On Sunday, May 17, 2020 at 9:06:12 PM UTC-5, Bruce wrote:
>
> On Mon, May 18, 2020 at 11:20 AM Lawrence Crowell <
> goldenfield...@gmail.com <javascript:>> wrote:
>
>> On Sunday, May 17, 2020 at 1:57:19 AM UTC-5, Bruce wrote:
>>>
>>> On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <
>>> goldenfield...@gmail.com> wrote:
>>>
>>>> There is nothing wrong formally with what you argue. I would though say
>>>> this is not entirely the Born rule. The Born rule connects eigenvalues
>>>> with
>>>> the probabilities of a wave function. For quantum state amplitudes a_i in
>>>> a
>>>> superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an
>>>> observable O obeys
>>>>
>>>> ⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.
>>>>
>>>> Your argument has a tight fit with this for O_i = ρ_{ii}.
>>>>
>>>> The difficulty in part stems from the fact we keep using standard ideas
>>>> of probability to understand quantum physics, which is more fundamentally
>>>> about amplitudes which give probabilities, but are not probabilities. Your
>>>> argument is very frequentist.
>>>>
>>>
>>>
>>> I can see why you might think this, but it is actually not the case. My
>>> main point is to reject subjectivist notions of probability: probabilities
>>> in QM are clearly objective -- there is an objective decay rate (or
>>> half-life) for any radioactive nucleus; there is a clearly objective
>>> probability for that spin to be measured up rather than down in a
>>> Stern-Gerlach magnet; and so on.
>>>
>>>
>> Objective probabilities are frequentism.
>>
>
>
> Rubbish. Popper's original propensity ideas may have had frequentist
> overtones, but we can certainly move beyond Popper's outdated thinking. An
> objective probability is one that is an intrinsic property of an object,
> such as a radio-active nucleus. One can use relative frequencies or
> Bayesian updating to estimate these intrinsic probabilities experimentally.
> But neither relative frequencies nor Bayesian updating of subjective
> beliefs actually define what the probabilities are in quantum mechanics.
>
>
Probability and statistics are in part an empirical subject. This is not
pure mathematics, and it is one reason why there is no single foundation.
It would be as if linear algebra or any area of mathematics had two
competing axiomatic foundation. There are also subdivisions in the
frequentist and subjectivist camps, particularly the first of these.
Quantum mechanics computes probabilities not according to some idea of
incomplete knowledge, but according to amplitudes. There is no lack of
information from the perspective of the observer or analyst, but it is
intrinsic. From what I can see this means it is irrelevant whether one
adopts a Bayesian or frequentist perspective. The division between these
two approaches to probability correlate somewhat with interpretations of QM
that are ψ-ontic (aligned with frequentist) and ψ-epistemic (aligned with
Bayesian).
The divisions between the two camps amongst statisticians is amazingly
sharp and almost hostile. I think the issues is somewhat irrelevant.
LC
>
> The idea from a probability perspective is that one has a sample space
>> with a known distribution.
>>
>
> These are consequences of the existence of probabilities -- not a
> definition of them.
>
> Your argument, which I agree was my first impression when I encountered
>> the Bayesian approach to QM by Fuchs and Schack, who I have had occasions
>> to talk to. My impression right way was entirely the same; we have
>> operators with outcomes and they have a distribution etc according to Born
>> rule. However, we have a bit of a sticking point; is Born's rule really
>> provable? We most often think in a sample space frequentist manner with
>> regards to the Born rule. However, it is at least plausible to think of the
>> problem from a Bayesian perspective, and where the probabilities have
>> become known is when the Bayesian updates have become very precise.
>>
>
> Again, you are confusing measuring or estimating the probabilities with
> their definition. Propensities are intrinsic properties, not further
> analysable. Whether or not the Born rule can be derived from simpler
> principles is far from clear. I don't think the attempts based on decision
> theory (Deutsch, Wallace, etc) succeed, and attempts based on self-location
> (Carroll, Zurek) are far from convincing, since they are probably
> intrinsically dualist.
