I'm a layperson fascinated with quantum mechanics and the MWI, and have reached a point where to obtain a better understanding of the qualitative descriptions (universes "splitting", "measure of a universe", etc.) I must learn the mathematical formalism. It appears that the popular descriptions of MWI use very loose terminology, and I suspect much has been lost in translation.

Digging through online sources such as MathWorld, Wikipedia, and CiteSeer, as well as reviving painful memories of matrix algebra from university (CS), I think I've learned enough to be dangerous. Below is a set of (possibly incorrect) statements and questions I have.

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Let |phi> represent the quantum mechanical state of a system S as a vector in Hilbert space. The state is determined by the angle of the vector, not it's length. So any state multiplied by a constant is the same physical state of the system. (Correct? Is this by decree or does it fall out of something more fundamental?)

Let A represent a Hermitian operator corresponding to some observable of the system S

`Let {l} represent the set of eigenvalues for operator A such that`

`A|phi> = l|phi>`

`And finally:`

`{|An>} is the set of eigenvectors for operator A corresponding to {l}`

This set of eigenvectors (if I understand correctly) form an orthonormal basis for the possible states of S, such that if S is in a state phi which is not an eigenvector of observable A, it may be represented as a linear combination of such eigenvectors:

`(1) |phi> = c1|A1> + c2|A2> + ... + cn|An>`

In the case where |phi> is indeed an eigenvector of A, then one of the constants cn is 1 while the remainder are 0.

`So far so good (I hope.) Here are my questions:`

A) What is the physical meaning of equation (1) above? Is this what is meant when a system is described as being in a "superposition" of states that are measured by A? Is "superposition" the accepted term in the MWI or is there another?

B) In the Copenhagen Interpretation (CI), the collapse postulate states that (somehow) as a result of a measurement, |phi> actually changes to one of {|An>} with a probability related to {cn}, though I'm not sure of the particulars. How do you describe the probability (within the CI) of obtaining measurement l from state |phi> based on equation (1) ? This is the Born rule, I think, but I haven't quite grasped the math.

C) In MWI, there is no collapse postulate. When a measurement occurs, the quantum mechanical state of the measuring device (and ultimately the observer) becomes a "superposition" as well, with each observer becoming a linear combination of states corresponding the effect the measured outcome has on the observer. Is this the technical meaning of "splitting universes"?

D) Even in the case where the spectrum of A is discrete, the set of constants {cn} in (1) can take on continuous values. When an observer "splits" as a result of measuring A on S, how many "splits" occur? Is there an infinity of them, each corresponding to a different set of constants {cn}? Or, is there a split only into the number of eigenvectors of A, since cn|An> represents the same physical state regardless of the numerical value of cn?

E) What is the "measure" associated with each of the "observer states" resulting from D? How is this "mathematically" related to the probability values from B)?

F) What happens when you use a different observable B? How do the answers to C), D), and E) change when observables A and B have different sets of eigenvectors? Is this the "preferred basis" problem?

`Struggling but determined to figure this out,`

`-Johnathan`