week 11

 A number of Fockspace computations require
exponentiating
functions of a and a* which are only essentially
selfadjoint, namely
selfadjoint upon operator closure (see, e.g.,
Exercise 41). A quick
recap on operator
closure can be find here
(from FA2 2011/2012).
 Discussion of the homework
assignment
11:
 E41  Coherent states in the bosonic Fock
space.
 E42  Group of Bogoliubov transformations
for the ideal
Bose gas.
 E43  Distributional formalism for the
(bosonic) Fock
space.
 E44  Twopoint function for a bosonic Gibbs
state.

week 10

 Further comments on BEC.
 Discussion of the homework
assignment
10:
 E37  Perierl's argument  part II.
 E38  Application of Renormalisation Group:
a quantum
flute – or a
bosonic string.
 E39  FermiDirac ideal gases: computation
of
thermodynamic quantities.
 E40  Variance of the particle number for a
FermiDirac
ideal gases.

week 9

 Discussion on Peierls' argument.
 Discussion on SchrammLoewner evolution.
 Discussion of the homework
assignment
08:
 E29  Explicit blockmatrix computations.
 E30  Computation of the free energy
density.
 E31  Peierls’ argument  Part I.
 Discussion of the homework
assignment
09:
 E33  Asymptotics of the complete elliptic
integral of
the first kind.
 E34  Computation of thermodynamic
quantities.
 E35/E36  Renormalisation Group analysis of
the
Sierpinski gasket.

week 8

 discussion of circulant matrices and plane wave
ansatz to
obtain their eigenvectors and eigenvalues
 realspace RG in Ising 1D, exact spin block
transformation,
decimation procedure and RG map
 realspace RG in Ising 2D, higher order
interaction terms
occur during spin block transformation,
approximate solution
 realspace RG in Ising on the Sierpinski gasket,
exact spin
block transformation because of the fractal
geometry; outline of how to
carry out computations in exercise 35 and 36

week 7

 Embedding of Ising model into the class of O(n)
vector models
and applications of these models to experimental
observations.
 Overview about different methods to analyse the
critical
behavior of the Ising model in different
dimensions (transfer matrix,
Peierls' argument, renormalization group, mean
field theory).
 Connection of mean field theory to
phenomenological
GinzburgLandau theory of phase transitions.
 Discussion of Ising 2D lattice simulations
(videos):
criticality and scale invariance at the critical
point.
 Discussion of the correlation function. Mean
field theory
means neglecting the correlations of the
fluctuations in the spin
variable (in thermodynamic equilibrium) around the
mean value (=
magnetization/site). At the critical point, this
assumptions is wrong:
at the critical temperature, fluctuations in the
spin variable around
the mean are correlated over all length scales
(i.e., correlation
length diverges).
 Discussion of the homework
assignment
07:
 E25  Representations of the Clifford
relations.
 E26  Spin representation of rotations.
 E27  Maximum Entropy Principle.
 E28  Ising model in mean field
approximation.

week 6

 Phenomenology of phase transitions.
 Smple liquid and ferromagnet.
 Phase diagrams, order parameter, discontinuous
and
continuous phase transitions, criticality.
 Concept of universality, scaling functions,
critical
exponents.
 Recapitulation of KMS
 Discussion of the homework
assignment
06:
 E21  A clustering state w.r.t. an
asymptotic abelian
dynamics exhibits
return to equilibrium in its folium.
 E22  Construction of KMS states as limit of
KMS states
(applicable for
a KMS state on a quasilocal algebra).
 E23  1D Ising: partition function (via
transfer
matrix), free energy,
magnetisation. Absence of phase transition at
T>0.
 E24  Tensor product of matrices.

week 5


week 4

 Generalisation from class: prescribing that the
map
A>Tr(qA) realised with a trace class operator
q is a state on the
C*algebra of bounded operators on a complex
Hilbert space, implies
that q is selfadjoint, positive, and with trace
equal to one.
 Revision: group of automorphisms and
C*dynamical systems.
In the
GNS representation of a C*algebra with respect to
a stationary state,
the dynamics is implemented by a strongly
continuous unitary group that
leaves the vacuum invariant. Equation of motion at
the level of the
algebra and at
the level of the GNS representation.
 Construction of a stationary state: in a
C*dynamical
system a stationary state always exists. The
problem of distinguishing
a stationary state from a KMS state.
 Discussion of the homework
assignment
04:
 E13  Quasilocal CAR algebra.
 E14  Equivalent conditions for a
representation to be
faithful. A
simple algebra has always faithful
representations.
 E15  Proof of irreducibility of the
Schroedinger
representation of the
Weyl algebra.
 E16  The GNS representation of C[0,1] on
L^2[0,1]

