Leech, and D. J. Newman. How to Use Groups. Barnes and Noble, 1969.
5. J. Rosen. Symmetry in Science: An Introduction to the General Theory. New York: Springer-Verlag, 1995.
6. M. Mizushima. Theoretical Physics: From Classical Mechanics to Group Theory of Microparticles. New York: John Wiley & Sons, 1972.
7. W. K. Tung. Group Theory in Physics. Singapore: Word Scientific Publishing, 1991.
8. J. F. Comwell. Group Theory in Physics. Vol. 1, 2. London: Academic Press, 1984.
9. G. Liubarskii. The Application of Group Theory in Physics. Oxford, New York, Pergamon Press, 1960.