Questions about MWI and mathematical formalism

From: Johnathan Corgan <jcorgan.domain.name.hidden>
Date: Mon, 03 May 2004 09:30:19 -0700

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
Received on Mon May 03 2004 - 12:41:51 PDT