KTH Fysik

Symmetries in Physics 5A1335

Course information 2005

Lecturer: Håkan Snellman - phone 5537 8172

When sending e-mail always put the course number 1335 or 'symmetries' in the subject line!


Lecture notes are sold in the student office for 50 SEK.
The introductory chapter will be available in html form here and will not be printed.

The first 8 chapters of the following book covers essentially the same material:
H.F. Jones: Groups, representations and physics. 2nd ed. IOP Publishing 1998.


Solution of homework problems: There will be two sets of 4 - 5 problems each.

More information is given in the first set

Contents of the lectures

These notes will be continuously updated.   Note the online introduction.

Lectures 1 - 2
Introduction to symmetries, with illustrations. Symmetry transformations
of geometrical figures and their group property.  C3v  as
symmetry group of NH3. Homomorphisms and isomorphisms of
groups. Groups acting on themselves by left and right  multiplication.
Abelian groups, additive notation. The real line R, unit circle S^1, and
the integers Z as Abelian groups.  Action of the group on functions
defined on the group and unitary representations for Abelian groups. The
Hilbert space of square integrable functions. Fourier series as expansions
in this Hilbert space for S^1. A review of operators in Hilbert space,
using the Dirac bra-ket notation.  Hermitian, unitary and normal
operators. The eigenvalues and  spectral representations of operators.
Functions of operators, the exponential.
Lecture notes: 1.1 - 1.4.  Appendix A.1-A.4.
Lectures 3 - 4
Simultaneous diagonalization of commuting operators. Consequences for
unitary representations of Abelian groups.  Discrete Fourier transform and
Fourier analysis as group representations. The dual group. Translations on
Z and the corresponding unitary operators.Simplest examples of symmetry of
quantum systems: mirror and inversion symmetry.  Definition of a parity 
operator in the Hilbert space of a Schrödinger particle. The
eigenvalues of the parity operator, its action on observables. The parity
selection rule.  The parity as a symmetry of the Hamiltonian. 
Schrödinger particle in a periodic  potential and Bloch's theorem.
The group of translations of the real line. Its action on the canonical
coordinate and momentum observables. Direct products of groups. Subgroups
and invariant subgroups.
Lecture notes: Appendix A.3,  1.4 - 2.4..

Lectures 5 - 6
Orthogonal transformations of R^2 as simplest non-Abelian group. Unitary
representations of SO(2) and O(2). The groups SO(3) and O(3).
Parametrization of SO(3). The spherical harmonics as a basis of reps of
O(3).  Action of the orthogonal group on the Hilbert space of a spinless
Schrödinger particle. Action on the coordinate and momentum vector
operators and on the Hamiltonian.  Central potentials, and spherical
symmetry. Separation of variables and radial Hamiltonian. Selection rules
and degeneracy of energy spectra due to spherical symmetry. 
    Conjugation of group elements and decomposition of a group into
conjugation classes. Example: the classes of SO(3).  Permutation groups.
Decomposition of a permutation into cycles. Young diagrams and the
conjugation classes of the permutation group. The parity of a
permutation.  The alternating group of even parity permutations. Cayley's
theorem. Example: the symmetry group of a regular tetrahedron as a
permutation group on the vertices.  The classes of the tetrahedral
Left cosets and  Lagrange's theorem.  Examples of Lagrange's theorem.
Lecture notes: 3.1 - 3.4, 3.6.  4.1 - 4.4. Take a look at 3.5 for 

Lectures 7 - 8
Normal subgroups and the factor (quotient) group.  The homomorphism
theorem. G-spaces, G-modules and representations. Direct sum and direct
product of representations.  Transitive action on G-spaces and orbits. The
isotropy (stabilizer) group. The orbit-stabilizer theorem = Theorem
5.1(1). Examples from regular polyhedra. The counting lemma = Theorem
5.1(2). Example: action of the  icosahedral group on the C60 molecule.
Lecture notes: 4.5 - 4.6, 5.3. Take a look at 4.7, 5.1 - 5.2.

Lectures 9 - 10
Euclidean group and semidirect products. Unitary reps defined by a
G-space. Unitary representations, unitary equivalence. Invariant subspaces
and reduction of reps. Irreps  and their equivalence classes. The
character of a rep and its properties. The orthogonality properties of
primitive charaters. Expansion of class functions into series of
characters. Reductiomn of a rep into irreps using  the primitive
characters. Application to some point groups.
Lecture notes: 5.4, 6.1 - 6.6

Lectures 11 - 12  
Proof of Schur's lemmas.  Consequences of Schur's lemmas: symmetry and
degeneracy, with some examples. Lifting of degeneracy when a symmetry is
broken.  Idea of the great orthogonality theorem, and the orthogonality of
primitive characters. The regular representation. Example: symmetry and 
the vibration modes of a molecule. The Clebsch-Gordan series.  Irreps of
the permutation groups.
Lecture notes: 6.7 - 6.9

Lectures 13 - 15
Lie groups. One-parameter Lie groups. Abelian Lie groups. Matrix groups
and continuity in matrix norm. Continuous paths in parameter space,
connected and simply connected groups. Contraction of closed paths and
homotopy, the fundamental group. SO(3) has a fundamental group of order 2.
Parameter space for SU(2), which is connected and simply connected.  The
homomorphism from SU(2) to SO(3) with kernel = the center of SU(2). The
infinitesimal generators and their commutation relations.  Lie algebras.
The identity of the Lie algebras su(2) and so(3). 
Matrix groups and matrix Lie algebras.
Lie groups generated by Lie algebras. The universal covering group.
Representations of Lie algebras  and the corresponding rep of the Lie
group. Multiple-valued reps of the group. The adjoint rep, example su(2).
Reps of Lie algebras: irreducibility, invariant subspaces, direct sum,
tensor product. Analysis on SU(2): classes, primitive characters,
invariant measure on the parameter space. Reps of the Lie algebra su(2)
and the relation with the representations of the Lie group. The invariant
measure  on SU(2) and the regular representation. Primitive characters
for  SU(2) and their orthogonality properties. The completeness properties
of  the matrix elements of the irreps and of the characters.  The 
Clebsch-Gordan series for SU(2). Definition  and orthogonality properties
of the Clebsch-Gordan coefficients. Transformation properties of
observables: scalars, vector operators. Definition of irreducible tensor
operators. The Wigner-Eckart theorem with applications.  Representations
of the permutation group.  Angular momentum and multi-particle states. 
SU(3) symmetry and the quark model. Space-time symmetry groups, Euclidean,
Lorentz and Poincaré groups. Induced irreps of the Euclidean group. 
Lecture notes: Ch 7 - 9

Modified Aug 18, 2005