### PHYSICAL APPLICATIONS OF GROUP THEORY

**1st module**

**(one semester, about 33 hours)**

**Prof. M. Toller**

**A.A. 1998/99**

- Elementary concepts about groups and group homomorphisms; examples; matrix groups; action of a group on a set; orbits, stability subgroup, homogeneous spaces; topological groups; compact, locally compact, connected, simply connected groups; monodromy theorem; universal covering; invariant measures.
- Linear and unitary representations of groups; invariant subspaces and subrepresentations; direct sum and tensor product of representations; reducible, irreducible and completely decomposable representations; intertwining operators; Schur lemma; tensor and spinor calculus.
- Symmetry operations in quantum mechanics; Wigner theorem; symmetry groups; projective (ray) representations; their reduction to unitary representations of a central extension of the group; one-parameter groups; time evolution.
- Direct integrals of Hilbert spaces; spectral measures; diagonalizable and decomposable operators; direct integral decompositions of unitary representations; commutative locally compact groups; Pontryagin duality, harmonic analysis, SNAG theorem.
- Imprimitivity systems; application to non relativistic quantum mechanics; regular and induced representations; imprimitivity theorem;
unitary representations of semi-direct products.
- Unitary representations of the Poincaré group; classification of the irreducible ones; applications to elementary systems; mass, spin, helicity.
- Relativistic wave equations; local transformation property; Klein-Gordon, Weyl, Maxwell and Dirac equations.

#### Textbooks:

- A. O. Barut and R. Raczka, Theory of Group Representations and Applications, PWN - Polish Scientific Publishers, Warszawa (1977).
- A. Kirillov, Elementary Theory of Representations, Springer Verlag, New York, (1976).
- A. Kirillov, Éléments de la théorie des représentations, Éditions MIR, Moscou (1974).
- S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York (1978).
- G. W. Mackey, Induced Representations of Groups and Quantum Mechanics, Benjamin, New York (1968).
- Lecture notes (in Italian).