Wick rotation in mathematics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:37:10Z http://mathoverflow.net/feeds/question/5443 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5443/wick-rotation-in-mathematics Wick rotation in mathematics Marcin Kotowski 2009-11-13T21:51:35Z 2009-11-14T15:58:18Z <p>In physics, esp. quantum field theory, Wick rotation (i.e. putting $t \mapsto i\tau$, imaginary time) is often used to simplify calculations, make things convergent or make connections between different models (e.g. quantum and statistical mechanics). Does the Wick rotation trick occur anywhere in mathematics not related to QFT (analysis, PDE etc.)?</p> http://mathoverflow.net/questions/5443/wick-rotation-in-mathematics/5446#5446 Answer by Mariano Suárez-Alvarez for Wick rotation in mathematics Mariano Suárez-Alvarez 2009-11-13T21:59:59Z 2009-11-13T21:59:59Z <p>One instance of (something similar to) that is the construction of models of hyperbolic geometry using a sphere of radius $i$.</p> http://mathoverflow.net/questions/5443/wick-rotation-in-mathematics/5476#5476 Answer by Aaron Bergman for Wick rotation in mathematics Aaron Bergman 2009-11-14T04:22:38Z 2009-11-14T04:22:38Z <p>Yes. It's called analytic continuation.</p> http://mathoverflow.net/questions/5443/wick-rotation-in-mathematics/5505#5505 Answer by Jose Brox for Wick rotation in mathematics Jose Brox 2009-11-14T14:12:39Z 2009-11-14T14:12:39Z <p>Well, since you are looking at it like a "trick", I think that it may be suitable to mention that it can be used to relate the trigonometric functions with the hyperbolic ones (and also to solve somewhat related integrals).</p> http://mathoverflow.net/questions/5443/wick-rotation-in-mathematics/5516#5516 Answer by David Bar Moshe for Wick rotation in mathematics David Bar Moshe 2009-11-14T15:22:24Z 2009-11-14T15:58:18Z <p>A general form of the Wick's rotation is the "Weyl's unitary trick". This construction allows to relate group actions of noncompact forms of a complex Lie group to those of the compact one by changing the signature of the Cartan-Killing form . Although, the representations of the compact and noncompact forms are different, the unitary trick introduces relations among their invariants and between the transition functions, hence the use in quantum field theory. Also, it introduces relations between their homogeneous spaces (see the example above of the sphere and the hyperboloid).</p>