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.

In the comments to his blog post announcing this lecture I asked Scott about Wick rotation.

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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