Over at Uncertain Principles, the uncertainty father has a couple of posts up about parallel universes the many-worlds interpretation of quantum theory. Which reminded me of a rant I’ve been needing to write (sorry a bit of technical jargon to follow.)

History adds that before or after dying he found himself in the presence of God and told Him: ‘I who have been so many men in vain want to be one and myself.’ The voice of the Lord answered from a whirlwind: ‘Neither am I anyone; I have dreamt the world as you dreamt your work, my Shakespeare, and among the forms in my dream are you, who like myself are many and no one.’“Everything and Nothing” by Jorge Luis Borges.

In the many-worlds interpretation of quantum theory one attempts to treat measurement in a way that doesn’t require an extra “measurement postulate” of quantum theory. One simply notes that if you measure a quantum system, you become entangled with this quantum system, and then any future behavior of you and the system you measure will be as if have done the measurement. Thus one could imagine that there is only unitary evolution of the wave function, and, since there is no reason to demote any of the universes in which your measurements come out different, one might as well think that these parallel branches of the wave function are equally valid realitys. Or something like that.

Interestingly upon reading a description of the many-worlds interpretation this is often followed by thoughts along the lines of “everything that can happen, does happen.” For example, here is the writer for a PBS special on Hugh Everett, the original proponent behind the many-worlds interpretation:

Byrne:…Actually, it’s very much in use in physics today. However, it has consequences to it that people were and remain uneasy with, which basically is that everything that is possible happens

But is this correct? Does the many-worlds interpretation mean that there is some probability that everything that can occur will occur?

Actually, as far as I can tell, it tells you nothing of the sort. In fact it tells you that there is a hell of a lot (to be defined technically below) which won’t occur. In fact in all quantum theories for systems of a minimal size quantum theory tells you that there exist an infinite number of worlds which **won’t** exist.

Suppose you have a quantum state of some large quantum system. This quantum state lives in a Hilbert space of some dimension. Now a projective nondegenerate measurement is given by a basis for this Hilbert space. In the many-worlds interpretation this is modeled by attaching a measuring system, unitarily evolving, and becoming entangled with the system being measured. But that really doesn’t matter much for our discussion. The important point is that every such basis corresponds to a particular measurement and in particular every such choice of basis corresponds to a different post measurement world. Now notice that if you make a measurement in which one of the basis elements points along the dimension of your quantum state then **any** measurement basis you chose all of the other measurement outcomes will never occur. And even if you chose to measure along a basis which doesn’t point exactly along your state, but such that some number of your basis states are orthogonal to you state, you will never observe these universes. In other words in quantum theory in any measurement setup (irrespective of the many worlds hypothesis) a quantum state can tell you that an infinite number of worlds do **not** exist after the measurement. But this all depends on the measurement basis choice, which, even in the many-worlds interpretation, is picked out by some physical interaction. Do we really expect all measurements outcomes to generically occur via any one of our basis choices? Not as far as I can tell.

I’m not sure were this meme that many-worlds implies all universes that can happen do happen came from, but I just don’t see how that could be true. Am I missing something here?

(And by the way, the above Borges story I link to is definitely worth reading and was such a favorite that I committed it to memory.)