Mathematical Statistical Physics

  spring term 2014 (SoSe 2014)

(This page is not being updated)


weekly schedule:



Alessandro's office hours
10-12, B-335





homework assignments:

  • Tutors (Tutoren): Johannes Alt, Dominik Schröder, Mykhaylo Panchenko.
  • Markers (Korrektoren): Mauro Miguel Monsalve Mercado & Toni Scharle (contact them if you need clarifications on your marked worksheets). By the way: markers are not graders...
  • Each new homework assignment is posted on this page every Thursday afternoon.
  • Hand-in deadline (Abgabe der Hausaufgaben): ordinarily, each Thursday by 12 p.m. into the designated drop box located on the first floor.
  • Pick up your marked worksheets from the designated return box on the first floor.
  • Please register through the LMU Maths Institute HOMEWORK WEBPAGE. When you're asked the year of your Studienhauptfach (=your major), that actually refers to the year when the Prüfungsordnung (=the exam regulations) was issued, NOT the year when you take the course. TMP students should select "TMP 2010".
week 02
assignment 02 solutions
week 03
assignment 03
week 04
assignment 04
week 05
assignment 05
week 06
assignment 06
week 07
assignment 07
week 08-09
assignment 08-09
week 10
assignment 10
week 11
assignment 11
week 12
assignment 12


  • Final exam (Endklausur): Fri 11 July 2014, 11-14, room B-004.
    final exam            solutions            results


  • O. Bratteli and D. Robinson. Operator Algebras and Quantum Statistical Mechanics I & II. Springer, 2nd edition, 1997.
  • D. Ruelle. Statistical Mechanics: Rigorous Results. World Scientific, 1999.
  • R. Haag. Local Quantum Physics: ‘Fields, Particles, Algebras’. Springer, second and enlarged edition, 2013.
  • W. Thirring. Quantum Mathematical Physics: Atoms, Molecules and Large Systems. Springer, 2nd edition, 2002.
  • B. Simon. The Statistical Mechnics of Lattice Gases, volume I. Princeton University Press, 1993.
  • K. Huang. Statistical Mechanics. Wiley, 2nd edition, 1987.
  • R.K. Pathria and P.D. Beale. Statistical Mechanics. Academic Press, 3rd edition, 2007.
  • J. Cardy. Scaling and Renormalization in Statistical Physics. Cambridge Lecture Notes in Physics. Cambridge University Press, 1996.
  • Sven Bachmann's lecture notes.

weekly diary of homework / tutorial sessions:

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 Epstein-Glaser renormalisation.
    • E34 - Duhamel's two-point function: thermal expectation and Bogolubov inequality.
    • E35 - Reflection positivity and the Laplacian.

week 12
23 June

  • 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.

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.

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 (-> 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.

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. 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.