You might be also be interested in “General Bayesian Theories and the Emergence of the Exclusivity Principle” by Chiribella et al. which claims that quantum theory is the most general theory which satisfies Bayesian consistency conditions.
By now, there are actually quite a few attempts to reconstruct quantum theory from more “reasonable” axioms besides Hardy’s. You can track the refrences in the paper above to find some more of them.
As you learn more about most systems, the likelihood ratio should likely go down for each additional point of evidence.
I’d be interested to see the assumptions which go into this. As Stuart has pointed out, it’s got to do with how correlated the evidence is. And for fat-tailed distributions we probably should expect to be surprised at a constant rate.
Note you can still get massive updates if B’ is pretty independent of B. So if someone brings in camera footage of the crime, that has no connection with the previous witness’s trustworthiness, and can throw the odds strongly in one direction or another (in equation, independence means that P(B’|H,B)/P(B’|¬H,B) = P(B’|H)/P(B’|¬H)).
Thanks, I think this is the crucial point for me. I was implicitly operating under the assumption that the evidence is uncorrelated which is of course not warranted in most cases.
So if we have already updated on a lot of evidence, it is often reasonable to expect that part of what future evidence can tell us is already included in these updates. I think I wouldn’t say that the likelihood ratio is independent of the prior anymore. In most cases, they have a common dependency on past evidence.
From the article:
At this point, I think I am somewhat below Nate Silver’s 60% odds that the virus escaped from the lab, and put myself at about 40%, but I haven’t looked carefully and this probability is weakly held.
Quite off-topic: what does it mean from a Bayesian perspective to hold a probability weakly vs. confidently? Likelihood ratios for updating are independent of the prior so a weakly-held probability should update exactly as a confidently-held one. Is there a way to quantifiy the “strongness” with which one holds a probability?
Thanks for your answer. Part of the problem might have been that I wasn’t that proficient with vim. When I reconfigured the clashing key bindings of the IDE I sometimes unknowingly overwrote a vim command which turned out to be useful later on. So I had to reconfigure numerous times which annoyed me so much that I abandoned the approach at the time.
A question for the people who use vim keybindings in IDEs: how do you deal with keybindings for IDE tasks which are not part of vim (like using the debugger, refactoring, code completion, etc.)? The last time I tried to use vim bindings in an IDE there were quite some overlaps with these so I found myself coming up with compromise systems which didn’t work that well because they weren’t coherent.
At least for me, I think the question of whether I’m buying too much for myself in a situation of limited supplies was more important for the decision than the fear of being perceived as weird. This depends of course on how limited the supplies actually were at the time of buying but I think it is generally important to distinguish between the shame because one might profit at the expense of others, and the “pure” weirdness of the action.
We have reason to believe that peptide vaccines will work particularly well here, because we’re targeting a respiratory infection, and the peptide vaccine delivery mechanism targets respiratory tissue instead of blood.
Just a minor point: by delivery mechanism, are you talking about inserting the peptides through the nose à la RadVac? If I understand correctly, Werner Stöcker injects his peptide-based vaccine.
You could also turn around this question. If you find it somewhat plausible that that self-adjoint operators represent physical quantities, eigenvalues represent measurement outcomes and eigenvectors represent states associated with these outcomes (per the arguments I have given in my other post) one could picture a situation where systems hop from eigenvector to eigenvector through time. From this point of view, continuous evolution between states is the strange thing.
The paper by Hardy I cited in another answer to you tries to make QM as similar to a classical probabilistic framework as possible and the sole difference between his two frameworks is that there are continuous transformations between states in the quantum case. (But notice that he works in a finite-dimensional setting which doesn’t easily permit important features of QM like the canonical commutation relations).
There are remaining open questions concerning quantum mechanics, certainly, but I don’t really see any remaining open questions concerning the Everett interpretation.
“Valid” is a strong word, but other reasons I’ve seen include classical prejudice, historical prejudice, dogmatic falsificationism, etc.
Thanks for answering. I didn’t find a better word but I think you understood me right.
So you basically think that the case is settled. I don’t agree with this opinion.
I’m not convinced of the validity of the derivations of the Born rule (see IV.C.2 of this for some critcism in the literature). I also see valid philosophical reasons for preferring other interpretations (like quantum bayesianism aka QBism).
I don’t have a strong opinion on what is the “correct” interpretation myself. I am much more interested in what they actually say, in their relationships, and in understanding why people hold them. After all, they are empirically indistinguishable.
Honestly, though, as I mention in the paper, my sense is that most big name physicists that you might have heard of (Hawking, Feynman, Gell-Mann, etc.) have expressed support for Everett, so it’s really only more of a problem among your average physicist that probably just doesn’t pay that much attention to interpretations of quantum mechanics.
There are other big name physicists who don’t agree (Penrose, Weinberg) and I don’t think you are right about Feynman (see “Feynman said that the concept of a “universal wave function” has serious conceptual difficulties.” from here). Also in the actual quantum foundations research community, there’s a great diversity of opinion regarding interpretations (see this poll).
I think it makes more sense to think of MWI as “first many, then even more many,” at which point questions of “when does the split happen?” feel less interesting, because the original state is no longer as special. [...] If time isn’t quantized, then this has to be spread across continuous space, and so thinking of there being a countable number of worlds is right out.
