How do probabilities arise in the Everett interpretation of quantum mechanics?
After a series of quantum experiments, there will be a version of the observer who sees a sequence of very unlikely outcomes. How to make sense of that?
Well, in MWI there is the space of worlds, each associated with a certain complex number (“amplitude”). Some worlds can be uniform hydrogen over all the universe, some contain humans, a certain subspace contains me (I mean, collection of particles moving in a way introspectively-identical to me) writing a LessWrong comment, etc.
It so happens that the larger magnitude of said complex number is, the more often we see such world; IIRC, that inequality allows to prove that likelihood of seeing any world is proportional to squared modulo of amplitude, which is Born’s rule.
The worlds space is presumably infinite-dimensional, and also expands over time (though not at exponential rate as is widely said, because “branches” must be merging all the time as well as splitting). That means that probability distribution assigns a very low likelihood to pretty much any world… but why do we get any outcomes then?
I’m not attempting to answer question why we experience things in the first place (preferring instead to seal it even for myself), but as for why we continue to do so conditional on experiencing something before: because of the “conditional on”. Conditional probability is the non-unitary operation over our observations of phase space, retaining some parts while zeroing all others, which are “incompatible with our observations”; also, as its formula is P(A|B)=P(AB)P(B), it can amplify likelihoods for small values of P(B). That doesn’t totally fix the issue, but I believe the right thing to do in improbable worlds is to continue updating on evidence and choosing best actions as usual. (To demonstrate the issue with small probabilities is not fixed, let’s divide likelihood of any single world by 2256; here’s a 256-bit random string: e697c6dfb32cf132805d38cf85a60c832247449749293054704ad56209d2440e).
You can say that probability comes from being calibrated—after many experiments where an event happens with probability 1⁄2 (e.g. spin up for a particle in state 1/√2 |up> + 1/√2 |down>), you’d probably have that event to happen half the time. The important word here is “probably”, which is what we are trying to understand in the first place.
I’m imagining the branch where a very unlikely outcome consistently happens (think winning a quantum lottery). Intelligent life in this branch would observe what seems like different physical laws. I just find this unsettling.
The worlds space is presumably infinite-dimensional, and also expands over time
If we take quantum mechanics, we have a quantum wavefunction in an infinite-dimensional hilbert space which is the tensor product of the hilbert space describing each particle. I’m not sure what you mean by “expands”, we just get decoherence over time. I don’t really know quantum field theory so I cannot say how this fits with special relativity. Nobody knows how to reconcile it with general relativity.
Well, in MWI there is the space of worlds, each associated with a certain complex number (“amplitude”). Some worlds can be uniform hydrogen over all the universe, some contain humans, a certain subspace contains me (I mean, collection of particles moving in a way introspectively-identical to me) writing a LessWrong comment, etc.
It so happens that the larger magnitude of said complex number is, the more often we see such world; IIRC, that inequality allows to prove that likelihood of seeing any world is proportional to squared modulo of amplitude, which is Born’s rule.
The worlds space is presumably infinite-dimensional, and also expands over time (though not at exponential rate as is widely said, because “branches” must be merging all the time as well as splitting). That means that probability distribution assigns a very low likelihood to pretty much any world… but why do we get any outcomes then?
I’m not attempting to answer question why we experience things in the first place (preferring instead to seal it even for myself), but as for why we continue to do so conditional on experiencing something before: because of the “conditional on”. Conditional probability is the non-unitary operation over our observations of phase space, retaining some parts while zeroing all others, which are “incompatible with our observations”; also, as its formula is P(A|B)=P(AB)P(B), it can amplify likelihoods for small values of P(B). That doesn’t totally fix the issue, but I believe the right thing to do in improbable worlds is to continue updating on evidence and choosing best actions as usual.
(To demonstrate the issue with small probabilities is not fixed, let’s divide likelihood of any single world by 2256; here’s a 256-bit random string: e697c6dfb32cf132805d38cf85a60c832247449749293054704ad56209d2440e).
You can say that probability comes from being calibrated—after many experiments where an event happens with probability 1⁄2 (e.g. spin up for a particle in state 1/√2 |up> + 1/√2 |down>), you’d probably have that event to happen half the time. The important word here is “probably”, which is what we are trying to understand in the first place.
I’m imagining the branch where a very unlikely outcome consistently happens (think winning a quantum lottery). Intelligent life in this branch would observe what seems like different physical laws. I just find this unsettling.
If we take quantum mechanics, we have a quantum wavefunction in an infinite-dimensional hilbert space which is the tensor product of the hilbert space describing each particle. I’m not sure what you mean by “expands”, we just get decoherence over time. I don’t really know quantum field theory so I cannot say how this fits with special relativity. Nobody knows how to reconcile it with general relativity.