## The quantum-classical transition

Given the excellent comments on my recent Eliztur-Vaidman bomb test post and a discussion about Berry’s phase on another site, I got to thinking about this topic. In particular, there was a bit of discussion surrounding something Aharonov and Rohrlich said in the context of a much larger discussion of Berry’s phase in their book <em>Quantum Paradoxes</em>. They said,

We can continuously observe any quantum state. If we continuously observe a decaying atom, it never decays; if we continuously check whether a particle has crossed a barrier, it never crosses. We can make a free particle scatter off a force-free region just by constantly checking whether the particle has entered the region.

Indeed, we can induce any evolution we want in a quantum system.

But can we really? I realize that what they’re trying to get at (I think) is that the act of observation in QM requires an interaction between observer and observed. I don’t deny that. But when you think about the mechanism by which an observer does this interacting on the most fundamental level – an exchange of some boson depending on the type of interaction – either the uncertainty principle limits the observer to a certain level of discreteness or the interaction must involve virtual bosons. As I am unaware of any device that has an unbroken chain of virtual exchanges from the observed quantum process to the observer’s mind, truly continuous observation must not be possible, at least given existing technology. Whether it is fundamentally prevented by QM is an open question. Perhaps things like entanglement are really just strings of virtual processes (which would make it possible for some people to interpret them as violating SR).

But it seems to me that this gets at the heart of whether anything is truly classical or not. Perhaps Bob Griffiths is right and the classical world – and continuity in particular – is simply an illusion (Freudian slip: I almost typed ‘allusion’) born of ‘coarse graining.’

August 15, 2010 at 12:28 am

Without infinite resources, I think the time resolution of any measurement must be finite, and hence the quantum Zeno effect can’t *completely* stop evolution. So I agree, I don’t think the statement you quote can be taken as literally true.

And while I am not unbiased (since Bob was my postdoc supervisor once upon a time), I do think the classical world is an illusion, in the sense that it is an approximation that is only valid for certain kinds of systems and requiring a high level of coarse-graining.

August 15, 2010 at 11:46 am

Ah, well it is good to know I am not alone in this belief. But then that calls into question the idea that continuously observing a quantum state forces that state to remain static, i.e. a watched pot

shouldboil.August 16, 2010 at 7:57 am

I too belong to the Griffiths/Brun coarse-graining school … what we teach engineering students is that it’s a good idea to learn quantum mechanics backwards-and-forwards.

The backwards way is the quick way — focus on the spectral properties of Hilbert space; specify all dynamical properties as axioms. This approach has the advantage of speed: most students are in a hurry to learn about the spectral mysteries of quantum mechanics, and so most textbooks are in a hurry to teach them!

The forward way is the slow way — focus on building-up a hierarchy of naturality; view classical and quantum dynamics

bothas symplectic flows; introduce Hilbert space only at the end … and even at the end, regard Hilbert space as a large-dimension approximation that happens to be convenient for coarse-grained spectral approximations.A really interesting question is, which way does

Naturedo it?Is quantum state-space exactly a linear Hilbert space, in the same way (we used to think) Newtonian space is exactly a linear Euclidean space?

Or is quantum state-space a curvilinear (possibly dynamical) Kählerian space, in the same way that (we now appreciate) the state-space of classical dynamics has Riemannian and symplectic structure, and is dynamic?

These are tough questions!

August 16, 2010 at 9:15 am

And as a postscript to the above, I will mention that it has been my growing impression that engineers and scientists sometimes show unnecessary diffidence to mathematical axioms, that sometimes are not as solidly founded in everyday reality as intuition would suggest … so much so, that I posted an essay upon this topic on Scott Aaronson’s weblog

Shtetl Optimized, that described both P≠NP and quantum simulation as mathematical Rabbits of Caerbannog.Undoubtedly these *are* mighty fearsome rabbits, however! 🙂

August 18, 2010 at 9:39 am

I think there’s more to the issue of transition to classical than just, when does a classic (not “al”, but typical in QM theory) measurement occur. How about, the reality of developing interactions. I thought of a problem where we would worry about slow transition to classical, and I don’t see a neat resolution (as I noted in EVBT thread):

Let a BS split a photon wave. Add to the MZ (like in EVBT) two parallel mirror tracks, one on each side. Let then the two channels involve repeated multiple reflections against the mirror walls surrounding the MZ as above (like zig-zag, running around the outer parts.) So classically, a wave is pushing outward against each side of the enhanced MZ. So, is it in outward tension and presumably inner compression too? There is enough momentum. Force would equal 2h*nu*(1/c)dn/dt for n reflections/second, in principle, or not? (The “two” from reversal of direction.) Or, “no effect” until we actually detect the photon one place or another?

See also my “Proposal Summary” thread at namelink.