Wick Rotation

WickRotation is a term which has come up in the course of discussion of CliffordAlgebra, see CliffordAlgebraDetails

It is sometimes reasonable to regard basis vectors that square to -1 as indicating negative curvature in the dimensions in question. See also WickRotation (e.g. http://en.wikipedia.org/wiki/Wick_rotation and discussion on blog of string theorist Luboš Motl (pronounce: "Loo-bosch Maw-tull") at http://motls.blogspot.com/2005/02/wick-rotation.html, and a different blog with discussion of Motl at http://www.math.columbia.edu/~woit/blog/archives/000160.html which includes the amusing comment, "Do you think it's statistically significant that the two most prominent string theorists with weblogs are both incredibly arrogant and incapable of admitting that anyone who disagrees with them might have a point?" -- DougMerritt

The WikiPedia reference gives some details of this, comparing the Minkowski metric

ds^2 = -(dt^2) + dx^2 + dy^2 + dz^2

with the four-dimensional Euclidian metric

ds^2 = dt^2 + dx^2 + dy^2 + dz^2

and pointing out that one is converted to the other via a rotation of the t axis through 90 degrees.

In CliffordAlgebra, this can be accommodated in a number of ways. The most general is to define an 8 dimensional algebra with four basis vectors with a positive square, e.g. e1 e2 e3 e4 and four with negative square f1 f2 f3 f4. These can be grouped in pairs e.g. e1 f1, e2 f2, etc., so that each pair defines a plane. The Minkowski and Euclidian metrics given in the previous paragraph represent choices of four axes from eight, and the relationship between them is a rotation in that space. This can also be thought of as a complexification of the four dimensional euclidean metric, by making each of its coefficients a complex number. Complexification is an approach used by some authors. Other authors want to be very clear that there may be several different objects in a calculation which square to -1, and want to distinguish carefully between them. -- JohnFletcher

On the other hand, things like WickRotation are very special and limited cases of complexification, which in general doubles the number of dimensions rather than simply rotating them, as with quantum mechanics complex probability amplitudes versus real probabilities, or complexified toroids in elliptic curve theory. -- DougMerritt

CategoryMath