Reading Schrödinger group
Time: 11:00–12:00 on Thursdays
of odd-numbered teaching weeks (12 & 26 Oct, 9 & 13 Nov, 7 Dec).
Place: MAGIC room.
The most prominent examples of three-dimensional Lie groups are: the
Heisenberg group ℍ and the group SL2(ℝ).
The Heisenberg group initiates the class of nilpotent Lie groups and
SL2(ℝ) is the smallest semisimple Lie group. In a sense, those two
classes are bearing many opposite properties.
For a wider community, ℍ and SL2(ℝ) are significant
for their applications:
The Heisenberg group appears in harmonic analysis, quantum
mechanics, signal processing and many other areas.
- The affine group—a part of SL2(ℝ)—is behind of numerous
techniques in real analysis, wavelet theory, etc. Furthermore, the
group SL2(ℝ) itself
is the group of holomorphic automorphisms of the unit disk (or
the upper half-plane) in complex analysis;
- has the Lie algebra sl2 spanned by the
Hamiltonian of the quantum harmonic oscillator and the respective
- has the discrete subgroup SL2(ℤ)
consisting of the integer 2× 2 matrices, which is crucial
for number theory.
The above two lists are partially overlapping and partially
complementing. There are deep reasons for this. The mentioned
generators of the Lie algebra sl2—the Hamiltonian of
the harmonic oscillator and the ladder operators—are quadratic
elements of the Heisenberg Lie algebra. In the opposite direction: the
group SL2(ℝ) naturally acts by outer automorphisms of the
Heisenberg group, thus one can build the semidirect product
S=ℍ⋊ SL2(ℝ), which is known as the
The Schrödinger group naturally covers any of the above applications of
ℍ and SL2(ℝ) and especially efficient in the areas
common to both: harmonic analysis, quantum mechanics and number
theory. It is enough to mention that:
S is the full group of symmetries of the wave, heat and
Schrödinger equations (hence its name).
- The discrete subgroup of S with integer components
defines the theta function.
Our aim is to learn fundamentals of the Schrödinger group in a
pedestrian way, which is suitable for young researchers. The initial
visitors to this page since 01/09/1999.
12 Oct The general and special linear groups, the projective line and
First PDF handouts.
What are GL(2,R) and SL(2,R)?
- What are relations between SL(2,R) and Möbius transformations?
- What is the relation between eigenvectors and fixed points?
- Are Möbius maps 3-transitive?
- What is the number of non-equivalent one-parameter subgroups
of SL(2,R)? (the classification is based on the number of fixed points).
- Can we represent the projective line as a homogeneous space SL(2,R)/H?
- What are the actions of SL(2,R) on the projective line, upper
half plane and unite disk?
- What are the similarities and differences, when we work in the
field of complex numbers or more complicated rings like, e.g.
dual or double numbers?
- 26 Oct Subgroups, homogeneous spaces, induced representations of the Heisenberg group, the Schrödinger
and Fock–Segal–Bargmann models:
Second PDF handouts.
The Heisenberg group and the corresponding Lie algebra.
- Classification of one- and two-dimensional continuous subgroups.
- The induction of representations from a subgroup.
- The induced representations of the Heisenberg group: the
Schrödinger representation and the Fock-Segal-Bargmann representation.
- The ladder operators and the vacuum state—the Gaussian.
- The Stone-von Neumann theorem, which classifies unitary irreducible
representations of the Heisenberg group.
- 9 Nov Automorphisms of the Heisenberg group, semidirect
product, the Schrodinger group, discrete subgroups, subgroup
averaging and the Gaussian.
- 23 Nov The classification of unitary irreducible representations of the
group SL2(ℝ), realisations in the Hardy, Bergman and Dirichlet spaces.
- 14 Feb Canonical commutation relations, the Hamiltonian of
harmonic oscillator, ladder (creation and annihilation) operators,
the vacuum vector and classification of unitary irreducible
representations of the Heisenberg group. Realisations in
Schrödinger, Fock and theta models.
- 21 Feb Bargmann classification of unitary irreducible
representations of the group SL2(ℝ) through raising-lowering
operators on the discrete spectrum of the maximal compact subgroup
- 7 March The realisations of unitary irreducible
representations of the group SL2(ℝ), in the Hardy, Bergman and
- ???? The metaplectic representation, theta functions, the heat
Last modified: February 18, 2018.
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