The positive evidence for WMI is that it’s already there inside quantum mechanics until you change quantum mechanics in some specific way to get rid of it!
MWI, as beautiful as it is, won’t fully convince me until it can explain the Born probability—other interpretations don’t do it more, so it’s not a point “against” MWI, but it’s still an additional rule you need to make the “jump” between QM and what we actually see. As long you need that additional rule, I’ve a deep feeling we didn’t reach the bottom.
I see two ways of resolving this. Both are valid, as far as I can tell. The first assumes nothing, but may not satisfy. The second only assumes that we even expect the theory to speak of probability.
1
Well, QM says what’s real. It’s out there. There are many ways of interpreting this thing. Among those ways is the Born Rule. If you take that way, you may notice our world, and in turn, us. If you don’t look at it that way, you won’t notice us, much as if you use a computer implementing a GAI as a cup holder. Yet, that interpretation can be made, and moreover it’s compact and yields a lot.
So, since that interpretation can be made, apply the generalized anti-zombie principle—if it acts like a sapient being, it’s a sapient being… And it’ll perceive the universe only under interpretations under which it is a sapient being. So the Born Rule isn’t a general property of the universe. It’s a property of our viewpoint.
2
Just from decoherence, without bringing in Born’s rule, we get the notion that sections of configuration space are splitting up and never coming back together again. If we’re willing to take from that the notion that this splitting should map onto probabilities, then there is exactly one way of mapping from relative weights of splits onto probabilities, such that the usual laws of probability apply correctly. In particular:
1) probabilities are not always equal to zero.
2) the probability of a decoherent branch doesn’t change after its initial decoherence (if it could change, it wouldn’t be decoherent), and the rules are the same everywhere, and in every direction, and at every speed, and so on.
The simplest way to achieve this is to go with ‘unitary operations don’t shift probabilities, just change their orientation in Hilbert Space’. If we require that the probability rule be simpler than the physical theory it’s to apply to (i.e. quantum mechanics itself), it’s the only one, since all of the other candidates effectively take QM, nullify it, and replace it with something else. Being able to freely apply Unitary operations implies that the probability is a function only of component amplitude, not orientation in Hilbert Space.
3) given exclusive possibilities A and B, P(A or B) = P(A) + P(B).
These three are sufficient.
Given a labeling b on states, we have | psi > = sum(b) [ A(b) |b>]
Define for brevity the capital letters J, K, and M as the vector component of |psi> in a particular dimension j, k, or m. For example, K = A(k) | k >
It is possible (and natural, in the language of decoherence) to choose the labeling b such that each decoherent branch gets exactly one dimension (at some particular moment—it will propagate into some other dimension later, even before it decoheres again). Now, consider two recently decohered components, K’ and M’. By running time backwards to before the split, we get the original K and M. Back at that time, we would have seen this as a different, single coherent component, J = K + M.
P ( J ) = P ( K + M) must be equal to P ( K ) + P ( M )
This could have occurred in any dimension, so we make this requirement general.
So, consider instead the ways of projecting a vector J into two orthogonal vectors, K and M. As seen above, the probability of J must not be changed by this re-projection. Let theta be the angle between J and M.
K = sin(theta) A(j) | k >
M = cos(theta) A(j) | m >
By condition (2), P(x) is a function of amplitude, not the vectors, so we can simplify the P ( J ) statement to:
1 and 2 together are pretty convincing to me. The intuition runs like this: it seems pretty hard to construct anything like an observer without probabilities, so there are only observers in as much as one is looking at the world according to the Born Rule view. So an easy anthropic argument says that we should not be surprised to find ourselves within that interpretation.
it seems pretty hard to construct anything like an observer without probabilities, so there are only observers in as much as one is looking at the world according to the Born Rule view
Even better than that—there can be other ways of making observers. Ours happens to be one. It doesn’t need to be the only one. We don’t even need to stake the argument on that difficult problem being impossible.
The positive evidence for WMI is that it’s already there inside quantum mechanics until you change quantum mechanics in some specific way to get rid of it!