What I called the “nice ontology” isn’t so much about the number of worlds or even countability but about whether the worlds are well-defined. The MWI gives up a unique reality for things. The desirable feature of the “nice ontology” is that the theory tells us what a “version” of a thing is. As we all seem to agree, the MWI doesn’t do this.
If it doesn’t do this, what’s the justification for speaking of different versions in the first place? I think pure MWI makes only sense as “first one, then one”. After all, there’s just the universal wave function evolving and pure MWI doesn’t give us any reason to take a part of this wavefunction and say there are many versions of this.
I agree that the question “how many worlds are there” doesn’t have a well-defined answer in the MWI. I disagree that it is a meaningless question.
From the bird’s-eye view, the ontology of the MWI seems pretty clear: the universal wavefunction is happily evolving (or is it?). From the frog’s-eye view, the ontology is less clear. The usual account of an experiment goes like this:
The system and the observer come together and interact
This leads to entanglement and decoherence in a certain basis
In the final state, we have a branch for each measurement outcome. i.e. there are now multiple versions of the observer
This seems to suggest a nice ontology: first there’s one observer, then the universe splits and afterwards we have a certain number of versions of the observer. I think questions like “When does the split happen?” and “How many versions?” are important because they would have well-defined answers if the nice ontology was tenable.
Unfortunately it isn’t, so the ontology is muddled. We have to use terms like “approximately zero” and “for all practical purposes” which takes us most of the way back to give the person who determines which approximations are appropriate and what is practical—aka the observer—an important part in the whole affair.
There isn’t a sharp line for when the cross-terms are negligible enough to properly use the word “branch”, but there are exponential effects such that it’s very clearly appropriate in the real-world cases of interest.
I agree that it isn’t a problem for practical purposes but if we are talking about a fundamental theory about reality shouldn’t questions like “How many worlds are there?” have unambiguous answers?
Right, but (before reading your post) I had assumed that the eigenvectors somehow “popped out” of the Everett interpretation.
This is a bit of a tangent but decoherence isn’t exclusive to the Everett interpretation. Decoherence is itself a measurable physical process independent of the interpretation one favors. So explanations which rely on decoherence are part of all interpretations.
I mean in the setup you describe there isn’t any reason why we can’t call the “state space” the observer space and the observer “the system being studied” and then write down the same system from the other point of view...
In the derivations of decoherence you make certain approximations which loosely speaking depend on the environment being big relative to the quantum system. If you change the roles these approximations aren’t valid any more. I’m not sure if we are on the same page regarding decoherence, though (see my other reply to your post).
What goes wrong if we just take our “base states” as discrete objects and try to model QM as the evolution of probability distributions over ordered pairs of these states?
You might be interested in Lucien Hardy’s attempt to find a more intuitive set of axioms for QM compared to the abstractness of the usual presentation: https://arxiv.org/abs/quant-ph/0101012
Ok, now comes the trick: we assume that observation doesn’t change the system
I think the basic point is that if you start by distinguishing your eigenfunctions, then you naturally get out distinguished eigenfunctions.
doesn’t sound correct to me.
The basis in which the diagonalization happens isn’t put in at the beginning. It is determined by the nature of the interaction between the system and its environment. See “environment-induced superselection” or short “einselection”.
I mean I could accept that the Schrödinger equation gives the evolution of the wave-function, but why care about its eigenfunctions so much?
I’m not sure if this will be satisfying to you but I like to think about it like this:
Experiments show that the order of quantum measurements matters. The mathematical representation of the physical quantities needs to take this into account. One simple kind of non-commutative objects are matrices.
If physical quantities are represented by matrices, the possible measurement outcomes need to be encoded in there somehow. They also need to be real. Both conditions are satisfied by the eigenvalues of self-adjoint matrices.
Experiments show that if we immediately repeat a measurement, we get the same outcome again. So if eigenvalues represent measurement outcomes the state of the system after the measurement must be related to them somehow. The eigenvectors of the matrix representing this state is a simple realization of this.
This isn’t a derivation but it makes the mathematical structure of QM somewhat plausible to me.
Do you see any technical or conceptual challenges which the MWI has yet to address or do you think it is a well-defined interpretation with no open questions?
What’s your model for why people are not satisfied with the MWI? The obvious ones are 1) dislike for a many worlds ontology and 2) ignorance of the arguments. Do you think there are other valid reasons?
Like yourself, people aren’t surprised by the outcome of your experiment. The surprising thing happens only if you consider more complicated situations. The easiest situations where surprising things happen are these two:
1) Measure the spins of the two entangled particles in three suitably different directions. From the correlations of the observed outcomes you can calculate a number known as the CHSH-correlator S. This number is larger than any model where the individual outcomes were locally predetermined permits. An accessible discussion of this is given in David Mermin’s Quantum Mysteries for Anyone. The best discussion of the actual physics I know of is by Travis Norsen in his book Foundations of Quantum Mechanics.
2) Measure the spins of three entangled particles in two suitably different directions. There, you get a certain combination of outcomes which is impossible in any classical model. So you don’t need statistics but just a single observation of the classically impossible event. This is discussed in David Mermin’s Quantum Mysteries Revistited.
No. The property which you are describing is not “mixedness” (technical term: “purity”). That the state vector in question can’t be written as a tensor product of state vectors makes it an *entangled* state.
Mixed states are states which cannot be represented by *any* state vector. You need a density matrix in order to write them down.