> On 25 Feb 2019, at 12:39, Lawrence Crowell <goldenfieldquaterni...@gmail.com>
> wrote:
>
> On Monday, February 25, 2019 at 2:44:14 AM UTC-6, Bruno Marchal wrote:
>
>> On 24 Feb 2019, at 15:24, Lawrence Crowell <goldenfield...@gmail.com
>> <javascript:>> wrote:
>>
>> On Friday, February 22, 2019 at 3:18:01 PM UTC-6, Brent wrote:
>>
>>
>> On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
>>> This sounds almost tautological. I have not read Masanes' paper, but he
>>> seems to be saying the Born rule is a matter of pure logic. In some ways
>>> that is what Born said.
>>>
>>> The Born rule is not hard to understand. If you have a state space with
>>> vectors |u_i> then a quantum state can be written as sum_ic_i|u_i>. For an
>>> observable O with eigenvectors o_i the expectation values for that
>>> observable is
>>>
>>> sum_{ij}<u_j|O|u_i> = sum_{ij}<u_j|o_i|u_i> = sum_ip_io_i.
>>>
>>> So the expectations of each eigenvalue is multiple of the probability for
>>> the system to be found in that state. It is not hard to understand, but the
>>> problem is there is no general theorem and proof that the eigenvalues of an
>>> operator or observable are diagonal in the probabilities.
>
> I am not sure I understand this.
>
>
>
>
>>> In fact this has some subtle issues with degeneracies.
>>
>> Doesn't Gleason's theorem show that there is no other consistent way to
>> assign probabilities to subspaces of a Hilbert space?
>>
>> Brent
>>
>> It is close. Gleason's theorem tells us that probabilities are a consequence
>> of certain measurements. So for a basis Q = {q_n} then in a span in Q =
>> P{q_n}, for P a projection operator that a measure μ(Q} is given by a trace
>> over projection operators. This is close, but it does not address the issue
>> of eigenvalues of an operator or observable. Gleason tried to make this work
>> for operators, but was ultimately not able to.
>
> It should work for the projection operator, that this is the
> yes-no-experiment, but that extends to the other measurement, by reducing (as
> usual) the question “what is the value of A” into the (many) question “does A
> measurement belong to this interval” … Gleason’s theorem assures that the
> measure is unique (on the subspaces of H with dim bigger or equal to 3), so
> the Born rule should be determined, at least in non degenerate case (but also
> in the degenerate case when the degeneracy is due to tracing out a subsystem
> from a bigger system. I will verify later as my mind belongs more to the
> combinator and applicative algebra that QM for now.
>
>
>
>>
>> Many years ago I had an idea that since the trace of a density matrix may be
>> thought of as constructed from projection operators with tr(ρ_n) = sum_n
>> |c_n|^2P_n, that observables that commute with the density matrix might have
>> a derived Born rule following Gleason. Further, maybe operators that do not
>> commute then have some dual property that still upholds Born rule. I was not
>> able to make this work.
>
> I will think about this. Normally the measure is determine by the “right"
> quantum logic, and the right quantum logic is determined by the any
> “provability” box accompanied by consistency condition (like []p & p, []p &
> <>t, …). The main difference to be expected, is that eventually we get a
> “quantum credibility measure”, not really a probability. It is like
> probability, except that credibility is between 0 and infinity (not 0 and 1).
>
> Bruno
>
>
> I think I ran into the issue of why Gleason's theorem does not capture the
> Born rule. Not all operators are commutative with the density matrix. So if
> you construct the diagonal of the density matrix, or its trace elements, with
> projector operators and off diagonal elements with left and right acting
> projectors (left acting hit bra vectors and right acting hit ket vectors) the
> problem is many operators are non-commutative. In particular the usual
> situation is for the Hamiltonian to have nontrivial commutation with the
> density matrix.
It seems to me that Gleason theorem takes this into account. It only means that
the probabilities does not make the same partition of the multiverse, but that
is not a problem for someone who use physics to see if it confirms or refute
the “observable” available to the universal numbers/machines in arithmetic.
I am not completely sure. You raise a doubt, and I’m afraid it will take some
time I come back to Gleason theorem. But I appreciate. My conversation with
Bruce and Brent makes me think that the notion of multiverse is far from clear.
At least with mechanism things are crystal clear! There is only the sigma_1
sentences, and the nuances imposed by incompleteness for the “Löbian number”
who “lives” through them (them for the sigma_sentences, which “realises” the
computations).
Of course I come from the other side, but if mechanism is correct, I can only
cross physics when and where physics is correct. For now, physics is not yet a
solved problem, as GR does not fit with QM. The very notion of “force” or
“interaction” seems conceptually very different in GR and QM. We can expect
surprise, but with Mechanism, the quantum weirdness is welcomed, and we are far
from having any notion of physical space, and why 3D or 11D or 26D. Mechanism
is a 0 dimension theory of the mind, à la Plato, where the ideas are the
numbers i, and the partial recursive function phi_i, and the operator
phi_phi_i, etc.
Space, like in Kant, is a universal pattern of the universal machine, although
this is not yet proved, only suspected, as it could still be that even space is
“geographical” and that consciousness can survive without it. Well, the
theology of the numbers is in its infancy, if not still an embryo: but the
propositional parts is given by the two arithmetical completeness theorem of
Solovay, leading to G and G* describing all what can be said on this. G gives
the part that all sound machine can justify, and G* gives the true, but non
justifiable part. In between the rational and the irrational there is a
“surrational part”: what science can learn from experience but never rationally
justify.
Bruno
>
> LC
>
>
>>
>> LC
>>
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