week 3

 Recap on representations of a C*algebra.
Automatically
continuous. Faithful representations. Irreducible
=> Cyclic =>
Nondegenerate. The GNS representation
relative to a given state is
irreducible iff the state is pure. The
representation of a simple
algebra is always faithful. Construction of the
Weyl C*algebra A_W:
von Neumann's theorem states that all the regular
and irreducible
representation of A_W are unitarily equivalent to
the customary
Schroedinger representation on R^d. (Homework
E10/E11 discusses
inequivalent representations instead.)
 Recap on Quasilocal Algebras. Physical meaning.
Quasilocal
algebra for an infinite quantum spin system. Note
that we defined a
quasiloc algebra A giving the algebra and its
family of local
subalgebras, whose union is dense in A.
Conversely, one can start from the family of local
subalgebras,
recognise that their union has all properties of a
C*algebra except
for completeness, then we take the "inductive
limit" (i.e., the
completion!) to construct A. Completion
is an intrinsic, "automatic"
construction: see
some standard facts on completeness from Problem
15, Problem
19, Exercise
13, Exercise
15, Exercise
16, Exercise
17, Exercise
19(ii), Exercise
21, Exercise
34, and Problem
28 from FA SoSe2012.
 Discussion of the homework
assignment
03:
 E09  Boundedness of creation/annihilation
operators in
the fermionic
Fock space. Irreducibility of the CAR algebra.
 E10 / E11  Another representation of the
anticommutation relation,
inequivalent to the Schroedinger
representation. (The groups are not
strongly continuous any longer!)
 E12  Characterisation of positive elements
in a
C*algebra. They are a
closed and convex cone.

week 2

 Recap on bosonic/fermionic Fock space: see this
excerpt from
BratteliRobinson.
 Further outlook:
 Heisenberg group and its representations
 The Weyl algebra
 in which sense the Schrödinger
representation is
unique (von Neumann's theorem)
 the BargmannFockSegal representation.
Reference on these topics: this chapter
from
Folland; see also this
informal note
for an outlook.
 Discussion of the homework
assignment
02:
 E05  A unital, commutative Banach
algebra with A^2=A^2 for every A is
necessarily commutative.
Proof through series expansion and holomorphic
functions methods.
 E06  An example of unital Banach algebra
that contains
nilpotent
elements. The Gelfand transform is in this
case noninjective.
Classification of its characters.
 E07  Weakly* closure of the space of
characters.
 E08  *isomorphism representing the
commutative
C*algebra generated
by 1,A,A* onto the continuous functions on the
spectrum of A.

week
1

 Revision of the algebraic setting for the
commutative case.
The characters (=nonzero multiplicative
functionals) of a unital
commutative Banach algebra form a weakly* compact
Hausdorff space X. A
commutative C*algebra is isomorphic to C(X)
(GelfandNaimark) >
ordinary CLASSICAL MECHANICS description is thus
retrieved.
 Revision of weak* topology in the dual B* of a
Banach
space B. It is indeed weaker then the norm
topology, sufficiently
weaker that the unit ball in the dual B* is
weakly* compact
(BanachAlaoglu), but still reach enough to be a
Hausdorff topology.
 Mention of the scenario in the noncommutative
case,
noncommutative GelfandNaimark theorem >
ordinary QUANTUM
MECHANICS is thus retrieved (needs heavier
representation
theory, see e.g. what
we
did
last
semester
in
MQM).
 Additional
problems
set 01, providing further examples and
applications to the
notions presented in class.
 Discussion of the homework
assignment
01:
 E01  Another example of a Banach*
algebra that fails to be C*: the case of
\ell^1(Z).
 E02  A unital commutative Banach algebra
that is
simple (i.e., no
proper ideals) is necessarily the complex
numbers.
 E03  The standard mechanism of adjoining a
unit to a
nonunital
C*algebra so to get a larger and unital
C*algebra.
 E04  The commutative C*algebra C(X), X
compact: the
spectrum of f in
the algebra is nothing but the range of f.

week 0