MWI, as beautiful as it is, won’t fully convince me until it can explain the Born probability—other interpretations don’t do it more, so it’s not a point “against” MWI, but it’s still an additional rule you need to make the “jump” between QM and what we actually see. As long you need that additional rule, I’ve a deep feeling we didn’t reach the bottom.
I see two ways of resolving this. Both are valid, as far as I can tell. The first assumes nothing, but may not satisfy. The second only assumes that we even expect the theory to speak of probability.
1
Well, QM says what’s real. It’s out there. There are many ways of interpreting this thing. Among those ways is the Born Rule. If you take that way, you may notice our world, and in turn, us. If you don’t look at it that way, you won’t notice us, much as if you use a computer implementing a GAI as a cup holder. Yet, that interpretation can be made, and moreover it’s compact and yields a lot.
So, since that interpretation can be made, apply the generalized anti-zombie principle—if it acts like a sapient being, it’s a sapient being… And it’ll perceive the universe only under interpretations under which it is a sapient being. So the Born Rule isn’t a general property of the universe. It’s a property of our viewpoint.
2
Just from decoherence, without bringing in Born’s rule, we get the notion that sections of configuration space are splitting up and never coming back together again. If we’re willing to take from that the notion that this splitting should map onto probabilities, then there is exactly one way of mapping from relative weights of splits onto probabilities, such that the usual laws of probability apply correctly. In particular:
1) probabilities are not always equal to zero.
2) the probability of a decoherent branch doesn’t change after its initial decoherence (if it could change, it wouldn’t be decoherent), and the rules are the same everywhere, and in every direction, and at every speed, and so on.
The simplest way to achieve this is to go with ‘unitary operations don’t shift probabilities, just change their orientation in Hilbert Space’. If we require that the probability rule be simpler than the physical theory it’s to apply to (i.e. quantum mechanics itself), it’s the only one, since all of the other candidates effectively take QM, nullify it, and replace it with something else. Being able to freely apply Unitary operations implies that the probability is a function only of component amplitude, not orientation in Hilbert Space.
3) given exclusive possibilities A and B, P(A or B) = P(A) + P(B).
These three are sufficient.
Given a labeling b on states, we have | psi > = sum(b) [ A(b) |b>]
Define for brevity the capital letters J, K, and M as the vector component of |psi> in a particular dimension j, k, or m. For example, K = A(k) | k >
It is possible (and natural, in the language of decoherence) to choose the labeling b such that each decoherent branch gets exactly one dimension (at some particular moment—it will propagate into some other dimension later, even before it decoheres again). Now, consider two recently decohered components, K’ and M’. By running time backwards to before the split, we get the original K and M. Back at that time, we would have seen this as a different, single coherent component, J = K + M.
P ( J ) = P ( K + M) must be equal to P ( K ) + P ( M )
This could have occurred in any dimension, so we make this requirement general.
So, consider instead the ways of projecting a vector J into two orthogonal vectors, K and M. As seen above, the probability of J must not be changed by this re-projection. Let theta be the angle between J and M.
K = sin(theta) A(j) | k >
M = cos(theta) A(j) | m >
By condition (2), P(x) is a function of amplitude, not the vectors, so we can simplify the P ( J ) statement to:
P( A(j) ) = P ( sin(theta) A(j) ) + P( cos(theta) A(j) )
this must be true as a function of theta, and for any A(j). The pythagorean theorem shows the one way to achieve this:
P(x) = C x* x for some C.
Since the probabilities are not identically zero, we know that C is not zero.
This, you may note, is the Born Probability Rule.
1 and 2 together are pretty convincing to me. The intuition runs like this: it seems pretty hard to construct anything like an observer without probabilities, so there are only observers in as much as one is looking at the world according to the Born Rule view. So an easy anthropic argument says that we should not be surprised to find ourselves within that interpretation.
Even better than that—there can be other ways of making observers. Ours happens to be one. It doesn’t need to be the only one. We don’t even need to stake the argument on that difficult problem being impossible.