On 20 Jan 2013, at 18:53, Bruno Marchal wrote:
On 19 Jan 2013, at 13:42, Telmo Menezes wrote:
On Thu, Jan 17, 2013 at 5:47 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 17 Jan 2013, at 16:01, Telmo Menezes wrote:
On Thu, Jan 17, 2013 at 3:01 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 17 Jan 2013, at 13:32, Telmo Menezes wrote:
Hi all,
Naive question...
Not being a physicists, I only have a pop-science level of
understanding of the MWI. I imagine the multi-verse as a tree,
where each time there is more than one possible quantum state we
get a branch. I imagine my consciousness moving down the tree.
Suppose Mary performs the Schrodinger's cat experiment in her
house and Joe does the same in his house. They both keep the
animals in the boxes and don't take a peak. Don't tell PETA. They
meet for a coffe in a nearby coffeeshop.
So now we have four possible universes where Mary and Joe can
meet. But from the double slit experiment we know that the cats
are both still dead+alive in the current universe. Right? So are
Mary and Joe meeting in the fours universes at the same time?
Let a = alive, d = dead, and the subscript 1 and 2 distinguishes
the two cats, which are independent. Both cats are in a superposed
state dead + alive:
(a1 + d1) and (a2 + d2),
so the two cats configuration is given by (a1 + d1) * (a2 + d2),
with "*" the tensor product.
This products is linear and so this give a1*a2 + a1*d2 + d1*a2 +
d2*a2.
Mary and Joe don't interact with any cats, so the global state is
also a direct tensor product M * J * (a1*a2 + a1*d2 + d1*a2 +
d2*a2), which gives:
M * J *a1*a2 + M * J *a1*d2 + M * J *d1*a2 + M * J *d2*a2
You can add the "normalization" constant, which are 1/sqrt(2)
times 1/sqrt(2) = 1/2=
1/2 M * J *a1*a2 + 1/2 M * J *a1*d2 + 1/2 M * J *d1*a2 + 1/2 M * J
*d2*a2
So the answer to your question is yes.
Nice. Thanks Bruno!
Welcome!
To be sure, the normalizing factor does not mean there are four
universes, but most plausibly an infinity of universes, only
partitioned in four parts with identical quantum relative measure.
Sure, I get that.
Am I a set of universes?
You can put it in that way. You can be identified by the set of the
universes/computations going through your actual states. But that
is really a logician, or category theoretician manner of speaking:
the identification is some natural morphism.
Well I think Bohr made the trick for the atoms. I think he defines
once an atom by the set of macroscopic apparatus capable of
measuring some set of observable.
That can be useful for some reasoning, but also misleading if taken
literally, without making clear the assumed ontology.
Ok. That mode of reasoning is weirdly appealing to me. Even Bohr's
take.
It is common in algebra, logic and exploited in category theory. As
long as we identify identity and morphism it is OK, in the applied
fields.
Don't confuse the price of a glass of beer with the set of all glass
of beers with the same price :)
Of course, I meant "As long as we DON'T identify identity and morphism
it is OK, in the applied fields.".
Bruno
Logicians often identify a world with a set of proposition (the
proposition true in that world).
But they identify also a proposition with the a set of worlds (the
worlds in which that proposition is true).
Doing both identification, you can see a world as a set of set of
worlds. That is useful for some semantics of modal logics.
What textbook would you recommend on modal logic? (I'm relatively
confortable with first-order logic from studying classical AI and
also from Prolog).
The two books by George Boolos (1979, 1993), on the self-referential
logics (G, G*, S4Grz) contains a quite good introduction to modal
logic.
The best textbook on modal logic is in my opinion is the book by
Brian Chellas: "Modal logic an introduction".
http://www.amazon.com/Modal-Logic-Introduction-Brian-Chellas/dp/0521295157
A recreative introduction to modal logic and self-reference (the
logic G) is "Forever Undecided" by Raymond Smullyan.
(A good book on first order logic, with the main theorems
(deduction, completeness and soundness, Löwenheim-Skolem,
incompleteness) is Elliott Mendelson.)
Those are examples of dualities, which abounds in logic, and which
can be very useful when used which much care, and very misleading
when forgetting that a morphism is not an identity relation.
To get the exact "number" of universes, we should first solve the
marriage of gravity with the quantum. And with comp, we should
also derive the Quantum from arithmetic (but that's not true,
actually: with comp we have directly the infinities of "universes").
Ok, sounds good but I have to dig deeper. (moving my own
understanding of what you're saying beyond the mushiness that it
currently is)
I can recommend the reading of the book by David Albert "Quantum
Mechanics and experience(*)". It is short and readable.
Nice. I bought it and I'm enjoying it so far.
Nice.
Best,
Bruno
To get all the quantum weirdness, and quantum computation, you
don't really need the Hilbert Space, a simple linear space, on the
complex numbers, is enough, with a good scalar product. It is about
infinitely easier to grasp quantum teleportation (and other very
weird quantum things) than to derive the structure of the Hydrogen
atom from the SWE. Quantum weirdness is simple!
I don't follow David Albert on Bohm, and he could have been less
quick on the Bell's inequality, ... and Everett, but it provides,
imo, the best simplicity/rigor tradeoff to get the main "conceptual
difficulties" of the QM theory.
Bruno
(*)
http://www.amazon.com/Quantum-Mechanics-Experience-David-Albert/dp/0674741137
http://iridia.ulb.ac.be/~marchal/
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