My Question About Quantum Mechanics

(And it's not what you think it is!)

Suppose you find a vector in the desert.  You define a coordinate
system based on (east, north, up) coordinates, and find that the
vector is a "superposition" of east, north, and up.
But the vector itself is unlikely to feel that it is a superposition
of anything.  Maybe it points to Vega.  Or maybe it points in a
direction that we cannot easily describe.  So we will have to
describe it in (east, north, up) terms.  But those are our terms,
not its.

If it were a quantum-mechanically inclined vector, we could interpret
east, north, and up as different possible states for the vector.
If we ask the vector which of those states it is in, it will tell
us which one.

note:
  Since this is a nice big real-life vector in the desert,
  we can look at which way it's pointing before we do our
  "experiment" (ask it whether it is pointing east, north,
  or up).
  In real atomic-scale experiments, we never get to see the
  vector; we only get to do "experiments" on it.  The vector
  is what we calculate with quantum mechanics, though, and
  it is believed to precisely represent the entire state of
  things.

We will notice, after experimenting with many such desert vectors,
that it will tell us that it points north with probability
proportional to how north/south-aligned it is, that it will tell
us it points east with probability proportional to how east/west-
aligned it is, and that it will claim to point up with probability
proportional to how up/down aligned it is.  All very reasonable!

The weird thing is that if it claims to be pointing north, then
all of a sudden it *is* pointing north!  In fact, that is how it
answers our question!  If we ask a whole bunch of vectors this
question, then they each change to be pointing either north, east,
or up.  Since they do it probabilistically according to how they
were already aligned, their collective average alignment isn't
particularly affected by us asking questions.  But their individual
alignments are.

When a vector replies "I'm pointing east", and then it suddenly
*is* pointing east, that's what people refer to as the "collapse"
of the vector.  It doesn't really collapse -- it's still just as
big -- it just changes its direction so as to be simply describable
according to our experiment.

Here's another observation:  We can ask a different question, such
as "Are you an up vector or a horizontal vector?"  In this case,
the vector will either answer "up" and become pointing straight
up, or it will reply "horizontal", and point in the direction
of its shadow when the sun is directly overhead.  When it says
"horizontal", it is getting to keep some of its previous information,
as we can see.  It gets to keep being "wishy-washy" (from the
north/east/up perspective) on whether it is pointing north or
east, since we haven't asked it about that yet.  Of course, we
can then ask it whether it is pointing north or east, and it
will snap to one of those positions.

But we can also ask it whether it is pointing northeast or
southeast, in which case it will snap to one of *those* two
positions, with the appropriate probabilities!

In fact, we can pick any coordinate system, and ask it whether
it is an x, y, or z vector in that system, and it will respond
according (probabilistically) to its alignment, and snap to be
exactly whatever its reply is.

note:
  This is why we believe in the reality of the vector, even though
  for atomic experiments we can only observe it in a clumsy way,
  by making it "snap to an answer".  But since it is ready to snap
  with the correct probabilities for whatever coordinate system
  you choose, it seems clear that it knew which way it was pointing.


Why am I going on and on about this, you ask?  This snapping
doesn't occur, you say.  The vectors just keep pointing in
their original directions (or swaying about over time, more
realistically).

But it does occur!  When you listen to a geiger counter, either
you hear it click at a certain time or you don't.  The vector has
snapped.  The vector is no longer pointing is some combination
of "clicked" and "didn't click".  At least, if you think it is
pointing in such a direction, you'll have to explain why it is
that we always either think we heard it or think we didn't.
What's so special about these two states (analogous to "north"
and "east"), that we always feel we're in exactly one of them?
If the vector didn't snap, then why should we possibly feel
compelled to interpret it as being in one of these two states?
The average person doesn't understand what a geiger counter is,
and they certainly don't try to do any experiments on it.  Yet
they too either hear it click or don't hear it click.  Why can't
it just be in the state it is naturally in?  Why do we never see
this and have no words for describing it?  We can only experience
the "snapped" states of "it clicked" or "it didn't click", even
though it was not naturally inclined to be in either of these
states.

The problem with Everett-style claims that there is no "collapse",
from my point of view, is that there is no explanation of how
nature chooses its "base states", the states that we wind up
being able to feel we have experienced.

According to the formalism, no base states are intrinsically
better than any others.  Physicists typically pick them so as
to simplify the computation, just like how mathematicians pick
coordinate systems for their problems.


The only time probabilities occur at all is when we ask a vector
an "inappropriate" question.  For example, it is quite appropriate
to ask a vector that is pointing straight up, "Are you pointing
east, north, or up?"  It will definitely (100% chance) reply that
it is pointing up, and it will not even have to move to accommodate
your question.

However, if you ask the same vector a different question, for
example, you stand a cube on a corner and ask which of the top
three faces the vector is perpendicular to, then it will have to
reply completely randomly, and noticeably snap over to its answer.

Snapping (and therefore probabilities) can only occur when you
ask an inappropriate question, when you "do an experiment".

The Everett-style interpretations seem to say that when you ask
the up-vector which cube face it's perpendicular to, the universe
splits into three branches, each of which shows the vector being
perpendicular to one of the faces.  But this involves snapping, too!
Why is there not just one universe, with the vector pointing as it
always did?

Some people say that the up-vector always existed as the "superposition"
of the three branches of the universe, and the copy of you that was
present in each branch naturally got a different answer in each branch,
and found that the vector was pointing in exactly the direction of
the answer.

