week 11
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- A number of Fock-space computations require
exponentiating
functions of a and a* which are only essentially
self-adjoint, namely
self-adjoint 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 - Two-point function for a bosonic Gibbs
state.
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week 10
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- 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 - Fermi-Dirac ideal gases: computation
of
thermodynamic quantities.
- E40 - Variance of the particle number for a
Fermi-Dirac
ideal gases.
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week 9
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- Discussion on Peierls' argument.
- Discussion on Schramm-Loewner evolution.
- Discussion of the homework
assignment
08:
- E29 - Explicit block-matrix 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.
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week 8
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- discussion of circulant matrices and plane wave
ansatz to
obtain their eigenvectors and eigenvalues
- real-space RG in Ising 1D, exact spin block
transformation,
decimation procedure and RG map
- real-space RG in Ising 2D, higher order
interaction terms
occur during spin block transformation,
approximate solution
- real-space 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
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week 7
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- 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
Ginzburg-Landau 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.
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week 6
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- 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 quasi-local algebra).
- E23 - 1D Ising: partition function (via
transfer
matrix), free energy,
magnetisation. Absence of phase transition at
T>0.
- E24 - Tensor product of matrices.
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week 5
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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 self-adjoint, 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 - Quasi-local 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]
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week 3
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- Recap on representations of a C*-algebra.
Automatically
continuous. Faithful representations. Irreducible
=> Cyclic =>
Non-degenerate. 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 Quasi-local Algebras. Physical meaning.
Quasi-local
algebra for an infinite quantum spin system. Note
that we defined a
quasi-loc algebra A giving the algebra and its
family of local
sub-algebras, whose union is dense in A.
Conversely, one can start from the family of local
sub-algebras,
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
anti-commutation 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.
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week 2
|
- Recap on bosonic/fermionic Fock space: see this
excerpt from
Bratteli-Robinson.
- Further outlook:
- Heisenberg group and its representations
- The Weyl algebra
- in which sense the Schrödinger
representation is
unique (von Neumann's theorem)
- the Bargmann-Fock-Segal 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 non-injective.
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.
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week
1
|
- Revision of the algebraic setting for the
commutative case.
The characters (=non-zero multiplicative
functionals) of a unital
commutative Banach algebra form a weakly-* compact
Hausdorff space X. A
commutative C*-algebra is isomorphic to C(X)
(Gelfand-Naimark) --->
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
(Banach-Alaoglu), but still reach enough to be a
Hausdorff topology.
- Mention of the scenario in the non-commutative
case,
non-commutative Gelfand-Naimark 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
non-unital
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.
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week 0
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