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 :)






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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