On 6/07/2017 5:55 pm, Russell Standish wrote:
On Thu, Jul 06, 2017 at 04:18:49PM +1000, Bruce Kellett wrote:
On 6/07/2017 2:33 pm, Russell Standish wrote:
Establishing linearity is key.
Yes, and you haven't made progress with that.
All I ask is to give me some more time on this. I have some further
ideas in this regard, but need some dedicated time to think about it.
Fair enough.
Establishing the resultant vector space
is a Hilbert space does follow fairly easily from Kolmogorov's axioms
(although its possible you have beef with those :).
I think the issue I would have here is that you assume that a
projection can give a range of results. I don't think that is
necessarily true even in a multiverse. Since projections are as
undefined as everything else here, it could be the case that a
projection gave a single value -- the same value in every world of
the multiverse, so that you had a version of classical mechanics.
Assuming a range of different values, assigning probabilities to
individual outcomes makes some sort of sense, but that is assuming a
lot of quantum mechanics at the start. Hence my worries about
circularity.
If there is only a single outcome, then the projection concerned
will be the certain event projection. With probability 1. That case is covered.
A more serious consideration occurred to me -- the projection might have
a range of possible outcomes, but these might be merely an illustration
of classical probability arising from ignorance. In quantum terms, you
have to distinguish between pure and mixed states. You appear to assume
that your projections are from a pure state, but that is circular: you
can only have pure states in that sense in a quantum theory. Pure states
can exhibit interference, mixed states cannot.
S=X+P is definitely an observable, and corresponds to measuring the sum
of position and momentum crisply. You can work out the equivalent
Heisenberg uncertainty relations too
ΔsΔx ≥ ℏ and ΔsΔp ≥ ℏ
So if you measure with S, you will have uncertainty in both position
and momentum.
So it is not a quantum operator or quantum observable. These are
defined to have precise values: eigenvalues corresponding to the
eigenfuntions of the Hermitian operator. And the eigenfunctions span
the corresponding Hilbert space.
I don't get your point. The operator S=X+P satisfies all of those.
No, position and momentum are dual in the sense I defined. The
observables are not compatible -- position and momentum are not
simultaneously observable.
My point here is that position
space, spanned by the eigenfunctions of the position operator, is
'dual' to momentum space, spanned by the eigenfunctions of the
momentum operator. These spaces are related by the Fourier transform
and each serves to give a distinct complete representation of the
underlying quantum state. (That is the sense in which I refer to
these operators and spaces as 'dual' -- it makes no quantum sense to
add them,)
Operators operate on the Hilbert state space, not position or momentum
space, which is more of classical mechanics concept. And yes, one can
add them (modulo an assumed unit conversion constant) mathematically,
and the sum X+P is meaningful physically.
Classically perhaps, but not in quantum mechanics.
In general, any bounded hermitian operator is an observable of some
sort (one can create a machine, albeit weirdly wonderful in a Heath
Robisonesque way, that will measure that particular quantity).
Certainly. If one has the Hilbert space for some quantum operator,
linear combinations of the eigenvector basis vectors that also span
the space will be the eigenvalues of some other hermitian operator
in the space. As you say, not all of the resultant operators
correspond to natural observables.
I don't say that. You're saying that, and I fail to see your point.
It is really the (dead-cat +- live-cat basis) issue. A superposition of
polarization states becomes entangled with a cat in Schrödinger's
thought experiment. We don't observe the superposition because the (dead
+- alive) basis does not correspond to a natural observable --
decoherence sees to that. Similarly for most other quantum variables.
The normal position basis, delta(x-x_i), can be superposed to give some
alternative basis, and there will be a corresponding operator on the
Hilbert space, but superpositions of position results are not given by
any achievable measurement -- again, decoherence leads to the preferred
stable basis: all other bases do not give results that are stable under
decoherence. Not all possible operators are possible measurements (ie,
not all operators give stable results).
The sum of two bounded hermitian operators is also a bounded hermition
operator. In fact a linear combination with real coefficients will
too, but not necessarily complex coeficients (since the hermitian
property may not be preserved).
Yes, but my concern here is that this generality does not extend to
tensor products of Hilbert spaces, and the product space is what one
has in quantum mechanics -- projections in one space are not
projections in another of the component spaces. You have also to
distinguish carefully between compatible and incompatible
observations -- commuting and non-commuting observables
(projections). I don't see how you could establish this distinction
in your approach without just assuming that it exists.
