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.)?
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4 Answers
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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).
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One instance of (something similar to) that is the construction of models of hyperbolic geometry using a sphere of radius $i$.
answered Nov 13, 2009 at 21:59
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$\begingroup$ I don't understand what you mean by this. $\endgroup$ Commented Nov 14, 2009 at 4:25
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$\begingroup$ I would bet that the reference is to the following fact: consider Minkowski space R^{n,1} with the Lorentzian metric x_1^2 + ... x_n^2 - t^2. The "sphere of radius -1" is the hyperboloid |x|^2 = t^2 + 1. The restriction of the Lorentzian metric to this hyperboloid is positive definite, and in fact has constant sectional curvature -1. Thus this hyperboloid gives a model of hyperbolic n-space H^n. [This exhibits the identification Isom^+(H^n) = SO^+(n,1).] But I cannot exactly see how to interpret the answer above in these terms. $\endgroup$ Commented Nov 14, 2009 at 7:22
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$\begingroup$ Typo: The "sphere of radius -1" is the hyperboloid |x|^2 = t^2 - 1. $\endgroup$ Commented Nov 14, 2009 at 7:23
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2$\begingroup$ Tom's correct, up to the following: the radius is $i$, not $-1$: the $-1$ is the square of the radius. This model of hyperbolic geometry is very very old, and nicely explains why hyperbolic trigonometry and spherical trogonometry are so similar. It is discussed, for example, in Kolmogorov's and Yushkievich´s book on 19th Centruty Geometry and Function theory. The idea is due to Lambert aroung the 1770s. $\endgroup$ Commented Nov 14, 2009 at 17:55
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Yes. It's called analytic continuation.
answered Nov 14, 2009 at 4:22
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$\begingroup$ Analytic continuation is much more general than Wick rotation. I was asking about the specific instances of $t \mapsto it$ substitution. $\endgroup$ Commented Nov 14, 2009 at 8:25
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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).
answered Nov 14, 2009 at 14:12