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
If one deals with N
alternative outcomes, the classical probablities p(i), i=1..N assigned
add up to 1. Quantum theory assigns a complex amplitude A(i) and
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.
In the comments to his blog post announcing this lecture I asked Scott
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
But if someone wants to defend the view that it has a more fundamental
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
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.
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
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
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