Can you give an example of a tensor product where the summation
failed? The way I see it, if observables A and B operated in separate
subspaces, AA and BB (think say X and Y positions),
Positions in different dimensions are compatible observables. You can't
do this with non-commuting operators.
then one can
extend the operators A and B to the product space AA⊗BB by making A a
constant along all axes of BB and vice-versa for B. In matrix
terminology, we create A' by copying A all down the diagonal,
similarly with B:
/a11 a12 a13 \
|a21 a22 a23 |
|a31 a32 a33 |
| a11 a12 a13 |
A' = | a21 a22 a23 |
| a31 a32 a33 |
| a11 a12 a13|
| a21 a22 a23|
\ a21 a32 a33/
/b11 b12 b13 \
| b11 b12 b13 |
| b11 b12 b13|
|b21 b22 b23 |
B' = | b21 b22 b23 |
| b21 b22 b23|
|b31 b32 b33 |
| b32 b32 b33 |
\ b33 b32 b33/
Then in the 9 dimensional product space AA⊗BB, the operators A' and B'
can be added, with the sum corresponding to a physical observable.
All bets are off with unbounded operators, of course, but my attitude
is that unbounded operators are stictly unphysical, albeit sometimes
convenient for computational purposes.
Thinking along those lines some more, I'm incorrect to say that the
vector space V is the set of all observer moments. It must be the set
of all successor observer moments to ψ, or all continuations as I
think you put it earlier. The clue lies in the linear span (D.8). OMs
that can't be reached from ψ just simply cannot be put in the linear
span of outcomes from an observation on ψ.
Any observer moment is the successor of some previous observer
moment, so concentrating on successor OMs of psi achieves nothing.
I don't believe so. What is the observer moment you experienced prior
to being born? It is not necessary for all OMs to have a predecessor.
Maybe your mother had some observer moments?
Yes, but they're not mine. My OMs did not succeed them.
They did, really. How else did you come to be?
But this just highlights one of the main reasons that I am out of
sympathy with the approach based on observer moments. The history of
quantum mechanics over the last hundred or so years has been an
attempt to find an account of the theory that removes the observer
from any central role. One of the concerns here, of course, is that
the universe showed quantum behaviour long before there were any
observers around -- long before any consciousness at all, in fact.
So physics must ultimately be independent of OMs. OMs might not even
be quantum in general -- we could have lived in a classical world --
we don't, but can you swear that consciousness is irreducibly
quantum? What about conscious digital machines that could be
entirely classical in construction and operation?
This is a whole other argument, which has been extensively rebutted on
this list. The antirealist approach is that the universe without
observers didn't/doesn't really exist, not in the same way as the
observed universe. That is why I make statements like we're in a
superposition of red/green T. Rexes etc. If there is no known fact
that determines the case, the world of dinosaurs is that bit more
nebulous than the one we inhabit. Obviously the dinosaurs will come to
same conclusion about out world - probably more so, as we have far
more information about the dinosaur world than they of us.
I, along with most other scientists, reject this antirealist approach.
Members of the list might have accepted the antirealist position, but
that does not mean it is true, or even a viable metaphysics.
Yes - I push back at the eliminativist approach of removing
subjectivity entirely - doing so appears to be the source of most of
the problems with defining complexity, and it also appears to cause
problems with QM, such as the preferred basis problem.
And assuming conscious classic digital machines, quantum phenomenology
appears at the observed level - a result in line with Bruno Marchal's
FPI result.
Prove it. Bruno has failed to do so -- his person duplication thought
experiments do not reproduce quantum behaviour.
The best one can do is operationalise the subjective, so that we can
shut up and get on with our calculations.
The union of the partitions is not another observer moment -- an
observation of a result is not the whole set. No observer sees all
possibilities simultaneously. Unitarity is not generally preserved
for observer moments because they change discontinuously.
The union of partitions is the original observer moment - the state
of the observer after having planned the experiment, but before
carrying it out.
Yes, that is one of the problems with purely unitary evolution, nothing
ever happens! The time development of the wave function is just a change
in an unmeasurable phase. If you are to actually get results that you
can compare with experiment out of the theory, you have to acknowledge
non-unitarity -- worlds must differentiate. This, along with the basis
problem, are two of the major problems with MWI. Observers are
non-unitary by nature -- there is no 0p or 3p view in QM -- there is no
super observer outside the branching process. Your OMs must branch, or
they do not correspond to any temporal development at all.
Bruce
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
