week 13
30 June

 Discussion of the homework assignment 11:
 E30  The lattice BCS model.
 E31  Application of RG: a quantum
flute (or a bosonic string).
 E32  Miscellaneous yes/no questions.
 Discussion of the homework assignment 12:
 E33  A toy model for the EpsteinGlaser
renormalisation.
 E34  Duhamel's twopoint function: thermal
expectation and Bogolubov inequality.
 E35  Reflection positivity and the
Laplacian.

week 12
23 June

 Outlook on the endofterm
calendar and deadlines.
 Discussion of the homework assignment 10:
 E27  The PerronFrobenius theorem.
 E28  Application of the PerronFrobenius
theorem: irreducible, aperiodic Markov
processes.
 E29  Mean field for a spin system on a
finite lattice. The gap equation.
 The lesson from E29 is the role played by the
Energy/Entropy Balance inequality. In that sense,
E29 shows that the EEB is sometimes an even more
useful characterization of equilibrium.
 Outlook on the homework assignment 12 (both
HW11 and HW12 will be discussed next week): E33
discusses a way to give meaning, via a
regularisation, to the product of two
distributions which would not make sense
otherwise. This is a typical situation that occurs
when discussing perturbatively the effect of an
interaction of a free Bose or Fermi gas. E33(iii)
gives a result which is precisely in the spirit of
the Renormalisation Group.

week 10
9 June 
 Discussion of the homework assignment 0809:
 E24 and E25  Thermodynamic limit of a free
Bose gas.
 E26  BoseEinstein statistics for Weyl's
operators.

week 09
2 June

 Discussion of the homework assignment 07:
 E20  Integral characterisation of KMS. Weak
limit of states each of which is KMS w.r.t. a
dynamics that converges strongly.
 E21  Return to equilibrium w.r.t. an
asymptotically abelian dynamics, under the
assumption that the state has the cluster
property.
 E22  Absence of spontaneously broken
symmetry in a KMS state.
 E23  Miscellaneous yes/no questions.

week 07
19 May

 Discussion of the homework assignment 06:
 E17  Finite volume Gibbs states maximise
the free energy and are equivalent to being
KMS.
 E18 and E19  Derivation on a UHF quantum
spin system algebra and associated
automorphism: from local to global.

week 06
12 May

 Discussion of the homework assignment 05:
 E13  The Schrödinger representation of the
Weyl algebra. Fock space realisation of L^2(R)
(harmonic oscillator).
 E14  Irreducibility of the Schrödinger
representation of the Weyl algebra
 E15  Coheren states in the bosonic Fock
space.
 E16  Bogolubov transformations in the Weyl
algebra.

week 05
5 May 
 General recap on CCR/CAR algebras. A reference
we suggest this year is Merkli's
note on The Ideal Quantum Gas.
 CAR(H) is seperable iff H is seperable, CCR(H)
is seperable iff H=0.
 CAR/CCR algebras are simple, hence all nonzero
representations of CAR/CCR algebras are faithful.
Also the converse is true: if all reps of a C*alg
are faithful, then the C*alg is simple.
 Regular representations.
 Relationship between Heisenberg and Weyl
relations (> BakerCampbellHausdorff).
 Position and momentum operator cannot be
bounded.
 Discussion of the homework assignment 03 and 04:
 E05  Holomorphic functions with value in a
C*algebra.
 E06  Explicit GNS representation of
C([0,1]) w.r.t. a nonpure state.
 E07  Explicit GNS representation of B(H)
w.r.t. a normal state.
 E08  Uniqueness of the GNS representation
up to unitary equivalence. A one parameter
weaklycontinuous group of *automorphisms in
a C*alg is realised, in the GNS
representation of an invariant state, by a
selfadjoint Hamiltonian generating the
unitary evolution.
 E09  Since every C*alg has a state
(HahnBanach), then averaging it in time
yields, in the limit of large average, an
invariant state w.r.t. a group of
*automorphisms.
 E10  Equipping CAR(h) with a structure of
quasilocal algebra w.r.t. a suitable net of
closed nonempty subspaces of h ordered by
inclusion and partially ordered w.r.t.
orthogonality.
 E11  Bogoliubov transformation on
CAR(L^2(R)) generated by space translations.
Asymptotic anticommutativity. Asymptotic
commutativity on the even elements.
 E12  Example of two inequivalent
representations, in the case of the
quasilocal alg. of an infinite chain of
spin1/2 systems.

week 03
21 April 
 Resolvent, spectrum, spectral radius, spectral
radius formula  for elements in a unital
C*algebra.
 Positive elements and states in a C*algebra.
 Pure states.
 To answer a number of questions on the topic:
quick revision on completeness
and completion and on the GNS representation.
 Discussion of the homework assignment 02:
 E01  Examples of algebras that are not C*
for various reasons.
 E02  The C*algebra generated by a 3x3
matrix. (Note: one decomposes the matrix in
Jordan blocks and inspects the space spanned
by the polynomial of each block.)
 E03  Two ideals in C([0,1]). One is closed,
hence the quotient inherits a C*algebraic
structure. The other is not, compute the
closure.
 E04  The group of invertible elements in a
C*algebra is open. The map that associates
each invertible element with its inverse is
continuous.

week 02
14 April 
 How the homework will be organised. Handin
deadline: Thursday before tutorials.
 Why we are intruducing C*. An abstract unifying
language for classical and quantum stat mechanics.
Radical change of perspective (observable come now
first, no Hilbert space or cotangent bundle).
Required a long process of inference to understand
the "right" structure of observables. Advantageous
for large (infinite) systems, equilibrium,
approach to equilibrium, phase transition.
Abstractness vs concrete models.
 A quick recap on how
we introduced C*. The fullfledged
definition, once again. Relevance of the C*
condition. States. Representations: irrep and
cyclic.
 Quotient of a C*algebra mod an ideal. The
algebraic structure is naturally inherited, but in
order to have an induced norm the ideal
needs to be closed. Mentioned, not proved:
this makes the quotient a C*algebra as well.
