Monday, February 09, 2015

Many Worlds: whence and what?

Where does the idea of the "Many Worlds Interpretation of Quantum Mechanics" actually come from? Everettians claim that it's simply a matter of taking the formalism seriously, in its own terms, as David Wallace explains from his paper: "A prolegomenon to the ontology of the Everett interpretation".
"To see how that works, let’s suppose we have a measurement device represented by a pointer, that can be in three states: pointing left, pointing right, and pointing nowhere. And suppose the measurement is set up so that if the electron is measured in position x the pointer moves so that it points left, and if it is measured in position y, it moves so that it points right. We can certainly find a state space suitable for such a pointer, and indeed can find wave-packet states φL (for the pointer pointing left), φR (for it pointing right), and φ0 (for it pointing nowhere). The idea of these states, as with the electron, is that φL (say) is a state such that, if we measure where the pointer is — with the naked eye, or otherwise — we’re pretty much guaranteed to get the result that it’s in the pointing-left position.

Given state spaces for the electron and for the pointer, quantum theory gives us a recipe to construct a state space (the so-called “tensor product space”) for the combined system of electron-plus-pointer. If φ is any state for the electron alone, and ψ any state of the pointer alone, there is then a combined state φ ⊗ ψ of both together, which gives the same experimental predictions as φ for measurements of the electron and the same experimental predictions as ψ for measurements of the pointer.

If the measurement device works as intended, the dynamics of measurement must look something like this:

ψx ⊗ φ =>  ψx ⊗ φL

ψy ⊗ φ0  =>  ψx ⊗ φR

In other words, if the electron starts off in a state such that its position is always found to be x, the pointer must reliably end up in a state such that its position is always found to be on the left (and similarly for y). But now, the linearity of the dynamics causes trouble: what if we measure the electron’s position when it is in the mysterious state αψx + βψy? The dynamics in this case have to give

( αψx + βψy) ⊗ φ0  =>   αψx ⊗ φL + βψy ⊗ φR

So now there seems to be a contradiction between our measurement algorithm and the actual physical process of measurement. The algorithm tells us that the measurement should give x a fraction |α |2 of the time and y the rest of the time, and hence that the pointer should point left a fraction |α |2 of the time and right the rest of the time. But the actual physical process never gives ‘left’ or ‘right’ as pointer states at all, and is not indeterministic at all: instead, it deterministically gives the strange, indefinite state

αψx ⊗ φL + βψy ⊗ φR,

in which the pointer seems to be pointing left and pointing right at the same time.


The immediate question one asks about the Everett interpretation — why do we only see one pointer, if actually there are two? — can be resolved by remembering that you too, dear reader, are a physical system, and if χ L and χ R are, respectively, states in your state space representing you seeing a pointer pointing left and you seeing it pointing right, then the same linearity argument used above predicts that the state of (you-plus-pointer-plus-electron), once you look at the pointer, will be

αψx ⊗ φL ⊗ χL + βψy ⊗ φR⊗ χR

In other words, you will be in a state of seeing left and seeing right at the same time, and this state (according to the Everett interpretation) should also be understood as telling us that there are two yous, one seeing the pointer pointing left and one seeing it pointing right.

Notice — crucially — that although the state above is the sum of two macroscopically very different state, in each term in the sum the results of the two measurements are correlated (in each term the electron has a particular position, the pointer records it as having that position, and you observe the pointer as so recording it.)

Once a system gets above a certain size, it cannot help being measured constantly — by chance collisions with the atmosphere and with sunlight, if by nothing else. In doing so, the multiplicity spreads to more and more systems, while the correlations in each term in the state remain. In due course, the state (schematically) evolves into something like

α (Whole planet is as if electron was found in position x) + β (Whole planet is as if electron was found in position y).

When this, too, is understood as representing both states of affairs simultaneously, the “many-worlds” label for the Everett interpretation starts to sound apposite."
Notice that we are doing nothing more here than taking the superposition seriously. But what kinds of mathematical (and physical) entities correspond to taking all the elements of the superposition as being concurrently 'present'? This is not at all obvious, and as I read Wallace's book and many papers, it seems that the Everettian community finds this stuff pretty opaque too. Sometimes, as above, each superposition branch is claimed to look like the 3 + 1 dimensional space-time which we appear to inhabit; other times the micro-physics underlying the world of appearances seems decidedly weird! Crudely: the quantum state lives in high-dimensional Hilbert space - and the world we inhabit doesn't.

And don't mention the g-word!


Gravity. A workable theory of quantum gravity might be strange in the micro-physics (i.e. at extremely small length scales - or in areas of extreme field strength) so that our placid large-scale experience of reality is once again emergent.