Wick-type rotations

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Quick description

When trying to compute a quantity (or find an identity) regarding an evolution equation or a field theory, try formally multiplying the time variable by $i$ . This can transform the theory to one which is easier to solve, e.g. from a dispersive PDE to a parabolic one, or from a hyperbolic PDE to an elliptic one. Then, divide the time variable by $i$ again to recover the solution to the original problem. (Actually establishing your answer is, in fact, a rigorous solution to the problem may however require additional effort.)

This method is widely used in relativistic physics, where it is called Wick rotation.

Prerequisites

Complex analysis; PDE.

Example 1

(fundamental solution of Schrodinger equation as Wick rotation of fundamental solution of heat equation.) More examples needed!

Example 2

Any identity involving trigonometric functions such as $\sin(x), \cos(x)$ can be converted into one involving the hyperbolic functions $\sinh(x), \cosh(x)$ using Osborn's rule, e.g.

$\cos^2(x) + \sin^2(x) = 1 \implies \cosh^2(x) - \sinh^2(x) = 1.$

This follows from the Wick-type rotation identities

$\cosh(x) = \cos(ix); \quad \sinh(x) = i \sin(ix)$

and analytic continuation.

Example 3

(Use Wick rotation to derive identities for the Laplace transform from those for the Fourier transform)

General discussion

This trick works well for algebraic identities (since the formal substitution is easily justified in this case), or for manipulations involving complex-analytic functions (as one can use analytic continuation). But it tends to fail terribly for establishing qualitative results, such as existence or uniqueness to a Cauchy problem, once one leaves the analytic category. For instance, the heat equation is solvable in the smooth category forward in time but not backwards in time, whereas the Schrodinger equation, which is a Wick rotation of the heat equation, is time-reversible.

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