In the discussion under the article The entropic principle about the recent paper by Ooguri, Vafa, and Verlinde, there has been a significant opposition of many participants against the concept of the Wick rotation - one invented by a renowned and virtually unknown physicist Gian-Carlo Wick. They were saying that this mathematical method can't be trusted; they were comparing the use of the Wick rotation to the idea that physical theories should not be tested experimentally. Because I believe that most of this criticism is unfair and the Wick rotation is a useful, and in many cases essential mathematical tool to calculate the physical predictions of a quantum theory, let me dedicate a special article to this issue. Peter Woit added his comments about the Wick rotation, too.
First of all, a summary
The Wick rotation is a calculational trick in quantum theory in which we assume that the energy or the time are pure imaginary. We do the calculations given these assumptions, which are often more well-defined, and then analytically continue the results back the usual real values of time and/or energy. It works. But let's now look at the situation a little bit more closely.
Behavior of path integrals
According to Feynman's approach to quantum mechanics, the probability amplitudes may be calculated as the sum (well, a path integral) over all conceivable classical histories of the physical system. Each of them is weighted by
- exp (i.S/hbar)
where "S" is the classical action calculated for this history. As you can see, the absolute value of this weight is always equal to one as long as "S" is real. From a naive viewpoint, that does not seem to be a good starting point for a convergent integral; the integral keeps on oscillating. Convergence is improved if we add a small negative real part to the exponent. Write the action as
- S = int dt L
and imagine that "dt" has a small imaginary part. You obtain the weight
- exp (i.(int dt (1+i.epsilon)).L/hbar).
Because of the term proportional to "-epsilon" in the exponent (i.e. because of the factor "exp(-epsilon.S)", roughly speaking, the contribution of the configurations with a large action will be exponentially damped, and the convergence will improve. This regularization is applied both to ordinary quantum mechanics as well as quantum field theory. In the latter case, it's the origin of the "i.epsilon" prescriptions for the propagators etc. While the naive Feynman's prescription is obviously reproduced for "epsilon" going to zero, a tiny nonzero value of "epsilon" is essential for making the path integral convergent.
The Wick rotation
This was not the Wick rotation yet, but I hope that the inevitability of this "epsilon" treatment is obvious to everyone: the simple prescription of Feynman is a heuristic inspiration, and the oscillating path integral must be regulated in some innocent way. The "i.epsilon" prescription is the way that preserves all symmetries. Not a big deal. Now let's look at the real Wick rotation.