Seminar : Hyperbolic versus elliptic problems -- Variations on the Wick rotation in physics

Veranstalter: Dorothea Bahns
Beginn: 2. November 2010 (Vorbesprechung).
Zeit und Ort: Dienstag, 12:30-14:00, Schlauch (Mathematisches Institut)
For a list of the talks, please visit the seminar's Stud.IP page
Special:     Related talks:
  • Nov 8, 2010, 14:15 - 15:15: Jochen Zahn, Göttingen, "Strohmaier's Noncommutative and semi-Riemannian Geometry". Location: physics department, for more info, see Born-Hilbert seminar.
  • Jan 24, 2011, 14:15 - 15:15: Snorre Christiansen, Oslo, who works on numerical simulations of wave equations, especially the discretization of problems from differential geometry and theoretical physics (with hyperbolic signature). Location: Mathem. Institut, for more info, see Runge-Herglotz-Seminar.


Different topics concerning elliptic vs. hyperbolic problems in physics.

A popular trick in physics is a formal manipulation, called the Wick rotation, where the time variable t is replaced by a complex time variable it. This trick allows one to map problems which are formulated in terms of hyperbolic PDEs to problems formulated in terms of elliptic PDEs (which are of course much easier to treat). Often, this manipulation can very well be understood in terms of mathematics, e.g. in terms of the fact that tempered distributions with certain support properties are boundary values of analytic functions.

We will start the seminar by briefly studying this latter point. We will moreover discuss the celebrated Osterwalder-Schrader theorem which tells us how an elliptic formulation of quantum field theory ('Euclidean QFT') can be related to the physically meaningful hyperbolic version. Moreover, we will study a more general version of the Wick rotation: Following Strohmaier, we will study a Wick rotation which allows one to generalize Connes' spectral triple formulation of (compact) Riemannian Spin manifolds to semi-Riemannian manifolds.

Among other things, if time allows, we will also study the Wick rotation used in quantum gravity and try to understand the mathematics behind it.


This is a Journal-club type seminar aimed at PhD students. Everyone -- not only the speaker -- is expected to prepare for the topic at hand, and active discussions among all participants during the talks are very welcome. Prior knowledge in physics is not necessary.


I will give more literature hints during the seminar. Here is an (uncomprehensive) list for some of the topics:

Osterwalder-Schrader and (the history of) Euclidean Quantum Field Theory

  • Osterwalder, K. and Schrader, R., Commun. Math. Phys. 31, 83 (1973); and 42, 281 (1975).
  • Constructive Quantum Field Theory, (Edited by G. Velo and A. S. Wightman), Lecture Notes in Physics 25, Springer (1973).
  • Jaffe, A., and Glimm, J., Quantum Physics: A Functional Integral Point of View, Springer 1987
Semi-Riemannian Spectral triples
  • Strohmaier, A., On Noncommutative and semi-Riemannian Geometry, J.Geom.Phys. 56 (2006) 175-195 [math-ph/0110001]
  • Paschke, M., and Verch, R., Local covariant quantum field theory over spectral geometries, Class.Quant.Grav. 21 (2004) 5299-5316 [gr-qc/0405057]
  • Moretti, V., Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes, Rev.Math.Phys. 15 (2003) 1171-1217, [gr-qc/0203095]
  • (related:) Franco, N., Global Eikonal Condition for Lorentzian Distance Function in Noncommutative Geometry, SIGMA 6 (2010) 064 [article]
  • Benedetti, R., and Bonsante, F., Canonical Wick rotations in 3-dimensional gravity, Memoirs of the American Mathematical Society, Vol 198 no. 926 (2009) [SUB K 2009 B 292]

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