But this view depends on selecting the "base states" according
to the three upper faces of the balanced cube.  And who selected
them?  They can be selected long after the vector has formed (or,
more convincingly, they can be selected at a spatially-separated
place from where the vector was formed, thus making it very
hard to believe that they influenced each other).  What if we had
asked the question "Are you an east, north, or up vector?"  How
does the branched-universe theory handle the fact that in this
case there isn't any branching, whereas in the other case we
were already thinking of the universe as having split into three
branches?  The same universe, mind you.


This whole idea seems to rest on people's inability to visualize
any vector besides due east, north, or up, for many situations.

For example, in this common situation:

> Schroedinger opens the box and looks inside.  Again the superposed
> state is amplified.  Schroedinger's mental state is consistent with
> the health of the cat.  Schroedinger ((believes the cat to be alive)
> + (believes the cat to be dead)) / 2.

people (myself included) are completely incapable of imagining
any other state besides "believes it's alive" and "believes it's
dead".  The only way we can imagine (alive + dead)/2 is as some
weird "superposition".

So we can happily believe that the universe is in two branches,
one of which has a healthy cat and one of which has a dead cat.
This is perfectly imaginable.  But this doesn't really correspond
to the quantum mechanics of it.

In the quantum mechanical description, we could perfectly well
change the base states around, to use base states "glorph" and
"hibble":

       glorph = (alive + dead) / sqrt(2)
       hibble = (alive - dead) / sqrt(2)

       alive = (glorph + hibble) / sqrt(2)
       dead  = (glorph - hibble) / sqrt(2)

Now, we can say that the system is absolutely (100% chance) in
the glorph state and absolutely not in the hibble state.

Nowhere does quantum mechanics say that the universe is really
based on the alive and dead states, and that "glorph" needs to
be represented in terms of "alive" and "dead".  In fact, we
humans (and the cat) seem to be the only ones really interested
in thinking in terms of the "alive" and "dead" states.

Asking whether the system is in the alive state or the dead state
is definitely an inappropriate question.  But if you do ask, it
will tell you one of those two things.

But why is it that opening the box amounts to asking this question?
Why do we not just see state glorph, jot down "another glorph" in
our notebook, and go on to the next experiment?  Why must we see
either state alive or state dead?  We never ever ever see a glorph
state, and thus we can't even imagine what such a state would be
like.  Why is that??

Some people would say that it is because we ourselves have become
a superposition; that really we are in a glorph state, a superposition
of saying "look, it's the alive state", and "look, it's the dead state".
But why must the glorph state be interpreted as a superposition of
live and dead?

Somewhere, it seems, the universe has decided that it must resolve
itself (some would say by wavefunction collapse, others would say
by branching into separate worlds) either into an alive state or
into a dead state, and not into a glorph or hibble state.  Why does
it decide this?

As long as you insist on saying that the glorph state is a
superposition of the alive and dead states, you will not see
my point.  My question is "Why must the glorph state be thought
of as a superposition of the alive and dead states?"  As long as
you keep on insisting that glorph is really just a superposition
of alive and dead, and thus we have a 50/50 chance of seeing the
cat alive or dead, I will keep on insisting that alive is really
a superposition of glorph and hibble, and that we have a 100%
chance of seeing glorph.  But when we open such boxes, you will
always be right and I will always be wrong.  Why?  Weren't our
arguments equally sound?

I see two camps of people:

1. One camp, including Schroedinger, feels that this "superposition"
   of states is absurd, and surely the universe resolves itself
   (collapses from a glorph-like state) into a "pure" life or death
   state long before you open the box.

My questions here are "When/why did it collapse?", and "How did it
know what its options were for what to collapse to?".  Most people
seem to concentrate on the former question, but the latter queston
is much more intriguing to me.

2. The other camp, including Everett, feels that this superposition
   is quite real, and the universe never resolves itself, but has
   multiple personalities, one of which contains a live cat, and
   another of which contains a dead cat.

My questions here are "Why is there any branching, instead of just
a pure glorph state?", and "Why are the live and dead states the
only ones that we ever experience?"  The latter question is much
more intriguing, just like last time.

3. There are yet other camps, such as that of Wigner, which claims
   that the universe resolves itself when a consciousness gets
   involved, but I think this is so silly I won't even address it,
   except to wonder how alert or intelligent you have to be before
   the universe takes you seriously enough to "collapse" on your
   behalf.

The question I think people should be asking is not when or
whether there is a collapsing wave function, but what is it
that makes some states (e.g. "alive") experienceable and others
(e.g. "glorph") not?  When does the universe decide to think
of itself as a superposition instead of as a pure state?  And
how does it decide what it should be a superposition of?

I'm sure that answering this question would be the best way to
figure out what "the collapse of the wave function" or "the
branching of the universe" really means.

Furthermore, this question should be answerable by experiment.
By investigating various circumstances, and seeing which
circumstances lead to behavior best explained by saying a
probabilistic choice was made (whether viewed in terms of
branches or in terms of a collapsing wave function), and
which circumstances lead to behavior best explained by saying
that the outcome corresponds to no choices having been made,
we should be able to clarify what "an experiment" is and how
the universe picks what base states to use when it needs to
"make a choice".  So far, experimental results have basically
supported the view that no choices are made until the end
of the experiment.  That's why "performing an experiment"
seems currently to be the appropriate metaphor for when the
universe "makes a choice".


-Matthew Cook
cook@wolfram.com

Some References

John Cramer's Transactional Interpretation of Quantum Mechanics

Michael Price's interpretation of Everett's "Many-Worlds" interpretation
... and my rebuttal of it.