>
> However, all this talk of probability theory may itself be wrong. Quantum
>> mechanics derives probabilities or distributions or spectra, but it really
>> is a theory of amplitudes or the density matrix. The probabilities come
>> with modulus square or the trace over the density matrix. Framing QM around
>> an interpretation of probability may be wrong headed to begin with.
>>
>
>
>
> The basic feature of quantum mechanics is that it predicts probabilities.
> You confuse the way this is expressed in the theory with the actuality of
> how probability is defined.
>
> The argument by Carroll and Sebens, using a concept of the wave function
>>>> as an update mechanism, is somewhat Bayesian.
>>>>
>>>
>>>
>>> It is this subjectivity, and appeal to Bayesianism, that I reject for
>>> QM. I consider probabilities to be intrinsic properties -- not further
>>> analysable. In other words, I favour a propensity interpretation. Relative
>>> frequencies are the way we generally measure probabilities, but they do not
>>> define them.
>>>
>>>
>> I could I suppose talk to Fuchs about this. He regards QM as having this
>> uncertainty principle, which we can only infer probabilities with a large
>> number of experiments where upon we update Bayesian priors. Of course a
>> frequentists, or a system based on relative frequencies, would say we just
>> make lots of measurements and use that as a sample space. In the end either
>> way works because QM appears to be a pure system that derives
>> probabilities. In other words, since outcome occur acausally or
>> spontaneously there are not meddlesome issues of incomplete knowledge.
>> Because of this, and I have pointed it out, the two perspective end up
>> being largely equivalent.
>>
>
>
> But these are ways of estimating probabilities -- probability is not
> necessarily defined in this way.
>
>
> This is curious since Fuchs developed QuBism as a sort of
>>>> ultra-ψ-epistemic interpretation, and Carroll and Sebens are appealing to
>>>> the wave function as a similar device for a ψ-ontological interpretation.
>>>>
>>>> I do though agree if there is a proof for the Born rule that is may not
>>>> depend on some particular quantum interpretation. If the Born rule is some
>>>> unprovable postulate then it would seem plausible that any sufficiently
>>>> strong quantum interpretation may prove the Born rule or provide the
>>>> ancillary axiomatic structure necessary for such a proof. In other words
>>>> maybe quantum interpretations are essentially unprovable physical axioms
>>>> that if sufficiently string provide a proof of the Born rule.
>>>>
>>>
>>>
>>> I would agree that the Born rule is unlikely to be provable within some
>>> model of quantum mechanics -- particularly if that model is deterministic,
>>> as is many-worlds. The mistake that advocates of many-worlds are making is
>>> to try and graft probabilities, and the Born rule, on to a
>>> non-probabilistic model. That endeavour is bound to fail. (In fact, many
>>> have given up on trying to incorporate any idea of 'uncertainty' into their
>>> model -- this is what is known as the "fission program".) One of the major
>>> problems people like Deutsch, Carroll, and Wallace encounter is trying to
>>> reconcile Everett with David Lewis's "Principal Principle", which is the
>>> rule that one should align one's personal subjective degrees of belief with
>>> the objective probabilities. When these people essentially deny the
>>> existence of objective probabilities, they have trouble reconciling
>>> subjective beliefs with anything at all.
>>>
>>> Bruce
>>>
>>
>> Maybe a part of the issue here is the role of counter-factual statements.
>>
>
>
> You are getting side-tracked. Quantum mechancs is not counterfactually
> definite, so that is a side issue as far as the origin of probability is
> concerned.
>
> I think that may be more of an issue here than with the objective vs
>> subjective perspective on QM. Most interpretations of QM are not
>> counter-factual definite. In fact the only one I think is CF definite is
>> deBroglie-Bohm. The Lewis idea was that CF statements are modal logical and
>> thus there exists these alternative worlds.
>>
>
>
> Lewis explicated counterfactual statements in terms of possible worlds,
> not in terms of existent alternative worlds.
>
> Bruce
>
> I am not sure to what degree MWI upholds that idea. However, that is also
>> not necessarily an argument against the idea of these branching worlds.
>>
> LC
>>
>
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