Wick rotation

2007-02-25

A few weeks ago, Scott Aaronson published his 9th lecture notes on Quantum Computing.
In his point of view, the main idea of quantum theory is the generalization of classical
probablitities to quantum amplitudes. (I happen to think that there is a bit more to it.)
If one deals with N alternative outcomes, the classical probablities p(i), i=1..N assigned to
each case add up to 1. Quantum theory assigns a complex amplitude A(i) and requires the
squares |A(i)|^2 to sum up to 1. According to Scott these two measures of chance are
indeed the only plausible choices, ruling out |A(i)|^q for general q except 1 and 2.

wb: How do you understand Wick rotation, which basically moves you from quantum theory
to (classical) thermodynamics and back ?

scott: Wick rotation always seemed to me like a mathematical trick (akin to many other
tricks for using complex numbers to simplify calculations involving real numbers).
But if someone wants to defend the view that it has a more fundamental meaning,
I’d be extremely interested to hear from them.

wb: but in this case it seems to be the other way around;
you rotate from complex amplitudes to real Boltzmann probabilities to (define and)
calculate expectation values…
[..]
I don’t know if Stephen Hawking reads your blog, but as far as I know he thinks that
the Euclidean sector is more ‘fundamental’ than the Lorentz sector.

scott: [..]
I’ve never understood the use of Wick rotation that you mention, but I’ve asked many
physicists about it over the years. My question always boiled down to some variant of
the following: if you really can translate any quantum problem into a classical
thermodynamics problem, then why couldn’t you exploit that, for example, to simulate
Shor’s algorithm in P? The answer I usually got was that Wick rotation only works in
certain special cases that are not that interesting for quantum computing — which
confirmed my view of it as a mathematical trick.

wb: Wick rotation of a simple 1-particle Schroedinger equ. should give you a diffusion
equation, i.e. a system of N classical particles, with N -> infty. Perhaps this is the reason
why one cannot exploit this.

Of course, Wick rotation is a familiar tool to all students and practitioners of quantum
field theory. But I find it fascinating how it connects quantum theory and statistical
mechanics via the special role of the time coordinate and I wonder if it can be used to
achieve a better understanding of the interpretation(s) of quantum theory, e.g. by
'rotating' the infamous measurement problem into an exercise of statistical mechanics.

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