week 13
30 June
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- 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 Epstein-Glaser
renormalisation.
- E34 - Duhamel's two-point function: thermal
expectation and Bogolubov inequality.
- E35 - Reflection positivity and the
Laplacian.
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week 12
23 June
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- Outlook on the end-of-term
calendar and deadlines.
- Discussion of the homework assignment 10:
- E27 - The Perron-Frobenius theorem.
- E28 - Application of the Perron-Frobenius
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
HW-11 and HW-12 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.
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week 10
9 June |
- Discussion of the homework assignment 08-09:
- E24 and E25 - Thermodynamic limit of a free
Bose gas.
- E26 - Bose-Einstein statistics for Weyl's
operators.
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week 09
2 June
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- 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.
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week 07
19 May
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- 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.
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week 06
12 May
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- 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.
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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 (-> Baker-Campbell-Hausdorff).
- 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 non-pure 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
weakly-continuous group of *-automorphisms in
a C*-alg is realised, in the GNS
representation of an invariant state, by a
self-adjoint Hamiltonian generating the
unitary evolution.
- E09 - Since every C*-alg has a state
(Hahn-Banach), 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
quasi-local algebra w.r.t. a suitable net of
closed non-empty 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 anti-commutativity. Asymptotic
commutativity on the even elements.
- E12 - Example of two inequivalent
representations, in the case of the
quasi-local alg. of an infinite chain of
spin-1/2 systems.
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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.
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week 02
14 April |
- How the homework will be organised. Hand-in
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 full-fledged
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.
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