# Questions about MWI and mathematical formalism

```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.```

-=-=-=-

```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

```

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• Questions about MWI and mathematical formalism Johnathan Corgan