Thanks Bruce, that actually makes a lot of sense ... and kind of completely trashes my previous understanding! It also makes QM weirder, and even makes me doubt MWI, which reading Deutsch had convinced me was the true account.
On Tuesday, August 11, 2015 at 2:42:50 PM UTC+10, Bruce wrote: > > Pierz wrote: > > I wonder if someone with a good understanding of QM can answer this > > question, which has been troubling me. Let us imagine the case of a > > single particle, and let us imagine we know its position at time 0. Now > > my understanding of the evolution of the wave function for the position > > of the particle, according to a Deutsch-ian version of MWI, is roughly > > as follows: > > > > The position wave function of the particle represents the distribution > > of universes in which the particle is at a certain position. As an > > observer, I don't know which of those universes I am in, until I make a > > measurement of the particle's position. There is one eigenvalue for each > > universe (or better, branch in the multiverse), and the probability of > > that eigenvalue is a measure of the underlying universe 'count', or > > proportion. > > This may be Deutsch's take on MWI, but it is idiosyncratic to the point > of incoherence. The wave function for a particle is, in general, a > superposition of eigenstates of the position (or momentum) operator. But > this does not represent a distribution of universes. The problem with > such an account is that the basis states for the superposition are > undetermined. For a single particle, there are an infinite number of > possible bases for position space, and an equal infinity of possible > bases for the representation in momentum space. This is known as the > basis problem -- why is it that when we make a measurement, one > particular basis is picked out, and the same basis is picked out > consistently every time. If it were merely a matter of self-location in > a multiverse, there would be no way of ensuring that the same basis > eigenfunctions were selected every time. > > The basis is actually determined dynamically by the process of > einselection, where the preferred basis is that which is stable against > environmental disturbances. Separate universe do not emerge until > decoherence has dissipated the interference terms into the environment > in an irreversible way. Thus universe emerge only after interaction > (measurement); they are not present beforehand. So the terms in the > superposition do not represent separate universes -- they are just > elements of a mathematical representation of the quantum state. > > As an aside, quantum probabilities as given by the Born Rule do not come > from branch counting -- the probability is not a measure of the > underlying universe count. > Then where do they come from? Without that notion of a measure, MWI seems not to be telling us all that much. > > > > So far so good. The explanation seems coherent, even if we > > haven't explained the distribution of those universes. However, I am > > puzzled by the case of spins. Consider a set-up in which a photon is > > polarized in the z direction, so that we know that the particle will, > > with probability 1, pass through another polarizer also oriented in the > > z direction. However what of the situation where the second polarizer is > > oriented at 45 degrees to the first one? In that case, the probability > > is 0.5 that the photon will pass through. If it does, then obviously the > > probability is 1 that it will also pass through a third polarizer also > > oriented at the same angle. > > This is a good illustration of the problem with thinking of the elements > of the superposition as separate worlds. Every state, even an eigenstate > of momentum or spin, is a superposition of eigenstates in different > bases -- in fact, an infinite number of different superpositions. When > you see it this way, you realize that the components of the > superposition can't be worlds, because then you would have to say that > the world in which we find ourselves is actually an infinite number of > different superpositions of different worlds -- a set for each possible > basis of the Hilbert space. The notion of a "world" then becomes > incoherent. > > The photon polarized in the z-direction is a superposition of > polarization states in any other direction. It is an equal mixture of 45 > degree polarizations, but unequal mixtures of polarizations at any other > angle. It is these coefficients of the expansion of the state in a > particular basis that give the probabilities (as absolute values of the > coefficients squared). Since any real value is possible, branch counting > is ruled out. > > Sorry can you clarify why branch counting is ruled out on this, ah, basis? Do you mean that there are an uncountable infinity of possible values and therefore you can't count anything? Actually this was going to be another part of my original question - surely "worlds" have to at least be countable, even if infinite? How much sense does a "continuum of worlds" make? > > > So what is going on in the multiverse in this scenario? Clearly, prior > > to the photon hitting the 45 degree polarizer, it can't be the case that > > half the universes have photons polarized at 45 degrees to the z axis, > > because in fact 100% of them are polarized in the z direction. Yet after > > the hitting the polarizer, half do. So in this case the discontinuity > > between quantum state and measurement, which MWI saves us from in the > > case of a continuous variable like position, seems to persist. What is > > going on at the point of the photon's interaction with the polarizer in > > an MWI account? Clearly, the multiverse differentiates into two branches > > corresponding to the two spin eigenvalues, each with measure 0.5. But > > does MWI have anything to say about the the weirdness of the jump > > between the z polarization and the 45-degree polarization? > > You have seen the difficulty in Deutsch's account, which is that he > makes worlds into effective hidden variables -- there is a hidden > variable which tells which world we are actually in. Deutsch is trying > to recover the idea that a quantum particle has definite values for each > observable before measurement, one value in each of the superposed > worlds. The "world" variable is then the operative hidden variable. > > It has long been understood that such hidden variable approaches to > quantum mechanics fail to account for the date. The notion of a "world" > cannot be regarded as a non-local hidden variable. > > Hmm, yes, that was another thing I'd wondered about. Thanks again for this helpful clarification, > Bruce > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
